Suppose that there exists $f : \mathbb{N} \rightarrow \mathcal{P}(\mathbb{N})$ bijective. By considering $A = \{x \in \mathbb{N} \mid x \notin f(x)\}$, establish a contradiction.
Suppose that there exists $f : \mathbb{N} \rightarrow \mathcal{P}(\mathbb{N})$ bijective. By considering $A = \{x \in \mathbb{N} \mid x \notin f(x)\}$, establish a contradiction.