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Ok, how are you defining a tie?

I believe the author is defining a tie as the following:

(a <= b) && (b <= a) && (a != b)

Then a and b are "tied". Where "!=" means a and b are different.

I'm fine with that restriction. But that isn't the axiom of antisymmetry. That's the axiom of antisymmetry plus a rule that holds that equality implies identity (which would be typically described in terms of the axiom of extensionality). My problem is with the implied claim that antisymmetry alone gives you that restriction, which is incorrect. Antisymmetry is entirely consistent with collections that contain equal but distinct elements.

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