  We investigate the combined effect of neutron and proton superfluidities on
the cooling of neutron stars whose cores consist of nucleons and electrons. We
consider singlet-state pairing of protons and triplet-state pairing of neutrons
in the cores of neutron stars. The critical superfluid temperatures T_c are
assumed to depend on the density of matter. We study two types of neutron
pairing with different components of the total angular momentum of Cooper pairs
along the quantization axis (|m_J| =0 or 2). Our calculations are compared with
observations of thermal emission from isolated neutron stars. We show that the
observations can be interpreted by using two classes of superfluidity models:
(1) strong proton superfluidity with a maximum critical temperature in the
stellar core T_c^{max} > 4 \times 10^9 K and weak neutron superfluidity of any
type (T_c^{max} < 2 \times 10^8 K); (2) strong neutron superfluidity (pairing
with |m_J|=0) and weak proton superfluidity. The two types of models reflect an
approximate symmetry with respect to an interchange of the critical
temperatures of neutron and proton pairing.
