Choice Function for Set does not imply Choice Function for Union of Set

Theorem

Let $S$ be a set of sets.

Let $\bigcup S$ denote the union of $S$.

Let there exist a choice function for $S$.

Then there does not necessarily exist a choice function for $\bigcup S$.

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 $4.3$