Von Neumann Hierarchy is Cumulative

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Theorem

For any two ordinals $x$ and $y$,

If $x < y$ then $V(x) \subsetneqq V(y)$.

Proof

By Von Neumann Hierarchy Comparison, $V(x) \in V(y)$. (1)

By (1) and the Axiom:Axiom of Foundation, $V(x) \ne V(y)$.

Furthermore, by (1) and Von Neumann Hierarchy is Supertransitive,

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