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

where $\left|{S}\right|$ denotes the cardinal number of $S$.

Proof
By Condition for Set Equivalent to Cardinal Number:
 * $S \sim \left|{S}\right|$

Therefore, by Equivalent Sets have Equal Cardinal Numbers:


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