I am writing a short (30-50 pages) report on AC for an exam. I really would like to include the proof that "Every vector space has a basis $\implies$ AC". Actually, every proof I could find proves that AMC (the Axiom of Multiple Choices) holds, not AC. Since ZF $\vdash$ "AC $\iff$ AMC", that would be formally fine, but then I would like to include the proof of "AMC $\implies$ AC". But every proof I could find proves it by passing through "$\cal P(\alpha)$ is well-orderable for every ordinal $\alpha \implies$ AC".
The proof of the latter seems to use von Neumann hierarchy. The professor I will present my report to is not a logician (he's an Algebra professor), and I'm pretty sure that he wouldn't like to read too much set-theory stuff.
Therefore my question is: do you know any way to prove "Every vector space has a basis $\implies$ AC" without mentioning von Neumann hierarchy?