Axiom of regularity

In set theory, the axiom of regularity, also known as the axiom of foundation, is that for every set S there is an element a in it which is disjoint from S. Under the axiom of choice, this axiom is equivalent to saying there is no infinite sequence {a_n} such that a_i+1 is a member of a_i. Some corollaries are that no set belongs to itself, since otherwise {S} would violate the axiom of regularity.