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In another post, the top answer says:

If there are sets at all, the axiom of subsets tells us that there is an empty set: If $x$ is a set, then $\{ y∈x ∣ y≠y \}$ is a set, and is empty, since there are no elements $y$ of $x$ for which $y≠y$. The axiom of extensionality then tells us that there is only one such empty set.”

I am having trouble understanding the move from 'there exists a set $x$' to '$\{ y∈x ∣ y≠y \}$ is a set'. Why is it the case that there is a subset of $x$, $y$, such that $y$ is distinct from itself?

I may be missing something basic here.

mjqxxxx
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YV1999
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    The Axiom Schema of Separation says (roughly) that if $\phi$ is a formula for sets, and $x$ is a set, the there is a set whose elements are precisely the $y\in x$ for which $\phi(y)$ is true. So if $\phi$ is the formula $y\neq y$ then ${y\in x\mid \phi(y)}$ is a set. – Arturo Magidin Aug 20 '23 at 19:53
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    @ArturoMagidin That can be an answer. – aschepler Aug 20 '23 at 19:57
  • +Some set exists either because model theory defines domains of discourse to be non-empty, or because of the axiom of infinity, or by adding an axiom asserting the existence of some set. – Chad K Aug 20 '23 at 20:10
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    Nobody said "there is a subset of $x$, $y$, such that $y$ is distinct from itself." Read the passge you quoted; that's not what it says. – bof Aug 20 '23 at 20:14
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    I'm just guessing, but it seems what you're missing is an understanding of what the notation ${ y∈x ∣ y≠y }$ means. – bof Aug 20 '23 at 20:16
  • @ArturoMagidin Thank you, that's extremely helpful. So the proof rests on y≠y counting as a formula and the Axiom Schema of Separation. – YV1999 Aug 20 '23 at 20:34
  • @bof That may be true--what's the right way to understand the notation then? – YV1999 Aug 20 '23 at 20:39
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    @YV1999 The problem with your interpretation "... there is a subset of $x$, $y$, such that $y$ is distinct from itself?" of $\ {y\in x,|,y\ne y}\ $ is in misidentifying a putative element $\ y\ $ of that subset of $\ x\ $ with the subset itself. A correct statement would be "...there is a subset of $\ x\ $ comprising those elements $\ y\ $ of $\ x\ $ which are distinct from themselves. – lonza leggiera Aug 20 '23 at 20:58
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    @YV1999 And of course, nothing is distinct from itself, so this subset of $x$ is empty. – spaceisdarkgreen Aug 20 '23 at 22:46

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