In my Discrete Math course, I encounter a question such as

$\mathrm{\exists}x\mathrm{\forall}y\phantom{\rule{thinmathspace}{0ex}}x<y\text{where}x,y\in \mathbb{Z}\phantom{\rule{thinmathspace}{0ex}},$,

and I explain it like this: "Let $x=1$ and $y=-2$. For an x in the universe not all y values make the statement $x<y$ true so the case is 1 is not equal -2" and I think this is sufficient for explaining but I wonder whether there is a way to indicate proof with symbols or mathematically. Thanks in advance.

$\mathrm{\exists}x\mathrm{\forall}y\phantom{\rule{thinmathspace}{0ex}}x<y\text{where}x,y\in \mathbb{Z}\phantom{\rule{thinmathspace}{0ex}},$,

and I explain it like this: "Let $x=1$ and $y=-2$. For an x in the universe not all y values make the statement $x<y$ true so the case is 1 is not equal -2" and I think this is sufficient for explaining but I wonder whether there is a way to indicate proof with symbols or mathematically. Thanks in advance.