# 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

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.10$