Intersection is Subset/General Result

Theorem
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

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