Class of Infinite Cardinals is Proper Class

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

Proof
Assume $\mathcal N’$ is a small class.

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

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

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

This contradicts Cardinal Class is Proper Class.

Therefore, $\mathcal N’$ is not a small class.