Ordinal Class is Ordinal
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Theorem
The Ordinal Class $\On$ is an ordinal.
Proof
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The epsilon relation is equivalent to the strict subset relation when restricted to ordinals by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.
It follows that:
- $\forall x \in \On: x \subset \On$
The initial segment of the class of ordinals is:
- $\set {x \in \On : x \subset \On}$
Therefore, by the definition of ordinal, $\On$ is an ordinal.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $7.12$