# Power Set of Empty Set

## Theorem

The power set of the empty set $\O$ is the set $\set \O$.

## Proof

$\O \in \powerset \O$
$S \subseteq \O \implies S = \O$

That is:

$S \in \powerset \O \implies S = \O$

Hence the only element of $\powerset \O$ is $\O$.

$\blacksquare$