Bijection and Isomorphim in Set Theory Let A and B be two well-ordered sets. Is it ture that A and B are isomorphic if A and B are bijective?
Motivation:
I am studying Bourbaki's Elements of Methematics and have just reached Sec 3.2, Chap 3 where the concept of $\mathbb{N}$ have not appeared yet but Bourbaki seems to have assumed Cantor–Bernstein theorem. I want to proof the theorem with the knowledge I have obtained from Bourbaki's work and if the above statement is ture the theorem is also ture.

The statement above is false. For example, consider:


*

*The natural numbers with the usual ordering.

*The set $\{1-{1\over n}: n\in\mathbb{N}_{>0}\}\cup\{1\}$ with the usual ordering.
(The corresponding ordinals are $\omega$ and $\omega+1$, respectively.)
These sets are each countable, so in bijection with each other. However, they're not isomorphic as well-orderings: the second has a greatest element but the first doesn't. 
