# Intersection with Empty Set

## Theorem

The intersection of any set with the empty set is itself the empty set:

$S \cap \O = \O$

## Proof

 $\ds S \cap \O$ $\subseteq$ $\ds \O$ Intersection is Subset $\ds \O$ $\subseteq$ $\ds S \cap \O$ Empty Set is Subset of All Sets $\ds \leadsto \ \$ $\ds S \cap \O$ $=$ $\ds \O$ Definition 2 of Set Equality

$\blacksquare$