Union of Class is Transitive if Every Element is Transitive
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Theorem
Let $A$ be a class.
Let $\bigcup A$ denote the union of $A$.
Let $A$ be such that every element of $A$ is transitive.
Then $\bigcup A$ is also transitive.
Proof
Let $A$ be such that every $y \in A$ is transitive.
Let $x \in \bigcup A$.
Then $x$ is an element of some element $y$ of $A$.
We have by hypothesis that $y$ is transitive.
Hence, by definition of transitive class:
- $x \subseteq y$
Because $y \in A$, by definition of union of class:
- $y \subseteq \bigcup A$
So:
- $x \subseteq \bigcup A$
As this is true for all $x \in A$, it follows by definition that $\bigcup A$ 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: $\S 5$ The union axiom: Exercise $5.3. \ \text {(d)}$