Consider an electronic circuit consisting of linear components plus a number of ideal diodes. By "ideal" I mean they can either be forward-biased (i.e. \$v_D=0\$ and \$i_D\geq 0\$ ) or reverse-biased (i.e. \$v_D\leq 0\$ and \$i_D=0\$).
These circuits can be calculated by arbitrarily declaring each diode either forward-biased or reverse-biased, and setting \$v_D=0\$ for every forward-biased diode and \$i_D=0\$ for every reverse-biased diode. After the resulting linear circuit has been calculated, we have to check whether at every forward-biased diode \$i_D\geq 0\$ and at every reverse-biased diode \$v_D\leq 0\$ is satisfied. If yes, that's our solution. If not, we have to try another set of choices for the diodes. So, for \$N\$ diodes, we can calculate the circuit by calculating at most \$2^N\$ linear circuits (usually much less).
Why does this work? In other words, why is there always one choice that leads to a valid solution and (more interestingly) why are there never two choices that both lead to valid solutions?
It should be possible to prove that on the same level of rigor with which e.g. Thevenin's theorem is proven in textbooks.
A link to a proof in the literature would also be an acceptable answer.