Let $\mathcal{M} : = \{ 0, 1 \}^X$ denote the D68: Set of maps from $X$ to $\{ 0, 1 \}$. Result R1841: Indicator function operator is a bijection shows that $$\begin{equation} |\mathcal{P}(X)| = |\mathcal{M}| \end{equation}$$ and result R4314: Number of boolean functions on a finite set shows that $$\begin{equation} |\mathcal{M}| = 2^{|X|} \end{equation}$$ $\square$