Injection from Set to Power Set

Theorem
For every set $S$, there exists an injection from $S$ to its power set $\mathcal P \left({S}\right)$.

Proof
If $S = \varnothing$, the empty mapping suffices, as it is vacuously an injection.

Let $f: S \to \mathcal P \left({S}\right)$ be the mapping defined as:
 * $\forall s \in S: f \left({s}\right) = \left\{{s}\right\}$

Let $s, t \in S$ such that $f \left({s}\right) = f \left({t}\right)$.

Then $\left\{{s}\right\} = \left\{{t}\right\}$.

By definition of set equality it follows directly that $s = t$.

Hence $f$ is the required injection, by definition.