I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am having trouble proving the other way. I want to prove, without choice, that any set is in bijection with a set of distinct cardinals. I don't know if this is true, but if it is could someone provide a proof for me? Thank you.
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