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 $x \in \bigcup \FF$:

$\bigcup \FF \subseteq \bigcup \GG$

$\blacksquare$