Power Set of Empty Set

Theorem
The power set of the empty set $\varnothing$ is the set $\left\{{\varnothing}\right\}$.

Proof
From Empty Set Element of Power Set and Set Element of its Power Set we have that $\varnothing \in \mathcal P \left({\varnothing}\right)$.

From Empty Set Subset of All it follows that $S \subseteq \varnothing \implies S = \varnothing$.

That is, $S \in \mathcal P \left({\varnothing}\right) \implies S = \varnothing$.

Hence the only element of $\mathcal P \left({\varnothing}\right)$ is $\varnothing$.