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well-ordering principle implies axiom of choice (Theorem)
Proof. Let $C$ be a collection of nonempty sets. Then $\displaystyle \bigcup_{S \in C} S$ is a set. By the well-ordering principle, $\displaystyle \bigcup_{S \in C} S$ is well-ordered under some relation $<$ Since each $S$ is a nonempty subset of $\displaystyle \bigcup_{S \in C} S$ each $S$ has a least member $m_S$ with respect to the relation $<$

Define $\displaystyle f \colon C \to \bigcup_{S \in C} S$ by $f(S)=m_S$ Then $f$ is a choice function. Hence, the axiom of choice holds. $ \qedsymbol$




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See Also: axiom of choice, Zermelo's well-ordering theorem

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Cross-references: choice function, subset, relation, well-ordered, collection, axiom of choice, implies, well-ordering principle
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This is version 4 of well-ordering principle implies axiom of choice, born on 2006-07-30, modified 2007-05-30.
Object id is 8200, canonical name is WellOrderingPrincipleImpliesAxiomOfChoice.
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Classification:
AMS MSC03E25 (Mathematical logic and foundations :: Set theory :: Axiom of choice and related propositions)

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