Cardinal Number Less than Ordinal

Theorem
Let $S$ be a set.

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

Let $x$ be an ordinal such that $S \sim x$.

Then:


 * $\vert S \vert \le x$.

Proof
Since $S \sim x$, it follows that $x \in \left\{ y \in \operatorname{On} : S \sim y \right\}$.

By Intersection Subset/General Result, it follows that $\bigcap \left\{ y \in \operatorname{On} : S \sim y \right\} \le x$.

Therefore, $\vert S \vert \le x$ by the definition of cardinal number.