Section doubleton_uniqueness_2.
Hypothesis k : EQ B C.
Let step1_1 : EQ C B := EQ_symm B C k.
(* Definition Big_Extensionality: (P: Ens-> Prop) (X,Y: Ens) (p: (P X)) (e : (EQ X Y)) (P Y) :=  big_extensionality_Out. *)
Let step1_2 : P B := Big_Extensionality P C B pC step1_1.
Definition doubleton_uniqueness_2_Out : False := nB step1_2.
End doubleton_uniqueness_2.
Let step3 : NEQ B C := doubleton_uniqueness_2_Out.
(* Definition Doubleton_th4_b : (X,Y: Ens) (Z: Ens) (k: (IN Z (Doubleton X Y))) (n: (not (EQ Z Y))) (EQ Z X) := doubleton_th4b_Out. th4a same but with NEQ Z X and conclusion EQ Z Y *)
Let conclusionAC : EQ A C := Doubleton_th4_b C D A iAdoub step2.
Let conclusionBD : EQ B D := Doubleton_th4_a C D B iBdoub step3.
Definition doubleton_uniqueness_Out : pairwise_EQ A B C D :=
Build_pairwise_EQ A B C D conclusionAC conclusionBD.
End doubleton_uniqueness_proof.

Theorem Doubleton_uniqueness :
forall (P : Ens -> Prop) (A B C D : Ens) (pA : P A)
(nB : ~ P B) (pC : P C) (nD : ~ P D)
(e : EQ (Doubleton A B) (Doubleton C D)), pairwise_EQ A B C D.