Cantor's Theorem: No Surjection X to P(X)

Every set is strictly smaller than its power set, $|X| < |\mathcal{P}(X)|$, so no surjection (and hence no bijection) $X \to \mathcal{P}(X)$ exists — which is why $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0} > \aleph_0$.