# Empty Set is Element of Power Set

## Theorem

The empty set is an element of all power sets:

$\forall S: \O \in \powerset S$

## Proof

 $\ds \forall S: \,$ $\ds \O$ $\subseteq$ $\ds S$ Empty Set is Subset of All Sets $\ds \leadsto \ \$ $\ds \forall S: \,$ $\ds \O$ $\in$ $\ds \powerset S$ Definition of Power Set

$\blacksquare$