#[local] Infix " \in " := E.In.
#[local] Infix " \subseteq " := E.isSubsetOf.
#[local] Notation In := List.In.
#[local] Hint Resolve E.insert_monotonic : core.

Lemma NegationE_preserves (Gamma : ensemble formula) (A : formula)
(ENTAILS1 : E.insert (NegationF A) Gamma ⊨ ContradictionF)
: Gamma ⊨ A.