Beware the difference between | and ||, between = and ==, and between {} and (). The following is closer to the textbook phrase:
ForAll[{n, m},
Element[n | m, Integers]
&& (n > 0 || n < 0 || n == 0)
&& (m > 0 || m < 0 || m == 0)
&& m >= n,
predicate]
For example
ForAll[{n, m},
Element[n | m, Integers]
&& (n > 0 || n < 0 || n == 0)
&& (m > 0 || m < 0 || m == 0)
&& m >= n,
2^m >= 2^n] // Reduce
