The transitive closure of a set $X$ is usually seen as a set, but it can also be seen as a graph $G(X)$ with $V(G)= TC({X})$ and $(x,y)\in E(G)$ iff $x\in y$. Such a (transitive closure) graph reveals irredundantly everything that is to know about the set ("*its hidden $\in$-structure*"). It is known that $G(X)\simeq G(Y)$ iff $X=Y$.

*G(X)* obviously 

 1. contains exactly one vertex with no out-arrows
 2. contains no two vertices with the same parents (→ axiom of extensionality)
 3. contains no directed loops (→ axiom of foundation)

Is this enough to characterize the class of transitive closure graphs of hereditarily finite sets: 

> Is there a 1:1 correspondence between
> the (isomorphism types of) finite
> digraphs with properties (1)-(3) and
> $V_\omega$?