sentences:  On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC.Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC).A first glance at the second proof of Tychnoff may suggest that the proof uses no more than (BPI), in contradiction to the above.