Cardinal Number Equivalence or Equal to Universe

Theorem
Let $S$ be a set.

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

Let $\mathbb U$ denote the universal class.

Then:


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

Proof
By Condition for Set Equivalent to Cardinal Number:

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


 * $S \sim \card S$

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

And thus by the definition of cardinal number:
 * $\card S = \mathbb U$