Set is Transitive iff Subset of Power Set
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Theorem
A set $a$ is transitive if and only if:
- $a \subseteq \powerset a$
where $\powerset a$ denotes the power set of $a$.
Proof
Necessary Condition
Let $a$ be transitive.
Let $x \in a$.
By definition of transitive set:
- $x \subseteq a$
Then by definition of power set:
- $x \in \powerset a$
Hence, by definition of subset:
- $a \subseteq \powerset a$
$\Box$
Sufficient Condition
Let $a \subseteq \powerset a$.
Let $x \in a$.
Then by definition of subset:
- $x \in \powerset a$
By definition of power set:
- $x \subseteq a$
As this is true for all $x \in a$, it follows by definition that $a$ 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 $10.4$