TITLE: How do you prove the trichotomy law for cardinal numbers?
QUESTION [10 upvotes]: Law of trichotomy is that for any two cardinals $a$ and $b$, exactly one of the
conditions $a<b$, $a=b$, or $a>b$ holds.

REPLY [2 votes]: If $A$, $B$ are any well-ordered sets then one of them is isomorphic to initial segment of the other one. Some authors call this fundamental theorem for well-ordered sets, e.g. Suppes: Axiomatic Set Theory, p.233, Theorem 80.
If you know from well-ordering theorem there for any two sets $A$, $B$ there is a well-order than the above implies that there is injective map either from $A$ to $B$ or the other way round. (Simply take arbitrary well-orderings of $A$ and $B$ and then there exists an isomorphism between one of the sets and a subset -- an initial segment -- of the other one. This isomorphism is necessarily a bijection.)