reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th36:
  A in B iff Rank A in Rank B
proof
  defpred OnP[Ordinal,Ordinal] means $1 in $2 implies Rank $1 in Rank $2;
A1: now
    let A;
    defpred P[Ordinal] means OnP[A,$1];
A2: for B st for C st C in B holds P[C] holds P[B]
    proof
      let B such that
A3:   for C st C in B holds OnP[A,C] and
A4:   A in B;
A5:   now
        given C such that
A6:     B = succ C;
A7:     A in C implies Rank A in Rank C by A3,A6,ORDINAL1:6;
     now
          assume
A8:       not A in C;
          A c= C & A <> C iff A c< C;
          hence Rank A = Rank C by A4,A6,A8,ORDINAL1:11,22;
        end;
        then Rank A c= Rank C by A7,ORDINAL1:def 2;
        hence thesis by A6,Th32;
      end;
  now
        assume
A9:    for C holds B <> succ C;
then A10:    B is limit_ordinal by ORDINAL1:29;
A11:    B <> succ A by A9;
    succ A c< B by A11,A4,ORDINAL1:21;
then A12:    succ A in B by ORDINAL1:11;
    Rank A in Rank succ A by Th32;
        hence thesis by A10,A12,Th31;
      end;
      hence thesis by A5;
    end;
    thus for B holds P[B] from ORDINAL1:sch 2(A2);
  end;
  hence OnP[A,B];
  assume that
A13: Rank A in Rank B and
A14: not A in B;
 B in A or B = A by A14,ORDINAL1:14;
  hence contradiction by A1,A13;
end;
