Assume that A is a set of sets.  Show that {x ∈ A|x ∉ x} ∉ A.

Proof:

  Let P(x) = x ∈ A ∧ x ∉ x, and F = {x|P(x)}
  We will show that F ∉ A

    Assume the contrary, F ∈ A
    There are only two possibilities: either F ∉ F or F ∈ F

    Suppose F ∉ F
    From F ∈ A and F ∉ F, we get P(F)
    From F = {x|P(x)}, it follows F ∈ F.  A contradiction.

    Suppose F ∈ F
    From F = {x|P(x)}, we get P(F)
    From P(F) = F ∈ A ∧ F ∉ F, it follows F ∉ F.  A contradiction.

  Thus F ∉ A, which means {x ∈ A|x ∉ x} ∉ A

# posted by rot13(Unafba Pune) @ 10:35 AM