Correct use of Axiom Schema of Comprehension

Hello all

I'm going through the book "introduction to set theory" by Hrbace. on page 11 Hrbacek introduces another way on defining a set that builds on the 'Axiom Schema or Comprehension' method

I'll post what he writes to explain his method... but to summarise he explains that

can be used as long as implies

Now when he first posted this I was unaware as the point of this. I started 'googling' axiom scheme of comprehension and came across these three posts...

Russell's paradox: how is it resolved?

ZFC and Russell's Paradox

[Russell's Paradox] Help with Notation

From here I learnt that the ASoC was developed by ZFC to get around this Russel Paradox. That is where

Here is what he writes...

Quote:

We conclude this section with another notational convention.

Let

be a property of

(and, possibly, of other parameters).

If there is a set

such that, for all

,

implies

. then

exists, and, moreover, does not depend on A. That means that if

is another set such that for all

, P(x) implies

. THen

Prove It

I don't follow his line of reasoning on why these sets are the same and why they exist and dont depend on their set A or A'.

Dylan