Union is Increasing

Theorem
Let $U$ be a set.

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

Then $\FF \subseteq \GG \implies \bigcup \FF \subseteq \bigcup \GG$.

That is, $\bigcup$ is an increasing 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 $x \in \bigcup \FF$.

Then by the definition of union:
 * $\exists S \in \FF: x \in S$

By the definition of subset:
 * $S \in \GG$

Thus by the definition of union:
 * $x \in \bigcup \GG$

Since this holds for all $x \in \bigcup \FF$:
 * $\bigcup \FF \subseteq \bigcup \GG$