reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;
reserve U1,U2,U for Universe;
reserve u,v for Element of U;

theorem Th63:
  Rank omega is Tarski
proof
  thus X in Rank omega & Y c= X implies Y in Rank omega by CLASSES1:41;
  thus X in Rank omega implies bool X in Rank omega
  proof
    assume X in Rank omega;
    then consider A such that
A1: A in omega and
A2: X in Rank A by Lm5,CLASSES1:31;
A3: bool X in Rank succ A by A2,CLASSES1:42;
    succ A in omega by A1,Lm5,ORDINAL1:28;
    hence thesis by A3,Lm5,CLASSES1:31;
  end;
  thus X c= Rank omega implies X,Rank omega are_equipotent or X in Rank omega
  proof
A4: 0 in omega;
    defpred P[object,object] means the_rank_of $2 c= the_rank_of $1;
    assume that
A5: X c= Rank omega and
A6: not X,Rank omega are_equipotent and
A7: not X in Rank omega;
A8: card X <> card omega by A6,Th62,CARD_1:5;
    card X c= card omega by A5,Th62,CARD_1:11;
    then card X in omega by A8,CARD_1:3;
    then reconsider X as finite set;
A9: for x,y being object holds P[x,y] or P[y,x];
A10: for x,y,z being object st P[x,y] & P[y,z] holds P[x,z] by XBOOLE_1:1;
    omega c= Rank omega by CLASSES1:38;
    then
A11: X <> {} by A7,A4;
    consider x being object such that
A12: x in X &
     for y being object st y in X holds P[x,y] from CARD_2:sch 2(A11,A9,A10);
    set A = the_rank_of x;
    for Y st Y in X holds the_rank_of Y in succ A by A12,ORDINAL1:22;
    then
A13: the_rank_of X c= succ A by CLASSES1:69;
    A in omega by A5,A12,CLASSES1:66;
    then succ A in omega by Lm5,ORDINAL1:28;
    then the_rank_of X in omega by A13,ORDINAL1:12;
    hence thesis by A7,CLASSES1:66;
  end;
end;
