Proof distinct_doubletons_have_geq_2_elements_Out.
(*******************************************)
(************* Big_Extensionality **********)
(*******************************************)
Section big_extensionality_proof.
Variable P : Ens -> Prop.
Variable X : Ens.
Variable Y : Ens.
Hypothesis m : P X.
Hypothesis k : EQ X Y.
Let D := Doubleton X Y.
Let iXD : IN X D := Doubleton_th1 X Y.
Let iYD : IN Y D := Doubleton_th2 X Y.
Let eYX : EQ Y X := EQ_symm X Y k.
Let S := Set_Of D P.
Let iXS : IN X S := Set_Of_th1 D P X iXD m.
Let iYS : IN Y S := Extensionality Y X S eYX iXS.
Definition big_extensionality_Out : P Y := Set_Of_th2 D P Y iYS.
End big_extensionality_proof.

Theorem Big_Extensionality :
forall (P : Ens -> Prop) (X Y : Ens) (p : P X) (e : EQ X Y), P Y.