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 cardinal number of $S$.


Then:

$\card S \in \On$

where $\On$ denotes the class of 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 minimal element is the cardinal number of $S$, by the definition of cardinal number.


Thus, it is an ordinal.

$\blacksquare$


Sources