Definition Its_empty (A : Ens) := forall X : Ens, ~ IN X A.
Section its_empty_implies_equals_emptyset_proof.
Variable A : Ens.
Hypothesis k : Its_empty A.
Section ieieep1.
Variable X : Ens.
Hypothesis i : IN X A.
Let step1_1 : False.
Proof k X i.

Lemma ieieep1_Out : IN X EmptySet.
Proof false_implies_everything (IN X EmptySet) step1_1.