Set is Subset of Union/General Result

Theorem
Let $S$ be a set.

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

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

Then:
 * $\ds \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

Proof
Let $x \in T$ for some $T \in \mathbb S$.

Then:

As $T$ was arbitrary, it follows that:
 * $\ds \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$