Many real-world complex systems, when hitting a tipping point, undergo irreversible sudden shifts that can eventually take a great toll on humanity and the natural world, such as ecosystem collapses, disease outbreaks, etc. Previous work has adopted approximations to predict the tipping points, but due to the nature of nonlinearity, this may lead to unexpected errors in predicting real-world systems. Here we obtain the rigorous bounds of the tipping points for general nonlinear cooperative networks. Our results offer two rigorous criteria that determine the collapse and survival of such a system. These two criteria are decided by the combined effect of dynamical parameters and interaction topology.