Logical equivalence - Russell's Paradox In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. However, based on what I learned from Velleman it appears that the set R is also equivalent to the logical statement: $\exists A (A \in U \land A \notin A \land A \in R )$. Is that correct? Is there a short way to show that the two logical statements are equivalent? Thanks in advance.
 A: The statement $(*)$ "$\exists A(A\in U\wedge A\not\in A\wedge A\in R)$" - that is, "$R$ is not empty" - is a consequence of one half of the paradox. It follows from the straightforward observation that $R\not\in R$ (since, if $R\in R$, then by definition of $R$ we know $R\not\in R$). Of course, the same reasoning, reversed, shows that $R\in R$. So $R\in R$ and $R\not\in R$, a contradiction.
The statement $(*)$ doesn't come up at all, specifically; I don't know why it's (equivalent to anything which is) relevant at all. This could be a typo; maybe the statement is supposed to read $$(**)\quad\exists A(A\in U\wedge A\in A\wedge A\in R),$$ which would be a contradiction with the definition of $R$? This would be another way to phrase the punchline of Russell's paradox. Note that $(**)$ is still not equivalent to anything in particular, the point is that it's a consequence of the existence of $R$: if $R\not\in R$, then $R\in R$ by definition of $R$, so by contradiction we know $R\in R$, and this implies $(**)$.
