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Theorems · Theorem · logic and foundations

FirstOrder.Language.MeetsDefinable.isElementary_closure

∀ {L : FirstOrder.Language} {M : Type u_1} [inst : L.Structure M] {A : Set M},
  FirstOrder.Language.MeetsDefinable A → ((FirstOrder.Language.Substructure.closure L).toFun A).IsElementary

The closure of a set meeting definable sets is an elementary substructure.

Defined in
Mathlib.ModelTheory.ElementarySubstructures
Cited by
0 results in Mathlib
Foundations
Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
FirstOrder.Language.Structure
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FirstOrder.Language.MeetsDefinable.toElementarySubstructure · cited by 0MeetsDefinable.toElementa…Set · cited by 53352SetSetLike.coe · cited by 8199SetLike.coeSet.Elem · cited by 7166Set.ElemSet.ofPred · cited by 6101Set.ofPredSet.Nonempty · cited by 2627Set.NonemptySet.ext · cited by 2266Set.extFirstOrder.Language · cited by 1084FirstOrder.LanguageFirstOrder.Language.Structure · cited by 775Language.StructureFirstOrder.Language.Substructure · cited by 242Language.SubstructureFirstOrder.Language.BoundedFormula · cited by 207Language.BoundedFormulaFin.snoc · cited by 113Fin.snocLowerAdjoint.toFun · cited by 105LowerAdjoint.toFunFirstOrder.Language.BoundedFormula.Realize · cited by 104BoundedFormula.RealizeFirstOrder.Language.Term.realize · cited by 81Term.realizeFirstOrder.Language.Substructure.closure · cited by 70Substructure.closureMeetsDefinable.isElementary_c…CITED BYCITES

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