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For example, is $A=\{-1\}$ a transitive set?

In wiki, it is said that:

A set $A$ is called transitive if either of the following equivalent conditions hold:

whenever $x ∈ A$, and $y ∈ x$, then $y ∈ A$.

whenever $x ∈ A$, and $x$ is not an urelement, then $x$ is a subset of $A$.

It seems $A$ satisfies the condition.

but we can also prove that set $A$ is transitive if and only if $A ⊆ P(A)$

And $A=\{-1\} ⊆ P(A)$ is $F$, because $P(A)$ is $\{\emptyset, \{-1\}\}$, so $A$ is not transitive

something was wrong, but I don't know what is it.

Heptapod
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ENE KaIku
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1 Answers1

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You have that $A$ is transitive if and only if $A \subset P(A)$ if $A$ does not contain any urelements, but your $A$ does contain an urelement.

Bram28
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