# Intersection is Idempotent

## Theorem

$S \cap S = S$

### Indexed Family

Let $\left\langle{ F_i }\right\rangle_{i \mathop \in I}$ be a non-empty indexed family of sets.

Suppose that all the sets in the family are the same.

That is, suppose that for some set $S$:

$\forall i \in I: F_i = S$

Then:

$\displaystyle \bigcap_{i \mathop \in I} F_i = S$

where $\displaystyle \bigcap_{i \mathop \in I} F_i$ is the intersection of $\left\langle{ F_i }\right\rangle_{i \in I}$.

## Proof

 $\displaystyle x$ $\in$ $\displaystyle S \cap S$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \in S$ $\land$ $\displaystyle x \in S$ Definition of Set Intersection $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle S$ Rule of Idempotence: Conjunction

$\blacksquare$