Element of Transitive Class

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $B$ be a transitive class.

Then:

$A \in B \implies A \subsetneq B$

where $\subsetneq$ denotes a proper subset).


Proof

NotZFC.jpg

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.


By the definition of a transitive class:

$A \in B \implies A \subseteq B$


But $A \ne B$ because $\left({A = B \land A \in B}\right) \implies A \in A$, which by No Membership Loops is a contradiction.

Therefore $A \subsetneq B$.

$\blacksquare$


Sources