Intersection is Subset/Family of Sets

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

$\ds x$

$\in$

$\ds \bigcap_{\alpha \mathop \in I} S_\alpha$

$\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

$\blacksquare$

1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Exercise $1 \ \text{(b)}$