Set is Subset of Intersection of Supersets/Set of Sets

Theorem
Let $T$ be a set.

Let $\mathbb S$ be a set of sets.

Suppose that for each $S \in \mathbb S$, $T \subseteq S$.

Then:


 * $T \subseteq \displaystyle \bigcap \mathbb S$

Proof
Let $x \in T$.

We are given that:
 * $\forall S \in \mathbb S: T \subseteq S$

Thus by definition of subset:
 * $\forall S \in \mathbb S: x \in S$

Hence by definition of intersection:
 * $x \in \displaystyle \bigcap \mathbb S$

Thus by definition of subset:
 * $T \subseteq \displaystyle \bigcap \mathbb S$