SET007 Axioms: SET007+76.ax


%------------------------------------------------------------------------------
% File     : SET007+76 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : A First Order Language
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : qc_lang1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  121 (  54 unit)
%            Number of atoms       :  402 (  79 equality)
%            Maximal formula depth :   18 (   4 average)
%            Number of connectives :  322 (  41 ~  ;   0  |; 111  &)
%                                         (  21 <=>; 149 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-3 arity)
%            Number of functors    :   62 (  21 constant; 0-4 arity)
%            Number of variables   :  146 (   0 singleton; 119 !;  27 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_qc_lang1,axiom,(
    ~ v1_xboole_0(k1_qc_lang1) )).

fof(fc2_qc_lang1,axiom,(
    ~ v1_xboole_0(k2_qc_lang1) )).

fof(fc3_qc_lang1,axiom,(
    ~ v1_xboole_0(k3_qc_lang1) )).

fof(fc4_qc_lang1,axiom,(
    ~ v1_xboole_0(k4_qc_lang1) )).

fof(fc5_qc_lang1,axiom,(
    ~ v1_xboole_0(k5_qc_lang1) )).

fof(fc6_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ~ v1_xboole_0(k7_qc_lang1(A)) ) )).

fof(t1_qc_lang1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,A)
         => r1_tarski(k2_zfmisc_1(k1_tarski(C),B),k2_zfmisc_1(A,B)) ) ) )).

fof(t2_qc_lang1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( m1_subset_1(D,A)
             => ! [E] :
                  ( m1_subset_1(E,A)
                 => r1_tarski(k2_zfmisc_1(k1_enumset1(C,D,E),B),k2_zfmisc_1(A,B)) ) ) ) ) )).

fof(d1_qc_lang1,axiom,(
    k1_qc_lang1 = k2_zfmisc_1(k1_enumset1(np__4,np__5,np__6),k5_numbers) )).

fof(t3_qc_lang1,axiom,(
    $true )).

fof(t4_qc_lang1,axiom,(
    r1_tarski(k1_qc_lang1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) )).

fof(d2_qc_lang1,axiom,(
    k2_qc_lang1 = k2_zfmisc_1(k1_tarski(np__4),k5_numbers) )).

fof(d3_qc_lang1,axiom,(
    k3_qc_lang1 = k2_zfmisc_1(k1_tarski(np__5),k5_numbers) )).

fof(d4_qc_lang1,axiom,(
    k4_qc_lang1 = k2_zfmisc_1(k1_tarski(np__6),k5_numbers) )).

fof(t5_qc_lang1,axiom,(
    $true )).

fof(t6_qc_lang1,axiom,(
    $true )).

fof(t7_qc_lang1,axiom,(
    $true )).

fof(t8_qc_lang1,axiom,(
    $true )).

fof(t9_qc_lang1,axiom,(
    $true )).

fof(t10_qc_lang1,axiom,(
    r1_tarski(k5_qc_lang1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) )).

fof(d6_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_qc_lang1)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k6_qc_lang1(A)
          <=> k1_mcart_1(A) = k1_nat_1(np__7,B) ) ) ) )).

fof(d8_qc_lang1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_qc_lang1)
         => ( m1_qc_lang1(B,A)
          <=> k3_finseq_1(B) = A ) ) ) )).

fof(d9_qc_lang1,axiom,(
    ! [A] :
      ( v1_qc_lang1(A)
    <=> ( m1_subset_1(A,k1_zfmisc_1(k13_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))))
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
               => ! [D] :
                    ( m1_qc_lang1(D,B)
                   => r2_hidden(k7_finseq_1(k12_finseq_1(k7_qc_lang1(B),C),D),A) ) ) )
        & r2_hidden(k9_finseq_1(k4_tarski(np__0,np__0)),A)
        & ! [B] :
            ( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
           => ( r2_hidden(B,A)
             => r2_hidden(k7_finseq_1(k9_finseq_1(k4_tarski(np__1,np__0)),B),A) ) )
        & ! [B] :
            ( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
           => ! [C] :
                ( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => r2_hidden(k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__2,np__0)),B),C),A) ) ) )
        & ! [B] :
            ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
           => ! [C] :
                ( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
               => ( r2_hidden(C,A)
                 => r2_hidden(k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__3,np__0)),k12_finseq_1(k2_qc_lang1,B)),C),A) ) ) ) ) ) )).

