Set Equivalence behaves like Equivalence Relation

Theorem
Set equivalence is an equivalence relation.

Proof
For two sets to be equivalent, there needs to exist a bijection between them.

In the following, let $\phi$, $\phi_1$, $\phi_2$ etc. be understood to be bijections.


 * Reflexive:

The identity mapping $I_S: S \to S$ is the required bijection.

Thus there exists a bijection from $S$ to itself and $S$ is therefore equivalent to itself.

Therefore set equivalence is reflexive.


 * Symmetric:

Therefore set equivalence is symmetric.


 * Transitive:

Therefore set equivalence is transitive.

So, from Relation Partitions a Set iff Equivalence, any family of sets can be partitioned into classes of equivalent sets.