Union of Subsets is Subset/Subset of Power Set
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Theorem
Let $S$ and $T$ be sets.
Let $\powerset S$ be the power set of $S$.
Let $\mathbb S$ be a subset of $\powerset S$.
Then:
- $\ds \paren {\forall X \in \mathbb S: X \subseteq T} \implies \bigcup \mathbb S \subseteq T$
Proof
Let $\mathbb S \subseteq \powerset S$.
Suppose that $\forall X \in \mathbb S: X \subseteq T$.
Consider any $\ds x \in \bigcup \mathbb S$.
By definition of set union, it follows that:
- $\exists X \in \mathbb S: x \in X$
But as $X \subseteq T$ it follows that $x \in T$.
Thus it follows that:
- $\ds \bigcup \mathbb S \subseteq T$
So:
- $\ds \paren {\forall X \in \mathbb S: X \subseteq T} \implies \bigcup \mathbb S \subseteq T$
$\blacksquare$