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.