The relation $\;\sqsubset\;\subseteq S\times S$ is asymmetric if
$$\forall a,b\in S:(a,b)\in\sqsubset\rightarrow (b,a)\notin\sqsubset$$
and it is antireflexive if
$$\forall a\in S:(a,a)\notin\;\sqsubset$$
I want to prove that
Now... it seem me obvious that is right, eve using some examples in my mind
but when I try to write the formal proof in this way I'm in confusion at the conclusion.
if $b=a$ I get
... this is a contraddiction..but I don't understand how it is the proof...
someone can explain me in easy words why this is a proof? I only see that the asymmetry lead to contraddiction when $b=a$ ... there is something I'm missing..