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> Is it consistent with $ZF$ to have a set $S$ and a function $F: P(S) \to S$ such that:
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$\forall X,Y \in P(S): X \subsetneq Y \implies F(X) \neq F(Y)$



Is it consistent to have a function that is sensitive to subset relation from the power set of a set to that set?
No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of $S$ that are well-orderable.


This is corollary 6 in



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> [MR0793235 (87d:03126)](https://mathscinet.ams.org/mathscinet-getitem?mr=793235). Todorčević, Stevo. *[Partition relations for partially ordered sets](https://projecteuclid.org/euclid.acta/1485890394)*. Acta Math. **155** (1985), no. 1-2, 1–25.
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The corollary is a consequence of theorem 5, stating that for any structure $\mathcal M$ with one binary relation, $\sigma \mathcal M$ is not "$\mathcal M$-embeddable". Here, if $\mathcal M=(M,R)$, then $\sigma\mathcal M=(\sigma M,\subseteq)$, where $\sigma M$ is the set of all one-to-one maps $s$ with domain an ordinal $\alpha$ such that whenever $\beta<\gamma<\alpha$, we have $s(\beta)\mathrel R s(\gamma)$. In Stevo's notation, that $(A,S)$ is $(B, T)$-embeddable means that there is a function $f\!:A\to B$ such that $f(a) \mathrel T f(b)$ and $f(a)\ne f(b)$ whenever $a,b\in A$, $a\mathrel S b$, and $a\ne b$.


