Set is Subset of Union/General Result

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Theorem

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:

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


Proof

Let $x \in T$ for some $T \in \mathbb S$.

Then:

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


As $T$ was arbitrary, it follows that:

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

$\blacksquare$