No Injection from Power Set to Set/Lemma

Theorem
Let $S$ be a set.

Let $\mathcal P(S)$ 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 $\mathcal P(S)$.

Proof
Suppose for the sake of contradiction that there is such a $B$.

Let $i: B \to S$ be an injection and let $f: B \to \mathcal P(S)$ be a surjection.

Let $i^*: \mathcal P(S) \to \mathcal P(B)$ be defined by letting $i^*(T)$ be the preimage of $T$ under $i$.

By Preimage Mapping of Injection is Surjection, $i^*$ is a surjection.

Let $f'': \mathcal P(B) \to \mathcal P \left({ \mathcal P(S) }\right)$ be the mapping induced by $f$.

By Image Mapping of Surjection is Surjection, $f''$ is a surjection:

Since $i^*: \mathcal P(S) \to \mathcal P(B)$ and $f: \mathcal P(B) \to \mathcal P \left({ \mathcal P(S) }\right)$ are surjective, their composition $f \circ i^* : \mathcal P(S) \to \mathcal P \left({ \mathcal P(S) }\right)$ is a surjection by Composite of Surjections is Surjection.

But this violates Cantor's Theorem, contradicting the assumption that such a $B$ exists.