Contradiction with Maxwell's demon thought experiment

There is an important inequality in thermodynamics which states that TdS>=dQ, where T is the temperature of the system in question (or, more generally, the temperature of the surroundings), dS is the (infinitesimal) change in entropy of the system, and dQ is the (infinitesimal) net amount of heat supplied to the system. The symbol >= means greater than or equal to. This is really just one form of the second law of thermodynamics. But it seems like with Maxwell’s demon, you can decrease the entropy of a gas (our "system" in this case) without interacting with the particles of the gas (you simply open or close a door), in which case no heat can be supplied to or taken out of the gas, meaning that dQ=0. But yet, the entropy of the gas decreases, which means that the inequality above is violated. What’s going on here? Although the entropy of the whole universe (system + demon) doesn’t decrease (because the demon acquires information about the states of the gas molecules, increasing its own entropy, which is greater than or equal to the entropy decrease in the gas), there is still this contradiction with that inequality above, it seems.