A "reverse" diagonal argument?

Suppose $f : \mathcal{P}(S) \to S$ were an injection. Consider the following recursive construction:

• $s_0 = f(\emptyset)$
• $s_\alpha = f(\{s_\beta\}_{\beta < \alpha})$

Because $f$ is injective, all the $s_\alpha$ are distinct. But this can't be otherwise we have an injection $\mathrm{Ord} \to S$, where $S$ is a set.

EDIT: Or, if we want to be frugal and not mention injections from $\mathrm{Ord}$, we can say that the fact that the $s_\alpha$ are all distinct contradicts Hartog's Lemma.