# Intersection of Power Sets

## Theorem

The intersection of the power sets of two sets $S$ and $T$ is equal to the power set of their intersection:

$\powerset S \cap \powerset T = \powerset {S \cap T}$

## Proof

 $\displaystyle X$ $\in$ $\displaystyle \powerset {S \cap T}$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle X$ $\subseteq$ $\displaystyle S \cap T$ Definition of Power Set $\displaystyle \leadstoandfrom \ \$ $\displaystyle X$ $\subseteq$ $\displaystyle S \land X \subseteq T$ Definition of Set Intersection $\displaystyle \leadstoandfrom \ \$ $\displaystyle X$ $\in$ $\displaystyle \powerset S \land X \in \powerset T$ Definition of Power Set $\displaystyle \leadstoandfrom \ \$ $\displaystyle X$ $\in$ $\displaystyle \powerset S \cap \powerset T$ Definition of Set Intersection

$\blacksquare$