# Cardinals form Equivalence Classes

## Theorem

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

Then $\operatorname{Card} \left({S}\right)$ forms an equivalence class which contains all sets which have the same cardinality as $S$.

## Proof

Follows directly from:

- The definition of a cardinal as $S \sim T \iff \operatorname{Card} \left({S}\right) = \operatorname{Card} \left({T}\right)$
- Set Equivalence is Equivalence Relation
- Relation Partitions Set iff Equivalence.

$\blacksquare$