Intersection of Subsets is Subset/Set of Sets

Theorem

Let $T$ be a set.

Let $\mathbb S$ be a non-empty set of sets.

Suppose that for each $S \in \mathbb S$:

$S \subseteq T$

Then:

$\bigcap \mathbb S \subseteq T$

Proof

Let $x \in \bigcap \mathbb S$.

Then by the definition of intersection:

$\forall S \in \mathbb S: x \in S$.

Since $\mathbb S$ is non-empty by the premise, it has some element $S$.

Then $x \in S$.

Since $S \in \mathbb S$, the premise shows that $S \subseteq T$.

By the definition of subset, $x \in T$.

Since this holds for each $x \in \bigcap \mathbb S$:

$\bigcap \mathbb S \subseteq T$

$\blacksquare$