Theorems · Theorem · logic and foundations
FirstOrder.Language.Theory.CompleteType.setOfPred_subset_eq_empty_iff
∀ {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (S : (L.withConstants α).Theory),
{p | S ⊆ ↑p} = ∅ ↔ ¬((L.lhomWithConstants α).onTheory T ∪ S).IsSatisfiable- Defined in
- Mathlib.ModelTheory.Types
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
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 and proof · cited by 8,199
- Set.ofPredstatement and proof · cited by 6,101
- LE.le.transproof · cited by 3,151
- FirstOrder.Languagestatement and proof · cited by 1,084
- Nonempty.someproof · cited by 340
- FirstOrder.Language.Theorystatement and proof · cited by 154
- Set.subset_union_leftproof · cited by 142
- FirstOrder.Language.Sentencestatement · cited by 127
- Set.subset_union_rightproof · cited by 123
- FirstOrder.Language.withConstantsstatement and proof · cited by 108
- Set.union_subsetproof · cited by 71
Cited by3
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Theory.CompleteType.setOfPred_mem_eq_univ_iffproof · cited by 1
- FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_empty_iffproof · cited by 0
- FirstOrder.Language.Theory.CompleteType.nonempty_iffproof · cited by 0