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 $F\colon X\to{\mathcal{P}}(X)$ from a set $X$ to its power set. Then we define the set $Z\subseteq X$ as follows:

Z=\{x\in X\mid x\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\Leftrightarrow x\not\in F(x)\Leftrightarrow x\not\in Z

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