For any sets 
and 
,
their cardinal numbers satisfy 
iff there is a one-to-one
function 
from 
into 
(Rubin 1967, p. 266; Suppes 1972, pp. 94 and 116). 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.
