Cardinal Number Less than Ordinal

Theorem
Let $S$ be a set.

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

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

Then:


 * $\card S \le x$

Proof
Since $S \sim x$, it follows that:
 * $x \in \set {y \in \On : S \sim y}$

By Intersection is Subset: General Result, it follows that:
 * $\ds \bigcap \set {y \in \On: S \sim y} \subseteq x$

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