Set Equivalence behaves like Equivalence Relation

Theorem
Set equivalence behaves like an equivalence relation.

That is:

where $S, T, S_1, S_2, S_3$ are sets.

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

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

Reflexive
Thus $\sim$ is seen to behave like a reflexive relation.

Symmetric
Thus $\sim$ is seen to behave like a symmetric relation.

Transitive
Thus $\sim$ is seen to behave like a transitive relation.

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