Representations of canonical commutation relations are fundamental for quantum theory. In quantum mechanics, the celebrated Stone-von Neumann theorem shows that, given mild regularity conditions, there is only one representation (up to unitary equivalence). While this is, in general, no longer true in quantum field theory, uniqueness may still be given in the presence of symmetries. So, in loop quantum gravity, diffeomorphism invariance is well known to imply uniqueness of states on the so-called holonomy-flux algebra. The corresponding proof by Lewandowski et al., however, has been very lengthy and technical. In our talk, we are going to present a drastically shorter proof of a more general uniqueness theorem we have obtained recently. This even applies to respective cosmological models.