Ordinal Number Equivalent to Cardinal Number

Theorem
Let $x$ be an ordinal.

Let $\left\vert{x}\right\vert$ denote the cardinal number of $x$.

Then:


 * $x \sim \left\vert{x}\right\vert$

where $\sim$ denotes set equivalence.

Proof
From Set Equivalence is Equivalence Relation:
 * $x \sim x$

Therefore, $x$ is equivalent to some ordinal.

By Condition for Set Equivalent to Cardinal Number:
 * $x \sim \left\vert{x}\right\vert$