Set is Subset of Intersection of Supersets/General Result

Theorem
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

Let $X$ be a set such that:
 * $\forall i \in I: X \subseteq S_i$

Then:
 * $\ds X \subseteq \bigcap_{i \mathop \in I} S_i$

where $\ds \bigcap_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$.

Proof
Let $X \subseteq S_i$ for all $i \in I$.

Then: