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.

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  1. Descending membership sequence
  2. Axiom of regularity
  3. Set theory
  4. Foundations of mathematics
  5. Area of mathematics
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