This paper investigates the problem of spectrally efficient operation of a multi-user uplink cognitive radio system in the presence of a single primary link. The secondary system applies opportunistic interference cancelation (OIC) and decode the primary signal when such an opportunity is created. We derive the achievable rate in the secondary system when OIC is used. This scheme has a practical significance, since it enables rate adaptation without requiring any action from the primary system. The approximated formulas and tight lower and upper bounds for the ergodic sum-rate capacity of the secondary network are found. Next, the power allocation is investigated in the secondary system for maximizing the sum-rate under an interference constraint at the primary system and it is shown that the optimal solution leads to a opportunistic user selection strategy. We propose a simple user selection strategy combined with OIC which achieves the optimal capacity given an interference constraint at the primary system. Finally, the analytical results are confirmed by simulations, indicating the fact that the low-complexity, spectral-efficient, flexible, and high-performing cognitive radio can be designed based on the proposed scheme.