Power Set of Transitive Set is Transitive

From ProofWiki
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