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 Th35:
  M is strongly_inaccessible implies card Rank M = M
proof
  consider L being Sequence such that
A1: dom L = M & for A st A in M holds L.A = f(A) from ORDINAL2:sch 2;
A2: rng L is c=-linear
  proof
    let X,Y be set;
    assume X in rng L;
    then consider x being object such that
A3: x in dom L and
A4: X = L.x by FUNCT_1:def 3;
    assume Y in rng L;
    then consider y being object such that
A5: y in dom L and
A6: Y = L.y by FUNCT_1:def 3;
    reconsider x,y as Ordinal by A3,A5;
A7: Y = Rank y by A1,A5,A6;
A8: x c= y or y c= x;
    X = Rank x by A1,A3,A4;
    then X c= Y or Y c= X by A7,A8,CLASSES1:37;
    hence thesis by XBOOLE_0:def 9;
  end;
  card M c= card Rank M by CARD_1:11,CLASSES1:38;
  then
A9: M c= card Rank M;
A10: Rank M = Union L by A1,CLASSES2:24
    .= union rng L by CARD_3:def 4;
  assume
A11: M is strongly_inaccessible;
  now
    let X be set;
    assume X in rng L;
    then consider x being object such that
A12: x in dom L and
A13: X = L.x by FUNCT_1:def 3;
    reconsider x as Ordinal by A12;
    card Rank x in M by A11,A1,A12,Th34;
    hence card X in M by A1,A12,A13;
  end;
  then card union rng L c= M by A2,CARD_3:46;
  hence thesis by A10,A9,XBOOLE_0:def 10;
end;
