Transitive Relation over Set - Feedbacks on proofwriting skills

[Kolmin]

Homework Statement

Assume a relation $P$ that is negatively transitive on a set $X$ that is not empty.
Define the binary relation $R$ on $X$ by $xRy$ iff $y P x$ is false.

Prove that $R$ is transitive.

Homework Equations

Negative Transitivity: $xPz \rightarrow xPy \vee yPz$

Like in the previous thread (Complete Relation), I think I have a proof, but I am really looking forward to any feedback on the style of this proof. I tried to incorporate the stylistic feedbacks of Michael Redei and pasmith who kindly gave me feedbacks on the previous attempt.

The Attempt at a Solution

Proof:
Let $x$, $y$ and $z$ be arbitrary and assume $xRy$ and $yRz$. By definition of $R$ it follows that $yPx$ and $zPy$ are both false. Since $P$ is negatively transitive, this implies that $zPx$ is false. Thus, again by definition of $R$, the falsity of $zPx$ implies $xRz$, which proves that the relation $R$ is transitive.

 

[haruspex]

Negative Transitivity: $xPy \rightarrow xPy \vee yPz$

I think you mean xPy → xPz V zPy, or somesuch.

 

[Kolmin]

I think you mean xPy → xPz V zPy, or somesuch.

I hate typos...

I edited the previous post: it is $xPz \rightarrow xPy \vee yPz$

 

[haruspex]

With that correction, your proof looks fine.