Here is a proof of the above statement that I would like to verify. I would like to know any possible errors or improvements possible.
Proof: Suppose tournament $T$ is transitive. Let $x$ and $y$ be any two vertices of $T$ and assume, without loss of generality, $(x,y)$ is an arc in $T$. Now let $U$ be the set of all vertices in $T$ that are adjacent from vertex $y$. Then it follows that $U$={$u \in V(T)$ | $(y,u)$ is an arc in $T$}, and the $outdegree (y) = |U|$. For each $u \in U$, the tansitivity in T implies that $(x,u)$ is an are in $T$. Thus the $outdegree(x) \ge |U|+1 \gt outdegree(y)$ and it is clear then that every two veritces of $T$ have distinct outdegrees.