Back when I was a new undergrad student taking my first analysis course, I was shown many proofs involving the cardinality of sets. I was introduced to the fact that the cardinality of the power set is always strictly greater than the set itself $$\text{card}(X) < \text{card}(\cP(X))$$ But I never bothered with looking into the proof, being scared of it looking complicated. Looking back, that was silly, because I finally got to properly read it, and it couldn't be simpler.

We can find an easy bijection $f : X \to \cP(X)$ where $f(x) = \{x\}$, Naturally, $x \ne y \implies {x} \ne {y}$. So we know $\text{card}(X) \le \text{card}(\cP(X))$. There are many other possible injective functions. We wish to prove that none of them are bijective. Define the set $$S = \set{ x \in X \given x \notin f(x) }$$ It follows that $S \subseteq X$. By contradiction, if $f$ was bijective, then there would be some $a \in X$ such that $f(a) = S$. By the law of excluded middle, either $a \in S$ or $a \notin S$. But by definition, $$a \in S \iff a \notin f(a) = S \iff a \notin S$$ which is a contradiction.

This is a type of self-reference proof, similar to Russell's paradox, except this one is a perfectly valid contradiction to disprove any set being bijective to its power set.