Quantum Collapse Requires Pre-and Immediate Post-measurement States to Belong to Disjoint Sets

We present a simple proof that the orthodox interpretation of quantum mechanics, due to its incorporation of quantum collapse, requires pre- and immediate post-measurement states to belong to disjoint sets of states. This requires a reformulation of the projection postulate as a transformation of the quantum state to one that ``looks like'' an eigenstate, because otherwise such projection implies that the two kinds of states belong to the same set of states. An attempt to render more precise what is meant by a state that ``looks like'' an eigenstate is presented.