Proof
fun (X Y : Ens) (P : Ens -> Prop) (e : EQ X Y) (p : P Y) =>
Big_Extensionality P Y X p (EQ_symm X Y e).
(*** a few conclusions using Big_Extensionality ***)
Section zero_different_from_one.
Variable X : Ens.
Variable Y : Ens.
Hypothesis m : has_leq_0_elements X.
Hypothesis n : has_geq_1_elements Y.
Hypothesis h : EQ X Y.
Let step1 : has_leq_0_elements Y :=
Big_Extensionality has_leq_0_elements X Y m h.
Definition zero_different_from_one_Out : False := n step1.
End zero_different_from_one.

(* Specifications: *)
Theorem counting_lem1 :
forall (X Y : Ens) (m : has_leq_0_elements X) (n : has_geq_1_elements Y),
~ EQ X Y.