Breaking Into Atoms

Let X be the set of all integers between 0 and n-1. Suppose we have a collection S1, S2, ..., Sm of subsets of X. Say an atom A is a subset of X such that for each Si we have either A is a subset of Si or A and Si do not have any common elements. Your task is to find a collection A1, ..., Ak of atoms such that every item in X is in some Ai and no two Ai, Aj with i ≠ j share a common item. Surely