I am having trouble understanding the following proof from Santambrogio's book Optimal Transport for Applied Mathematicians. Here $\mathcal{J}_c(\mu,\nu)$ denotes the cost to transport $\mu$ to $\nu$ and $\operatorname{Lip}_1$ denote the $1$-Lipschitz maps.
Proposition : If $c : X\times X\to\mathbb{R}$ is a distance and $\mu,\nu$ are such that $\mathcal{J}_c(\mu,\nu)=0$, then $\mu=\nu$.
Proof : The duality formula gives $\int ud(\mu-\nu)=0$ for every $u\in\operatorname{Lip}_1$, which is enough to guarantee $\mu=\nu$
In case $c : X\times X\to\mathbb{R}$ is a distance the duality formula states $$\mathcal{J}_c(\mu,\nu)=\max\left\{\int ud(\mu-\nu)\;:\; u\in\operatorname{Lip}_1\right\}.$$ However, if $\mathcal{J}_c(\mu,\nu)=0$, couldn't it still be that for some Lipschitz map $u$ we have $$\int ud(\mu-\nu)<0?$$ but the maximum over all Lipschitz maps is zero?