Cardinal Number Equivalence or Equal to Universe

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Theorem

Let $S$ be a set.

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

Let $\mathbb U$ denote the universal class.


Then:

$S \sim \left\vert{S}\right\vert \lor \left\vert{S}\right\vert = \mathbb U$


Proof

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

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

$S \sim \left\vert{S}\right\vert$


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

\(\displaystyle \bigcap \left\{ {x \in \operatorname{On} : S \sim x}\right\}\) \(=\) \(\displaystyle \bigcap \varnothing\)
\(\displaystyle \) \(=\) \(\displaystyle \mathbb U\) by Intersection of Empty Set

And thus by the definition of cardinal number:

$\left\vert{S}\right\vert = \mathbb U$

$\blacksquare$


Sources