Set is Subset of Union/General Result

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

Then $\displaystyle \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

Proof
We wish to show that $\forall T: \forall x \in T: x \in \bigcup \mathbb S$

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

Then:

Thus for each $T \in \mathbb S$,

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