Set Intersection Preserves Subsets/Families of Sets

Theorem
Let $\left({A_i}\right)_{i \in I}, \left({B_i}\right)_{i \in I}$ be collections of sets.

Suppose that for all $i \in I: A_i \subseteq B_i$.

Then:


 * $\displaystyle \bigcap_{i \in I} A_i \subseteq \bigcap_{i \in I} B_i$

Corollary
Suppose that for all $i \in I: A_i = A$ for some set $A$.

Then:


 * $\displaystyle A \subseteq \bigcap_{i \in I} B_i$