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Proposition: Every quasi order is antisymmetric:
forall S, < : < is quasi order on S => < is antisymmetric on S.
Proof: Take arbitrary S and quasi order < on S. Assume there exist x in S and y in S with x != y such that x < y and y < x. By transitivity, we have x < x which contradicts the irreflexivity of < .
Only difference between partial orders and quasi orders is reflexivity versus irreflexivity.