Set Intersection Preserves Subsets/Families of Sets

Theorem
Let $I$ be an indexing set.

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

Let:
 * $\forall \beta \in I: A_\beta \subseteq B_\beta$

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

Proof
By definition of subset:
 * $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$