There are lots of equivalent statements to the axiom of choice, Zorn's lemma being a famous example. Another interesting one is the fact that every vector space has a basis. We can always find a set of vectors which are linearly independent and generate the entire space.

The proof that every vector space has a basis is simple enough if you already understand Zorn's lemma. You take all the sets which have linearly independent vectors, and the union of any chain of those sets will give a set of linearly independent vectors, so there must exist a maximal linearly independent set, which will be a basis.

Interestingly though, this is an equivalence, we can prove that if every vector space has a basis, then the axiom of choice holds. However, if you try to look for a proof anywhere on the internet, every single possible source will say "This was proven by Andreas Blass and you can see it in this pdf". He did this in 1984 and since then, not a single person bothered to make an alternate proof? Or rewrite it in a more digestible way?