[FOM] Why ZF as foundation of mathematics?

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs,

ZFC is been mentioned as the foundation theory of mathematics:

Some quotes:
 
"Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics" [Wikipedia]

"Russell's Paradox can be avoided by a careful choice of construction principles, so that one has the expressive power needed for *usual mathematical arguments* while preventing the existence of paradoxical sets. See the supplement on Zermelo-Fraenkel Set Theory" [Thomas Jech: Stanford Encyclopedia of Philosophy]

Also Thomas Jech in his "Introduction to Set Theory" makes similar remarks about ZF role in codifying mainstream mathematics.

On the other hand it is well known that most of mainstream mathematics can be formalized in second order arithmetic, which is way weaker than ZF. 

So why ZF is mentioned for that purpose if much weak class\set theories can do the job of formalizing mainstream mathematics in set theory?

I suggest a simple alternative class\set theory in mono-sorted first order logic language with a single extra-logical symbol, that of class membership, all objects are classes. The followings are the axioms:

[1] Given a property there exists a class of all sets(i.e., elements of classes) that satisfy that property.
[2] for any "singleton or empty" classes A,B; the class 
{x|set(x) & (x=A or x=B)} is a set.
[3] Extensionality over all classes.

This has the strength of second order arithmetic, has much simpler structure than ZF (proves only classes of hereditarily 'empty or singleton' sets and classes of pairs of those sets), and the axioms are dead simple.

Best Regards,

Zuhair Al-Johar