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Having:

$$ O \in \{{A,B,C}\} $$

How to stress that $O$ has to be one of types defined in collection? Does inclusion already does it?

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    what do you mean by 'one of types defined in collection?' Can it not be any of the three?2011-06-07
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    @mixedmath: it has to be _only_ one of A,B,C. Does inclusion already define that or should I write it otherwise?2011-06-07
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    If your question is if saying that $O \in \{ A, B, C \}$ is equivalent to say that either $O = A$, $O = B$ or $O = C$, then the answer is yes, you can say it in that way.2011-06-07
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    @Adrián: that was what I meant.2011-06-07

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To say that $ x \in \{A, \, B, \, C \}$ already incorporates the idea that $x$ has to be exactly one of $A,\; B, $ or $C$.

Perhaps you wonder, how might one allow it to be both $A$ and $C$ ? One way of notating this would be to write $x \in \{A, \, B, \, C, \, \{A,\, C \} \, \}$, allowing $x$ to be the set containing both $A$ and $C$ (which is one way of notating that it's both).

If you really are just interested in stressing that it's exactly one of $A,\; B$ or $C$, you could just write that line out.

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    If $A = C$ then we could have $x \in \{A,B,C\}$, $x = A$, and $x = C$. I agree, though, that this sort of thing is easiest to deal with in natural language.2011-06-07
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    @Carl: That's a good point, too.2011-06-07