Ordinal is Subset of Class of All Ordinals
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Theorem
Suppose $A$ is an ordinal.
Then:
- $A \subseteq \On$
where $\On$ represents the class of all ordinals.
Proof
By Ordinal is Member of Class of All Ordinals:
- $A \in \On \lor A = \On$
In either case:
- $A \subseteq \On$
since $\On$ is transitive.
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Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $7.15$