Set Union Preserves Subsets/Families of Sets

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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:

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


Proof

\(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} A_\alpha\)
\(\ds \leadsto \ \ \) \(\ds \exists \alpha \in I: \, \) \(\ds x\) \(\in\) \(\ds A_\alpha\) Definition of Union of Family
\(\ds \leadsto \ \ \) \(\ds \exists \alpha \in I: \, \) \(\ds x\) \(\in\) \(\ds B_\alpha\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} B_\alpha\) Definition of Union of Family

By definition of subset:

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

$\blacksquare$


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