A fundamental set-theoretic principle: if every chain (totally ordered subset) of a partially ordered set has an upper bound in the set, then the set has at least one maximal element.
Zorn's Lemma is logically equivalent to the Axiom of Choice — you can prove either from the other given the axioms of set theory (minus Choice). It is the preferred form for applications in algebra, analysis, and topology because its hypothesis is easy to verify in practice.
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