# Class is Not Element of Itself

Jump to navigation
Jump to search

## Theorem

Let $A$ be a class.

Then $A$ is not an element of itself:

- $A \notin A$

## Proof

*This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.*

*If you see any proofs that link to this page, please insert this template at the top.*

*If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.*

$A \in A$ would create a relational loop for the class membership sign.

By No Membership Loops, this is a contradiction.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 5.21$