Set is Subset of Union/Family of Sets

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

Then:
 * $\displaystyle \forall i \in I: S_i \subseteq \bigcup_{j \mathop \in I} S_j$

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

Proof
Let $x \in S_i$ for some $i \in I$.

Then:

As $i$ was arbitrary, it follows that:


 * $\displaystyle \forall i \in I: S_i \subseteq \bigcup_{j \mathop \in I} S_j$