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
From Identity Mapping is Bijection, 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.

Also see
The definition of a cardinal of a set as the equivalence class of that set under set equivalence.