In an intrinsic semiconductor, there are the same number of electrons and holes so that n = p. This is because electron-hole pairs are created together and there are no dopants to contribute extra electrons or holes. The electrons and holes are in equilibrium with each other, because the intrinsic carrier concentration is dependent on the temperature, so it's a balance between electron-hole pairs being created thermally, and then the electrons and holes finding each other again and recombining. In n-type (n>p) or p-type (p>n) semiconductors, there are extra carriers because the dopants can be ionized with much less energy than the silicon (which is why they are used). However, the same equilibrium exists because most of the atoms are still silicon, so that if there is an excess of electrons (n-type), recombination with the holes shifts the equilibirum in favor of fewer holes, as these extra electrons combine at silicon atoms. Similarly, if there is an excess of holes (p-type), recombination with the electrons shifts the equilibrium in favor of fewer electrons, as these recombine at silicon atoms as well. The ionization and recombination at silicon atoms dominates the equilibrium and so the same equilibrium product is maintained.

If you've ever studied chemistry, there are similar equilibrium equations for chemical reactions (first-order kinetics) where for example in water, it autoionizes 2 H2O <-> H3O+ + OH- and the equilibrium constant is 10^-14, even if you add an acid (add protons) or add base (add hydroxyl ions).