This result could not be intuitively simpler, yet it depends on the axiom of choice for its truth!
Letbe the set of all equivalence classes under the relation ~ on A defined by . Without the axiom of choice, we would not be able to take from each member of one and only one element to form our subset C.
(More precisely, it is the axiom of choice that guarantees the existence of an injective function, from which we set .)
Zorn's lemma is obviously true, the Well-ordering principle is obviously false, and who knows about the Axiom of Choice.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."