Cardinal Number Equivalence or Equal to Universe

Theorem
Let $S$ be a set.

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

Let $\mathbb U$ denote the universal class.

Then:


 * $S \sim \vert S \vert \lor \vert S \vert = \mathbb U$

Proof
By Condition for Set Equivalent to Cardinal Number, it follows that:

If $\exists x \in \operatorname{On}: S \sim x$, then:


 * $S \sim \vert S \vert$

If $\neg \exists x \in \operatorname{On}: S \sim x$, then:

And thus $\vert S \vert = \mathbb U$ by the definition of cardinal number.