Cardinal of Cardinal Equal to Cardinal/Corollary

Theorem

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

Let $x$ be an ordinal.

Then:

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

Proof

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 $x \in \NN$.

$\Box$

Sufficient Condition

 $\ds x$ $\in$ $\ds \NN$ by hypothesis $\ds \leadsto \ \$ $\ds x$ $=$ $\ds \card y$ Definition of Class of Cardinals $\ds \leadsto \ \$ $\ds \card x$ $=$ $\ds \card y$ Cardinal of Cardinal Equal to Cardinal $\ds \leadsto \ \$ $\ds x$ $=$ $\ds \card x$ Equality is Transitive

$\blacksquare$