Phase transitions in layered systems

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_\gamma(x,y)=\gamma J(\gamma(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $\gamma^{A}$ (where $A\ge 2$ is fixed) and prove that for any $\beta$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $\gamma$ small enough.