reserve X for set;
reserve UN for Universe;

theorem Th43:
  omega c< FinSETS
  proof
A1: omega c= FinSETS by Th16;
    now
      assume
A2:   omega = FinSETS;
      then reconsider x1 = 1 as Element of FinSETS;
      {x1} in FinSETS;
      then
A3:   {1} is epsilon-transitive by A2;
A4:   1 in {1} by TARSKI:def 1;
      not 1 c= {1}
      proof
        assume
A5:     1 c= {1};
        0 in 1 by CARD_1:49,TARSKI:def 1;
        hence thesis by A5;
      end;
      hence thesis by A4,A3;
    end;
    hence thesis by A1;
  end;
