Cardinal of Cardinal Equal to Cardinal

Theorem
Let $S$ be a set such that $S \sim x$ for some ordinal $x$.

If the axiom of choice holds, then this condition holds for any set.

Then:


 * $| \left({ |S| }\right) | = |S|$

Proof
By Condition for Set Equivalent to Cardinal Number, $S \sim |S|$

Therefore, by Equivalent Sets have Equal Cardinal Numbers:


 * $|S| = | \left({ |S| }\right) |$