reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem
  [{x},{y}] is Surreal iff x < y
proof
  hereby assume [{x},{y}] is Surreal;
    then reconsider xy=[{x},{y}] as Surreal;
    x in L_xy << R_xy & y in {y} by SURREAL0:45,TARSKI:def 1;
    hence x < y;
  end;
  assume A1:x < y;
  consider M be Ordinal such that
  A2: for o st o in {x}\/{y} ex A be Ordinal st A in M & o in Day A
    by SURREAL0:47;
  [{x},{y}] in Day M by A2,A1,Th21,SURREAL0:46;
  hence thesis;
end;
