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    $\begingroup$ You don’t need intersection if you have complement: $X\cap Y=X\setminus(X\setminus Y)$. $\endgroup$
    – Emil Jeřábek
    Commented Apr 9, 2020 at 14:11
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    $\begingroup$ Good point! Thank you. But anyway the smallest number of (elementary) Godel's operations at the moment remains 8. $\endgroup$
    – Taras Banakh
    Commented Apr 9, 2020 at 14:23
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    $\begingroup$ Yes, but it means you can drop one of your axioms of NBG. $\endgroup$
    – Emil Jeřábek
    Commented Apr 9, 2020 at 14:34
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    $\begingroup$ This cannot be right, as it would also make these axioms redundant in ZFC by the usual expand-a-model-with-definable-classes argument. Obviously, the operations $\bigcup X$ and $\mathcal P(X)$ are definable using only set quantifiers, hence the existence of $\bigcup X$ and $\mathcal P(X)$ as classes follows from the class existence axioms. But the point of the axioms of union and powerset is that if $x$ is a set, then the (already existing classes) $\bigcup x$ and $\mathcal P(x)$ are also sets. $\endgroup$
    – Emil Jeřábek
    Commented Apr 9, 2020 at 15:05
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    $\begingroup$ @PaceNielsen Very good! Thank you. Following your suggestions I made the simplifications of the proofs. Now all of them are almost trivial. $\endgroup$
    – Taras Banakh
    Commented Apr 9, 2020 at 20:40