Theorems · Theorem · logic and foundations
FirstOrder.Language.Theory.IsComplete.models_not_iff
∀ {L : FirstOrder.Language} {T : L.Theory},
T.IsComplete → ∀ (φ : L.Sentence), T ⊨ᵇ FirstOrder.Language.Formula.not φ ↔ ¬T ⊨ᵇ φ- Defined in
- Mathlib.ModelTheory.Satisfiability
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- FirstOrder.Languagestatement and proof · cited by 1,084
- Nonempty.someproof · cited by 340
- FirstOrder.Language.Theorystatement and proof · cited by 154
- FirstOrder.Language.Sentencestatement and proof · cited by 127
- FirstOrder.Language.Theory.ModelType.Carrierproof · cited by 59
- FirstOrder.Language.Theory.ModelsBoundedFormulastatement and proof · cited by 30
- FirstOrder.Language.Formula.notstatement and proof · cited by 27
- FirstOrder.Language.Theory.IsCompletestatement and proof · cited by 11
- FirstOrder.Language.Theory.models_sentence_iffproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Theory.IsComplete.realize_sentence_iffproof · cited by 3
- FirstOrder.Field.ACF_zero_realize_iff_infinite_ACF_prime_realizeproof · cited by 2