Theorem: LO.Modal.Standard.MaximalParametricConsistentTheory.neg_imp (Logic/Modal/Standard/Kripke/Completeness.lean)

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
⊢ ~p ∈ Ω₁.theory → ~q ∈ Ω₂.theory
Tactic:
contrapose
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
⊢ ~q ∉ Ω₂.theory → ~p ∉ Ω₁.theory

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
⊢ ~q ∉ Ω₂.theory → ~p ∉ Ω₁.theory
Tactic:
intro hq
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ ~p ∉ Ω₁.theory

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ ~p ∉ Ω₁.theory
Tactic:
apply iff_mem_neg.mp
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ ~~p ∈ Ω₁.theory

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ ~~p ∈ Ω₁.theory
Tactic:
apply mem_dn_iff.mp
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ p ∈ Ω₁.theory

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ p ∈ Ω₁.theory
Tactic:
apply h
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ q ∈ Ω₂.theory

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p q : Formula α
h : q ∈ Ω₂.theory → p ∈ Ω₁.theory
hq : ~q ∉ Ω₂.theory
⊢ q ∈ Ω₂.theory
Tactic:
exact mem_dn_iff.mpr $ iff_mem_neg.mpr hq
State after:
no goals