Section Union_th1_proof.
Variable F : Ens.
Variable X : Ens.
Variable U : Ens.
Hypothesis h : IN X U.
Hypothesis k : IN U F.
Let step1 : nested_IN X U F.
Proof Build_nested_IN X U F h k.
Let step2 : EXISTS (fun U1 : Ens => nested_IN X U1 F).
Proof EXISTS_th1 (fun U1 : Ens => nested_IN X U1 F) U step1.
Proof Union_pr1 F X step2.
End Union_th1_proof.
Proof union_th1_OutA.
(************************************************)
(**********     Elementary Counting  ************)
(************************************************)
Definition has_leq_0_elements := Its_empty.
(*** recall that this is equal to [A: Ens] (X: Ens) (not (IN X A)) ***)

Theorem zero_has_leq_0_elements : has_leq_0_elements Zero.