I am reading a book Axioms and Set Theory - A first course in Set Theory by Robert André. Here the auther writes as follows.
Let L = { x : set(x) }, By axiom of class construction L is a class of all sets.
Later he proves that L is a Proper Class. That is also fine.
Then he stated that elements of L are pure sets and every element x of set that belongs to L is a set.
I need a proof for above statement.
Also he stated that x is a pure set if and only if y belongs to x belongs to L implies that y belong to L.
I need a proof that why elements of L satisfies this property.
Also want to know if set C = { 1 , 2 } is an element of L. C is a set so should be an element of L (Class of all sets). But then all elements of L are pure sets and C is not a pure set. So I am confused.
I've read this particular book before, and I think some clarification is required.
When discussing formal theories, it is important to state the language in which the theory is being presented. ZFC is a first-order theory, meaning that the language of ZFC is a type of first-order logic. The syntax of first-order logic, which is explained here, distinguishes between logical and nonlogical symbols, and identifies certain symbols as terms.
ZFC is a pure set theory, which means that all of its terms are sets, and there is one nonlogical symbol $(\in)$. You cannot prove that the class of sets contains only sets, or that the elements of every member of the class of sets is a set. Rather, the only objects in the language of ZFC are defined as sets.
In other set theories - KPU, for example - there are object called urelements which are not sets. These are not present in ZF set theory because such objects can be encoded as sets using a variety of techniques; the purely set-theoretic account of the natural numbers is explained in Chapter V. As explained there, the set $\{1,2\}$ is defined as $\{\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$.
I would also note that classes are not objects in ZFC set theory. André's use of classes extends the usual axioms of ZFC. In standard ZFC set theory, proper classes like the class of all sets are merely "implied" rather than stated.
Edit/Correction:
André's explanation does treat classes and sets as objects, and provides that any element of a set is a set. So the proof of $y\in x\in L\implies y\in L$ is $y\in x\implies set(y)$.