For any sets A and B, their cardinal numbers satisfy left bracketing bar A right bracketing bar <= left bracketing bar B right bracketing bar iff there is a one-to-one function f from A into B. It is easy to show this satisfies the reflexive and transitive axioms of a partial order. However, it is difficult to show the antisymmetry property, whose proof is known as the Schröder-Bernstein theorem. To show the trichotomy property, one must use the axiom of choice. Although an order type can be defined similarly, it does not seem usual to do so.

Schröder-Bernstein theorem