You're describing an open circuit.
A double-balanced mixer, when it's terminated to be a unidirectional device and driven nice and hard, should produce something like
y_IF(t) = x_RF(t) * signum(sin(w_LO * t)),
where signum(x) = -1 for x < 0, signum(x) = 1 for x > 0 and don't ask me about x = 0. So there is most definitely a signal coming out of y_IF for any signal x_RF going in.
You probably mean there isn't a _component_ of y_IF at the _frequency_ of x_RF without a peculiar combination of RF and local oscillator frequency. That is the case but that's not what the definition speaks to.