# Cardinal Number is Ordinal

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## Theorem

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

Let $\card S$ denote the cardinality of $S$.

Then:

- $\card S \in \On$

where $\On$ denotes the class of all ordinals.

## Proof

If $S \sim x$, then $\set {x \in \On: S \sim x}$ is a non-empty set of ordinals.

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

This article, or a section of it, needs explaining.In particular: Needs a link to the fact that $\set {x \in \On: S \sim x}$ is the subset of a well-ordered set, and also that the intersection is that minimal element.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

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*: $\S 10.8$