Is the symbol ">" antisymmetric? For example, if I say $(x>y) \wedge(y>x) \rightarrow \exists(x,y)|x=y$ is vacuously true since the premise cannot be true. This means that $x>y$ is antisymmetric? I do know that $\geq$ is antisymmetric because there is a case where x=y.
Thanks!
Yes, it is vacuously antisymmetric.
However, note that in your formulation "$(x>y)\land(y>x)\to \exists(x,y)\mid x=y$" the symbols $\exists(x,y)$ make no sense and should not be there. If you want quantifiers in that statement (which is not a bad idea), it should be as $$ \forall x\;\forall y\;\bigl[ (x>y)\land(y>x)\to x=y \bigr]$$