Cardinals form Equivalence Classes

Theorem
Let $\map \Card S$ denote the cardinal of the set $S$.

Then $\map \Card S$ induces an equivalence class which contains all sets which have the same cardinality as $S$.

Proof
Follows directly from:
 * The definition of a cardinal as $S \sim T \iff \map \Card S = \map \Card T$
 * Set Equivalence is Equivalence Relation
 * Relation Partitions Set iff Equivalence.