Cardinal Number Equivalence or Equal to Universe

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

$\card S = \mathbb U$

$\blacksquare$


Sources