fof(d10_qc_lang1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( A = k8_qc_lang1
      <=> ( v1_qc_lang1(A)
          & ! [B] :
              ( ~ v1_xboole_0(B)
             => ( v1_qc_lang1(B)
               => r1_tarski(A,B) ) ) ) ) ) )).

fof(t11_qc_lang1,axiom,(
    $true )).

fof(t12_qc_lang1,axiom,(
    $true )).

fof(t13_qc_lang1,axiom,(
    $true )).

fof(t14_qc_lang1,axiom,(
    $true )).

fof(t15_qc_lang1,axiom,(
    $true )).

fof(t16_qc_lang1,axiom,(
    $true )).

fof(t17_qc_lang1,axiom,(
    $true )).

fof(t18_qc_lang1,axiom,(
    $true )).

fof(t19_qc_lang1,axiom,(
    $true )).

fof(t20_qc_lang1,axiom,(
    $true )).

fof(t21_qc_lang1,axiom,(
    v1_qc_lang1(k8_qc_lang1) )).

fof(d11_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_qc_lang1)
     => ! [B] :
          ( m2_finseq_1(B,k1_qc_lang1)
         => ( k6_qc_lang1(A) = k3_finseq_1(B)
           => k9_qc_lang1(A,B) = k7_finseq_1(k12_finseq_1(k5_qc_lang1,A),B) ) ) ) )).

fof(t22_qc_lang1,axiom,(
    $true )).

fof(t23_qc_lang1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
         => ! [C] :
              ( m1_qc_lang1(C,A)
             => k9_qc_lang1(B,C) = k7_finseq_1(k12_finseq_1(k7_qc_lang1(A),B),C) ) ) ) )).

fof(d12_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => k10_qc_lang1(A) = A ) )).

fof(d13_qc_lang1,axiom,(
    k11_qc_lang1 = k9_finseq_1(k4_tarski(np__0,np__0)) )).

fof(d14_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => k12_qc_lang1(A) = k7_finseq_1(k9_finseq_1(k4_tarski(np__1,np__0)),k10_qc_lang1(A)) ) )).

fof(d15_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => k13_qc_lang1(A,B) = k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__2,np__0)),k10_qc_lang1(A)),k10_qc_lang1(B)) ) ) )).

fof(d16_qc_lang1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => k14_qc_lang1(A,B) = k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__3,np__0)),k12_finseq_1(k2_qc_lang1,A)),k10_qc_lang1(B)) ) ) )).

fof(d17_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v2_qc_lang1(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
                & ? [D] :
                    ( m1_qc_lang1(D,B)
                    & A = k9_qc_lang1(C,D) ) ) ) ) ) )).

fof(d18_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v3_qc_lang1(A)
      <=> ? [B] :
            ( m1_subset_1(B,k8_qc_lang1)
            & A = k12_qc_lang1(B) ) ) ) )).

fof(d19_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v4_qc_lang1(A)
      <=> ? [B] :
            ( m1_subset_1(B,k8_qc_lang1)
            & ? [C] :
                ( m1_subset_1(C,k8_qc_lang1)
                & A = k13_qc_lang1(B,C) ) ) ) ) )).

fof(d20_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v5_qc_lang1(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
            & ? [C] :
                ( m1_subset_1(C,k8_qc_lang1)
                & A = k14_qc_lang1(B,C) ) ) ) ) )).

fof(t24_qc_lang1,axiom,(
    $true )).

fof(t25_qc_lang1,axiom,(
    $true )).

fof(t26_qc_lang1,axiom,(
    $true )).

