Element of Transitive Class

Theorem
Let $B$ be a transitive class.

Then:
 * $A \in B \implies A \subsetneq B$

where $\subsetneq$ denotes a proper subset).

Proof
By the definition of a transitive class:


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

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

Therefore $A \subsetneq B$.