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(Set.{u} α) a b)\n    (And (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) a b) (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) b a))","statement_sha256":"ab5dced81759df452489f192d8ab7f94346fea5494ad0ba312d5b444cfa105be","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ac24fed231c6e71322d3911deef288b2584dda09250bee5f2bb01e2ef29ef800","qualified_name":"Set.Subset.refl","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":8200,"column":57,"line":253},"start":{"byte":8135,"column":0,"line":252}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (a : Set.{u} α), @LE.le.{u} (Set.{u} α) 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(@Set.instLE.{u} α) a c","statement_sha256":"9212eba081259d883e8907ad9d77a169b6a05694ba435570920b30f845c3c3b1","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3cf979d433bc446230658611d16848dbacb769151f54deb23e5de8b63ce7e759","qualified_name":"Set.bot_eq_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":2920,"column":5,"line":95},"start":{"byte":2860,"column":0,"line":93}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u},\n  @Eq.{u + 1} (Set.{u} α)\n    (@Bot.bot.{u} (Set.{u} α)\n      (@OrderBot.toBot.{u} (Set.{u} α) (@Set.instLE.{u} α)\n        (@BoundedOrder.toOrderBot.{u} (Set.{u} α) (@Set.instLE.{u} α) (@Set.instBoundedOrder.{u} α))))\n    (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} 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x)","statement_sha256":"dc82d348f6e861b3403f829c747e1dd3040eed72897db3412087982880bdcbd8","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3e0ca5b3abbcaac9d3801a3b7c7ba03394a5ba5c33927fb4c1b03ac9520fd179","qualified_name":"Set.coe_setOf","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":4828,"column":75,"line":149},"start":{"byte":4753,"column":0,"line":149}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : α → Prop),\n  @Eq.{u + 2} (Type u) (@Set.Elem.{u} α (@Set.ofPred.{u} α fun x => p x)) (@Subtype.{u + 1} α fun x => p x)","statement_sha256":"dc82d348f6e861b3403f829c747e1dd3040eed72897db3412087982880bdcbd8","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-804310c80112181b84afac4751eefea6475f23227570341c4c8dc35ec98a9c11","qualified_name":"Set.compl_ssubset_univ","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":21098,"column":6,"line":617},"start":{"byte":20970,"column":0,"line":615}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s : Set.{u} α},\n  Iff\n    (@LT.lt.{u} (Set.{u} α)\n      (@Preorder.toLT.{u} (Set.{u} α)\n        (@PartialOrder.toPreorder.{u} (Set.{u} α)\n          (@SemilatticeInf.toPartialOrder.{u} (Set.{u} α)\n            (@Lattice.toSemilatticeInf.{u} (Set.{u} α)\n              (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))))\n      (@Compl.compl.{u} (Set.{u} α) (@Set.instCompl.{u} α) s) (@Set.univ.{u} α))\n    (@Set.Nonempty.{u} α s)","statement_sha256":"b0c5dda0751c2c8fb344296d423ac209591f5d3d34e128014af8dacebf598438","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-c6bdd7d67a9e0854c9d1f3bfbf62ae80047c2ab3ec8e15da66282a71611da46a","qualified_name":"Set.decidableCompl","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":39227,"column":39,"line":1143},"start":{"byte":39114,"column":0,"line":1142}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u} →\n  (s : Set.{u} α) →\n    (a : α) →\n      [Decidable (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) s a)] →\n        Decidable\n          (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n            (@Compl.compl.{u} (Set.{u} α) (@Set.instCompl.{u} α) s) a)","statement_sha256":"2ce4392daea55431159795337f57b58e863e5d44e6acc1786e1d2f749956c32a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-f8afb36f22bbe10b2b8b15a1c13dc4426a5a07a25273cb45651b02415c2932ca","qualified_name":"Set.decidableEmptyset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":39321,"column":92,"line":1145},"start":{"byte":39229,"column":0,"line":1145}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u} →\n  (a : α) →\n    Decidable\n      (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n        (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)) 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α)","statement_sha256":"3927aff13ca700a346f5e18cb59cec5453cf235f2209696ab9f0674a8d20a892","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-ad9e2c0a940836138b61b9b69461c5b3831f1d61f3a384d3214596f7af5c792b","qualified_name":"Set.decidableInsert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":39542,"column":39,"line":1150},"start":{"byte":39402,"column":0,"line":1149}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u} →\n  (s : Set.{u} α) →\n    (a b : α) →\n      [Decidable (@Eq.{u + 1} α a b)] →\n        [Decidable (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) s a)] →\n          Decidable\n            (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n              (@Insert.insert.