Set is Subset of Power Set of Union

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Theorem

Let $x$ be a set of sets.

Let $\displaystyle \bigcup x$ denote the union of $x$.

Let $\powerset {\displaystyle \bigcup x}$ denote the power set of $\displaystyle \bigcup x$.


Then:

$x \subseteq \powerset {\displaystyle \bigcup x}$


Proof

Let $z \in x$.

By Element of Class is Subset of Union of Class:

$z \subseteq \displaystyle \bigcup x$

By definition of power set:

$z \in \powerset {\displaystyle \bigcup x}$

The result follows by definition of subset.

$\blacksquare$


Sources