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:

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

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

$\blacksquare$