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I have some trouble witch the axiom of regularity. I wolud like to show that

$$x = (x,y)$$

for any y, not exists.

As example pair definition based on set I use Kuratowski definition - so:

$$x=\{\{x\},\{x,y\}\}$$

When I read axiom of regularity it is easy to show that $x=\{x\}$ - not exists. But here e.g. for $y=x$ we have something more nested: $x=\{\{x\}\}$ - and actually I'm stuck - I really have no idea...

CiaPan
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Kamil Kiełczewski
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    I'm not sure you're question is a duplicate, but at least it's related to this old question; you may find some of the answers there helpful: https://math.stackexchange.com/questions/2537526/axiom-of-regularity-allows-for-this-set-be-an-element-of-itself – bof Feb 26 '24 at 09:29

2 Answers2

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Hint

From Regularity we have that, for each pair of sets, only one can be element of the other.

Apply it to $x$ and $\{ x \}$.

Mauro ALLEGRANZA
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The axiom of regularity says that every nonempty set is disjoint from one of its elements. If $x=\{\{x\},\{x,y\}\}$ then the set $\{x,\{x\}\}$ has nonempty intersection with each of its elements, contradicting the axiom of regularity. If $x=\{\{x\}\}$ then the set $\{x,\{x\},\{\{x\}\}\}$ has nonempty intersection with each of its elements, contradicting the axiom of regularity.

user14111
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