reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th68:
  No_Ordinal_op A < No_Ordinal_op B iff A in B
proof
  thus No_Ordinal_op A < No_Ordinal_op B implies A in B
  proof
    assume
A1: No_Ordinal_op A < No_Ordinal_op B;
    assume not A in B;
    then per cases by ORDINAL1:16,XBOOLE_0:def 8;
    suppose A=B;
      hence thesis by A1,SURREALO:3;
    end;
    suppose B c< A;
      then No_Ordinal_op B <= No_Ordinal_op A by Lm23,ORDINAL1:11;
      hence thesis by A1;
    end;
  end;
  thus thesis by Lm23;
end;
