Every vector space has a basis. This follows from Zorn's lemma, an equivalent formulation of the Axiom of Choice. Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice. The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. "Dimension theorem for vector spaces"). It is called the "dimension" of the vector space, denoted by dim "V". If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.