Past exam of the logic and set theory course of the University of Oxford / 2010 / B / 4 / a / Solution

True in ZF. For a set $A$, the singleton map injects $A$ into its power set. No map $f:A\to \mathcal{P}(A)$ is onto: the diagonal setdiffers from $f(a)$ at the element $a$ for every $a$. Thus there is no bijection between $A$ and its power set. The injection together with this failure proves $\boxed{\kappa <2^{\kappa }}$.

If strict comparison is formulated as injection in one direction but not the other, the Cantor-Schröder-Bernstein theorem, which requires no choice, excludes an injection of the power set back into $A$. This is Cantor's theorem in a choice-free form.