[Siren-Of-Step]

See this $$\sim[\exists x \forall y [xy<0 \rightarrow (x<0 \vee y < 0)]] \equiv \sim [\exists x\forall y[\sim (xy < 0) \vee (x < 0 \vee y<0)]]$$
$$\equiv \forall x\exists y[(xy<0) \wedge \sim (x<0 \vee y<0)]$$
$$\equiv \forall x\exists y[(xy <0) \wedge (x \geqslant 0 \wedge y \geqslant 0)]$$

I confused the opposite of $>$ is $\leqslant$ and $<$ is $\geqslant$ ?

normally opposite of $< is >$ Example Inequality