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
  for A,X st X in Rank A holds
  not X,Rank A are_equipotent & card X in card Rank A
proof
  defpred OnP[Ordinal] means
  for X st X in Rank $1 holds not X,Rank $1 are_equipotent;
A1: for A st for B st B in A holds OnP[B] holds OnP[A]
  proof
    let A such that
A2: for B st B in A holds OnP[B];
    let X;
    assume
A3: X in Rank A;
A4: now
      given B such that
A5:   A = succ B;
A6:   B c= A by A5,ORDINAL1:6,def 2;
A7:   Rank succ B = bool Rank B by Lm2;
then A8:   not Rank B,Rank A are_equipotent by A5,CARD_1:13;
  Rank B c= Rank A by A6,Th37;
      hence thesis by A3,A5,A7,A8,CARD_1:24;
    end;
 now
      assume that
A9:  A <> {} and
A10:  for B holds A <> succ B;
  A is limit_ordinal by A10,ORDINAL1:29;
      then consider B such that
A11:  B in A and
A12:  X in Rank B by A3,A9,Th31;
      A13:  (
 not X,Rank B are_equipotent)& X c= Rank B by A2,A11,A12,ORDINAL1:def 2;
  Rank B c= Rank A by A11,Th36,ORDINAL1:def 2;
      hence thesis by A13,CARD_1:24;
    end;
    hence thesis by A3,A4,Lm2;
  end;
A14: for A holds OnP[A] from ORDINAL1:sch 2(A1);
  let A,X;
  assume
A15: X in Rank A;
A16: card X c= card Rank A by A15,CARD_1:11,ORDINAL1:def 2;
 card X <> card Rank A by A14,A15,CARD_1:5;
  hence thesis by A14,A15,A16,CARD_1:3;
end;
