We study classes of ordered partitions into p parts of an n-element set, in which for each k the family of k-th parts is an antichain (that is, a family of sets in which there are no subset relations). Meshalkin's theorem, a generalization of Sperner's Theorem, states that the maximum size of such a class is the maximum size of a multinomial coefficient \binom{n}{a_1,...,a_p}. We generalize this and a stronger inequality called an LYM inequality. Our proof is simpler as well as more general than all previous proofs and extends to analogous statements--with a twist--about flats in a projective geometry.