# 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

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

$S \sim \card S$

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

 $\ds \bigcap \set {x \in \On : S \sim x}$ $=$ $\ds \bigcap \O$ $\ds$ $=$ $\ds \mathbb U$ Intersection of Empty Set

And thus by the definition of cardinal number:

$\card S = \mathbb U$

$\blacksquare$