Existence of bases implies the axiom of choice, a foundation-free proof

We prove that, in Zermelo–Fraenkel set theory with the axiom of Foundation removed, the statement that every vector space has a basis implies the Axiom of Choice, concluding that the classical equivalence between $\text{AC}$ and the existence of bases does not require the Axiom of Foundation. More specifically, we prove that if every vector space over a field of characteristic zero has a basis, then $\text{AC}$ holds. This result extends to set theory with atoms.