Section first_uniqueness.
Variable A : Ens.
Variable B : Ens.
Hypothesis m : EQ (First A) (First B).
Let F := First A.
Let G := First B.
Let FsubG : SUB F G := EQ_forwards F G m.
Let iAF : IN A F := Singleton_pr1 A.
Let iBF : IN A G := FsubG A iAF.
Let step1 : EQ B A := Singleton_pr2 B A iBF.
Definition first_uniqueness_Out : EQ A B := EQ_symm B A step1.
End first_uniqueness.
Proof first_uniqueness_Out.

Theorem Singleton_uniqueness :
forall (A B : Ens) (m : EQ (Singleton A) (Singleton B)), EQ A B.