Power-set fibres force well-orderability

Fix the injection $j$. For each $x\in X$, the image of $\{x\}\times \mathcal{P}(X)$ cannot lie wholly in the $X$-summand. Otherwise it gives an injection $\mathcal{P}(X)\to X$; its inverse, extended by $\mathrm{\varnothing }$ off the range, would surject $X$ onto its power set.

The ordinal part of that fibre is therefore nonempty. Let $b(x)$ be its least ordinal. Different fibres have disjoint images by injectivity of $j$, so $b:X\to \alpha$ is injective. Pull back the ordinal well-order along $b$: totality follows from injectivity and ordinal trichotomy, and a nonempty subset has the unique point whose image is its image set’s least ordinal. For $X=\mathrm{\varnothing }$ this is the empty order; if $\alpha =0$ the first step forces $X=\mathrm{\varnothing }$.