Cardinal Class is Subset of Ordinal Class

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Theorem

Let $\mathcal N$ denote the class of all cardinal numbers.

Let $\operatorname{On}$ denote the class of all ordinals.

Then:

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

Proof

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By definition of the cardinal class:

$\mathcal N = \left\{ x \in \operatorname{On} : \exists y: x = \left|{ y }\right| \right\}$

Every element of $\mathcal N$ is thus an element of $\operatorname{On}$.

$\blacksquare$