Von Neumann Hierarchy is Supertransitive

Theorem
Let $V$ denote the Von Neumann Hierarchy.

Let $x$ be an ordinal.

Then $V \left({ x }\right)$ is supertransitive.

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

Basis for the Induction

 * $V_0 = \varnothing$ and $\varnothing$ is supertransitive by the very fact that it has no elements.

This proves the basis for the induction.

Induction Step
Suppose $V \left({x}\right)$ is supertransitive.

First, to prove transitivity.

Suppose $V \left({x}\right)$ is transitive:

Next, to prove supertransitivity:

This proves the induction step.

Limit Case
Suppose $x$ is a limit ordinal.

Furthermore, suppose $V \left({y}\right)$ is transitive for all $y \in x$.

Then:

This proves transitivity.

Now, to prove supertransitivity:

This proves the limit case.