Natural Number is Superset of its Union

Theorem

Let $n \in \N$ be a natural number as defined by the von Neumann construction.

Then:

$\bigcup n \subseteq n$

Proof

Let $n \in \N$.

From Natural Number is Transitive Set, $n$ is transitive.

From Class is Transitive iff Union is Subclass it follows directly that:

$\bigcup n \subseteq n$

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 3$ Derivation of the Peano postulates and other results: Exercise $3.2 \ \text {(b)}$