Class of All Cardinals is Proper Class

Theorem
The class $\mathcal N$ of all cardinal numbers is a proper class.

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

$\mathcal N \subseteq \operatorname{On}$ by Cardinal Class Subset of Ordinal Class.

Therefore, $\bigcup \mathcal N$ is an ordinal by Union of Subset of Ordinals is Ordinal.

Take $x = \left\{ y \in \operatorname{On} : y \preccurlyeq \bigcup \mathcal N \right\}$.

By Cardinal Equal to Collection of All Dominated Ordinals, $x$ is a cardinal number and there is no injection $f : x \to \bigcup \mathcal N$.

But since $x$ is a cardinal number, $x \in \mathcal N$, so $x \subseteq \bigcup \mathcal N$ by Subset of Union/General Result.

Therefore, the identity map $I_x : x \to \bigcup \mathcal N$ is an injection.

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