Union of Transitive Class is Transitive/Proof 1
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Theorem
Let $A$ be a class.
Let $\bigcup A$ denote the union of $A$.
Let $A$ be transitive.
Then $\bigcup A$ is also transitive.
Proof
Let $A$ be transitive.
By Class is Transitive iff Union is Subclass:
- $\bigcup A \subseteq A$
By Union of Subclass is Subclass of Union of Class:
- $\map \bigcup {\bigcup A} \subseteq \bigcup A$
Then by Class is Transitive iff Union is Subclass:
- $\bigcup A$ is transitive.
$\blacksquare$
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 $10.2$