Union of Elements of Power Set

Theorem

Let $S$ be a set.

Then:

$\ds S = \bigcup_{X \mathop \in \powerset S} X$

where $\powerset S$ denotes the power set of $S$.

Proof

By Subset of Union:

$\ds \forall X \in \powerset S: X \subseteq \bigcup_{X \mathop \in \powerset S} X$

From Set is Subset of Itself, $S \subseteq S$ and so $S \in \powerset S$.

So:

$\ds S \subseteq \bigcup_{X \mathop \in \powerset S} X$

From Union is Smallest Superset:

$\ds \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$

where $\mathbb S \subseteq \powerset S$.

But as $\powerset S \subseteq \powerset S$ from Set is Subset of Itself:

$\ds \paren {\forall X \in \powerset S: X \subseteq S} \iff \bigcup \powerset S \subseteq S$

The left hand side is no more than the definition of the power set, making it a tautological statement, and so:

$\ds \bigcup \powerset S \subseteq S$

The result follows by definition of set equality.

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers