We study the mixing inequalities which were introduced by Gunluk and Pochet (2001). We show that a mixing inequality which mixes n MIR inequalities has MIR rank at most n if it is a type I mixing inequality and at most n-1 if it is a type II mixing inequality. We also show that these bounds are tight for n=2. We define mixing inequalities for a general mixed-integer set and show that the elementary mixing closure can be described using a bounded number of mixing inequalities, each of which has a bounded number of terms. This implies that the elementary mixing closure is a polyhedron. Finally, we show that any mixing inequality can be derived via a polynomial length MIR cutting plane proof. Combined with results of Dash (2006) and Pudlak (1997), this implies that there are valid inequalities for a certain mixed-integer set that cannot be obtained via a polynomial-size mixing cutting-plane proof.