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 $\family {S_\alpha}_{\alpha \mathop \in I}$ 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 $\family {S_\alpha}_{\alpha \mathop \in I}$.
Proof
\(\ds x\) | \(\in\) | \(\ds \bigcap \mathbb S\) | ||||||||||||
\(\ds \implies \ \ \) | \(\, \ds \forall T \in \mathbb S: \, \) | \(\ds x\) | \(\in\) | \(\ds T\) | Definition of Set Intersection | |||||||||
\(\ds \implies \ \ \) | \(\, \ds \forall T \in \mathbb S: \, \) | \(\ds \bigcap \mathbb S\) | \(\subseteq\) | \(\ds T\) | Definition of Subset |
$\blacksquare$