Section distinct_doubletons_have_geq_2_elements_proof.
Variable X : Ens.
Variable Y : Ens.
Hypothesis n : ~ EQ X Y.
Let D := Doubleton X Y.
Hypothesis h : has_leq_1_elements D.
Let iXD : IN X D := Doubleton_th1 X Y.
Let iYD : IN Y D := Doubleton_th2 X Y.
Let step1 : EQ X Y := h X Y iXD iYD.
Definition distinct_doubletons_have_geq_2_elements_Out : False := n step1.
End distinct_doubletons_have_geq_2_elements_proof.

Theorem distinct_Doubletons_have_geq_2_elements :
forall (X Y : Ens) (n : ~ EQ X Y), has_geq_2_elements (Doubleton X Y).