Cardinal of Cardinal Equal to Cardinal

Theorem
Let $S$ be a set such that $S$ is equivalent to its cardinal.

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

Then:
 * $\card {\paren {\card S} } = \card S$

where $\card S$ denotes the cardinal number of $S$.

Proof
By Condition for Set Equivalent to Cardinal Number:
 * $S \sim \card S$

Therefore, by Equivalent Sets have Equal Cardinal Numbers:


 * $\card S = \card {\paren {\card S} }$