Union of Subsets is Subset/Family of Sets

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Theorem

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.


Then for all sets $X$:

$\displaystyle \paren {\forall i \in I: S_i \subseteq X} \implies \bigcup_{i \mathop \in I} S_i \subseteq X$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ is the union of $\family {S_i}$.


Proof

Suppose that $\forall i \in I: S_i \subseteq X$.

Consider any $\displaystyle x \in \bigcup_{i \mathop \in I} S_i$.

By definition of set union:

$\exists i \in I: x \in S_i$

But as $S_i \subseteq X$ it follows that $x \in X$.

Thus it follows that:

$\displaystyle \bigcup_{i \mathop \in I} S_i \subseteq X$

So:

$\displaystyle \paren {\forall i \in I: S_i \subseteq X} \implies \bigcup_{i \mathop \in I} S_i \subseteq X$

$\blacksquare$


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