The well ordering principle, the axiom of choice, and Zorn's Lemma are all "equivalent", meaning you can pick any one as an axiom and prove the other two.

So some text books may pick one as the axiom and others pick a different axiom.

The crazy thing about the well-ordering principle: It states that a well ordering exists on the reals, which means that you can find an ordering such that any open set has a minimum. Apparently, elsewhere in mathematics, they've proven that even though it exists, you cannot articulate that ordering.

There's a common joke:

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"