# Cardinal Number is Ordinal

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## Theorem

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

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

Then:

- $\left|{S}\right| \in \operatorname{On}$

where $\operatorname{On}$ denotes the class of ordinals.

## 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 $S$, by the definition of cardinal number.

Thus, it is an ordinal.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.8$