# Sets containing unions, and vice versa

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• May 20th 2011, 06:26 AM
confuzzled12
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).
• May 20th 2011, 07:27 AM
MoeBlee
3. U{0} e {0}.

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