Under general hypotheses on the target set S and the dynamics of the system, we show that the minimal time function T-S(.) is a proximal solution to the Hamilton-Jacobi equation. Uniqueness results are obtained with two different kinds of boundary conditions. A new propagation result is proven, and as an application, we give necessary and sufficient conditions for T-S(.) to be Lipschitz continuous near S. A Petrov-type modulus condition is also shown to be sufficient for continuity of T-S (.) near S.