{u, u} α (Set.{u} α) (@Set.instInsert.{u} α) b s) 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(@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) t a)] →\n          Decidable\n            (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n              (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) s t) 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(@Set.Nonempty.{u} α s)","statement_sha256":"2974dfdedd1e031ff9327c689b33e3b28f1f1c26641b1ffe5770fded1d3014fa","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f1b40a36dabc62825f25093eb5e7c939e74c637c22ad25b27bde0bd0fe2616ae","qualified_name":"Set.eq_of_forall_subset_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":9683,"column":95,"line":294},"start":{"byte":9588,"column":0,"line":294}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  (∀ (u : Set.{u} α),\n      Iff (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s u) (@LE.le.{u} (Set.{u} α) 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(@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) a b) c)\n    (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) a (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) b 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(@Set.instInter.{u} α) t u₁) →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) u₂ u₁ →\n      @Eq.{u + 1} (Set.{u} α) (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) s u₂)\n        (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) t 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s t)","statement_sha256":"7da0f3c79640b1859c2d129c988ca69fe31cfb3dd1dbc89c828bde29242ccad4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4565433292226dab34554946f4a311b4815ee13f712296fb5473a5d4252f373f","qualified_name":"Set.left_eq_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":28253,"column":68,"line":832},"start":{"byte":28185,"column":0,"line":832}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  Iff (@Eq.{u + 1} (Set.{u} α) s (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) s t))\n    (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s t)","statement_sha256":"0b3ef6319a46a3fd605da0e39271a3b024f1c9e2e33efa26dd4ebc87a57725d4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c717f5179bc9c71cd84aa390870cc646835aa9843459cefd84811bb12edae81c","qualified_name":"Set.lt_eq_ssubset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":3477,"column":5,"line":111},"start":{"byte":3301,"column":0,"line":109}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u},\n  @Eq.{u + 1} (Set.{u} α → Set.{u} α → Prop)\n    (fun x1 x2 =>\n      @LT.lt.{u} (Set.{u} α)\n        (@Preorder.toLT.{u} (Set.{u} α)\n          (@PartialOrder.toPreorder.{u} (Set.{u} α)\n            (@SemilatticeInf.toPartialOrder.{u} (Set.{u} α)\n              (@Lattice.toSemilatticeInf.{u} (Set.{u} α)\n                (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))))\n        x1 x2)\n    fun x1 x2 =>\n    @LT.lt.{u} (Set.{u} α)\n      (@Preorder.toLT.{u} (Set.{u} α)\n        (@PartialOrder.toPreorder.{u} (Set.{u} α)\n          (@SemilatticeInf.toPartialOrder.{u} (Set.{u} α)\n            (@Lattice.toSemilatticeInf.{u} (Set.{u} α)\n              (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))))\n      x1 x2","statement_sha256":"0d8d97bf704c9741be0174d566d320a927e9bcd2b7002bf91b080d01d9c45816","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f709157ea14a18c181a4dafb38bc08728a1af3163ccf7b6fef0c013da0788695","qualified_name":"Set.lt_iff_ssubset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":3770,"column":9,"line":119},"start":{"byte":3626,"column":0,"line":117}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  Iff\n    (@LT.lt.{u} (Set.{u} α)\n      (@Preorder.toLT.{u} (Set.{u} α)\n        (@PartialOrder.toPreorder.{u} (Set.{u} α)\n          (@SemilatticeInf.toPartialOrder.{u} (Set.{u} α)\n            (@Lattice.toSemilatticeInf.{u} (Set.{u} α)\n              (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))))\n      s t)\n    (@LT.lt.{u} (Set.{u} α)\n      (@Preorder.toLT.{u} (Set.{u} α)\n        (@PartialOrder.toPreorder.{u} (Set.{u} α)\n          (@SemilatticeInf.toPartialOrder.{u} (Set.{u} α)\n            (@Lattice.toSemilatticeInf.{u} (Set.{u} α)\n              (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))))\n      s t)","statement_sha256":"507a09fc5b5b99fa4261131da2d09ce777c2db8f1b4c8f4a9c471e99a9ee54c8","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-5b155ebb08063db12a8315cfafc03ee0ba075e01c1041ad0a9ebaafd6c9b2f3d","qualified_name":"Set.mem_bijective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":2444,"column":101,"line":75},"start":{"byte":2343,"column":0,"line":75}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u},\n  @Function.Bijective.{u + 1, u + 1} (Set.