How is Zorn’s lemma equivalent to the axiom of choice?

In set theory, Zorn’s lemma and the axiom of choice are mathematically equivalent, meaning that assuming the validity of one allows for the formal proof of the other. Along with the well-ordering theorem, these three principles form a triad of equivalent statements; accepting any one of them as an axiom enables the derivation of the remaining two.123

The equivalence signifies that they are different ways of asserting the same underlying logical power, particularly regarding infinite sets where no explicit selection rule exists. While the axiom of choice focuses on the ability to simultaneously select elements from an infinite collection of sets, Zorn’s lemma provides a tool to prove the existence of a "maximal" object within a collection that is closed under the unions of chains.132

Because these principles assert the existence of sets or objects without providing a constructive method to build them, they are often grouped together in the ZFC (Zermelo-Fraenkel set theory with the axiom of choice) framework. Their equivalence is fundamental to modern mathematics, serving as an indispensable tool for proofs in fields like algebra and analysis.132