Cardinal of Cardinal Equal to Cardinal/Corollary

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

Let $x$ be an ordinal.

Then:


 * $x \in \NN \iff x = \card x$

Necessary Condition
Suppose $x = \card x$.

Then $x = \card y$ for some $y$ by Existential Generalisation.

By definition of cardinal class:


 * $\NN = \set {x \in \On: \exists y: x = \card y}$

It follows that $x \in \NN$.