Infinite Cardinal Class is Proper Class

Theorem
The infinite cardinal class $\NN’$ is a proper class.

Proof
$\NN’$ is a small class.

By Union of Small Classes is Small, $\NN’ \cup \omega$ is a small class.

By definition of the infinite cardinal class, $\NN \subseteq \NN’ \cup \omega$.

But by Axiom of Subsets Equivalents, this means that $\NN$ is a small class.

This contradicts Cardinal Class is Proper Class.

Therefore, $\NN’$ is not a small class.