Section doubleton_th4_sub2.
Hypothesis u : IN Z sY.
Let step2_1 : EQ Y Z := Singleton_pr2 Y Z u.
Let step2_2 : EQ Z Y := EQ_symm Y Z step2_1.
Definition doubleton_th4_sub2_Out : False := n step2_2.
End doubleton_th4_sub2.
Let step2 : ~ IN Z sY := doubleton_th4_sub2_Out.
Let step3 : ~ IN Z D := union_th3 sX sY Z step1 step2.
Definition doubleton_th4_Out : False := step3 k.
End doubleton_th4_proof.

Theorem Doubleton_th4 :
forall (X Y Z : Ens) (k : IN Z (Doubleton X Y)) (m : ~ EQ Z X)
(n : ~ EQ Z Y), False.