Theorems · Theorem · logic and foundations
FirstOrder.Language.Theory.models_iff_not_satisfiable
∀ {L : FirstOrder.Language} {T : L.Theory} (φ : L.Sentence),
T ⊨ᵇ φ ↔ ¬(T ∪ {FirstOrder.Language.Formula.not φ}).IsSatisfiable- Defined in
- Mathlib.ModelTheory.Satisfiability
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Set.mem_singletonproof · cited by 183
- FirstOrder.Language.Theorystatement and proof · cited by 154
- Set.subset_union_leftproof · cited by 142
- FirstOrder.Language.Sentencestatement and proof · cited by 127
- Set.subset_union_rightproof · cited by 123
- FirstOrder.Language.Formulastatement · cited by 93
- FirstOrder.Language.Sentence.Realizeproof · cited by 62
- FirstOrder.Language.Theory.ModelTypeproof · cited by 62
- FirstOrder.Language.Theory.ModelType.Carrierproof · cited by 59
- FirstOrder.Language.Theory.ModelsBoundedFormulastatement · cited by 30
Cited by4
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Theory.ModelsBoundedFormula.realize_sentenceproof · cited by 5
- FirstOrder.Language.Theory.IsMaximal.mem_of_modelsproof · cited by 2
- FirstOrder.Language.Theory.CompleteType.setOfPred_mem_eq_univ_iffproof · cited by 1
- FirstOrder.Language.Theory.CompleteType.mem_of_modelsproof · cited by 0