Class is Transitive iff Union is Subclass
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Theorem
A class $A$ is transitive if and only if:
- $\bigcup A \subseteq A$
Proof
Necessary Condition
Let $A$ be transitive.
Let $x \in \bigcup A$.
Then by definition:
- $\exists y \in A: x \in y$
By definition of transitive class:
- $x \in y \land y \in A \implies x \in A$
and so:
- $x \in A$
Hence the result by definition of subclass.
$\Box$
Sufficient Condition
Let $\bigcup A \subseteq A$.
Let $x \in \bigcup A$.
Then by definition:
- $\exists y \in A: x \in y$
By definition of subclass:
- $x \in A$
Thus we have that:
- $x \in y \land y \in A \implies x \in A$
It follows by definition that $A$ is a transitive class.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 10$ Some useful facts about transitivity: Theorem $10.1$