TITLE: How to prove relation is asymmetric if it is both anti-symmetric and irreflexive
QUESTION [5 upvotes]: Prove a relation is asymmetric if it is both anti-symmetric and irreflexive (anti-reflexsive). 

I tried to go from the definitions of the relations:
Anti symmetric: $\forall x,y \, (xRy \land yRx \Rightarrow x=y )$
Irreflexsive: $\forall x\in A \ ,((x,x)\notin R)  $
Assymetric: $\forall x,y \in A \,(xRy \Rightarrow \lnot yRx ) $
But it doesn't get me anywhere... I also tried to think about proof by contraposition but I can't seem to be able to connect the definitions.
Any help would be appreciated.

REPLY [8 votes]: Proof by contradiction will work here.
Assume $R$ is antisymmetric and irreflexive:

Let $R$ be irreflexive: $$\forall x \in A, (x, x)\notin A$$ which
means alternatively, $$\forall x \in A, \lnot( xRx)$$
Let $R$ be antisymmetric: $$\forall x \in A, \forall y \in A, \Big(x
   R y \land yRx \rightarrow (x = y)\Big)$$

And assume, for contradiction, that $R$ is not asymmetric. The negation of asymmetry is given by $$\exists x \in A, \exists y \in A\,\Big(x R y \land yRx\Big)$$
Now show that this assumption contradicts antisymmetry or irreflexivity:
Can you see that this last assumption implies, by the definition of antisymmetry, that $x = y$? 
But if $x = y$, then  $xRy \implies xRx$. 
But this contradicts irreflexivity! Contradiction. $\square$