Intersection with Empty Set

Theorem

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

$S \cap \O = \O$

Proof

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

$\blacksquare$