I'm looking for a proof that, given two sets $A$ and $B$, the cardinality of $A$ is less than $B$ implies the cardinality of the powerset of $A$ is less than the cardinality of the powerset of $B$.
In short, does $\overline{\overline {A}}<\overline{\overline {B}}\ \ $ imply $ \ \ \overline{\overline {P(A)}} < \overline{\overline {P(B)}}$?
There seems to exist a clear method of proof for when $A$ and $B$ are finite (nonempty) sets, since there cannot exist a bijection between a finite set and a proper subset of itself, and hence $\overline{\overline {A}}<\overline{\overline {B}}$ implies there exists a bijection between $A$ and a proper subset of $B$, which one could use to construct a bijection between $P(A)$ and a proper subset of $P(B)$ and then apply the fact that there exists no bijection from $P(B)$ to a proper subset of itself.
But what about when $A$ and $B$ are infinite? How would we know that the proper subset of $P(B)$ which is bijective to $P(A)$ is not also bijective to $P(B)$?
