Bourbaki Proof of Zorn's Lemma in Lang's Algebra Serge Lang in his book Algebra has a nice appendix on set theory at the end of the book. In particular, in paragraph 2, pp. 881-884 he provides a proof of Zorn's Lemma from "other properties of sets which everyone would immediately grant as acceptable psychologically" (see middle of p. 881). As i understand Zorn's Lemma is equivalent to the Axiom of Choice and to the Well Ordering Principle. However, in Lang's proof i can not identify a point where either of the above two axioms is used. At the top of p. 882 he gives an argument according to which "we can assume that the set under consideration has a least element", but this, "without loss of generality". So, even though this resembles the "well ordering principle", it does not seem to be it. My question is: Are indeed the axiom of choice or the well-ordering principle not used in Lang's proof? If yes where? If not, then what is the subtle set-theoretic axiom that this proof uses to deliver Zorn's Lemma?
Edited: In the appendix that i am referring to, Zorn's Lemma appears as Corollary 2.5. However, when by "proof of Zorn's Lemma", i mean all the material that Lang proves to get to Corollary 2.5, i.e. Theorem 2.1, Lemma 2.2, Lemma 2.3, Corollary 2.4. 
 A: The proof I saw was using, essentially, the Hausdorff Maximality Principle.

Hausdorff Maximality Principle (HMP). Every partially ordered chain has a $\subseteq$-maximal chain.

Clearly if $(P,\leq)$ is a partially ordered set satisfying the conditions for Zorn's lemma, and $C\subseteq P$ is a maximal chain then $C$ has an upper bound $c$ and by the maximality of $C$ we have to have $c\in C$ and that it is a maximal element.
The proof seems to go through other means to prove HMP, and it uses the Bourbaki-Witt theorem.
The Bourbaki-Witt theorem does not rely on the axiom of choice at all. It is provable in ZF, as remarked by Brian here. The reason it is provable in ZF is that it assumes the existence of a least upper bound of every chain, and therefore allows a canonical choice for $a_1$ when moving from $D_0$ to $D_1$ (and so on).
The axiom of choice comes in, full-blown power, in the proof of Corollary 2.4:

Suppose that $A$ does not have a maximal element. Then for each $x\in A$ there is an element $y_x\in A$ such that $x<y_x$. Let $f\colon A\to A$ be the map such that $f(x)=y_x$ for all $x\in A$.

Here we actually say that $f$ is a choice function. In the proof of the Bourbaki-Witt theorem we needed to know what is the next step in "growing up", but we assumed that $f$ was given to us. From that it was just a matter of finishing the proof.
But in this corollary we need to actually define $f$. And this requires the axiom of choice.

I should add that there are books full of equivalents to the axiom of choice, and there are a lot of them which are very not-set theoretical axioms, or very unrelated to well-ordering (e.g. "Every non-empty set can be made into a group"). Since they are all equivalent to the axiom of choice, all of them are equivalent to Zorn's lemma. 
