However, we know how negation interacts with quantifiers: we can pass a negation over a quantifier by switching the quantifier type (between universal and existential). So the statement above should be logically equivalent to
Notice that \(y \lt x\) is the negation of \(x \le y\text{.}\) This literally says, “for every number \(x\) there is a number \(y\) which is smaller than \(x\text{.}\)” We see that this is another way to make our original claim.
in-context