This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.
So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.
Joel David Hamkins
- 236.4k
- 44
- 777
- 1.4k