A basic, basic question about the definition of a superstructure

Question

A superstructure $V(X)$ over a set $X$ is defined as:

1. $V_0(X) =X$
2. $V_{i+1}(X) = V_i(X) \cup P(V_i(X))$
3. $V(X) = ⋃_{i=0}^{\infty}V_i(X)$

My question is in regard the line item 2, where the set $V_{i+1}(X)$ is defined as the union of the set $V_i(X)$ with its own power set $P(V_i(X))$, right?

But does not the Power Set of a set includes the set itself? Is not $S \cup P(S) = P(S)$ ?

Thank you for your answers.

Answer 1

Let $X=\{a,b\}$. Then $P(X)$ does not contain $a$ and it doesn't contain $b$. It contains $\{a\}$ and $\{b\}$, but those are different. So $$V_1(X) = \{a,b,\emptyset,\{a\},\{b\},\{a,b\}\}$$