Set Theory: Separation Axiom and Garling's Theorem 1.2.2

[Math Amateur]

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ...

At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...

The Separation Axiom and Theorem 1.2.2 read as follows:

Garling argues that the Separation Axiom needs to be in place before we can prove Theorem 1.2.2 ... ... but I cannot see where the Separation Axiom is needed in the proof of Theorem 1.2.2 ...

Can someone give a clear explanation of exactly why we need the Separation Axiom in order to prove Theorem 1.2.2.

Help will be much appreciated ... ...

Peter

 

[fresh_42]

He uses the separation axiom for the existence of $b$.
Can you define $A$ and $Q(x)$ for the usage in the proof of Theorem 1.2.2? This is needed for otherwise $b$ simply couldn't exist, and then the contradiction became meaningless.