fof(t27_qc_lang1,axiom,(
    $true )).

fof(t28_qc_lang1,axiom,(
    $true )).

fof(t29_qc_lang1,axiom,(
    $true )).

fof(t30_qc_lang1,axiom,(
    $true )).

fof(t31_qc_lang1,axiom,(
    $true )).

fof(t32_qc_lang1,axiom,(
    $true )).

fof(t33_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ~ ( A != k11_qc_lang1
          & ~ v2_qc_lang1(A)
          & ~ v3_qc_lang1(A)
          & ~ v4_qc_lang1(A)
          & ~ v5_qc_lang1(A) ) ) )).

fof(t34_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => r1_xreal_0(np__1,k3_finseq_1(k10_qc_lang1(A))) ) )).

fof(t35_qc_lang1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
         => k6_qc_lang1(B) = A ) ) )).

fof(t36_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__0
         => A = k11_qc_lang1 )
        & ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__1
         => v3_qc_lang1(A) )
        & ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__2
         => v4_qc_lang1(A) )
        & ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__3
         => v5_qc_lang1(A) )
        & ( ? [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
              & m2_subset_1(k1_funct_1(k10_qc_lang1(A),np__1),k5_qc_lang1,k7_qc_lang1(B)) )
         => v2_qc_lang1(A) ) ) ) )).

fof(t37_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( k10_qc_lang1(A) = k7_finseq_1(k10_qc_lang1(B),C)
               => k10_qc_lang1(A) = k10_qc_lang1(B) ) ) ) ) )).

fof(d21_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v2_qc_lang1(A)
       => ! [B] :
            ( m1_subset_1(B,k5_qc_lang1)
           => ( B = k15_qc_lang1(A)
            <=> ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & ? [D] :
                      ( m1_qc_lang1(D,C)
                      & ? [E] :
                          ( m2_subset_1(E,k5_qc_lang1,k7_qc_lang1(C))
                          & B = E
                          & A = k9_qc_lang1(E,D) ) ) ) ) ) ) ) )).

fof(d22_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v2_qc_lang1(A)
       => ! [B] :
            ( m2_finseq_1(B,k1_qc_lang1)
           => ( B = k16_qc_lang1(A)
            <=> ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & ? [D] :
                      ( m2_subset_1(D,k5_qc_lang1,k7_qc_lang1(C))
                      & ? [E] :
                          ( m1_qc_lang1(E,C)
                          & B = E
                          & A = k9_qc_lang1(D,E) ) ) ) ) ) ) ) )).

fof(d23_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v3_qc_lang1(A)
       => ! [B] :
            ( m1_subset_1(B,k8_qc_lang1)
           => ( B = k17_qc_lang1(A)
            <=> A = k12_qc_lang1(B) ) ) ) ) )).

fof(d24_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v4_qc_lang1(A)
       => ! [B] :
            ( m1_subset_1(B,k8_qc_lang1)
           => ( B = k18_qc_lang1(A)
            <=> ? [C] :
                  ( m1_subset_1(C,k8_qc_lang1)
                  & A = k13_qc_lang1(B,C) ) ) ) ) ) )).

fof(d25_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v4_qc_lang1(A)
       => ! [B] :
            ( m1_subset_1(B,k8_qc_lang1)
           => ( B = k19_qc_lang1(A)
            <=> ? [C] :
                  ( m1_subset_1(C,k8_qc_lang1)
                  & A = k13_qc_lang1(C,B) ) ) ) ) ) )).

fof(d26_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v5_qc_lang1(A)
       => ! [B] :
            ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
           => ( B = k20_qc_lang1(A)
            <=> ? [C] :
                  ( m1_subset_1(C,k8_qc_lang1)
                  & A = k14_qc_lang1(B,C) ) ) ) ) ) )).

fof(d27_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v5_qc_lang1(A)
       => ! [B] :
            ( m1_subset_1(B,k8_qc_lang1)
           => ( B = k21_qc_lang1(A)
            <=> ? [C] :
                  ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
                  & A = k14_qc_lang1(C,B) ) ) ) ) ) )).

