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We have that any subset of P(A) is transitive.

And if {x} ∈ P(A) then {x} is a transitive set, then are these following steps true?

∴ x ∈ {x} ∴ x ⊆ {x} 

then:

if {x} ∈ P(A) ∴ x ∈ P(A) since x ⊆ {x} ∈ P(A) 

Is this true?

Thanks

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I'm not sure what you are trying to prove, however I do need to point out the following:

Claim: $x\subseteq\{x\}$ if and only $x=\varnothing$.

Proof. If $x=\varnothing$ then this is trivial. On the other hand if $x\subseteq\{x\}$ then either $x=\varnothing$ or for every $y\in x$, $y\in\{x\}$. From this follows that $y=x$ and therefore $x\in x$ which is a contradiction to the axiom of regularity. Therefore $x\subseteq\{x\}$ implies $x=\varnothing$.

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    Yes, I k$n$ow it's trivial si$n$ce a$n$y set is subset of itself, but I'm more on the side of proving things through unconventional and orthodox means. But, thanks s lot.2012-09-27