{u} α) (α → Prop)\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α))","statement_sha256":"0cb78214bf458eb3d9616e7e232ba3f914e43571f5446a1beb6fc3fef2b174e7","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c01f206f8f72b4ffce9f634526a68bca9df4ee7fb064f027412c3edcfb7f7f74","qualified_name":"Set.mem_dite_empty_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":37433,"column":103,"line":1090},"start":{"byte":37118,"column":0,"line":1087}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : Prop) [inst : Decidable p] (t : Not p → Set.{u} α) (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@dite.{u + 1} (Set.{u} α) p inst\n        (fun h => @EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)) fun h => t h)\n      x)\n    (∃ (h : Not p), @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) (t h) x)","statement_sha256":"5c63042da5aa1b5aea4b405c0fbab72e100dd6fb6d790233a8bb453439e9fabd","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4031c8dd80fc725b63324579df1541e56998d40a8400eaf40ebbc089aa9cfee3","qualified_name":"Set.mem_dite_empty_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":36929,"column":92,"line":1080},"start":{"byte":36619,"column":0,"line":1077}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : Prop) [inst : Decidable p] (t : p → Set.{u} α) (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@dite.{u + 1} (Set.{u} α) p inst (fun h => t h) fun h =>\n        @EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α))\n      x)\n    (∃ (h : p), @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) (t h) x)","statement_sha256":"45f57d3417c044d7982119967bdf5cf747d6837796b229267fe63a3cc63b8402","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-01c5c35c28cfc22f823614f29ed219ca1f3b60781cfba319428478d773ab78c1","qualified_name":"Set.mem_dite_univ_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":36445,"column":24,"line":1070},"start":{"byte":36264,"column":0,"line":1068}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : Prop) [inst : Decidable p] (t : Not p → Set.{u} α) (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@dite.{u + 1} (Set.{u} α) p inst (fun h => @Set.univ.{u} α) fun h => t h) x)\n    (∀ (h : Not p), @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) (t h) x)","statement_sha256":"635e18a83b66a33d7b8b269f8d52d3f9605109ea6ec6517497cea8e464d0aa95","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-dd3d5c9be18a72edfb3646167c7b198b06c8907c9f7edf1c59c7300ec47f0e7f","qualified_name":"Set.mem_dite_univ_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":36090,"column":17,"line":1061},"start":{"byte":35919,"column":0,"line":1059}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : Prop) [inst : Decidable p] (t : p → Set.{u} α) (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@dite.{u + 1} (Set.{u} α) p inst (fun h => t h) fun h => @Set.univ.{u} α) x)\n    (∀ (h : p), @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) (t h) x)","statement_sha256":"0ff091e88a87ea0ce5562ecf91e7c0fe6ea1d59c5dc92b015396022701221bd1","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b565e2b48f0228a3eb54c173be08bf13478107b0980c32c6a73f507a17a9211e","qualified_name":"Set.mem_empty_iff_false","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":14336,"column":9,"line":423},"start":{"byte":14231,"column":0,"line":421}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)) x)\n    False","statement_sha256":"b7f21db916a02fb08203b95938d0beeba7f4af1c9d0eca4c0c9f0c2f9d113a83","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-038c7a178747db9847b2a9e6eb9e69b22badcb18747e1d845baa7f05af16bf99","qualified_name":"Set.mem_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":2237,"column":101,"line":73},"start":{"byte":2136,"column":0,"line":73}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u},\n  @Function.Injective.{u + 1, u + 1} (Set.{u} α) (α → Prop)\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α))","statement_sha256":"13387af85b9dbf2d6ff1338dcb6a9f15695c750b4857955981d5e342fce37edc","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-90673a72844fa3a11d12c32c48a0ef15a98669642d9f3adb2ae1cf9b36a8803b","qualified_name":"Set.mem_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":26467,"column":14,"line":777},"start":{"byte":26362,"column":0,"line":776}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {x : α} {a b : Set.{u} α},\n  @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) a x →\n    @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) b x →\n      @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n        (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) a b) x","statement_sha256":"ba66f1e3bc10eea9a5355709a81c298b62f8cdb817c7962eab0a59ce3fbd669f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-817687b079a79b80d31b1b07f5cbbb81d3c3be2a135052d93b59508d27c062d1","qualified_name":"Set.mem_inter_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":26360,"column":9,"line":774},"start":{"byte":26227,"column":0,"line":772}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (x : α) (a b : Set.