Axiom:Axiom of Choice/Formulation 4

Axiom

Let $A$ be a non-empty set.

Then there exists a mapping $f: \powerset A \to A$ such that:

for every non-empty proper subset $x$ of $A$: $\map f x \in x$

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

Also see

• Equivalence of Formulations of Axiom of Choice

• Results about the Axiom of Choice can be found here.

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: $\S 1$ Introduction to well ordering: Exercise $1.6$
• 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