fof(t38_qc_lang1,axiom,(
    $true )).

fof(t39_qc_lang1,axiom,(
    $true )).

fof(t40_qc_lang1,axiom,(
    $true )).

fof(t41_qc_lang1,axiom,(
    $true )).

fof(t42_qc_lang1,axiom,(
    $true )).

fof(t43_qc_lang1,axiom,(
    $true )).

fof(t44_qc_lang1,axiom,(
    $true )).

fof(t45_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ~ ( v3_qc_lang1(A)
          & r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k17_qc_lang1(A)))) ) ) )).

fof(t46_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v4_qc_lang1(A)
       => ( ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k18_qc_lang1(A))))
          & ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k19_qc_lang1(A)))) ) ) ) )).

fof(t47_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ~ ( v5_qc_lang1(A)
          & r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k21_qc_lang1(A)))) ) ) )).

fof(t48_qc_lang1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
         => ( k1_mcart_1(B) != np__0
            & k1_mcart_1(B) != np__1
            & k1_mcart_1(B) != np__2
            & k1_mcart_1(B) != np__3 ) ) ) )).

fof(t49_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( k1_mcart_1(k1_funct_1(k10_qc_lang1(k11_qc_lang1),np__1)) = np__0
        & ~ ( v2_qc_lang1(A)
            & ! [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => ~ m2_subset_1(k1_funct_1(k10_qc_lang1(A),np__1),k5_qc_lang1,k7_qc_lang1(B)) ) )
        & ( v3_qc_lang1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__1 )
        & ( v4_qc_lang1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__2 )
        & ( v5_qc_lang1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__3 ) ) ) )).

fof(t50_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v2_qc_lang1(A)
       => ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__0
          & k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__1
          & k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__2
          & k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__3 ) ) ) )).

fof(t51_qc_lang1,axiom,
    ( ~ v2_qc_lang1(k11_qc_lang1)
    & ~ v3_qc_lang1(k11_qc_lang1)
    & ~ v4_qc_lang1(k11_qc_lang1)
    & ~ v5_qc_lang1(k11_qc_lang1)
    & ! [A] :
        ( m1_subset_1(A,k8_qc_lang1)
       => ( ~ ( v2_qc_lang1(A)
              & v3_qc_lang1(A) )
          & ~ ( v2_qc_lang1(A)
              & v4_qc_lang1(A) )
          & ~ ( v2_qc_lang1(A)
              & v5_qc_lang1(A) )
          & ~ ( v3_qc_lang1(A)
              & v4_qc_lang1(A) )
          & ~ ( v3_qc_lang1(A)
              & v5_qc_lang1(A) )
          & ~ ( v4_qc_lang1(A)
              & v5_qc_lang1(A) ) ) ) )).

fof(d30_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v6_qc_lang1(A)
      <=> k24_qc_lang1(A) = k1_xboole_0 ) ) )).

fof(s1_qc_lang1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
             => ! [C] :
                  ( m1_qc_lang1(C,A)
                 => p1_s1_qc_lang1(k9_qc_lang1(B,C)) ) ) )
      & p1_s1_qc_lang1(k11_qc_lang1)
      & ! [A] :
          ( m1_subset_1(A,k8_qc_lang1)
         => ( p1_s1_qc_lang1(A)
           => p1_s1_qc_lang1(k12_qc_lang1(A)) ) )
      & ! [A] :
          ( m1_subset_1(A,k8_qc_lang1)
         => ! [B] :
              ( m1_subset_1(B,k8_qc_lang1)
             => ( ( p1_s1_qc_lang1(A)
                  & p1_s1_qc_lang1(B) )
               => p1_s1_qc_lang1(k13_qc_lang1(A,B)) ) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
         => ! [B] :
              ( m1_subset_1(B,k8_qc_lang1)
             => ( p1_s1_qc_lang1(B)
               => p1_s1_qc_lang1(k14_qc_lang1(A,B)) ) ) ) )
   => ! [A] :
        ( m1_subset_1(A,k8_qc_lang1)
       => p1_s1_qc_lang1(A) ) )).

