How come asymmetry implies antisymmetry

Question

It is said that asymmetry implies antisymmetry, but how come so?

If $aRb \Rightarrow \neg(bRa)$, isn't this also the case for $a = b$, which indicates $aRa \Rightarrow \neg(aRa)$, which is a contradiction.

Answer 1

Asymmetry says that given $aRb$, we can't have $bRa$. Antisymmetry has both $aRb$ and $bRa$ as hypothesis, so in this case it is true by vacuity, since both hypothesis can never be satisfied simultaneously.