Intersection is Decreasing

Theorem
Let $\mathcal F$ and $\mathcal G$ be sets of sets.

Then $\mathcal F \subseteq \mathcal G \implies \bigcap \mathcal G \subseteq \bigcap \mathcal F$.

Proof
Let $\mathcal F \subseteq \mathcal G$.

Let $x \in \bigcap \mathcal G$.

Then for each $S \in \mathcal F$, $S \in \mathcal G$.

By the definition of intersection, $x \in S$.

Since this holds for all $S \in \mathcal F$, $x \in \bigcap \mathcal F$.

Since this holds for all $ x \in \bigcap \mathcal G$:
 * $\bigcap \mathcal G \subseteq \bigcap \mathcal F$.