Proof that an ordinal number does not contain itself

Question

Hrbacek and Jech gives the following definition for ordinal numbers:

However, the following proof seems to rely on the fact that an ordinal number does not contain itself (argument circled in red).

It isn't clear to me why the red circle is true. Why is it clear that an ordinal does not contain itself?

Answer 1

No set (including an ordinal) is permitted to include itself due to the Axiom of regularity.