reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;
reserve U1,U2,U for Universe;
reserve u,v for Element of U;

theorem
  FinSETS = Rank omega
proof
A1: omega c= Rank omega by CLASSES1:38;
  then reconsider M = Rank omega as non empty set;
  0 in omega;
  then M is_Tarski-Class_of {} by A1,Th63;
  hence FinSETS c= Rank omega by CLASSES1:def 4;
A2: {} in On FinSETS by Th51,ORDINAL3:8;
A3: FinSETS = Rank On FinSETS by Th50;
  On FinSETS is limit_ordinal by Th51;
  then omega c= On FinSETS by A2,ORDINAL1:def 11;
  hence thesis by A3,CLASSES1:37;
end;
