I think I solved Ex. (12) in Chapter I of Kunen's book. It states that ZF sans Foundation proves: For every set $X$, $$ \aleph(X) < \aleph(\mathcal{P}^3(X)), $$ where $\aleph(X):= \sup\{\alpha \in \mathrm{Ord} : \alpha \text{ injects into } X\}$ is Hartogs' function.
This has as a consequence that there is no infinite sequence $\{X_n : n\in \omega\}$ such that for every $n$, there is an injection from $\mathcal{P}(X_{n+1})$ into $X_n$. I was delighted to find this last statement, because it is
an elementary (or easy-to-state) problem, solvable with axioms that do not defy any conservative intuition, but requires a serious development of Set Theory to be attacked.
My question: Is my last assertion correct? I mean, do you know any proof that there isn't such a sequence that does not require ordinals and eventually the argument using Hartogs?
I understand that the easier assertion with 4 replacing 3 in the displayed inequation also suffices, but it seems that I still need the same machinery.