It is an assumption independent of the other axioms in Zermelo-Fraenkel set theory.  
It is equivalent to the statement that the cartesian product of an infinite family of sets is non-empty; to Zorn's Lemma; and to the well-ordering principle.  It is an assumption independent of the other axioms in Zermelo-Fraenkel set theory.  FTP, what name is given to the assertion that from any infinite family of sets a new set may be created containing exactly one element from each set in the family?
It is equivalent to the statement that the cartesian product of an infinite family of sets is non-empty; to Zorn's Lemma; and to the well-ordering principle.  
axiom of choice (accept Zorn's Lemma and the well-ordering principle before they are mentioned in the question)