The continuum hypothesis was shown to be independent of ZFC plus this statement by Paul Cohen, and it is equivalent to Zorn's Lemma and the well-ordering principle.
Elliot Mendelson proved the independence of a weak form of this statement and a consequence of it is Vitali's theorem. A proof using this statement involves showing that the intersection of all towers is a chain. This statement is used in proving the LowenheimSkolem theorem by implying the existence of a function that maps M to the n to M. Zermelo first used it to prove a theorem that for every set there is some relation such that every subset of that set has a least element. The continuum hypothesis was shown to be independent of ZFC plus this statement by Paul Cohen, and it is equivalent to Zorn's Lemma and the well-ordering principle. For 10 points, name this set-theoretic axiom that informally states that we can come up with a function that will pick things for us.
Elliot Mendelson proved the independence of a weak form of this statement and a consequence of it is Vitali's theorem.
axiom of choice [accept Zorn's lemma or well ordering principle before mention]