Element of Transitive Class

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Let $B$ be a transitive class.


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

where $\subsetneq$ denotes a proper subset).



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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 $A \subsetneq B$.