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 $f:C\to {\displaystyle \bigcup _{S\in C}}S$ 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
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