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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.

Use the Kuratowski ordered pairThe axiom of pairing first constructs $\{x\}$ and $\{x,y\}$, then constructs the displayed pair of those sets, proving that it is a set.

For any $a,b$, the intersection of the members of this pair is $\{a\}$, and their union is $\{a,b\}$. Thus equality of two ordered pairs first gives $\{a\}=\{c\}$, hence $a=c$, and then $\{a,b\}=\{a,d\}$. If $b=a$, this latter set is a singleton and forces $d=a$. If $b\ne a$, its unique member distinct from $a$ is $b$, forcing $d=b$. Hence equality of ordered pairs gives equality of both coordinates.

For a set $R$ of ordered pairs, both coordinates of every pair belong towhich is a set by two applications of the Axiom of union. Axiom of separation inside $U$ givesThese are sets, including when $R$ is empty.

The Axiom of foundation asserts that every nonempty set $A$ has an element $a$ with $a\cap A=\varnothing$. If a descending membership sequence $f:\omega \to V$ existed, replacement would give the nonempty range set $S=\{f(n):n\in \omega \}$. Foundation would give some $f(n)$ disjoint from $S$. But $f(n^{+})\in f(n)$ and $f(n^{+})\in S$, contradicting that disjointness. Thus foundation rules out every such sequence.

Conversely suppose there is no descending membership sequence, but foundation fails. There is then a nonempty set $A$ such that $a\cap A\ne \varnothing$ for every $a\in A$. The axiom of choice selects a function $h$ on $A$ with $h(a)\in a\cap A$. Choose $a_0\in A$ and use recursion to defineEvery value remains in $A$, and $f(n^{+})\in f(n)$ for every $n$. This is the forbidden sequence. Therefore, with the stated choice assumption, foundation is equivalent to the absence of descending membership sequences. The selection step explains exactly where choice is used; dependent choice would suffice.