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.

$\blacksquare$


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