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Math Help - Axiom of Regularity Problem

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    Axiom of Regularity Problem

    In conjunction with my other post on checking the regularity of a database, I am interested in solving Problem 1 on page 56 of Suppes' Axiomatic Set Theory:

    Prove that for all sets A, B, and C it is not the case that

    A\in B\land B\in C\land C\in A.

    Now, the proof of Theorem 2.106 on page 54 is bound to be similar. Here's the theorem statement:

    \neg(A\in B\land B\in A).

    Proof: (Quoting directly from Suppes) Suppose that A\in B\land B\in A. Then

    (1)\quad A\in\{A,B\}\cap B\quad\text{and}\quad B\in\{A,B\}\cap A.

    By the axiom of regularity there is an x in \{A,B\} such that

    \{A,B\}\cap x=0

    and by Theorem 2.43 [Ackbeet: on page 31. It states that z\in\{x,y\}\iff z=x\vee z=y.]

    x=A\quad\text{or}\quad x=B.

    Hence

    \{A,B\}\cap A=0\quad\text{or}\quad\{A,B\}\cap B=0,

    which contradicts (1). QED.

    So here's my attempt at a proof of the non-existence of the 3-cycle:

    Suppose the assertion were false. Then the following hold:

    (1)\quad A\in\{A,B,C\}\cap B,
    (2)\quad B\in\{A,B,C\}\cap C,
    (3)\quad C\in\{A,B,C\}\cap A.

    By the axiom of regularity there is an x\in\{A,B,C\} such that

    \{A,B,C\}\cap x=0.

    By the obvious extension of Theorem 43,

    x=A\quad\text{or}\quad x=B\quad\text{or}\quad x=C.

    Hence, at least one of \{A,B,C\}\cap A or \{A,B,C\}\cap B or \{A,B,C\}\cap C is zero, contradicting (1), (2), or (3). QED.

    Is this a valid proof?
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  2. #2
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    Quote Originally Posted by Ackbeet View Post
    Is this a valid proof?
    Took me a little while to decipher, but I say yes, it is valid.
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