Intersection is Subset/General Result

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Theorem

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.


Then:

$\displaystyle \forall T \in \mathbb S: \bigcap \mathbb S \subseteq T$


Family of Sets

In the context of a family of sets, the result can be presented as follows:


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

\(\displaystyle x\) \(\in\) \(\displaystyle \bigcap \mathbb S\)
\(\displaystyle \implies \ \ \) \(\, \displaystyle \forall T \in \mathbb S: \, \) \(\displaystyle x\) \(\in\) \(\displaystyle T\) Definition of Set Intersection
\(\displaystyle \implies \ \ \) \(\, \displaystyle \forall T \in \mathbb S: \, \) \(\displaystyle \bigcap \mathbb S\) \(\subseteq\) \(\displaystyle T\) Definition of Subset

$\blacksquare$