Given an arbitrary finite dimensional Hamiltonian H_0, we consider the model H=H_0+Delta H, where Delta H is a generic fully connected interaction. By using the strong law of large numbers we easily prove that, for all such models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional mean-field models the term H_0 gives rise to short-range correlations and, furthermore, when H_0 has negative couplings, first-order phase transitions and inverse transition phenomena may take place even when only two-body interactions are present. The dependence from a non uniform external field and finite size effects are also explicitly calculated. Partially, these results were derived long ago by using min-max principles but remained almost unknown.