Are the two following equivalent under ZF :

  1. Axiom of Choice/Zorn's Lemma
  2. Given any two sets A, B, there exists an injection from A to B or there exists an injection from B to A

I could prove that (1) implies (2) and I wonder if they are actually equivalent but I couldn't prove it. I read somewhere that (2) cannot be proven with only ZF, that's why the equivalence seems plausible to me.

Thank you :)


marked as duplicate by Asaf Karagila axiom-of-choice Nov 3 '17 at 14:16

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    $\begingroup$ There are probably a few other rogue duplicates, and otherwise related threads. But I'm too lazy to find them now. $\endgroup$ – Asaf Karagila Nov 3 '17 at 14:18