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 $A \not = B$ because $( A = B \land A \in B ) \implies A \in A$, which by No Membership Loops is a contradiction.

Therefore, $A \subsetneq B$

Source

 * : $7.2$