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:

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

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:

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

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

Proof

 $\ds x$ $\in$ $\ds \bigcap \mathbb S$ $\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

$\blacksquare$