A chain is a subset c satisfying x ≺ y ∨ x = y ∨ y ≺ x for all x y ∈ c.

Hausdorff's maximality principle

There exists a maximal totally ordered subset of α. Note that we do not require α to be partially ordered by r.

Zorn's lemma

If every chain has an upper bound, then there is a maximal element