#[local] Infix " \in " := E.In.
#[local] Infix " \subseteq " := E.isSubsetOf.
#[local] Notation In := List.In.
#[local] Hint Resolve E.insert_monotonic : core.
Proof with (simpl in *; tauto) || eauto with *.
ii. eapply ENTAILS1; done!.
Qed.

Lemma ImplicationI_preserves (Gamma : ensemble formula) (A : formula) (B : formula)
(ENTAILS1 : E.insert A Gamma ⊨ B)
: Gamma ⊨ ImplicationF A B.