The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence ("an") such that "ai+1" is an element of "ai" for all "i". With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.