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

theorem Th4:
     Y in Rrank X iff ex Z st Z in X & Y in bool Rrank Z
proof
  thus Y in Rrank X implies
    ex Z st Z in X & Y in bool Rrank Z
  proof
    assume Y in Rrank X;
    then consider B be Ordinal such that
A1:   B in the_rank_of X & Y in bool Rank B by Th3;
    consider a be set such that
A2:   a in X & B c= the_rank_of a by A1,CLASSES1:70;
    Rank B c= Rrank a by A2,CLASSES1:37;
    then Y c= Rrank a by A1;
    hence thesis by A2;
  end;
  given Z be set such that
A3: Z in X & Y in bool Rrank Z;
  succ the_rank_of Z c= the_rank_of X by A3,CLASSES1:68,ORDINAL1:21;
  then
A4: Rank succ the_rank_of Z c= Rrank X by CLASSES1:37;
  Y in Rank succ the_rank_of Z by A3,CLASSES1:30;
  hence thesis by A4;
end;
