. intro hp;
  suffices ~p ∉ Ω.theory by apply or_iff_not_imp_right.mp $ (either_mem Ω p); assumption;
  by_contra hC;
  have hnp : Ω.theory *⊢[L]! ~p := Context.by_axm! hC;
  have := hnp ⨀ hp;
  have := not_provable_falsum Ω.consistent;
  contradiction;
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p : Formula α
⊢ Ω.theory *⊢[L]! p → p ∈ Ω.theory
no goals