In the book Set theory, Chapter 3 N.Bourbaki, I would like to understand how Bourbaki proves ZL. I wrote the proof. It uses Zermelo's principle (which is okay since they are equivalent), so I tried to understand how Zermelo's principle is proved but I can't find any occurrence of the axiom of choice or anything equivalent in the pages before. How is it possible? It seems that Bourbaki proved Zermelo's principle from nowhere, so where am I wrong?
I have only the third chapter of Set Theory so maybe the set theory he is using is not ZF, which could explain a proof of such unprovable statement. Would anyone help me understand this?
Thank you.
The proof uses the fact that there is a function $p$ such that $p(X)\notin X$, that is $p(X)\in E\setminus X$. This is exactly a choice function, whose existence is asserted by the axiom of choice.