# well-ordering principle implies 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. ∎

Title well-ordering principle implies axiom of choice WellorderingPrincipleImpliesAxiomOfChoice 2013-03-22 16:07:46 2013-03-22 16:07:46 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Theorem msc 03E25 AxiomOfChoice ZermelosWellOrderingTheorem