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.