# Intersection is Subset/Family of Sets

## Theorem

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Then:

$\displaystyle \forall \beta \in I: \bigcap_{\alpha \mathop \in I} S_\alpha \subseteq S_\beta$

where $\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha$ is the intersection of $\family {S_\alpha}_{\alpha \mathop \in I}$.

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \beta \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle S_\beta$ Definition of Intersection of Family $\displaystyle \leadsto \ \$ $\displaystyle \forall \beta \in I: \ \$ $\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha$ $\subseteq$ $\displaystyle S_\beta$ Definition of Subset

$\blacksquare$