fof(s2_qc_lang1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,k8_qc_lang1)
       => ( ( v2_qc_lang1(A)
           => p1_s2_qc_lang1(A) )
          & p1_s2_qc_lang1(k11_qc_lang1)
          & ( ( v3_qc_lang1(A)
              & p1_s2_qc_lang1(k17_qc_lang1(A)) )
           => p1_s2_qc_lang1(A) )
          & ( ( v4_qc_lang1(A)
              & p1_s2_qc_lang1(k18_qc_lang1(A))
              & p1_s2_qc_lang1(k19_qc_lang1(A)) )
           => p1_s2_qc_lang1(A) )
          & ( ( v5_qc_lang1(A)
              & p1_s2_qc_lang1(k21_qc_lang1(A)) )
           => p1_s2_qc_lang1(A) ) ) )
   => ! [A] :
        ( m1_subset_1(A,k8_qc_lang1)
       => p1_s2_qc_lang1(A) ) )).

fof(s3_qc_lang1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k8_qc_lang1,f1_s3_qc_lang1)
      & m2_relset_1(A,k8_qc_lang1,f1_s3_qc_lang1)
      & k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k11_qc_lang1) = f2_s3_qc_lang1
      & ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => ( ( v2_qc_lang1(B)
             => k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f3_s3_qc_lang1(B) )
            & ( v3_qc_lang1(B)
             => k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f4_s3_qc_lang1(k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k17_qc_lang1(B))) )
            & ( v4_qc_lang1(B)
             => k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f5_s3_qc_lang1(k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k18_qc_lang1(B)),k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k19_qc_lang1(B))) )
            & ( v5_qc_lang1(B)
             => k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f6_s3_qc_lang1(B,k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k21_qc_lang1(B))) ) ) ) ) )).

fof(dt_m1_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ! [B] :
          ( m1_qc_lang1(B,A)
         => m2_finseq_1(B,k1_qc_lang1) ) ) )).

fof(existence_m1_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ? [B] : m1_qc_lang1(B,A) ) )).

fof(dt_k1_qc_lang1,axiom,(
    $true )).

fof(dt_k2_qc_lang1,axiom,(
    m1_subset_1(k2_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) )).

fof(dt_k3_qc_lang1,axiom,(
    m1_subset_1(k3_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) )).

fof(dt_k4_qc_lang1,axiom,(
    m1_subset_1(k4_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) )).

fof(dt_k5_qc_lang1,axiom,(
    $true )).

fof(dt_k6_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_qc_lang1)
     => m2_subset_1(k6_qc_lang1(A),k1_numbers,k5_numbers) ) )).

fof(dt_k7_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_subset_1(k7_qc_lang1(A),k1_zfmisc_1(k5_qc_lang1)) ) )).

fof(dt_k8_qc_lang1,axiom,(
    ~ v1_xboole_0(k8_qc_lang1) )).

fof(dt_k9_qc_lang1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_qc_lang1)
        & m1_finseq_1(B,k1_qc_lang1) )
     => m1_subset_1(k9_qc_lang1(A,B),k8_qc_lang1) ) )).

fof(dt_k10_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m2_finseq_1(k10_qc_lang1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ) )).

fof(dt_k11_qc_lang1,axiom,(
    m1_subset_1(k11_qc_lang1,k8_qc_lang1) )).

fof(dt_k12_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k12_qc_lang1(A),k8_qc_lang1) ) )).

fof(dt_k13_qc_lang1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k8_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k13_qc_lang1(A,B),k8_qc_lang1) ) )).

fof(dt_k14_qc_lang1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k14_qc_lang1(A,B),k8_qc_lang1) ) )).

fof(dt_k15_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k15_qc_lang1(A),k5_qc_lang1) ) )).

