Set is Equivalent to Itself

Theorem
Let $S$ be a set.

Then:
 * $S \sim S$

where $\sim$ denotes set equivalence.

Proof
From Identity Mapping is Bijection, the identity mapping $I_S: S \to S$ is a bijection from $S$ to $S$.

Thus there exists a bijection from $S$ to itself

Hence by definition $S$ is therefore equivalent to itself.