Set is Subset of Power Set of Union

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.