reserve Al for QC-alphabet,
     PHI for Consistent Subset of CQC-WFF(Al),
     PSI for Subset of CQC-WFF(Al),
     p,q,r,s for Element of CQC-WFF(Al),
     A for non empty set,
     J for interpretation of Al,A,
     v for Element of Valuations_in(Al,A),
     m,n,i,j,k for Element of NAT,
     l for CQC-variable_list of k,Al,
     P for QC-pred_symbol of k,Al,
     x,y for bound_QC-variable of Al,
     z for QC-symbol of Al,
     Al2 for Al-expanding QC-alphabet;
reserve J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th1:
 ex s being set st for p,x holds not [s,[x,p]] in QC-symbols(Al)
proof
  assume
A1: for s being set holds ex p,x st [s,[x,p]] in QC-symbols(Al);
  for s being set holds s in union union QC-symbols(Al)
  proof
    let s be set;
    consider p,x such that
A2:  [s,[x,p]] in QC-symbols(Al) by A1;
A3: {s} in {{s,[x,p]},{s}} by TARSKI:def 2;
A4: s in {s} by TARSKI:def 1;
    {{s,[x,p]},{s}} c= union QC-symbols(Al) by A2,ZFMISC_1:74;
    then {s} c= union union QC-symbols(Al) by A3,ZFMISC_1:74;
    hence thesis by A4;
  end;
  then union union QC-symbols(Al) in union union QC-symbols(Al) &
  for X being set holds not X in X;
  hence contradiction;
end;
