well-ordering principle implies axiom of choiceWellOrderingPrincipleImpliesAxiomOfChoice2006-07-30 03:55:572007-05-30 22:14:06Theorem\usepackage{amssymb}
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\begin{thm*}
The well-ordering principle implies the axiom of choice.
\end{thm*}
\begin{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.
\end{proof}