Most of the explanation of the theorem on wikipedia states the same as the video with the exception is the following: two systems of electrons with two potentials U_1(r) and U_2(r) that have the same ground-state density. Then the difference betwee the two potentials is constant, U_1(r)-U_2(r) is a constant. This is a small detail especially if you consider that the energy scale may be renormalized, shift it by a constant value.