MHB ZFC and the Axiom of Power Sets ....

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I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...

I am currently focused on Searcoid's treatment of ZFC in Chapter 1: Sets ...

I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ...

The relevant text from Searcoid is as follows:
Searcoid - The Axioms ... Page 6 .png

At the end of the above text we read the following:

" ... ... Thus every set is a subset of itself and no set is a proper subset of itself. But we do not yet now that any set has a proper subset. ... ... "My question is as follows:

Can someone explain exactly why/how it is that we do not yet now that any set has a proper subset. ... ..?

My thinking is that surely we do know that any set has a proper subset ... for example if $$b = \{ s, t, r \}$$ then $$a = \{ s, t \}$$ is a proper subset of $$b$$ ... and the existence of a is guaranteed by the Axiom of Power Sets ...

Help will be much appreciated ...

Peter
 
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The set \[ \{s\} \] has power set \[\{\varnothing, \{s\}\}.\] Empty set is proper by the definition, but it is not yet introduced. So we don’t know if \[\{s\}\] has any proper subsets.
 
Thanks for the help AndreI ...

... still reflecting on how what you have written answers my question ...

Thanks again ...

Peter
 
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