Set theory : Anti-symmetric but transitive (proof)

Question

The exam practice question is as follows:

• We call a relation $R$ anti-reflexive iff $\forall a \in A : (a,a) \notin R$ and anti-symmetric iff $\forall a,b \in A : (a,b) \in R \rightarrow (b,a) \notin R$. Prove or refute that any anti-reflexive and transitive relation is also anti-symmetric.

I know it works because: $A = {(1,2),(2,3),(1,3)}$ is anti-reflexive, transitive and anti-symmetric - but I am struggling to prove this using only the definitions of the set relations.

Answer 1

Suppose that there are $a,b \in Dom(R)$. Such that $(a, b) \in R$ and $(b, a) \in R$. If the relation is transitive then it follows that $(a, a) \in R$. Here is the contradiction.

Answer 2

Try to put $c=a$ into the definition of transitivity...