Set equals Union of Power Set

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Theorem

Let $x$ be a set of sets.

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

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


Then:

$x = \map \bigcup {\powerset x}$


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:

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

$\Box$


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

Then by definition of union|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:

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

$\Box$


We have that:

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

and:

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

Hence by definition of set equality:

$x = \map \bigcup {\powerset x}$

$\blacksquare$


Sources