To prove that there exists a unique set such that for all is it as simple as choosing and so: ?
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I guess its right then?
Not quite. You have shown existence but not uniqueness.
To show uniqueness: We want to find a set such that . This means that and so that .
By definition, (which is unique). And so it follows that and .
Would this show uniqueness?
This what must be done.
Suppose that and . Then and . It follows that .
Is this correct?
No that does not work.
I think it is safe to say that most of us use this equivalence: .
With that we can say if then that means .
From which we can conclude that .
Why doesn't it work? ?
Oh ok: Its apparent that if then . Sort of like a contradiction if we didn't have the empty set.
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