Equivalent Statements for Transitive Set
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Theorem
Let $a$ be a set.
The following statements are equivalent:
- $(1): \quad \bigcup a \subseteq a$
- $(2): \quad a \subseteq \powerset a$
- $(3): \quad a$ is a transitive set
Proof
$(1) \iff (3)$
This is shown in Class is Transitive iff Union is Subclass.
$\Box$
$(2) \iff (3)$
This is shown in Set is Transitive iff Subset of Power 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: Remark