Power Set of Transitive Set is Transitive
Jump to navigation
Jump to search
Theorem
Let $x$ be a transitive set.
Then its power set $\powerset x$ is also a transitive set.
Proof
Let $x$ be transitive.
By Set is Transitive iff Subset of Power Set:
- $x \subseteq \powerset x$
Then by Power Set of Subset:
- $\powerset x \subseteq \powerset {\powerset x}$
Thus by Set is Transitive iff Subset of Power Set:
- $\powerset x$ is a transitive set.
$\blacksquare$
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 $10.5$