I am trying to prove a small proposition that got me completely stumped, and I cannot find a single counterexample.
(ZF) Suppose that $E$ is such that for every $A\subseteq\mathcal P(E)$ either $|E|\leq|A|$ or $|A|\leq|E|$, then $E$ can be well-ordered.
It is not a biconditional statement since we have models of ZF (e.g. Solovay's model) where $\omega$ serves as a counterexample to this, but I still make true or false of the above statement.
Is this result known, or known to be false?
If the above is indeed false, how about a stronger requirement:
(ZF) Suppose $E$ is such that the cardinalities below $|\mathcal P(E)|$ are linearly ordered, then $E$ can be well-ordered.