The prose "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me" is inaccurate for describing antisymmetry. In the same short section, you first state the correct condition:
You have x ≤ y and y ≤ x only if x = y
from which it doesn't follow that "It also means that no ties are permitted". The "no ties" idea belongs to a stronger notion such as a strict total order, not to antisymmetry.
You (presumably) aren't your grandmother, so we have x=/=y. Therefore by the biimplication, (x ≤ y and y ≤ x) is false i.e. either x ≤ y (I am better than my grandmother) or y ≤ x (my grandmother is better than me). The "neither" case is excluded by the law of totality.
We literally said the same thing. It doesn't follow from antisymmetry.
My point is precisely that:
(x <= y /\ y <= x) -> x = y
does not entail
x <= y \/ y <= x
The second statement is totality/comparability, not antisymmetry.