Section doubleton_th3_proof.
Variable X : Ens.
Variable Y : Ens.
Variable A : Ens.
Hypothesis iXA : IN X A.
Hypothesis iYA : IN Y A.
Let D := Doubleton X Y.
Let sXsubA : SUB (Singleton X) A := Singleton_th1 X A iXA.
Let sYsubA : SUB (Singleton Y) A := Singleton_th1 Y A iYA.
Definition doubleton_th3_Out : SUB D A :=
union_pr3 (Singleton X) (Singleton Y) A sXsubA sYsubA.
End doubleton_th3_proof.

Theorem Doubleton_th3 :
forall (X Y A : Ens) (iXA : IN X A) (iYA : IN Y A), SUB (Doubleton X Y) A.