Intersection is Subset/General Result

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:

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

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 $\family {S_\alpha}_{\alpha \mathop \in I}$.

$\displaystyle x$

$\in$

$\displaystyle \bigcap \mathbb S$

$\displaystyle \implies \ \ $

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

$\displaystyle x$

$\in$

$\displaystyle T$

$\displaystyle \implies \ \ $

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

$\displaystyle \bigcap \mathbb S$

$\subseteq$

$\displaystyle T$

Definition of Subset

$\blacksquare$