Section B_sub_V_proof1.
Hypothesis h3 : EQ X Z.
Let step1_1 : IN X B.
Proof Extensionality X Z B h3 h2.
Proof k step1_1.
End B_sub_V_proof1.
Let step2 : ~ EQ X Z. Proof B_sub_V_proof1_Out.
Let step3 : IN Z U.
Proof union_th2 A B Z h2.
Let step4 : union_th3_property Z.
Proof Build_union_th3_property Z step3 step2.
Definition B_sub_V_proof_Out : IN Z V := V_pr1 Z step4.
End B_sub_V_proof.
Let B_sub_V : SUB B V. Proof B_sub_V_proof_Out.
Let U_sub_V : SUB U V.
Proof union_pr3 A B V A_sub_V B_sub_V.
Let X_in_V : IN X V.
Proof U_sub_V X m.
Let step5 : union_th3_property X.
Proof V_pr2 X X_in_V.
Let step6 : ~ EQ X X.
Proof union_th3_property_not_equal_X X step5.
Let step7 : EQ X X. Proof EQ_refl X.
Proof step6 step7.
End union_th3_proof.

Theorem union_th3 :
forall (A B X : Ens) (h : ~ IN X A) (k : ~ IN X B), ~ IN X (union A B).