Suppes calls xϵy a primitive atomic formula without further definition: pg 15

How Quine (Set Theory and its Logic) gets there is quite interesting: You can skip everything between ----- but it’s educational.

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“The schematic predicate (what is said about something) letters ‘F’, ‘G’,…attach to variables to make dummy clauses ‘Fxy’, ‘Gx,…’ pg 1

Class undefined but very roughly an open sentence. see *

“In subsequent chapters we shall be concerned not with theories in general, but with theories specifically of classes; and these will regularly be conceived in such a form as to involve a single primitive predicate, the two-place predicate ‘ϵ’ of class membership.”

*“The basic trick consists of defining ‘ϵ’ with the form of notation of class abstraction ‘{x:F(x)}’ which purports to designate the class of all objects x such that Fx.” pgs 1-2

“In the eliminable combination (?) here envisaged ‘ϵ’ occurs only before class abstracts and class abstracts occur only after ‘ϵ’.” pg 15

This almost looks like we’re on the way to addressing the OP, but now we enter rough seas:

“The whole combination ‘y ϵ {x:F(x)}’ reduces, by what I call the law of concretion, to F(y). Turning the tables and speaking not of elimination (?) but of introduction, we may view ‘ϵ’ and class abstraction simply as fragments of a combination in toto thus:

2.1) ‘y ϵ {x:F(x)}’ for ‘Fy’.”

ie, what???? But let’s move on. He notes next y is undefined, and there follows a series of other requirements which leads to:

‘y ϵ {x:F(x)}’ for ‘(Ez)(z=y . (Ex)(x = z . Fx))’E-for some.

“Having noted these refinements I shall for brevity keep to 2.1.” pgs16-17.

Then we move on to Real Classes. Summarized, pg 29

“We could adopt one style of quantifiable variables for classes and another for individuals and allow only the one style after and the other before….but the acceptance of classes as members of further classes adds very considerably to what can be said about numbers and classes. Let us therefore accept:

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primitive predicate ‘ϵ’ and ‘yϵz’ as significant equally for individuals and classes as values of y and z.

Halmos cuts to the chase: (pg 2)

“The principle concept of set theory, the one that in completeley axiomatic studies is the principle (undefined) concept, is that of belonging. If x belongs to A (if x is an element of A, x is contained in A) we shall write:

xϵA”

The notation is misleading because it implies x is a member and A is the container. Subsequently no distinction is made.

COMMENTS ON ABOVE

In none of the above is the stipulation made that xϵy only if x=’y. Without that, I would suspect a general derivation would be rife with paradoxes.

Certainly if you want to draw real conclusions from your results by giving them real interpretations, you have to start with real interpretations of the foundation and show that the derivation of your results was correct starting from real interpretation of the foundation:

ie xϵy only if x=’y