reserve U for Universe;
reserve x for Element of U;

theorem Th17:
  union {NAT} c= FinSETS & not union {NAT} in FinSETS &
  not {NAT} c= FinSETS & not {NAT} in FinSETS
  proof
    thus union {NAT} c= FinSETS by CLASSES4:17;
    thus not union {NAT} in FinSETS;
    thus not {NAT} c= FinSETS
    proof
      assume
A1:   {NAT} c= FinSETS;
      NAT in {NAT} by TARSKI:def 1;
      hence contradiction by A1;
    end;
    assume
A2: {NAT} in FinSETS;
    NAT in {NAT} by TARSKI:def 1;
    then NAT in union FinSETS by A2,TARSKI:def 4;
    then NAT in FinSETS by CLASSES4:81;
    hence contradiction;
  end;
