Intersection is Subset/Family of Sets

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Theorem

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


Then:

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

where $\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha$ is the intersection of $\left \langle{S_\alpha}\right \rangle_{\alpha \mathop \in I}$.


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \forall \beta \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S_\beta\) $\quad$ Definition of Intersection of Family $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \forall \beta \in I: \ \ \) \(\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha\) \(\subseteq\) \(\displaystyle S_\beta\) $\quad$ Definition of Subset $\quad$

$\blacksquare$


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