Subset of Union

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Theorem

The union of two sets is a superset of each:

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:

$\forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \in S$	$\implies$	$\displaystyle x \in S \lor x \in T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Addition
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle x \in S \cup T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Union
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle S \subseteq S \cup T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Subset

Similarly for $T$.

$\blacksquare$

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall T \in \mathbb S: x \in T$	$\implies$	$\displaystyle x \in \bigcup \mathbb S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Union
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Subset

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\implies\)	\(\displaystyle x \in S \lor x \in T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Addition
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \in S \cup T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Union
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle S \subseteq S \cup T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Subset

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall T \in \mathbb S: x \in T\)	\(\implies\)	\(\displaystyle x \in \bigcup \mathbb S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Union
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Subset