{u} α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) a b) x)\n    (And (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) a x)\n      (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) b x))","statement_sha256":"ef04d5d14d30806c8326fccfcff3b323c04518acdb5c9539b326482b820333a0","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4c4dfdb69191573ffa2fb8703a0466df89dc34e60af01ceb4ae13cbf5d44f8f8","qualified_name":"Set.mem_ite_empty_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":37620,"column":56,"line":1095},"start":{"byte":37435,"column":0,"line":1092}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : Prop) [inst : Decidable p] (t : Set.{u} α) (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@ite.{u + 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x))","statement_sha256":"eb5989b48ac9b19589f1b56e482158338427dfd51fe6c9484b93955b7379ce7d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-82a9c748efbd1b786240c5c45c30509680081bc7dc7db80cd6ce906bb5c3ad80","qualified_name":"Set.mem_ite_empty_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":37116,"column":57,"line":1085},"start":{"byte":36931,"column":0,"line":1082}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (p : Prop) [inst : Decidable p] (t : Set.{u} α) (x : α),\n  Iff\n    (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@ite.{u + 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(@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) a b) x →\n    @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) a x","statement_sha256":"a566760a5d21c8aed0da68dfce1ef58e030f5e3f6ad21d37448685b1dea39153","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-535b884aff2373dcdf330af7fa2c09f906fa4f1db1d2d4df12fc25d42c374831","qualified_name":"Set.mem_of_mem_inter_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":26663,"column":9,"line":783},"start":{"byte":26566,"column":0,"line":782}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {x : α} {a b : Set.{u} α},\n  @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) a b) x →\n    @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) b 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   @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) t x","statement_sha256":"66418336b064b5b98d6d63710a78f8d30bb82c4cba4fa424df663be82de2d379","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-65835cae6c90d1f34cacecd79609b9cb87e12e53ac4c39e8c587fe69f6786db1","qualified_name":"Set.mem_of_subset_of_mem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":9045,"column":6,"line":276},"start":{"byte":8919,"column":0,"line":275}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s₁ s₂ : Set.{u} α} {a : α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s₁ s₂ →\n    @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) s₁ a →\n      @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) s₂ a","statement_sha256":"256546a23f4b0f551a9d0baa7e19ddb967a3034f33108c5aed84116ae1499a7d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-1b665bd980d5454b4dd9f49354bfb9f5e11875916da7b711b56f2a6ba321b073","qualified_name":"Set.mem_or_mem_of_mem_union","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":21712,"column":3,"line":636},"start":{"byte":21608,"column":0,"line":635}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {x : α} {a b : Set.{u} α},\n  @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) a b) x →\n    Or (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) a x)\n      (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) b 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(Set.{u} α) (@Set.instMembership.{u} α) b x →\n    @Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) a b) 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@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) (@Set.ofPred.{u} α fun a => p a) (@Set.ofPred.{u} α fun a => q a)","statement_sha256":"7fbb040250413461c230e8251760da6378f23dbae593435a0393ae728ede41dc","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f5ea7c5384a8827b015071d0c39880848a9adc8703fda904c2951599dd621d2c","qualified_name":"Set.setOf_top","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":18898,"column":29,"line":559},"start":{"byte":18831,"column":0,"line":558}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u},\n  @Eq.{u + 1} (Set.{u} α)\n    (@Set.ofPred.{u} α fun _x 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u}, @Eq.{u + 1} (Set.{u} α) (@Set.ofPred.{u} α fun _x => True) (@Set.univ.{u} α)","statement_sha256":"2d5dbe4e06d7ed0db8272ad342e9861b8fb80d3e92eff9880f301a673aa20565","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b36357ae5ce31fe2c4569cfff9a5b1fdb840184a85ab2a0d695e30522aad6cf4","qualified_name":"Set.ssubset_def","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":8133,"column":5,"line":250},"start":{"byte":8056,"column":0,"line":248}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  @Eq.