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Every family of nonempty sets has a choice function.
Axiom of Choice: For every indexed family of nonempty sets, there exists a function such that for all .
Equivalently, every indexed family of sets of nonempty sets admits a choice function selecting one element from each set. In the presence of the other ZF axioms, this axiom is equivalent to Zorn's lemma and to the well-ordering theorem.
An indexed family of sets is a function where is an index set and is a set whose elements are sets. One writes the family as , where .
Zorn's lemma: Let be a partially ordered set. If every chain has an upper bound in , then has a maximal element.
Here a chain means a subset on which the restriction of is a total order, and an element is maximal if there is no with (i.e., and ).
Well-ordering theorem: For every set , there exists a relation such that is a well-ordered set.