The local-GCH Hartogs dichotomy

Put $P=\mathcal{P}(X)$ and $h=h(X)$. Suppose $h⪯P$. Then $X\prec X\bigsqcup h⪯P\bigsqcup P\approx P$. Strictness holds because $X\bigsqcup h\approx X$ would embed $h$ into $X$. For the last bijection, $P\bigsqcup P\approx \mathcal{P}(X\bigsqcup 1)\approx P$, using the omega shift. Local GCH therefore gives $X\bigsqcup h\approx P$.

Singleton injection in the first coordinate and square absorption give $X\times P⪯P^2\approx P$, while $P⪯X\times P$ using one fixed point of $X$. Thus $X\times P\approx P\approx X\bigsqcup h$, and the fibre lemma well-orders $X$.

Let $\lambda$ be the least ordinal equipotent with this well-orderable $X$. Hartogs $h$ is an infinite initial ordinal: an equipotent smaller ordinal would contradict its least-nonembedding property. Moreover $\lambda <h$. Both $X$ and $h$ embed in $h$, so $X\bigsqcup h$ injects into $h\times h\approx h$, and the reverse injection is immediate. Hence $P\approx h$.

If instead $h(X)$ does not inject into $P$, its least nonembedding ordinal satisfies $h(P)\le h(X)$. Since $X$ injects into $P$ by singletons, every ordinal embedding into $X$ embeds into $P$, so $h(X)\le h(P)$. This yields the second branch.