Models of extensionality and comprehension without union

Question

I want to find a model that satisfy comprehension and extensionality axiom but that it does not satisfy union axiom. I think that $M:=\{a,b,c,d,e,g\}$ with $\in^M=\{(a,b),(a,c),(b,e),(c,d),(e,g),(d,g)\}$ is one of such models. It is easy to see that $M\models extensionality$ and $M\nvDash union$, but I can't formalize a proof to show that $M\vDash comprehension$. Can someone give me a hint to complete my task?

Answer 1

Comprehension (which, as Asaf points out, should be called subset comprehension or separation to distinguish from the inconsistent unrestricted comprehension) is pretty straightforward for finite models, since there’s never any question of the definability of a subset. Furthermore, here, every set has at most one member, so at most two possible subsets, itself and the empty set. We have an empty set in $M$ (the element $a$), and of course the set itself is in $M,$ so we’re done.

Really we should just stare at $$ \forall x,z,\vec w\;\exists y\;\forall x (x\in y\leftrightarrow x\in z \land \phi(x,z,\vec w)).$$ As I mentioned, the “arbitrary formula” part is trivial since we can make it behave however we want as a function of $x.$