The power set of a set $X$ is
\[\powerset(X) = \{Y : Y \subseteq X\}.\]This is a set by Axiom of Power Set.
Proposition 2. Cantor’s Theorem.
For any set $X$,
\[\lvert X \rvert < \lvert \powerset(X) \rvert.\]
Proof. If $f : X \to \powerset(X)$ were a surjection, there would be an $a \in X$ such that
\[f(a) = \{x \in X : x \notin f(x)\}.\]Consider whether $a \in f(a)$.
For any $X$,
\[\lvert \powerset(X) \rvert = 2^{\lvert X \rvert}.\]
Proof.
\[\varphi : \powerset(X) \to \fun(X, 2), \, A \mapsto f_A\]is a bijection, where
\[f_A : X\to 2, \, x \mapsto \begin{cases} 1, & \text{if $x\in A$}; \\ 0, & \text{if $x\notin A$}. \end{cases}\]
As a corollary of Cantor’s Theorem and Proposition 3:
For any cardinal $\kappa$,
\[\kappa < 2^\kappa.\]