Section singletons_have_leq_1_elements_proof.
Variable A : Ens.
Let S := Singleton A.
Variable X : Ens.
Variable Y : Ens.
Hypothesis iXS : IN X S.
Hypothesis iYS : IN Y S.
Let step1 : EQ A X := Singleton_pr2 A X iXS.
Let step2 : EQ A Y := Singleton_pr2 A Y iYS.
Let step3 : EQ X A := EQ_symm A X step1.
Definition singletons_have_leq_1_elements_Out : EQ X Y :=
EQ_trans X A Y step3 step2.
End singletons_have_leq_1_elements_proof.

Theorem Singletons_have_leq_1_elements :
forall A : Ens, has_leq_1_elements (Singleton A).