Is the following true for integers $p_{x\leq y}\iff \lnot p_{y<x}$

Question

$p_{x \leq y}$ is the variable that is true when a given x is less than or equal to a given y ,false otherwise.

My book says that $p_{x\leq y}\implies \lnot p_{y<x}$ is true for integers but i feel like they could be even equivalent to one another hence $p_{x\leq y}\iff \lnot p_{y<x}$. Am i wrong ?

Answer 1

For any linear order <=, such as the integers,
x <= y iff not (y < x). The use of P makes no sense.