Cardinal Number Equivalence or Equal to Universe

Theorem
Let $S$ be a set.

Let $\operatorname{Card} \left({ S }\right)$ denote the cardinal number of $S$.

Let $\mathbb U$ denote the universal class.

Then:


 * $S \sim \operatorname{Card} \left({ S }\right) \lor \operatorname{Card} \left({ S }\right) = \mathbb U$

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

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


 * $S \sim \operatorname{Card} \left({ S }\right)$

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

And thus $\operatorname{Card} \left({ S }\right) = \mathbb U$ by the definition of cardinal number.