Set is Subset of Union/Set of Sets

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Theorem

Let $\mathbb S$ be a set of sets.


Then:

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


Proof

Let $T$ be any element of $\mathbb S$.

We wish to show that $T \subseteq S$.

Let $x \in T$.

Then:

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

Since this holds for each $x \in T$:

\(\displaystyle T\) \(\subseteq\) \(\displaystyle \bigcup \mathbb S\) Definition of Subset


As $T$ was arbitrary, it follows that:

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

$\blacksquare$