# Set Union Preserves Subsets/Families of Sets

## Theorem

Let $I$ be an indexing set.

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

Let:

$\forall \beta \in I: A_\beta \subseteq B_\beta$

Then:

$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle \exists \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Definition of Union of Family $\displaystyle \leadsto \ \$ $\displaystyle \exists \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle B_\alpha$ Definition of Subset $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcup_{\alpha \mathop \in I} B_\alpha$ Definition of Union of Family

By definition of subset:

$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$

$\blacksquare$