Equal Sets and Identical Sets

We have A =B iff Ax [x e A <--> x e B] .
However, the relation 'e' itself is undefined,
and so can be interpreted in different ways.
For example,we could take a e b to mean
'a is a child of b '.Thus, if A and B have children
p and q , ( who have no children) we have

p e A and q e A , as also p e B and q e B .

Then , we have P =Q, though P and Q are
not identical.

Additionally, we also have
Ax ~ x e p as also Ax ~x e q , that is,
both p and q have no members. So, we have
p=q , but again p and q are not identical.

What allows us to assert that the 'e' we
have in set theory ensures that equal sets are
identical sets ? How do we assert that there is
only one null set ( although all null sets may be equal )?
How do we infer from A = B that {A} ={B}, unless we
assume that A and B are identical ?

-apoorv
.

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