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Is it possible to get a set by a intersection of 2 (non-set) classes?

I can see it is possible from a set and a class, simply by a set contained in the class. Also, I think that the union/product of two classes can't be a set, because then the inclusion/projection function shows that also the classes are sets.

Am I right?

user160823
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    Well let $k(1)$ denote the class of all $1$-element sets, and $k(2)$ denote the class of all $2$-element sets. What can you say about $(k(1) \cup {{0,1,2}}) \cap (k(2) \cup {{0,1,2}})$ ? – goblin GONE Jul 10 '14 at 02:52
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    @user18921: Easier still: $k(1) \cap k(2)$ is a set... – Nate Eldredge Jul 10 '14 at 07:08
  • @NateEldredge, true! – goblin GONE Jul 10 '14 at 07:09
  • To see that $k(1)$ and $k(2)$ are indeed proper classes, see this question: http://math.stackexchange.com/questions/730974/proving-that-for-any-cardinal-number-there-doesnt-exist-a-set-containing-conta (and other questions listed there among linked questions). – Martin Sleziak Jul 10 '14 at 12:41
  • Here's a cute observation: $\mathrm{Ord}\cap{\mathcal P(x)\mid x=x}=\langle 0,1\rangle$. – Asaf Karagila Jul 10 '14 at 17:16

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