The proof of this theorem is fairly simple using the following construction, which is central to Cantor's diagonal argument.

Consider a function $F\colon X\to \P(X)$ from a set $X$ to its power set. Then we define the set $Z\sse X$ as follows: $$ Z = \{x\in X \mid x\not\in F(x)\ $$

Suppose that $F$ is a bijection. Then there must exist an $x\in X$ such that $F(x)=Z$ . Then we have the following contradiction: $$ x\in Z \equiv x\not\in F(x) \equiv x\not\in $$

Hence, $F$ cannot be a bijection between $X$ and $\P(X)$ .