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.
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