Union of Subsets is Subset/Subset of Power Set

Theorem
Let $S$ and $T$ be sets.

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S$ be a subset of $\powerset S$.

Then:
 * $\ds \paren {\forall X \in \mathbb S: X \subseteq T} \implies \bigcup \mathbb S \subseteq T$

Proof
Let $\mathbb S \subseteq \powerset S$.

Suppose that $\forall X \in \mathbb S: X \subseteq T$.

Consider any $\ds x \in \bigcup \mathbb S$.

By definition of set union, it follows that:
 * $\exists X \in \mathbb S: x \in X$

But as $X \subseteq T$ it follows that $x \in T$.

Thus it follows that:
 * $\ds \bigcup \mathbb S \subseteq T$

So:
 * $\ds \paren {\forall X \in \mathbb S: X \subseteq T} \implies \bigcup \mathbb S \subseteq T$