proof of Cantor’s theorem

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

Consider a function $F:X\to 𝒫(X)$ from a set $X$ to its power set. Then we define the set $Z\subseteq X$ as follows:

$$Z=\{x\in X\mid x\notin F(x)\}$$

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:

$$x\in Z\iff x\notin F(x)\iff x\notin Z$$

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