Intersection is Subset

From ProofWiki
Jump to navigation Jump to search


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

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


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

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 $I$.


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

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


\(\ds x\) \(\in\) \(\ds S \cap T\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds S \land x \in T\) 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 $T$.