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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ...

I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ...

I need help with an aspect of the proof of Proposition 1.7.5 ...

Proposition 1.7.5 reads as follows:

In the above proof we read the following:

"By the foundation axiom, there exists ##n \in \mathbb{Z}^+## such that no member of ##f(n)## is in ##f( \mathbb{Z}^+ )##. ... ... "

Can someone please explain how/why the foundation axiom implies that there exists ##n \in \mathbb{Z}^+## such that no member of ##f(n)## is in ##f( \mathbb{Z}^+ )##. ... ... ?

Peter

NOTE:

To enable readers to follow the above post I am providing Garling's text on the foundation axiom and the axiom of infinity ... ...

I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ...

I need help with an aspect of the proof of Proposition 1.7.5 ...

Proposition 1.7.5 reads as follows:

In the above proof we read the following:

"By the foundation axiom, there exists ##n \in \mathbb{Z}^+## such that no member of ##f(n)## is in ##f( \mathbb{Z}^+ )##. ... ... "

Can someone please explain how/why the foundation axiom implies that there exists ##n \in \mathbb{Z}^+## such that no member of ##f(n)## is in ##f( \mathbb{Z}^+ )##. ... ... ?

Peter

NOTE:

To enable readers to follow the above post I am providing Garling's text on the foundation axiom and the axiom of infinity ... ...