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# well-ordering principle implies axiom of choice

###### Theorem.

The well-ordering principle implies the axiom of choice.

###### 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. ∎

Related:

AxiomOfChoice, ZermelosWellOrderingTheorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

03E25*no label found*

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