Set equals Union of Power Set

Theorem
Let $x$ be a set of sets.

Let $\powerset x$ denote the power set of $x$.

Let $\displaystyle \map \bigcup {\powerset x}$ denote the union of $\powerset x$.

Then:
 * $x = \displaystyle \map \bigcup {\powerset x}$

Proof
From Set is Element of its Power Set:
 * $x \in \powerset x$

From Element of Class is Subset of Union of Class it follows that:
 * $x \subseteq \displaystyle \map \bigcup {\powerset x}$

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

Then by definition of union:
 * $\exists y \in \powerset x: z \in y$

By definition of $\powerset x$
 * $\exists y \subseteq x: z \in y$

But by definition of subset, that means:
 * $z \in x$

Thus, again by definition of subset:
 * $\displaystyle \map \bigcup {\powerset x} \subseteq x$

We have that:
 * $x \subseteq \displaystyle \map \bigcup {\powerset x}$

and:
 * $\displaystyle \map \bigcup {\powerset x} \subseteq x$

Hence by definition of set equality:
 * $x = \displaystyle \map \bigcup {\powerset x}$