Inspired by the answer to this question, is there a model of $\mathsf{ZFC{-}FA}{+}\exists\{x:x\in x\}$ such that $\{x:x\in x\}$ is not empty and does not belong to itself?
Some context
There is a minor debate in philosophy about the status of so-called hypodoxes, which are usually only defined by instances. Two major instances are: co-Russell's set $\exists\{x:x\in x\}$ and Truth-teller sentence "This sentence is true.". The common theme of these instances is that there is some sort of naive under-determinancy around assertions about them (as opposite to paradoxes, which are over-determined in a sense): does co-Russell's set belong to itself? is Truth-teller sentence true?
If you find the concept of a hypodox confusing and unclear, you are not alone. I am now collecting info on the mathematical determinants of the co-Russell's set belonging (or not belonging) to itself. That is also why I ask this question.
