Set is Subset of Union/Family of Sets

Theorem
Let $\left \langle{S_\alpha}\right \rangle_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Then:
 * $\displaystyle \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$

where $\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha$ is the union of $\left \langle{S_\alpha}\right \rangle$.

Proof
Let $x \in S_\beta$ for some $\beta \in I$.

Then:

As $\beta$ was arbitrary, it follows that:


 * $\displaystyle \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$