Given any arbitrary binary relation $\geq$ defined on some set $S$, we define a new binary relation $>$ on $S$ by: $$ x > y \quad\text{iff}\quad (x \geq y) \wedge \neg(y \geq x) $$ In accordance with our usual intuition for inequalities, I would like to prove that:
Claim: $$ \neg(x > y) \quad\text{iff}\quad y \geq x $$
Unfortunately, all I could conclude was that: $$ \neg(x > y) \quad\text{iff}\quad \neg(x \geq y) \vee (y \geq x) $$
Is my claim even true? If so, could somehow help me finish off my work? If not, what additional hypotheses should be added in order to salvage the claim?
In general this is not true. Consider $\mathcal P(\Bbb N)$ with $x\leq y$ being $x\subseteq y$. If $\lnot(x\subsetneq y)$ then it's not necessarily that $y\subseteq x$. It might be that neither is a subset of the other.
That is to say, if we don't assume that $\leq$ is a total relation to begin with, i.e. $\forall x\forall y(x\leq y\lor y\leq x)$, then there is no way to infer what you want to conclude.