# Proper Subsets of Ordinals ... ... Searcoid, Theorem 1.4.4 ... ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ...

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

I need some help in fully understanding Theorem 1.4.4 ...

In the above proof by Searcoid we read the following:

"... ... Now, for each $$\displaystyle \gamma \in \beta$$ , we have $$\displaystyle \gamma \in \alpha$$ by 1.4.2, and the minimality with respect to $$\displaystyle \in$$ of $$\displaystyle \beta$$ in $$\displaystyle \alpha \text{\\} x$$ ensures that $$\displaystyle \gamma \in x$$. ... ...

Ca someone please show formally and rigorously that the minimality with respect to $$\displaystyle \in$$ of $$\displaystyle \beta$$ in $$\displaystyle \alpha \text{\\} x$$ ensures that $$\displaystyle \gamma \in x$$. ... ...

Help will be appreciated ...

Peter

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It may help MHB readers of the above post to have access to the start of Searcoid's section on the ordinals (including Theorem 1.4.2 ... ) ... so I am providing the same ... as follows:

It may also help MHB readers to have access to Searcoid's definition of a well order ... so I am providing the text of Searcoid's Definition 1.3.10 ... as follows:

Hope that helps ...

Peter