Cardinal Number is Ordinal

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

Let $\vert S \vert$ denote the cardinal number of $S$.

Then:


 * $\vert S \vert \in \operatorname{On}$

Proof
If $S \sim x$, then $\left\{ x \in \operatorname{On} : S \sim x \right\}$ is a nonempty set of ordinals.

It follows that this set has a minimal element, its intersection.

This minimal element is the cardinal number of $S$.

Thus, it is an ordinal.