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I have the following statements:

  • Definition of set. $y$ is a set if and only if there is $z$ such that $y\in z$; otherwise $y$ is a proper class
  • The comprehension axiom. Given a formula $\psi(x)$ where $x$ occurs free in $\psi$ and a class $y$, we have that $y\in\{x:\psi(x)\}$ if and only if $y$ is a set and $\psi(y)$
  • Definition of the empty set. $\emptyset$ is defined as $\{x:x\neq x\}$
  • Definition of the universe. $V$ is defined as $\{x:x=x\}$

At this point I managed to prove that:

  • no set belongs to $\emptyset$
  • every set belongs to $V$
  • $\{x:x\notin x\}$ is a proper class

How can I prove that $\emptyset$ is a set and that $V$ is a proper class? How can I prove that $\{x:x\notin x\}$ is equal to $V$?

Do I need additional statements perhaps?

Thank you.

3 Answers 3