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Two different sets cannot have the same reason for membership, so if beauty is the reason why a painting is in the set of beautiful paintings, then beauty cannot be the reason why the painting is in any other set, such as the set of good paintings.
Is that fair?

### No.

No.
There is a set of even numbers. There is also a set of numbers that are even or prime. (Note, by the way: something can be even and prime: the number 2.) The number 8 is in the first set because it's even. It's also in the second set because it's even, hence even or prime.
Not all good paintings are beautiful, but for present purposes, we can still assume that all beautiful paintings are good paintings. A beautiful painting clearly fits the membership condition for the set of beautiful paintings. But it also fits the membership condition for being in the set of paintings that are beautiful or good and it fits it by virtue of being beautiful.
There's nothing peculiar here at all. If X and Y are both sets, their union is also a set. That's elementary set theory, and it's so whether or not X and Y are mutually exclusive.

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