Set is Subset of Union/Family of Sets

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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:

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

$\blacksquare$


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