information:classification
Proof from axiom of choice

The well-ordering theorem follows from the axiom of choice as follows.

Let the set we are trying to well-order be A {\displaystyle A}, and let f {\displaystyle f} be a choice function for the family of non-empty subsets of A {\displaystyle A}. For every ordinal α {\displaystyle \alpha }, define an element a α {\displaystyle a_{\alpha }} that is in A {\displaystyle A} by setting a α = f (A ∖ { a ξ ∣ ξ < α }) {\displaystyle a_{\alpha }\ =\ f(A\smallsetminus \{a_{\xi }\mid \xi <\alpha \})} if this complement A ∖ { a ξ ∣ ξ < α } {\displaystyle A\smallsetminus \{a_{\xi }\mid \xi <\alpha \}} is nonempty, or leave a α {\displaystyle a_{\alpha }} undefined if it is. That is, a α {\displaystyle a_{\alpha }} is chosen from the set of elements of A {\displaystyle A} that have not yet been assigned a place in the ordering (or undefined if the entirety of A {\displaystyle A} has been successfully enumerated). Then the order < {\displaystyle <} on A {\displaystyle A} defined by a α < a β {\displaystyle a_{\alpha }<a_{\beta }} if and only if α < β {\displaystyle \alpha <\beta } (in the usual well-order of the ordinals) is a well-order of A {\displaystyle A} as desired, of order type sup { α ∣ a α is defined } {\displaystyle \sup\{\alpha \mid a_{\alpha }{\text{ is defined}}\}}.