I made a mistake in my previous post to this Usenet titled as Interpreting ZFC.
The correct formulation must be the following:
BI is the closure of all sentences entailed by FOL(e) from the following axioms:
(1) Boundedness: if phi is a formula, then:
[EB: (Vy in B(Ex C A:phi)) & (Vx C A ((Ey:phi) ->(Ey in B:phi)))]
is an axiom.
Where C is a modified subset relation defined as:
x C A iff Vm in x (En: n in A & m in n)
V,E signifies universal and existential quantification respectively.
2) Infinity. /
Now clearly BI is a sub-theory of ZFC. Yet BI interpret the whole of ZFC!
BI depicts a marvelous use of the property of transitivity of sets, BI interprets ZFC over the realm of the cumulative hierarchy using the properties of transitive sets which constitutes the stages of that hierarchy.
It is a nice experience to try interpret the whole of ZFC inside BI.