Set Intersection Preserves Subsets/Families of Sets/Corollary

Theorem
Let $\left({B_i}\right)_{i \mathop \in I}$ be a collection of sets.

Let $A$ be a set such that $A \subseteq B_i$ for all $i \in I$.

Then:


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

Proof
For each $i \in I$, define $A_i := A$.

Then by Intersection is Idempotent, it follows that:


 * $\displaystyle \bigcap_{i \mathop \in I} A_i = A$

Since $A \subseteq B_i$ for all $i \in I$, the premises of Set Intersection Preserves Subsets are satisfied.

Applying this theorem gives:


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

which is precisely the desired result.