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Math Help - Sets containing unions, and vice versa

  1. #1
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    Sets containing unions, and vice versa

    I've been going through some old exam papers, and I came upon an interesting part of a question that I'm not sure how to solve. Throughout the question I've been proving things like b\in b is false, at most one of b\in c and c\in b is true, and so on. I've been asked whether any of the following can occur, using only axioms (ZF1)-(ZF4) and (ZF9). (Extensionality, null set, pairs, unions, foundation.)

    1. x\in\bigcup x
    2. x\in\bigcup \bigcup x
    3. \bigcup x \in x

    I think (1) is false, by extension I think (2) is false (but can't get my reasoning right in my head), and I'm not sure about (3).
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  2. #2
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    3. U{0} e {0}.

    Indeed, for any ordinal x, we have U(x+) = x and x e x+.
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