reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;
reserve E for set;

theorem
  for A being non empty set, b being object st A <> {b}
  ex a being Element of A st a <> b
proof
  let A be non empty set, b be object such that
A1: A <> {b};
  assume
A2: for a being Element of A holds a = b;
  now
    set a0 = the Element of A;
    let a be object;
    thus a in A implies a = b by A2;
    assume
A3: a = b;
    a0 = b by A2;
    hence a in A by A3;
  end;
  hence contradiction by A1,TARSKI:def 1;
end;
