Union is Increasing
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Theorem
Let $U$ be a set.
Let $\FF$ and $\GG$ be sets of subsets of $U$.
Then $\FF \subseteq \GG \implies \bigcup \FF \subseteq \bigcup \GG$.
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 $x \in \bigcup \FF$:
- $\bigcup \FF \subseteq \bigcup \GG$
$\blacksquare$