Union of Subsets is Subset/Set of Sets

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Theorem

Let $T$ be a set.

Let $\mathbb S$ be a set of sets.

Suppose that for each $S \in \mathbb S$, $S \subseteq T$.


Then:

$\displaystyle \bigcup \mathbb S \subseteq T$


Proof

Let $x \in \displaystyle \bigcup \mathbb S$.

By the definition of union, there exists an $S \in \mathbb S$ such that $x \in S$.

By premise, $S \subseteq T$.

By the definition of subset, $x \in T$.

Since this result holds for each $x \in \displaystyle \bigcup \mathbb S$:

$\displaystyle \bigcup \mathbb S \subseteq T$

$\blacksquare$