TITLE: Any Set of Ordinals is Well-Ordered
QUESTION [5 upvotes]: Is this a theorem of ZF or a theorem of ZFC?
My suspicion is that it is a theorem of ZFC (the proof I've seen in P. L. Clark's online notes on countable ordinals requires selecting an element from each set of a countable collection of sets).
Does the situation change if the ordinals in the collection are themselves countable?

REPLY [5 votes]: It is not hard to show directly from the definitions:

If $(A,<)$ is a well-ordered set, then for every $B\subseteq A$, $(B,<\restriction_B)$ is also well-ordered.
Every set of ordinals is a subset of an ordinal.

So without using the axiom of choice we have that every set of ordinals is well-ordered.