It seems intuitive that the power set of a union of sets P(XunionY) is not a subset of the union of the two respective power sets P(X)unionP(Y). For finite sets the former will have more elements than the latter. However, I can't figure out what is wrong with the following line of reasoning: For a set S ( S in P(XunionY) ) implies ( S is a subset of XunionY ) implies (1) for some x ( x in S implies x in XunionY ) implies (2) ( x not in S OR x in X OR x in Y ) implies (3) ( ( x in S implies x in X ) OR (x in S implies x in Y) ) implies (4) ( ( S is a subset of X ) or ( S is a subset of Y ) ) (5) I could continue, but I'm nearly certain I have made a mistake somewhere already. I can't figure out where. I suspect that (3) is incorrect, but I don't see why I can't make the substitution I made namely: (A implies B) iff (notA OR B) Also, I know that (5) is incorrect. Perhaps (5) doesn't follow from (4).