You'd need to make your claim more precise, in order to prove it, but a proof might go something like this: assume that an anti-bloom filter starts empty, and that it over time sees some small fraction of the possible objects (because objects are 64+ bits, say). If the anti-bloom filter usually reports "yes" for a significant fraction (even just 1%) of the seen objects, then an information theoretic argument tells you it must have size around the number of objects times the object size.