No Injection from Power Set to Set/Proof 2

Theorem

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then there is no injection from $\powerset S$ into $S$.

Proof

Lemma

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then there does not exist a set $B$ such that there is an injection from $B$ into $S$ and a surjection from $B$ onto $\powerset S$.

$\Box$

The identity mapping $I_{\powerset S}: \powerset S \to \powerset S$ is a surjection by Identity Mapping is Surjection.

Thus, by the lemma, there can be no injection from $\powerset S$ into $S$.

$\blacksquare$