Power Set of Empty Set

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

Proof
From Empty Set is Element of Power Set and Set is Element of its Power Set:
 * $\varnothing \in \mathcal P \left({\varnothing}\right)$

From Empty Set is Subset of All Sets:
 * $S \subseteq \varnothing \implies S = \varnothing$

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

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