Sufficient Statistics and Discrete Exponential Families

_θ\ is a set of probabilities on a countable set _χ such that P_θ(x) > 0 for each x and θ. We prove that if _θ\ is not an exponential family, then each sufficient statistic for n independent observations must be one-to-one, modulo permutations, on an infinite product set (which does not depend on the sufficient statistic).