Power Set of Empty Set

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

Proof
From Empty Set is Element of Power Set and Set is Element of its Power Set:
 * $\O \in \powerset \O$

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

That is:
 * $S \in \powerset \O \implies S = \O$

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