yet been assigned a place in the ordering (or undefined if the entirety of has been successfully enumerated). Then is a well-order of as desired. Proof of axiom of choice The axiom of choice can be proven from the well-ordering theorem as follows. To make a choice function for a collection of non-empty sets, , take the union of the sets in and call it . There exists a well-ordering of ; let be such an ordering. The function that to each set of associates the smallest element of , as ordered by (the restriction to of) , is a choice function for the collection . An essential point of this proof is that it involves only a single arbitrary choice, that of ; applying the well-ordering theorem to each member of separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each a well-ordering would require just as many choices as simply choosing an element from each . Particularly, if contains uncountably many sets, making all uncountably many choices is not allowed under the axioms of Zermelo-Fraenkel set theory without the axiom of choice. Notes External links Mizar system proof: http://mizar.org/version/current/html/wellord2.html Axiom of choice Theorems in the foundations of mathematics Axioms of set theory