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An axiom of set theory, typically included alongside the Zermelo-Fraenkel axioms (from which it is independent) as one of the foundational axioms of set theory. Its statement is ''For any collection of nonempty, disjoint, there exists a set which contains precisely each element from each member of the collection.'' In symbols: \(\forall x:(\emptyset\notin x\land \forall a,b\in x:a\cap b=\emptyset)\Rightarrow\exists c:\forall a\in x:\exists!y\in a:y\in c\). The axiom of choice is equivalent to Zorn's Lemma, and statements such as ''Every set has a total order'', ''every vector space has a basis''.