# Intersection with Universe

## Theorem

The intersection of a set with the universe is the set itself:

$\mathbb U \cap S = S$

## Proof

 $\displaystyle S$ $\subseteq$ $\displaystyle \mathbb U$ $\quad$ Definition of Universe (Set Theory) $\quad$ $\displaystyle \iff \ \$ $\displaystyle \mathbb U \cap S$ $=$ $\displaystyle S$ $\quad$ Intersection with Subset is Subset‎ $\quad$

$\blacksquare$