reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;
reserve M for non countable Aleph;
reserve X for Subset of M;
reserve N,N1 for cardinal infinite Element of M;
reserve A for Ordinal;
reserve x,y,X,Y for set;

theorem Th34:
  M is strongly_inaccessible & A in M implies card Rank A in M
proof
  assume that
A1: M is strongly_inaccessible and
A2: A in M;
  defpred P[Ordinal] means $1 in M implies card Rank $1 in M;
A3: for B1 st P[B1] holds P[succ B1]
  proof
    let B1 such that
A4: P[B1];
    assume succ B1 in M;
    then succ B1 c= M by ORDINAL1:def 2;
    then
A5: exp(2,card Rank B1) in M by A1,A4,CARD_FIL:def 14,ORDINAL1:21;
    Rank succ B1 = bool Rank B1 by CLASSES1:30;
    hence thesis by A5,CARD_2:31;
  end;
A6: cf M=M by A1,CARD_5:def 3;
A7: for B1 st B1 <> 0 & B1 is limit_ordinal & for B2 st B2 in B1 holds P[
  B2] holds P[B1]
  proof
    let B1 such that
    B1 <> 0 and
A8: B1 is limit_ordinal and
A9: for B2 st B2 in B1 holds P[B2];
    consider L being Sequence such that
A10: dom L = B1 & for A st A in B1 holds L.A = f(A) from ORDINAL2:sch
    2;
A11: card rng L c= card B1 by A10,CARD_1:12;
    assume
A12: B1 in M;
    then card B1 in M by CARD_1:9;
    then
A13: card rng L in cf M by A6,A11,ORDINAL1:12;
A14: for Y st Y in rng L holds card Y in M
    proof
      let Y;
      assume Y in rng L;
      then consider x being object such that
A15:  x in dom L and
A16:  Y = L.x by FUNCT_1:def 3;
      reconsider x1=x as Element of B1 by A10,A15;
      x1 in M & Y = Rank x1 by A12,A10,A15,A16,ORDINAL1:10;
      hence thesis by A9,A10,A15;
    end;
    Rank B1 = Union L by A8,A10,CLASSES2:24
      .= union rng L by CARD_3:def 4;
    hence thesis by A13,A14,Th33;
  end;
A17: P[0] by CLASSES1:29;
  for B1 holds P[B1] from ORDINAL2:sch 1(A17,A3,A7);
  hence thesis by A2;
end;
