Distinct solutions to generated Jacobian equations cannot intersect

We prove that if two $C^{1,1}(\mathrm{\Omega })$ solutions of the second boundary value problem for the generated Jacobian equation intersect in $\mathrm{\Omega }$ then they are the same solution. In addition we extend this result to $C^2(\bar{\mathrm{\Omega }})$ solutions intersecting on the boundary, via an additional convexity condition on the target domain.