The question here is about the consistency of a rather very simply presented theory and if it is equivalent to ZFC.
The theory is a first order theory of classes, so it has its primitives being equality and membership, with a new primitive one place predicate added that is "set" to denote "..is a set". Now the axioms are those of Extensionality written exactly as in ZFC. An axiom stating that every class is a set if and only if it is a class of sets. A comprehension axiom schema stating that whenever a formula holds strictly of sets without using the predicate "set", then it defines a set. The last axiom is that of infinity stating that every natural number is a set, where natural number is defined in the customary way as a finite von Neumann ordinal.
FORMAL EXPOSITION
To the language of set theory (first order logic with equality and membership) add a primitive one place predicate symbol $``set"$, denoting "is a set".
Axioms:
Extensionality: $\forall x \forall y [\forall z (z \in x \leftrightarrow z \in y) \to x=y]$
Sethood: $\forall x [set(x) \leftrightarrow \forall y \in x (set(y))]$
Comprehension: if $\phi$ is a formula in the langauge of set theory (i.e. doesn't use the symbol $``set"$), in which the symbol $``x"$ is not free, then all closures of:$$ \forall y (\phi \to set(y)) \to \exists x \forall y \ (y \in x \leftrightarrow \phi)$$; are axioms.
Infinity: $\forall n \ [natural(n) \to set(n)]$
Where $natural$ is defined as finite von Neumann ordinal, like as "well founded transitive sets of transitive sets, that are successors and every non empty element of them is a successor"
Questions:
Is this theory consistent?
If it is consistent, is it interpretable in ZFC?
If 2, would it interpret ZFC?
This theory [if consistent] does interpret and prove the consistency of Zermelo set theory, over set $V_{\omega+\omega}$. I'd conjecture that it is equi-interpretable with ZFC also?! However, this theory might be inconsistent. Although this theory does prove existence of non-set classes, but it doesn't stipulate comprehension axioms about them. This is deliberately done here as to avoid set theoretic paradoxes since Sethood axiom is more powerful than the two completeness axioms of Ackermann's set theory, and also comprehension is not restricted to set parameters as it is the case with Ackermann's. So this theory is hazardous. It would be nice to see if it is consistent! and also it would be nice to see what its exact strength relative to Ackermann's and ZFC!