reserve U for Universe;

theorem Th3:
  for x be set st x in FinSETS holds x is finite
proof
A1: omega is limit_ordinal by ORDINAL1:def 11;
  let x be set;
  assume x in FinSETS;
  then consider n being Ordinal such that
A2: n in omega and
A3: x in Rank n by A1,CLASSES1:31,CLASSES2:64;
  reconsider n as Nat by A2;
  x is Element of Rank n by A3;
  hence thesis;
end;
