Suppose n_p and n_e represent the numbers of holes and conduction...

In an intrinsic semiconductor, charge carriers are generated by the thermal excitation of electrons from the valence band to the conduction band.

Each excitation creates exactly one free electron in the conduction band and leaves behind exactly one hole in the valence band.

Therefore, the number of conduction electrons is always equal to the number of holes, which gives $n_p = n_e$.