reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;

theorem Th9:
  H is being_equality or H is being_membership or H is negative or
  H is conjunctive or H is universal
proof
A1: H is Element of WFF by Def9;
  assume
A2: not thesis;
  then x.0 '=' x.1 <> H;
  then
A3: not x.0 '=' x.1 in { H } by TARSKI:def 1;
A4: now
    let x,y;
    x '=' y <> H by A2;
    then
A5: not x '=' y in { H } by TARSKI:def 1;
    x '=' y in WFF by Def8;
    hence x '=' y in WFF \ { H } by A5,XBOOLE_0:def 5;
    x 'in' y <> H by A2;
    then
A6: not x 'in' y in { H } by TARSKI:def 1;
    x 'in' y in WFF by Def8;
    hence x 'in' y in WFF \ { H } by A6,XBOOLE_0:def 5;
  end;
A7: now
    let x,p;
    assume
A8: p in WFF \ { H };
    then reconsider H1 = p as ZF-formula by Def9;
    All(x,H1) <> H by A2;
    then
A9: not All(x,p) in { H } by TARSKI:def 1;
    All(x,p) in WFF by A8,Def8;
    hence All(x,p) in WFF \ { H } by A9,XBOOLE_0:def 5;
  end;
A10: now
    let p,q;
    assume
A11: p in WFF \ { H } & q in WFF \ { H };
    then reconsider F = p, G = q as ZF-formula by Def9;
    F '&' G <> H by A2;
    then
A12: not p '&' q in { H } by TARSKI:def 1;
    p '&' q in WFF by A11,Def8;
    hence p '&' q in WFF \ { H } by A12,XBOOLE_0:def 5;
  end;
A13: now
    let p;
    assume
A14: p in WFF \ { H };
    then reconsider H1 = p as ZF-formula by Def9;
    'not' H1 <> H by A2;
    then
A15: not 'not' p in { H } by TARSKI:def 1;
    'not' p in WFF by A14,Def8;
    hence 'not' p in WFF \ { H } by A15,XBOOLE_0:def 5;
  end;
  x.0 '=' x.1 in WFF by Def8;
  then
A16: WFF \ { H } is non empty set by A3,XBOOLE_0:def 5;
  for a st a in WFF \ { H } holds a is FinSequence of NAT by Def8;
  then WFF c= WFF \ { H } by A16,A4,A13,A10,A7,Def8;
  then H in WFF \ { H } by A1;
  then not H in { H } by XBOOLE_0:def 5;
  hence contradiction by TARSKI:def 1;
end;
