Proof doubleton_th3_Out.
Section doubleton_th4_proof.
Variable X : Ens.
Variable Y : Ens.
Variable Z : Ens.
Hypothesis k : IN Z (Doubleton X Y).
Hypothesis m : ~ EQ Z X.
Hypothesis n : ~ EQ Z Y.
Let D := Doubleton X Y.
Let sX := Singleton X.
Let sY := Singleton Y.
Section doubleton_th4_sub1.
Hypothesis u : IN Z sX.
Let step1_1 : EQ X Z := Singleton_pr2 X Z u.
Let step1_2 : EQ Z X := EQ_symm X Z step1_1.
Definition doubleton_th4_sub1_Out : False := m step1_2.
End doubleton_th4_sub1.
Let step1 : ~ IN Z sX := doubleton_th4_sub1_Out.
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.