Set is Subset of Union/Family of Sets

Theorem

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ 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 $\family {S_\alpha}$.

Proof

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

Then:

 $\displaystyle x$ $\in$ $\displaystyle S_\beta$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \set {x: \exists \alpha \in I: x \in S_\alpha}$ Definition of Indexed Family of Sets $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha$ Definition of Union of Family $\displaystyle \leadsto \ \$ $\displaystyle S_\beta$ $\subseteq$ $\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha$ Definition of Subset

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

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

$\blacksquare$