I want to show that the multivariate Frechet-Hoeffding lower bound given by $C^-(u_1,\dots, u_n) = \left(\sum_{i=1}^{n}{u_i} - (n-1)\right)_+$ is not a copula when $n \geq 3$.
My question is:
Is it possible to prove the result by considering that for all $i \neq j$, $(U_i, U_j)$ has the same distribution of $(U, 1-U)$ with $U \sim \mathcal{U}([0,1])$?
Suppose that $C^-$ is a copula, i.e. it is the cdf of $(U_1,\cdots,U_n)$ where $U_i\sim\mathcal U((0,1))$. Then for $i\neq j$ and $u\in(0,1)$, $$ \mathbb P(U_i\le u,U_j\le u)=\left(u+u+\sum_{i=1}^{n-2}1-(n-1)\right)_+=(2u-1)_+=\mathbb P(U_1\le u,1-U_1\le u). $$
As the cdf characterises the distribution, we deduce that $(U_i,U_j)$ has the same distribution as $(U_1,1-U_1)$, hence $U_j=1-U_i$ almost surely.
Then for $n\ge3$ the contradiction comes naturally out: by the preceding reasoning we must have almost surely $U_1=1-U_2$, $U_2=1-U_3$ hence $U_1=U_3$, but $U_3=1-U_1$ as well, which is of course impossible to satisfy.