I'm asked to prove that if C is a copula such that $C(t,1-t) = 0$ for all $t$ in $[0,1]$ then $C = max(u+v-1,0)$. So far I have this:
Let $u,v$ be in $[0,1]$. If $u+v-1 < 0$, i.e. $u < 1-v$. Given that $C$ is nondecreasing $$0 \leq C(u,v) \leq C(1-v,v) = 0$$ Therefore $C(u,v) = 0$ when $u+v-1 < 0$. On the other hand, suppose $u+v-1 \geq 0$, then $u \geq v-1$. It follows from the Fréchet-Hoeeffding theorem that $$max(u+v-1,0)=u+v-1 \leq C(u,v)$$
That's all I have come up with. Obviously, I need to show that $u+v-1$ is also an upper bound of C(u,v) but I can't find a way to do that. My best attempt has been:
It follows from the nondeacreasing property of C that $$C(u,v) = C(u,v)-C(u, 1-u) \leq C(u,v) - C(1-v,1-u)$$ Additionally, I noticed that $min(u,v) - min(1-u,1-v) = u + v -1$. So, my main focus has been trying to proof $$C(u,v) - C(1-v, 1-u) \leq min(u,v) - min(1-u,1-v)$$ I don't know if I should try to solve this problem using a different approach or maybe I'm ignoring some known properties of copulas that might help me in this particular situation. Any suggestions?