Intersection is Subset/Family of Sets

From ProofWiki
Jump to navigation Jump to search

Theorem

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

\(\ds x\) \(\in\) \(\ds \bigcap_{\alpha \mathop \in I} S_\alpha\)
\(\ds \leadsto \ \ \) \(\ds \forall \beta \in I: \, \) \(\ds x\) \(\in\) \(\ds S_\beta\) Definition of Intersection of Family
\(\ds \leadsto \ \ \) \(\ds \forall \beta \in I: \, \) \(\ds \bigcap_{\alpha \mathop \in I} S_\alpha\) \(\subseteq\) \(\ds S_\beta\) Definition of Subset

$\blacksquare$


Sources