{1} Prop\n    (@LT.lt.{u} (Set.{u} α)\n      (@Preorder.toLT.{u} (Set.{u} α)\n        (@PartialOrder.toPreorder.{u} (Set.{u} α)\n          (@SemilatticeInf.toPartialOrder.{u} (Set.{u} α)\n            (@Lattice.toSemilatticeInf.{u} (Set.{u} α)\n              (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))))\n      s t)\n    (And (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s t) (Not (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t 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s)))","statement_sha256":"6b065b5d0380f715c3db9d6a2945593522727d1e823f126d3862838237c2b2cb","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-743e57aaf74d0984ecaa53a735e094b739608220bf39dee7f8d14294c3671caa","qualified_name":"Set.ssubset_iff_of_subset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":10265,"column":82,"line":308},"start":{"byte":10086,"column":0,"line":307}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s t →\n    Iff\n      (@LT.lt.{u} (Set.{u} α)\n        (@Preorder.toLT.{u} (Set.{u} 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(Set.{u} α) (@Set.instLE.{u} α) s\n      (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)))\n    (@Eq.{u + 1} (Set.{u} α) s (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} 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Set.{u} α}, Iff (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s t) (∀ ⦃a : α⦄, a ∉ t → a ∉ s)","statement_sha256":"dd983838f1d506c7fdfd8f22d51d88516aa1dffdafbca0afe22cf981a64c0aaf","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-de14ae33842c7e546191f7802b48f221f8fc7a40a342a0dac9033872ab095ef5","qualified_name":"Set.subset_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":27895,"column":18,"line":822},"start":{"byte":27779,"column":0,"line":821}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t r : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) r s 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@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t)","statement_sha256":"7194e034330529ae62d7f83131003d81e2e6dce643e678304fc967bd451a3da6","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-abab0a014f794a60571d05528e837fa7e2e688fc957211193ea5933be6f83460","qualified_name":"Set.subset_union_of_subset_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":24333,"column":27,"line":715},"start":{"byte":24209,"column":0,"line":714}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) 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(@Lattice.toSemilatticeSup.{u} (Set.{u} α)\n            (@DistribLattice.toLattice.{u} (Set.{u} α) (@Set.instDistribLattice.{u} α))))\n        x1 x2)\n    fun x1 x2 => @Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) x1 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(@Set.instUnion.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) a b) c)\n    (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) a (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) b 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(@Set.instLE.{u} α) u (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t) →\n      @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t)\n        (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u)","statement_sha256":"12c21958dc34571ce8930f36e60e6d8e354f8204a43cf4240f5fc378e7260a05","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-19ec2e1776c4cdf2e165cc71311ef51bc14c1ecd8cc7586bf8ab1261bbf01e7a","qualified_name":"Set.union_congr_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":24691,"column":23,"line":724},"start":{"byte":24577,"column":0,"line":723}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t u : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) t u) →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u) →\n      @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u)\n        (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) t u)","statement_sha256":"7009966f19fa8b07f37b80ae59cc04459a3f4255907055cf12f62d031c574402","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-18b24d63522e38c5dd4b0e43676fb11a22a8746a51a4efb3eb7b86b37d0c6b2d","qualified_name":"Set.union_def","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":21407,"column":5,"line":627},"start":{"byte":21308,"column":0,"line":626}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s₁ s₂ : Set.{u} α},\n  @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ s₂)\n    (@Set.ofPred.{u} α fun a =>\n      Or (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) s₁ a)\n        (@Membership.mem.{u, u} α (Set.{u} α) (@Set.instMembership.{u} α) s₂ a))","statement_sha256":"1127711708aadd41928646b707691237771bb705c0f00e30ddb86dfb357681cd","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e9a54ce936b90926f86714e927c3e291be6788aa5716edcf0a6c7b59efde853d","qualified_name":"Set.union_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":22178,"column":37,"line":652},"start":{"byte":22081,"column":0,"line":650}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (a : Set.