Union is Increasing

Theorem
Let $U$ be a set.

Let $\mathcal F$ and $\mathcal G$ be sets of subsets of $U$.

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

That is, $\bigcup$ is an increasing mapping from $(\mathcal P(\mathcal P(U)), \subseteq)$ to $(\mathcal P(U), \subseteq)$, where $\mathcal P(U)$ is the power set of $U$.

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

Let $x \in \bigcup \mathcal F$.

Then by the definition of union, for some $S \in \mathcal F$, $x \in S$.

By the definition of subset, $S \in \mathcal G$.

Thus by the definition of union, $x \in \bigcup \mathcal G$.

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