reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

theorem Th7:
  X is closed_wrt_A1-A7 implies omega c= X
proof
  assume
A1: X is closed_wrt_A1-A7;
  assume not thesis;
  then consider o being object such that
A2: o in omega and
A3: not o in X;
  defpred P[Ordinal] means $1 in omega & not $1 in X;
A4: ex K st P[K] by A2,A3;
  consider L such that
A5: P[L] & for M st P[M] holds L c= M from ORDINAL1:sch 1(A4);
  L<>{} by A1,A5,Th3;
  then
A6: {} in L by ORDINAL3:8;
  not omega c= L by A5;
  then not L is limit_ordinal by A6,ORDINAL1:def 11;
  then consider M such that
A7: succ M = L by ORDINAL1:29;
A8: M in L by A7,ORDINAL1:6;
A9: L c= omega by A5;
A10: now
    assume not M in X;
    then
A11: L c= M by A5,A9,A8;
    M c= L by A8,ORDINAL1:def 2;
    then M=L by A11;
    hence contradiction by A7,ORDINAL1:9;
  end;
  then {M} in X by A1,Th2;
  then M \/ {M} in X by A1,A10,Th4;
  hence contradiction by A5,A7;
end;