{u} α),\n  @Eq.{u + 1} (Set.{u} α)\n    (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) a\n      (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)))\n    a","statement_sha256":"48b02d2ca91060ce7a057765d0ff6eda6aee8b39591ecdf5bb1bb16a51896c9c","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-49d71f1c25301309e659dbbbeed2ee04fa1a197abc623309a594654eaa1c3db0","qualified_name":"Set.union_empty_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":25411,"column":24,"line":743},"start":{"byte":25259,"column":0,"line":740}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  Iff\n    (@Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t)\n      (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)))\n    (And (@Eq.{u + 1} (Set.{u} α) s (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α)))\n      (@Eq.{u + 1} (Set.{u} α) t (@EmptyCollection.emptyCollection.{u} (Set.{u} α) (@Set.instEmptyCollection.{u} α))))","statement_sha256":"9e5512c03a6baa14cb79b83d50279ac6bce13e0fc9ec536a70bca122bca8c1e6","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ba3a45496078232dd804c4b6dda8e7638418084673576f316c1f42db02f35c11","qualified_name":"Set.union_eq_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":22998,"column":13,"line":678},"start":{"byte":22911,"column":0,"line":676}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  Iff (@Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t) s)\n    (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t s)","statement_sha256":"64b88013323f664ab0ec470687f1c057ccd64dd1d640cdec6c3ecfd14d35c790","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-62cd01971b15baca7e455c1840302cf4a8a935498d5d680c6f7c0e21802e25c1","qualified_name":"Set.union_eq_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":23089,"column":14,"line":682},"start":{"byte":23000,"column":0,"line":680}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  Iff (@Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t) t)\n    (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s t)","statement_sha256":"6738f4bd2dfbd5e1208cce0a1a6b16822425ed10a31cceb2308375850b309703","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-cfed998cf497285e1d3869c50fb0859551a3bab82575c47a07b275b157ab0917","qualified_name":"Set.union_eq_self_of_subset_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":23196,"column":22,"line":685},"start":{"byte":23091,"column":0,"line":684}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s t →\n    @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t) t","statement_sha256":"37c96889296ca7fa9e6852c30eb6f295b1797a7f5d1a97ad8548bea4eec80b64","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3ed228de6e5a2e3d90fb29d9bd9fc5779730828136e13d81170c6ae3beebc662","qualified_name":"Set.union_eq_self_of_subset_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":23303,"column":21,"line":688},"start":{"byte":23198,"column":0,"line":687}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t s →\n    @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t) s","statement_sha256":"4eb80aab2c3a629ed0861ba55d0f123be8dac935bb552294c6f5df042feccd4f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4de3d44e43aa3956b1ac67e3c7409624a14fbe47c6568eefc50576d6884f7716","qualified_name":"Set.union_eq_union_iff_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":24805,"column":21,"line":727},"start":{"byte":24693,"column":0,"line":726}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t u : Set.{u} α},\n  Iff\n    (@Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u))\n    (And (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u))\n      (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) u (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t)))","statement_sha256":"74f4bd7ed36a12d1ac5f1bf20ca086c1758e89a922c05dc101404adc36d34c40","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ee19492bb828169bb931a53b17f59ca918afafda1d521fa664aab8350d3ee730","qualified_name":"Set.union_eq_union_iff_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":24921,"column":22,"line":730},"start":{"byte":24807,"column":0,"line":729}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t u : Set.{u} α},\n  Iff\n    (@Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) t u))\n    (And (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) t u))\n      (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u)))","statement_sha256":"59ec16191328d2faf47715b8b8e1be524f9b998afc1e7c87708cb6515c9a72fc","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e9de7547a7c16f8ba76bb1bb4f8c5671e1c54c2faf6115ff2161923af6312a08","qualified_name":"Set.union_eq_union_mono_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":25088,"column":27,"line":734},"start":{"byte":24923,"column":0,"line":732}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {t u s₁ s₂ : Set.