Proof singletons_have_leq_1_elements_Out.
Definition has_geq_2_elements (A : Ens) := ~ has_leq_1_elements A.
Section when_given_two_distinct_elements.
Variable A : Ens.
Variable X : Ens.
Variable Y : Ens.
Hypothesis k : IN X A.
Hypothesis l : IN Y A.
Hypothesis m : ~ EQ X Y.
Hypothesis h : has_leq_1_elements A.
Definition when_given_two_distinct_elements_Out : False := m (h X Y k l).
End when_given_two_distinct_elements.

Theorem has_geq_2_elements_th1 :
forall (A X Y : Ens) (k : IN X A) (l : IN Y A) (m : ~ EQ X Y),
has_geq_2_elements A.