# 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 $A$.

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Of course, how silly of me! thanks – user52534 Jan 16 '13 at 8:45
I would think the other direction, requiring replacement in an essential way, and building bijections via approximations, is the one that is not immediate... Maybe "clear" simply means "proved in the book". – Andrés Caicedo Jan 16 '13 at 19:16
@Andres: Yeah, that's what Hagen's comment said and that was what I thought when I started reading the question as well... – Asaf Karagila Jan 16 '13 at 19:22