{u} α},\n  @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ t)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ u) →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s₁ s₂ →\n      @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₂ t)\n        (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₂ 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(@Set.instUnion.{u} α) t u₁) →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) u₁ u₂ →\n      @Eq.{u + 1} (Set.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s u₂)\n        (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) t 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(@Set.instUnion.{u} α) s (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) t u))\n    (@Inter.inter.{u} (Set.{u} α) (@Set.instInter.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s 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(@Set.instUnion.{u} α) s₂ s₃))\n    (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₂ (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ s₃))","statement_sha256":"1d084f9b26f979c6004d563420acb624df74c9c909f5173cfc2937d6c024a606","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-cf69479fb84f83114ff366a09b5377404d657c3ccd860d0e3ab34c4caff10a1b","qualified_name":"Set.union_nonempty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":12754,"column":11,"line":377},"start":{"byte":12658,"column":0,"line":375}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t : Set.{u} α},\n  Iff (@Set.Nonempty.{u} α (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t))\n    (Or (@Set.Nonempty.{u} α s) (@Set.Nonempty.{u} α t))","statement_sha256":"26e38b2e6e20c65e9907c09e84f42f2f1c5f4a7b372787cd2bf9d492d55852f7","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3a8eed2bd2c4359f9b90de15b4f059f38e99fb4da71a9183821437ef35435423","qualified_name":"Set.union_right_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":22909,"column":28,"line":674},"start":{"byte":22777,"column":0,"line":673}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (s₁ s₂ s₃ : Set.{u} α),\n  @Eq.{u + 1} (Set.{u} α)\n    (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ s₂) s₃)\n    (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ s₃) 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a","statement_sha256":"cc2743b875fccf94cf0bfe7ea35cf3a5fab612c329f0ef0c65e7ac2158fda704","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b6d1af111b149299db08abf12afbd429a72519bb7b0e45f2c5f25955da2bd44b","qualified_name":"Set.union_subset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":23596,"column":24,"line":697},"start":{"byte":23476,"column":0,"line":696}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s t r : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s r →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t r →\n      @LE.le.{u} (Set.{u} α) 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(@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t) u)\n    (And (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s u) (@LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t u))","statement_sha256":"97d3aa05cfd08c87a44da07c25d228f5d79e081adcf94f0ebe1f133a29fe910e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-abd9848b7ee01f61412ac3706a80df84494e1e8aab1c95186fb808e39258527e","qualified_name":"Set.union_subset_union","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":23920,"column":22,"line":706},"start":{"byte":23743,"column":0,"line":703}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} {s₁ s₂ t₁ t₂ : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) s₁ s₂ →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t₁ t₂ →\n      @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ t₁)\n        (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₂ 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(@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₁ t)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s₂ t)","statement_sha256":"361cf9ced568fa9f40c18c87fd4fdce04e3d037f6b2145e20577a421f8c5385c","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a7fe400c0b57fd9cf761926d686c45d74f13f5960ddb4a5825a5b0aa9750a2fb","qualified_name":"Set.union_subset_union_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Set.Basic","path":"Mathlib/Data/Set/Basic.lean","range":{"end":{"byte":24207,"column":33,"line":712},"start":{"byte":24065,"column":0,"line":711}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u} (s : Set.{u} α) {t₁ t₂ : Set.{u} α},\n  @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) t₁ t₂ →\n    @LE.le.{u} (Set.{u} α) (@Set.instLE.{u} α) (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s t₁)\n      (@Union.union.{u} (Set.{u} α) (@Set.instUnion.{u} α) s 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