reserve X for set,
        A for Subset of X,
        R,S for Relation of X;
reserve QUS for Quasi-UniformSpace;

theorem
  for X being non empty set,cB being empty Subset-Family of [:X,X:] holds
  not cB is axiom_UP1
  proof
    let X be non empty set,
    cB be empty Subset-Family of [:X,X:];
    assume
A1: cB is axiom_UP1;
    {} is Element of cB by SUBSET_1:def 1;
    then id X c= {} by A1;
    hence thesis;
  end;
