Intersection is Subset/General Result

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Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.


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

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 $I$.


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

where $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is the intersection of $\family {S_\alpha}_{\alpha \mathop \in I}$.


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