Is an irreflexive and transitive set an anti symmetric set?

Question

I have read that a simple ordered set is a total ordered set which is irreflexive and transitive. I want to know if irreflexivity and transitivity implies antisymmetry?

Answer 1

Suppose a relation $R$, which is anti-reflexive and transitive, is also symmetric. Therefore $xRy$ and $yRx$. By transitivity, $xRy \wedge yRx \Longrightarrow xRx$, but this is impossible.

So far $R$ is not symmetric. Moreover, $xRy\wedge yRx \Longrightarrow x=y$ is necessary and sufficient condition for $R$ to be antisymmetric. But we have just shown that $xRy\wedge yRx$ can't hold for any $x,y$ in the set on which $R$ is defined. Therefore it follows that for every $x,y$ in that set on which $R$ is defined if $xRy\wedge yRx \Longrightarrow x=y$. This is true because the premise is false, therefore the expression is true, therefore the relation is antisymmetric.