Let be a nonempty set partially ordered by . Then the following are equivalent.
- Every nonempty subset of contains a minimal element.
- If is a sequence in such that for all , then there exists such that for all , . (The descending chain condition.
Suppose every nonempty subset of contains a minimal element, and let be a descending chain in . In particular, is a nonempty subset of , which contains a minimal element . We have for all as desired.
Suppose satisfies the descending chain condition, and let be a nonempty subset. Suppose does not have a minimal element. Define a descending sequence inductively as follows. Let be any element of . (We can do this since is not empty.) Given that is defined, since is not minimal, there exists . Now is a descending chain, and for all , ; a violation of the descending chain condition.