# Intersection is Subset

## Theorem

The intersection of two sets is a subset of each:

$S \cap T \subseteq S$
$S \cap T \subseteq T$

### General Result

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

## Proof

 $\ds x$ $\in$ $\ds S \cap T$ $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds S \land x \in T$ Definition of Set Intersection $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds S$ Rule of Simplification $\ds \leadsto \ \$ $\ds S \cap T$ $\subseteq$ $\ds S$ Definition of Subset

Similarly for $T$.

$\blacksquare$