# 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 $\bigcup _{S\in C}}S$ is a set. By the well-ordering principle, $\bigcup _{S\in C}}S$ is well-ordered under some relation^{} $$. Since each $S$ is a nonempty subset of $\bigcup _{S\in C}}S$, each $S$ has a least member ${m}_{S}$ with respect to the relation $$.

Define $f:C\to {\displaystyle \bigcup _{S\in C}}S$ by $f(S)={m}_{S}$. Then $f$ is a choice function. Hence, the axiom of choice holds. ∎

Title | well-ordering principle implies axiom of choice |
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Canonical name | WellorderingPrincipleImpliesAxiomOfChoice |

Date of creation | 2013-03-22 16:07:46 |

Last modified on | 2013-03-22 16:07:46 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 03E25 |

Related topic | AxiomOfChoice |

Related topic | ZermelosWellOrderingTheorem |