Intersection is Subset/General Result

Theorem

Let $S$ be a set.

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

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

Then:

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

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:

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

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: \, $

$\ds x$

$\in$

$\ds T$

$\ds \leadsto \ \ $

$\ds \forall T \in \mathbb S: \, $

$\ds \bigcap \mathbb S$

$\subseteq$

$\ds T$

Definition of Subset

$\blacksquare$