Descending membership sequence ($f(n+1)\in f(n)(n\in \omega )$)

A descending membership sequence is a function on the natural numbers with each next value a member of the preceding value. Foundation excludes it: the range is a nonempty set with no member disjoint from that range.

Foundation implies that no descending membership sequence exists. Conversely, with dependent choice, failure of foundation gives a nonempty set in which every element contains another element of that set. Successively selecting such members produces a descending membership sequence. Thus the two conditions are equivalent with dependent choice; full choice is sufficient.