Intersection is Subset/General Result

Theorem
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.

Then:
 * $\ds \forall T \in \mathbb S: \bigcap \mathbb S \subseteq T$

Family of Sets
In the context of a family of sets, the result can be presented as follows: