Proof
fun (F : Ens -> Ens) (X Y : Ens) (i : IN X (Intersection F)) =>
Intersection_pr2 F X i Y.
(********************************)
(********** Singletons **********)
(********************************)
Section Singleton_th1_proof.
Variable X : Ens.
Variable A : Ens.
Hypothesis h : IN X A.
Let B := Singleton X.
(* now prove SUB B A *)
Section Singleton_th1_sub1.
Variable C : Ens.
Hypothesis h2 : IN C B.
Let step1 : EQ X C.
Proof Singleton_pr2 X C h2.
Let step2 : EQ C X.
Proof EQ_symm X C step1.
Definition Singleton_th1_sub1_Out : IN C A := Extensionality C X A step2 h.
End Singleton_th1_sub1.
Definition Singleton_th1_proof_Out : SUB B A := Singleton_th1_sub1_Out.
End Singleton_th1_proof.

Theorem Singleton_th1 : forall (X A : Ens) (h : IN X A), SUB (Singleton X) A.
Proof Singleton_th1_sub1_Out.