Is it possible that $\mathsf{P} \not = \mathsf{NP}$ and the cardinality of $\mathsf{P}$ is the same as the cardinality of $\mathsf{NP}$?  Or does $\mathsf{P} \not = \mathsf{NP}$ mean that $\mathsf{P}$ and $\mathsf{NP}$ must have different cardinalities?
Does $\mathsf{P} \ne \mathsf{NP}$ imply that $|\mathsf{NP}| > |\mathsf{P}|$?
complexity classes;p vs np