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 , , and it is not the case that

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

Proof: (Quoting directly from Suppes) Suppose that Then

By the axiom of regularity there is an in such that

and by Theorem 2.43 [Ackbeet: on page 31. It states that ]

Hence

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:

By the axiom of regularity there is an such that

By the obvious extension of Theorem 43,

Hence, at least one of or or is zero, contradicting (1), (2), or (3). QED.

Is this a valid proof?