Von Neumann Hierarchy is Supertransitive

Theorem
Let $V$ denote the Von Neumann Hierarchy.

Let $x$ be an ordinal.

Then $\map V x$ is supertransitive.

Proof
The proof shall proceed by Transfinite Induction on $x$.

Basis for the Induction
We have that:
 * $V_0 = \O$

and $\O$ is supertransitive by the very fact that it has no elements.

This proves the basis for the induction.

Induction Step
Let $\map V x$ be supertransitive.

First, to prove transitivity.

Let $\map V x$ be transitive.

Then:

Next, to prove supertransitivity:

This proves the induction step.

Limit Case
Let $x$ be a limit ordinal.

Furthermore, let $\map V y$ be transitive for all $y \in x$.

Then:

This proves transitivity.

Now, to prove supertransitivity:

This proves the limit case.