# A set bijective with an ordinal

Lemma I.10.10 in Kunen's set thoery (2011) claims that a set $A$ is well-orderable iff $A$ is bijective with an ordinal.

He does not prove this, and one direction is clear to me.

However, I can't see why $A$ bijective with an ordinal implies $A$ is well-orderable. By the way he develops this within ZFC except foundation, power set, and choice.

Any help is appreciated -Thanks!

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Oh, I'd have thiought that this is the "one direction is clear to me". The well-order of the cardinal induces a well-order on $A$ via the bijection. –  Hagen von Eitzen Jan 16 '13 at 8:40

If $f\colon A\to \alpha$ is a bijection then $a\prec b\iff f(a)\in f(b)$ is a well ordering of $A$.