Successor Set of Ordinal is Ordinal/Proof 1

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Let $\On$ denote the class of all ordinals.

Let $\alpha \in \On$ be an ordinal.

Then its successor set $\alpha^+ = \alpha \cup \set \alpha$ is also an ordinal.


We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.

Hence $\On$ is a fortiori a superinductive class with respect to the successor mapping.

Hence, by definition of superinductive class:

$\On$ is closed under the successor mapping.

That is:

$\forall \alpha \in \On: \alpha^+ \in \On$