Union of Subsets is Subset/Set of Sets

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:

$\ds \bigcup \mathbb S \subseteq T$

Proof

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

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 $x \in \ds \bigcup \mathbb S$:

$\ds \bigcup \mathbb S \subseteq T$

$\blacksquare$