Cardinal of Cardinal Equal to Cardinal/Corollary

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

Let $x$ be an ordinal.

Then:


 * $x \in \mathcal N \iff x = \left|{ x }\right|$

Necessary Condition
Suppose $x = \left|{ x }\right|$.

Then $x = \left|{ y }\right|$ for some $y$ by Existential Generalisation.

By definition of the cardinal class:


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

It follows that $x \in \mathcal N$.