i could be wrong (i'm sort of skimming here) but the problem is not with the latter part of your statement, but with the beginning bit.

the original statement starts out: "if three numbers x,y,z are chosen between 0 and 1 with 0 < x < y < z < 1....."

this is equivalent to: for all (x,y,z) such that 0 < x < y < z < 1....

the negation of which is: there exists some (x,y,z) with 0 < x < y < z < 1 such that (negation of rest of statement).

x,y and z can be chosen arbitrarily in the original sentence. the negation of an arbitrary choice is a specific choice, which is an assertion of existence.