Antireflexive and Transitive Relation is Asymmetric

Theorem
Every relation which is antireflexive and transitive is also asymmetric.

Proof
Let $$\mathcal{R} \subseteq S \times S$$ be antireflexive and transitive.

That is:

Now suppose $$\left({x, y}\right) \in \mathcal{R} \land \left({y, x}\right) \in \mathcal{R}$$.

Then from the transitivity of $$\mathcal{R}$$ that would mean $$\left({x, x}\right) \in \mathcal{R}$$.

But that would contradict the antireflexivity of $$\mathcal{R}$$.

Therefore $$\left({x, y}\right) \in \mathcal{R} \Longrightarrow \left({y, x}\right) \notin \mathcal{R}$$ and $$\mathcal{R}$$ has been shown to be asymmetric.