Today I was reading this paper, which includes discussion of essential undecidability of various weak theories. On page 24, I was surprised to find out that Vaught has showed that set theory with the following two axioms is essentially undecidable:
$$\forall x\exists y\neg(y\in x)$$ $$\forall x,y\exists u\forall z(z\in u\Leftrightarrow(z\in x\lor z\in y))$$
I was surprised as I have found in some recent paper (which I can't remember) that ST is the simplest known set theory which is essentially undecidable, while (arguably) Vaught's theory is simpler. I wasn't able to track down Vaught's work which would talk of this theory.
On the second thought, this theory seems unlikely to be essentially undecidable, because, first, it vacuously has an empty model, and second, less trivially, a finite model with $\in$ being empty relation seems to satisfy the theory as well, thus giving a decidable extension of the theory.
Can anyone provide a reference for where Vaught proves essential undecidability of this theory?
Thanks in advance.
Edit: Since the result seems to be false, the question arises whether the article I mention on the beginning has a typo. Finding a work of Vaught which the paper appears to be quoting might help clarify that, but again - I couldn't find anything.