reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem Th20:
  A in B iff aleph A in aleph B
proof
  defpred P[Ordinal] means for A st A in $1 holds aleph A in aleph $1;
A1: for B st P[B] holds P[succ B]
  proof
    let B such that
A2: P[B];
    let A;
A3: aleph succ B = nextcard aleph B by Lm1;
A4: now
      assume A in B;
      then
A5:   aleph A in aleph B by A2;
      aleph B in nextcard aleph B by Th17;
      hence thesis by A3,A5,ORDINAL1:10;
    end;
A6: A c< B iff A c= B & A <> B;
    assume A in succ B;
    hence thesis by A3,A6,A4,Th17,ORDINAL1:11,22;
  end;
A7: for B st B <> 0 & B is limit_ordinal & for C st C in B holds P[C]
  holds P[B]
  proof
    let B such that
A8: B <> 0 and
A9: B is limit_ordinal and
    for C st C in B for A st A in C holds aleph A in aleph C;
    let A;
    consider S such that
A10: dom S = B & for C st C in B holds S.C = f(C) from ORDINAL2:sch 2;
    assume A in B;
    then succ A in B by A9,ORDINAL1:28;
    then
A11: S.succ A in rng S & S.succ A = aleph succ A by A10,FUNCT_1:def 3;
    sup rng S = sup S by ORDINAL2:26;
    then
A12: aleph succ A c= sup S by A11,ORDINAL1:def 2,ORDINAL2:19;
A13: card aleph succ A = aleph succ A;
A14: aleph succ A = nextcard aleph A & aleph A in nextcard aleph A
     by Th17,Lm1;
    aleph B = card sup S by A8,A9,A10,Lm1;
    then aleph succ A c= aleph B by A12,A13,Th10;
    hence thesis by A14;
  end;
A15: P[0];
A16: for B holds P[B] from ORDINAL2:sch 1(A15,A1,A7);
  hence A in B implies aleph A in aleph B;
  assume
A17: aleph A in aleph B;
  then
A18: aleph A <> aleph B;
  not B in A by A16,A17;
  hence thesis by A18,ORDINAL1:14;
end;
