Hi all! I am trying to understand Specker (1953)'s proof (found here) that the axiom of choice is false in New Foundations. I am stuck on the following point. At 3.5 Specker writes:

3.5. The cardinal numbers are well ordered by the relation "there are sets $a,b$ such that $a \in n, b \in m$ and $a \subseteq b$" (axiom of choice).

I am assuming that this is a consequence of the axiom of choice, which he is using to derive a contradiction. Is that true? If so, how is it a consequence of the axiom of choice?

Another, broader question: can anybody give an intuitive explanation of why AC fails in NF?

Thank you!