By the way, the Shen’s book takes a different route to the Zorn’s lemma: first he introduces well-ordered sets, then uses tranfinite recursion to prove Zermelo’s theorem (that any set can be well-ordered), then he uses Zermelo’s theorem and tranfinite recursion to prove Zorn’s lemma. Thus the proof of Zorn’s lemma is reduced from two pages to a few lines. I personally found it easier to follow and remember.
By the way, the Shen’s book takes a different route to the Zorn’s lemma: first he introduces well-ordered sets, then uses tranfinite recursion to prove Zermelo’s theorem (that any set can be well-ordered), then he uses Zermelo’s theorem and tranfinite recursion to prove Zorn’s lemma. Thus the proof of Zorn’s lemma is reduced from two pages to a few lines. I personally found it easier to follow and remember.