Intersection is Decreasing

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Theorem

Let $U$ be a set.

Let $\FF$ and $\GG$ be sets of subsets of $U$.

Then $\FF \subseteq \GG \implies \bigcap \GG \subseteq \bigcap \FF$, where by convention $\bigcap \O = U$.

That is, $\bigcap$ is a decreasing mapping from $\struct {\powerset {\powerset U}, \subseteq}$ to $\struct {\powerset U, \subseteq}$, where $\powerset U$ is the power set of $U$.

Proof

Let $\FF \subseteq \GG$.

Let $x \in \bigcap \GG$.

Then for each $S \in \FF$, $S \in \GG$.

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

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

Since this holds for all $ x \in \bigcap \GG$:

$\bigcap \GG \subseteq \bigcap \FF$

$\blacksquare$