If the core is empty, then the last primal constraint is not binding, and so $\eta^*=0$ in every optimal dual solution. Thus for any optimal dual solution,$\sum_{S\ni i}\lambda^*_S=1$, and so the weights are balanced. Consequently $\sum_S\lambda^*_Sv(S)> v(\nn)$, and so the characteristic function $v$ is not balanced.

If $v$ is not balanced, then there exists a feasible solution to the primal with $\sum_S\lambda_Sv(S)>v(\nn)$. This must be true of the optimum as well, and so the value of the dual, and the primal, exceeds $v(\nn)$. Consequently, the core is empty. ∎