Show $x^x+y^y=z^z+t^t$ has no solution in integers for distinct $x,y,z,t$. 
Prove that there does not exist pairwise distinct positive integers $x,y,x,t$ such that $$x^x+y^y=z^z+t^t\;.$$

I was trying to find some inequalities by assuming a solution does exist but cant get anywhere.

Let $x,y,z,t$ be pairwise distinct positive integers. Without loss of generality assume that $t$ is greater than $x,y,z$. Then $t>2$ and $(t-1)^{t-1}$ is not smaller than both $x^x, y^y$. Therefore
$$z^z + t^t > t^t > t \cdot (t-1)^{t-1} > 2(t-1)^{t-1} \ge x^x + y^y$$
so $x^x+y^y\neq z^z+t^t$.
