It’s not obvious to me that what you wrote guarantees that A is undefeated.

It’s definitely not obvious! a sketch is in footnote 10

10The proof is by induction on the number of candidates. If A wins, then there is some B who does not defeat A such that A wins without B in the election. Hence by the inductive hypothesis, A is undefeated in the election without B. Then since B does not defeat A in the full election, it is easy to see from the definition of defeat that A is still undefeated.

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