Theorems · Theorem · logic and foundations
FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_empty_iff
Deprecated since 2026-07-09Use FirstOrder.Language.Theory.CompleteType.setOfPred_subset_eq_empty_iff instead.
∀ {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (S : (L.withConstants α).Theory),
{p | S ⊆ ↑p} = ∅ ↔ ¬((L.lhomWithConstants α).onTheory T ∪ S).IsSatisfiableAlias of FirstOrder.Language.Theory.CompleteType.setOfPred_subset_eq_empty_iff.
- Defined in
- Mathlib.ModelTheory.Types
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- SetLike.coestatement · cited by 8,199
- Set.ofPredstatement · cited by 6,101
- FirstOrder.Languagestatement · cited by 1,084
- FirstOrder.Language.Theorystatement · cited by 154
- FirstOrder.Language.Sentencestatement · cited by 127
- FirstOrder.Language.withConstantsstatement · cited by 108
- FirstOrder.Language.lhomWithConstantsstatement · cited by 39
- FirstOrder.Language.Theory.CompleteTypestatement · cited by 35
- FirstOrder.Language.LHom.onTheorystatement · cited by 27
- FirstOrder.Language.Theory.IsSatisfiablestatement · cited by 25
- FirstOrder.Language.Theory.CompleteType.setOfPred_subset_eq_empty_iffproof · cited by 3
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