reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;

theorem
  M is infinite implies ex A st M = aleph A
proof
  defpred P[set] means $1 is infinite implies ex A st $1 = aleph A;
A1: P[K] implies P[nextcard K]
  proof
    assume that
A2: P[K] and
A3: not nextcard K is finite;
    now
      assume K is finite;
      then reconsider K9 = K as finite set;
      nextcard Segm card K9 = card Segm(card K9 + 1) by NAT_1:42;
      hence contradiction by A3;
    end;
    then consider A such that
A4: K = aleph A by A2;
    take succ A;
    thus thesis by A4,CARD_1:19;
  end;
A5: K <> {} & K is limit_cardinal & (for N st N in K holds P[N]) implies P[
  K]
  proof
    deffunc a(Ordinal) = aleph $1;
    defpred R[object,object] means ex A st $1 = aleph A & $2 = A;
    assume that
    K <> {} and
A6: K is limit_cardinal and
A7: for N st N in K holds P[N] and
A8: not K is finite;
    defpred P[object] means ex N st N = $1 & not N is finite;
    consider X such that
A9: for x being object holds x in X iff x in K & P[x] from XBOOLE_0:sch 1;
A10: for x being object st x in X ex y being object st R[x,y]
    proof
      let x be object;
      assume
A11:  x in X;
      then consider N such that
A12:  N = x and
A13:  not N is finite by A9;
      N in K by A9,A11,A12;
      then ex A st x = aleph A by A7,A12,A13;
      hence thesis;
    end;
    consider f such that
A14: dom f = X &
for x being object st x in X holds R[x,f.x] from CLASSES1:sch 1(
    A10);
    now
      let x be set;
      assume x in rng f;
      then consider y being object such that
A15:  y in X and
A16:  x = f.y by A14,FUNCT_1:def 3;
      consider A such that
A17:  y = aleph A and
A18:  x = A by A14,A15,A16;
      thus x is Ordinal by A18;
      thus x c= rng f
      proof
        let z be object;
        assume
A19:    z in x;
        then reconsider z9 = z as Ordinal by A18;
A20:    aleph z9 in aleph A by A18,A19,CARD_1:21;
        aleph A in K by A9,A15,A17;
        then
A21:    aleph z9 in K by A20,ORDINAL1:10;
        not aleph z9 is finite by Th3;
        then
A22:    aleph z9 in X by A9,A21;
        then ex A st aleph z9 = aleph A & f.(aleph z9) = A by A14;
        then z9 = f.(aleph z9) by CARD_1:22;
        hence thesis by A14,A22,FUNCT_1:def 3;
      end;
    end;
    then reconsider A = rng f as epsilon-transitive epsilon-connected set
by ORDINAL1:19;
    consider L being Sequence such that
A23: dom L = A & for B st B in A holds L.B = a(B) from ORDINAL2:sch 2;
    now
      let B;
      assume B in A;
      then consider x being object such that
A24:  x in X and
A25:  B = f.x by A14,FUNCT_1:def 3;
      consider C such that
A26:  x = aleph C and
A27:  B = C by A14,A24,A25;
A28:  aleph succ C = nextcard aleph C by CARD_1:19;
      aleph C in K by A9,A24,A26;
      then
A29:  aleph succ C c= K by A28,CARD_3:90;
      aleph succ C <> K by A6,A28;
      then
A30:  aleph succ C in K by A29,CARD_1:3;
      not aleph succ C is finite by Th3;
      then
A31:  aleph succ C in X by A9,A30;
      then consider D being Ordinal such that
A32:  aleph succ C = aleph D and
A33:  f.(aleph succ C) = D by A14;
      succ C = D by A32,CARD_1:22;
      hence succ B in A by A14,A27,A31,A33,FUNCT_1:def 3;
    end;
    then A is limit_ordinal by ORDINAL1:28;
    then
A34: A = {} or aleph A = card sup L by A23,CARD_1:20;
    take A;
    sup L c= K
    proof
      let x be Ordinal;
      assume
A35:  x in sup L;
      reconsider x9 = x as Ordinal;
      consider C such that
A36:  C in rng L and
A37:  x9 c= C by A35,ORDINAL2:21;
      consider y being object such that
A38:  y in dom L and
A39:  C = L.y by A36,FUNCT_1:def 3;
      reconsider y as Ordinal by A38;
A40:  C = aleph y by A23,A38,A39;
      consider z being object such that
A41:  z in dom f and
A42:  y = f.z by A23,A38,FUNCT_1:def 3;
      ex D being Ordinal st z = aleph D & y = D by A14,A41,A42;
      then C in K by A9,A14,A40,A41;
      hence thesis by A37,ORDINAL1:12;
    end;
    then card sup L c= card K by CARD_1:11;
    then
A43: card sup L c= K;
    now
      per cases;
      case
        A = {};
        then not omega in X by A14,RELAT_1:42;
        then not omega in K by A9;
        then
A44:    K c= omega by CARD_1:4;
        omega c= K by A8,CARD_3:85;
        hence K = omega by A44;
      end;
      case
A45:    A <> {};
        assume K <> aleph A;
        then
A46:    card sup L in K by A34,A43,A45,CARD_1:3;
        not aleph A is finite by Th3;
        then
A47:    card sup L in X by A9,A34,A45,A46;
        then consider B such that
A48:    card sup L = aleph B and
A49:    f.(card sup L) = B by A14;
        A = B by A34,A45,A48,CARD_1:22;
        then A in A by A14,A47,A49,FUNCT_1:def 3;
        hence contradiction;
      end;
    end;
    hence thesis by CARD_1:46;
  end;
A50: P[{}];
  for M holds P[M] from CARD_1:sch 1(A50,A1,A5);
  hence thesis;
end;
