Section Bounded_NN_proof.
Hypothesis u : DOESNT_EXIST (fun A : Ens => Bounded_By A Element_of_NN).
Let step1 : forall A : Ens, ~ Bounded_By A Element_of_NN := u.
Let B : Ens := Set_containing_NN.
Let v : Contains_NN B := Set_containing_NN_pr.
Section Bounded_NN_proof2.
Variable Z : Ens.
Hypothesis w : Element_of_NN Z.
Definition Bounded_NN_proof2_Out : IN Z B := w B v.
End Bounded_NN_proof2.
Let step2 : Bounded_By B Element_of_NN := Bounded_NN_proof2_Out.
Definition Bounded_NN_proof_Out : False := step1 B step2.
End Bounded_NN_proof.

Theorem Bounded_NN : Bounded Element_of_NN.
Proof Bounded_NN_pre.