Successor of Ordinal Smaller than Limit Ordinal is also Smaller/Proof 2
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Theorem
Let $\On$ denote the class of all ordinals.
Let $\lambda \in \On$ be a limit ordinal.
Then:
- $\forall \alpha \in \On: \alpha < \lambda \implies \alpha^+ < \lambda$
Proof
Because $\lambda$ is a limit ordinal:
- $\lambda \ne \alpha^+$
Moreover, by Successor of Element of Ordinal is Subset:
- $\alpha \in \lambda \implies \alpha^+ \subseteq \lambda$
Therefore by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:
- $\alpha^+ \subset \lambda$ and $\alpha^+ \in \lambda$
$\blacksquare$