Intersection is Subset/Family of Sets

Theorem

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

Then:

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

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

Proof

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

$\blacksquare$