If $\succsim$ is transitive but irreflexive, then it is asymmetric.
this is my proof:
Suppose $\succsim$ is not asymmetric, which means that for any $x,y \in X$ $x\succsim y \rightarrow y \succsim x$. By definition $\succsim$ is transitive, i.e. for any $x, y, z \in X$ we have $x \succsim y$ & $y\succsim z$ $\rightarrow$ $x \succsim z$. So since we are supposing $\succsim$ is symmetric: $y\succsim x \rightarrow x \succsim$ y. Now since $\succsim$ is transitive $y \succsim x$ & $x\succsim y \rightarrow y \succsim y$ but $\succsim$ is irreflexive so it's a contradition and $\succsim$ is asymmetric.
What do you think of my proof, is it wrong or not clear enough?