reserve X,Y,Z for set,
        x,y,z for object,
        A,B,C for Ordinal;
reserve U for Grothendieck;

theorem Th3:
  X in Rank A iff ex B st B in A & X in bool Rank B
proof
  thus X in Rank A implies ex B st B in A & X in bool Rank B
  proof
    assume
A1: X in Rank A;
    per cases;
    suppose
      A is limit_ordinal;
      then consider B such that
A3:   B in A & X in Rank B by A1,CLASSES1:29,CLASSES1:31;
      take B;
      B c= B\/{B} by XBOOLE_1:7;
      then B c= succ B by ORDINAL1:def 1;
      then Rank B c= Rank succ B by CLASSES1:37;
      then X in Rank succ B by A3;
      hence thesis by A3,CLASSES1:30;
    end;
    suppose not A is limit_ordinal;
      then consider B be Ordinal such that
A4:     A= succ B by ORDINAL1:29;
      take B;
      thus thesis by A1,A4,ORDINAL1:6,CLASSES1:30;
    end;
  end;
  thus thesis by CLASSES1:36,41;
end;
