Proof that a^a + b^b = c^c has no solutions for any natural numbers

Question

I have to prove that there is no natural numbers a,b and c such that a^a + b^b = c^c.

I assumed that a $\geq b $\ Thus a+1 $\ge b $\ and (a+1)^(a+1) $\ $\ge b^b

But I have no idea if I’m thinking correctly.