A short convex-hull proof for the all-different system with the inclusion property

An all-different constraint on some discrete variables imposes the condition that no two variables take the same value. A linear-inequality description of the convex hull of solutions to a system of all-different constraints is known under the so-called inclusion property: the convex hull is the intersection of the convex hulls of each of the all-different constraints of the system. We give a short proof of this result, which in addition shows the total dual integrality of the linear system.