Element of Transitive Class
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Theorem
Let $B$ be a transitive class.
Then:
- $A \in B \implies A \subsetneq B$
where $\subsetneq$ denotes a proper subset).
Proof
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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 $A \subsetneq B$.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.2$