The Axiom of Dependent Choice and No Infinite Descending Sequence of Sets Implies Regularity
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.
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