Condition for Set Equivalent to Cardinal Number

Theorem
Let $S$ be a set.

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

That is, let $\left\vert{S}\right\vert$ be the intersection of all ordinals equivalent to $S$.

Note that in the absence of the Axiom of Choice, $\left\vert{S}\right\vert$ may be the class of all sets.

Then the following are equivalent:


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

where $A \sim B$ means that there is a bijection from $A$ onto $B$, and the quantification over $y$ is unbounded (so $y$ may be any set).

$2 \implies 1$
If $\exists x \in \operatorname{On}: S \sim x$, then by Ordinal Class is Strongly Well-Ordered by Subset 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 is 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 $S \not \sim \left\vert{S}\right\vert$ by the definition of cardinal number.

The result follows by contraposition.