The successor of a set x is the union of x and the set containing x, which exist by pairing and union. Its members are x and all the members of x.
Let r be a recursive set that contains only r. The successor of r is a set that contains r and the members of r, but remember, r is the only member of r. Thus the successor of r contains only r. By axiom 1, there can be only one set with the single member r, namely r. Thus successor(r) = r.
If x is the set containing y and y is the set containing x, the successor of x contains x and y, just like the successor of y. Suppose successor(x) = successor(y), and x ≠ y. Now x and the members of x must be the same as y and the members of y. Since x is not y, x contains y and y contains x. For any j in x other than y, j is also in y. Thus x and y contain each other, and (perhaps) other common elements. This is the only way two sets can have the same successor, but we usually try to avoid these self-referencing or mutually-referencing sets.
Let's use the successor operator to build the integers from the ground up.
By convention, the empty set represents 0, successor(0) = (0) represents 1, successor(1) = (0,1) represents 2, successor(2) = (0,1,2) represents 3, and so on. The successor of n is (0,1,2,…n). If you expand 3, it looks like this:
(∅,(∅),(∅,(∅)))
Obviously (0,1,2) is clearer.
All these "integer" successors are transitive. (We'll prove this later.) For example, 17 contains 8 and 8 contains 3, hence 17 contains 3. Also, the members of 17 are well ordered by membership:
0∈1∈2∈3∈…∈16