Cardinal Class is Proper Class

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Theorem

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


Proof

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Suppose $\mathcal N$ is a small class.

By Cardinal Class is Subset of Ordinal Class:

$\mathcal N \subseteq \operatorname{On}$

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 Set is 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.

$\blacksquare$


Sources