Set is Subset of Union/Set of Sets

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

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

Proof
Let $T$ be any element of $\mathbb S$.

We wish to show that $T \subseteq S$.

Let $x \in T$.

Then:

Since this holds for each $x \in T$:

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