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I'm currently working through the book "Introduction to Set Theory: Third Edition" by Hrbacek and Jech and I came up with the following proof to $3.3(b)$ above (p12 in the book):

Suppose for a contradiction that there exists a set $A$ such that $\nexists$ $x$ with $x \notin A$. Then by the Comprehension Schema (also shown above), there exists a set $B$ such that $x\in B$ if any only if $x \in A$ and $x$ is a set. For any set $x$, we have that $x \in A$ and hence $x \in B$ and conversely, $x \in B$ implies that $x$ is a set. Therefore, $B$ is a set of all sets, contradicting $3.3(a)$

My question is, is "$x$ is a set" a valid property $P(x)$ since it's not really a precise mathematical statement?

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  • If your axiom theory says that all objects are sets then $P(x)$ is an identity property. Actually, 3.3(b) states that every set has a complement. – E.Papapetros Feb 29 '24 at 13:46

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