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On a past exam paper in a set theory module I am taking I am asked the question:

Express as a first-order sentence in the language of set theory, the instance of the Axiom Schema of Separation for the property "$x = \{x\}$"

I am slightly confused since in the notes we are only really given the definition of the Axiom Schema of Separation and then there are no real examples or explanation how to use it. So, in our notes the axiom is defined by:

$\forall w\forall y_1,y_2,...,y_n \exists z\forall x(x\in z \iff \phi(x,y_1,...,y_n) \wedge(x\in w))$

So for this case I guess that $\phi(x,y_1,...,y_n)$ relates to our property $x = \{x\}$. Then we are told $w$ is a bounding set, so does it follow in this case $w = \mathcal{P}(x)$ since $\{x\}\in\mathcal{P}(x)$? And in this case our axiom is:

$\exists z\forall x(x\in z \iff (x=\{x\}) \wedge(x\in \mathcal{P}(x))$

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The separation schema takes a formula as a parameter, and says that for every $z$ there is a $w$ which is made of exactly the elements of $z$ satisfying the formula.

Here the formula is given informally as $x=\{x\}$, and you are tasked with writing the relevant axiom in details.

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$\forall x\exists y(z\in y\iff (z\in x\wedge \forall w(w\in z\iff w=z)))$

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  • $\begingroup$ Maybe $z=\{z\}$ is expected to be also spelled out as $\forall w\colon w\in z\leftrightarrow w=z$? $\endgroup$ Dec 6 '14 at 13:24

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