Power Set of Empty Set

Theorem

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

Proof

$\varnothing \in \mathcal P \left({\varnothing}\right)$
$S \subseteq \varnothing \implies S = \varnothing$

That is:

$S \in \mathcal P \left({\varnothing}\right) \implies S = \varnothing$

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

$\blacksquare$