Union of Subset of Family is Subset of Union of Family

Theorem

Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.

Let $J \subseteq I$

Then:

$\ds \bigcup_{\alpha \mathop \in J} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} A_\alpha$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.

$\ds x$

$\in$

$\ds \bigcup_{\alpha \mathop \in J} A_\alpha$

$\ds \leadsto \ \ $

$\ds \exists \alpha \in J: \, $

$\ds x$

$\in$

$\ds A_\alpha$

$\ds \leadsto \ \ $

$\ds \exists \alpha \in I: \, $

$\ds x$

$\in$

$\ds A_\alpha$

Definition of Subset: $J \subseteq I$

$\ds \leadsto \ \ $

$\ds x$

$\in$

$\ds \bigcup_{\alpha \mathop \in I} A_\alpha$

$\blacksquare$

1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $3 \ \text{(a)}$
1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Exercise $3 \ \text{(b)}$