Condition for Set Equivalent to Cardinal Number

Theorem
Let $S$ be a set.

Let $\vert S \vert$ denote the cardinal number of $S$.

Then the following are equivalent:


 * 1) $S \sim \vert S \vert$
 * 2) $\exists x \in \operatorname{On}: S \sim x$
 * 3) $\exists x \in \operatorname{On}: \exists y: \left({ y \subseteq x \land S \sim y }\right)$

$2 \implies 1$
If $\exists x \in \operatorname{On}: S \sim x$, then there is a smallest ordinal $x$ such that $S \sim x$.

This smallest ordinal $x$ is the cardinal number of $S$, by definition.

$3 \implies 2$
Suppose that $y \subseteq x$ and $S \sim y$ for some ordinal $x$.

Since $y \subseteq x$, it follows that $y \sim z$ for some $z \in \operatorname{On}$ by Unique Isomorphism between Ordinal Subset and Unique Ordinal.

Therefore, by Set Equivalence an Equivalence Relation, $S \sim z$.

$1 \implies 3$
Suppose that $3$ is not true.

It follows that $S \not \sim x$ for any ordinal $x$.

But $S \not \sim \mathbb U$, so $S \not \sim \vert S \vert$ by the definition of cardinal number.

The result follows by contraposition.