What set theory is Pinter's Axioms? I'm about to study Pinter's "A book of Set Theory", and I just want to know if the Axioms from the book forms ZFC or NBG or etc. There are 13 axioms in the book, which are
A1. Axiom of Extent 
A2. Axiom of Class Construction
A3. Every subclass of a set is a set
A4. $\emptyset$ is a set
A5. $a,b$ are sets then $\{a,b\}$ is a set.
A6. $\mathscr{A}$ is set of sets then $\bigcup_{A\in\mathscr{A}} A$ is set
A7. The power set of a set is a set
A8. Axiom of foundation
A9. Axiom of replacement
A10. Axiom of Choice
A11. Axiom of Infinity
A12. Axiom of Cardinality
A13. Axiom of Ordinality
I'm new to set theory, and after some internet surfing, I concluded that these don't form ZFC since the book speaks about proper classes, and not NBG either since it talks only about Axiom of choice, not Axiom of global choice. So what is this system?
 A: In my experience, "NBG" doesn't necessarily mean global choice... oftentimes it does, but conventions vary. So it's good form to be explicit. So you could just say "NBG with local choice".
However, that's assuming that, choice aside, this is NBG. The last two axioms I've never heard of, but I'll just assume they are innocuous quirks of the presentation.
(edit: I looked it up and they are unnecessary axioms the author introduces on a provisional basis before proving.)
The main issue has to do with axiom A2. Are the formulas for class construction restricted so that you can't quantify over proper classes? If not, then you aren't dealing with an NBG-like theory, but rather a Morse-Kelley-like one.
(edit: I could be mistaken cause the presentation of logic is too long for me to read right now, but it doesn't look like the author is making this restriction. So seems like Morse-Kelley. Although it’s possible the author never uses the stronger comprehension instances so is effectively working in NBG.)
