Intersection of Family is Subset of Intersection of Subset of Family

Theorem
Let $I$ be an indexing set.

Let $\left \langle {A_\alpha} \right \rangle_{\alpha \mathop \in I}$ be a family of subsets of a set $S$.

Let $J \subseteq I$

Then:
 * $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in J} A_\alpha$

where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\left \langle {A_\alpha} \right \rangle_{\alpha \mathop \in I}$.

Also see

 * Union of Subset of Family is Subset of Union of Family