# Cardinal Number Less than Ordinal

## Theorem

Let $S$ be a set.

Let $\left|{S}\right|$ denote the cardinal number of $S$.

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

Then:

$\left|{S}\right| \le x$

### Corollary

Let $x$ be an ordinal.

Let $\left|{x}\right|$ denote the cardinal number of $x$.

Then:

$\left|{x}\right| \le x$

## Proof

Since $S \sim x$, it follows that:

$x \in \left\{{y \in \operatorname{On} : S \sim y}\right\}$

By Intersection is Subset: General Result, it follows that:

$\displaystyle \bigcap \left\{{y \in \operatorname{On} : S \sim y}\right\} \subseteq x$

Therefore $\left|{S}\right| \le x$ by the definition of cardinal number.

$\blacksquare$