Choice Function for Power Set implies Choice Function for Set

Theorem

Let $S$ be a set.

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

Let there exists a choice function for $\powerset S$.

Then there exists a choice function for $S$.

Proof

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 4$ Well ordering and choice: Exercise $4.2$