# 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

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

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.

$\Box$

### Induction Step

Let $\map V x$ be supertransitive.

First, to prove transitivity.

Let $\map V x$ be transitive.

Then:

 $\displaystyle y$ $\in$ $\displaystyle \map V {x^+}$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $\subseteq$ $\displaystyle \map V x$ $\displaystyle \leadsto \ \$ $\, \displaystyle \forall z: \,$ $\displaystyle z \in y$ $\implies$ $\displaystyle z \subseteq \map V x$ Transitivity of $\map V x$ $\displaystyle \leadsto \ \$ $\, \displaystyle \forall z: \,$ $\displaystyle z \in y$ $\implies$ $\displaystyle z \in \map V {x^+}$ Definition of Von Neumann Hierarchy $\displaystyle \leadsto \ \$ $\displaystyle y$ $\subseteq$ $\displaystyle \map V {x^+}$ Definition of Subset

Next, to prove supertransitivity:

 $\displaystyle y$ $\in$ $\displaystyle \powerset {\map V x}$ Hypothesis $\displaystyle \leadsto \ \$ $\displaystyle y$ $\subseteq$ $\displaystyle \map V x$ Definition of Power Set $\displaystyle \leadsto \ \$ $\displaystyle \powerset y$ $\subseteq$ $\displaystyle \powerset {\map V x}$ Power Set Preserved Under Subset $\displaystyle$ $=$ $\displaystyle \map V {x^+}$ Definition of Von Neumann Hierarchy

This proves the induction step.

$\Box$

### Limit Case

Let $x$ be a limit ordinal.

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

Then:

 $\displaystyle z$ $\in$ $\displaystyle \map V x$ Hypothesis $\displaystyle \leadsto \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\in$ $\displaystyle \map V y$ Definition of Von Neumann Hierarchy $\displaystyle \leadsto \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\subseteq$ $\displaystyle \map V y$ Transitivity of $V_y$ $\displaystyle$ $\subseteq$ $\displaystyle \map V x$ Set is Subset of Union

This proves transitivity.

Now, to prove supertransitivity:

 $\displaystyle z$ $\in$ $\displaystyle \map V x$ Hypothesis $\displaystyle \leadsto \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\in$ $\displaystyle \map V y$ Definition of Von Neumann Hierarchy $\displaystyle \leadsto \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\subseteq$ $\displaystyle \map V y$ Transitivity of $V_y$ $\displaystyle \leadsto \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle \powerset z$ $\subseteq$ $\displaystyle \map V {y + 1}$ Power Set Preserved Under Subset $\displaystyle$ $\subseteq$ $\displaystyle \map V x$ Set is Subset of Union

This proves the limit case.

$\blacksquare$