fof(dt_k16_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m2_finseq_1(k16_qc_lang1(A),k1_qc_lang1) ) )).

fof(dt_k17_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k17_qc_lang1(A),k8_qc_lang1) ) )).

fof(dt_k18_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k18_qc_lang1(A),k8_qc_lang1) ) )).

fof(dt_k19_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k19_qc_lang1(A),k8_qc_lang1) ) )).

fof(dt_k20_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m2_subset_1(k20_qc_lang1(A),k1_qc_lang1,k2_qc_lang1) ) )).

fof(dt_k21_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k21_qc_lang1(A),k8_qc_lang1) ) )).

fof(dt_k22_qc_lang1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_qc_lang1)
     => m1_subset_1(k22_qc_lang1(A),k1_zfmisc_1(k2_qc_lang1)) ) )).

fof(dt_k23_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_qc_lang1)
     => m1_subset_1(k23_qc_lang1(A),k1_zfmisc_1(k2_qc_lang1)) ) )).

fof(redefinition_k23_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_qc_lang1)
     => k23_qc_lang1(A) = k1_tarski(A) ) )).

fof(dt_k24_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k24_qc_lang1(A),k1_zfmisc_1(k2_qc_lang1)) ) )).

fof(d5_qc_lang1,axiom,(
    k5_qc_lang1 = a_0_0_qc_lang1 )).

fof(d7_qc_lang1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k7_qc_lang1(A) = a_1_0_qc_lang1(A) ) )).

fof(d28_qc_lang1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_qc_lang1)
     => k22_qc_lang1(A) = a_1_1_qc_lang1(A) ) )).

fof(d29_qc_lang1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k2_qc_lang1))
         => ( B = k24_qc_lang1(A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
                & m2_relset_1(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
                & B = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,A)
                & ! [D] :
                    ( m1_subset_1(D,k8_qc_lang1)
                   => ( k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k11_qc_lang1) = k1_xboole_0
                      & ( v2_qc_lang1(D)
                       => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = a_1_2_qc_lang1(D) )
                      & ( v3_qc_lang1(D)
                       => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k17_qc_lang1(D)) )
                      & ( v4_qc_lang1(D)
                       => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k4_subset_1(k2_qc_lang1,k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k18_qc_lang1(D)),k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k19_qc_lang1(D))) )
                      & ( v5_qc_lang1(D)
                       => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k6_subset_1(k2_qc_lang1,k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k21_qc_lang1(D)),k23_qc_lang1(k20_qc_lang1(D))) ) ) ) ) ) ) ) )).

fof(fraenkel_a_0_0_qc_lang1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_qc_lang1)
    <=> ? [B,C] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & m2_subset_1(C,k1_numbers,k5_numbers)
          & A = k4_tarski(B,C)
          & r1_xreal_0(np__7,B) ) ) )).

fof(fraenkel_a_1_0_qc_lang1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_0_qc_lang1(B))
      <=> ? [C] :
            ( m1_subset_1(C,k5_qc_lang1)
            & A = C
            & k6_qc_lang1(C) = B ) ) ) )).

fof(fraenkel_a_1_1_qc_lang1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_qc_lang1)
     => ( r2_hidden(A,a_1_1_qc_lang1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k1_funct_1(B,C)
            & r1_xreal_0(np__1,C)
            & r1_xreal_0(C,k3_finseq_1(B))
            & r2_hidden(k1_funct_1(B,C),k2_qc_lang1) ) ) ) )).

fof(fraenkel_a_1_2_qc_lang1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k8_qc_lang1)
     => ( r2_hidden(A,a_1_2_qc_lang1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k1_funct_1(k16_qc_lang1(B),C)
            & r1_xreal_0(np__1,C)
            & r1_xreal_0(C,k3_finseq_1(k16_qc_lang1(B)))
            & r2_hidden(k1_funct_1(k16_qc_lang1(B),C),k2_qc_lang1) ) ) ) )).
%------------------------------------------------------------------------------