The axiom of infinity ("there exists an infinite set") is what gives you permission to carry out the construction you describe.

The usual construction of the naturals via set theory has us define...

1. 0 as the empty set ∅

2. succ(n) as n ∪ {n}

So you get...

    0 := ∅
    1 := {0} = {∅}
    2 := {0, 1} = {∅, {∅}}
    3 := {0,1,2} = {∅, {∅}, {∅, {∅}}}
    4 := {0,1,2,3} = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}

    ... etc. ...

But how do we know the following?

1. The empty set ∅ is a set

2. If T is a set then {T} is also a set

3. If T is a set then T ∪ {T} is a set

4. There exists a set X such that ∅ is in X and T ∪ {T} is in X whenever T is in X

These all follow from the axioms of ZFC and (4) is actually itself one of the axioms — the Axiom of Infinity. It says that an infinite set like X exists.

But you can see that the essence of it is just: "There exists a set X which includes the natural numbers." Note that this set X might contain many things beyond the set-theoretic copy of the natural numbers — the axiom just guarantees that a set containing the natural numbers exists ("some infinite set").

You can then use some of the other axioms to remove the unwanted elements, creating the unique, smallest set that satisfies this property.

In other words, the axioms of ZFC were designed to be Peano-friendly. It wasn't as if we arrived at ZFC and then said, "Oh, look at that, we can represent the natural numbers as sets. What a surprise!":D