SET007 Axioms: SET007+72.ax


%------------------------------------------------------------------------------
% File     : SET007+72 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of ZF Models
% 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    : zfmodel1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   28 (   4 unit)
%            Number of atoms       :  265 (  16 equality)
%            Maximal formula depth :   22 (  10 average)
%            Number of connectives :  281 (  44 ~  ;   3  |; 100  &)
%                                         (  17 <=>; 117 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-3 arity)
%            Number of functors    :   32 (  13 constant; 0-4 arity)
%            Number of variables   :  101 (   0 singleton;  95 !;   6 ?)
%            Maximal term depth    :    7 (   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(t1_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => r2_zf_model(A,k6_zf_model) ) ) )).

fof(t2_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k7_zf_model)
        <=> ! [B] :
              ( m1_subset_1(B,A)
             => ! [C] :
                  ( m1_subset_1(C,A)
                 => r2_hidden(k2_tarski(B,C),A) ) ) ) ) ) )).

fof(t3_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k7_zf_model)
        <=> ! [B,C] :
              ( ( r2_hidden(B,A)
                & r2_hidden(C,A) )
             => r2_hidden(k2_tarski(B,C),A) ) ) ) ) )).

fof(t4_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k8_zf_model)
        <=> ! [B] :
              ( m1_subset_1(B,A)
             => r2_hidden(k3_tarski(B),A) ) ) ) ) )).

fof(t5_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k8_zf_model)
        <=> ! [B] :
              ( r2_hidden(B,A)
             => r2_hidden(k3_tarski(B),A) ) ) ) ) )).

fof(t6_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k9_zf_model)
        <=> ? [B] :
              ( m1_subset_1(B,A)
              & B != k1_xboole_0
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => ~ ( r2_hidden(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,A)
                         => ~ ( r2_xboole_0(C,D)
                              & r2_hidden(D,B) ) ) ) ) ) ) ) ) )).

fof(t7_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k9_zf_model)
        <=> ? [B] :
              ( r2_hidden(B,A)
              & B != k1_xboole_0
              & ! [C] :
                  ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ~ ( r2_xboole_0(C,D)
                          & r2_hidden(D,B) ) ) ) ) ) ) )).

fof(t8_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k10_zf_model)
        <=> ! [B] :
              ( m1_subset_1(B,A)
             => r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) ) ) ) ) )).

fof(t9_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( r2_zf_model(A,k10_zf_model)
        <=> ! [B] :
              ( r2_hidden(B,A)
             => r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) ) ) ) ) )).

fof(t10_zfmodel1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_zf_lang)
     => ! [B] :
          ( ( v1_zf_lang(B)
            & m2_finseq_1(B,k5_numbers) )
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k1_zf_lang,C)
                    & m2_relset_1(D,k1_zf_lang,C) )
                 => ( r1_zf_model(C,D,B)
                   => ( r2_hidden(A,k2_zf_model(B))
                      | r1_zf_model(C,D,k8_zf_lang(A,B)) ) ) ) ) ) ) )).

fof(t11_zfmodel1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_zf_lang)
     => ! [B] :
          ( m2_subset_1(B,k5_numbers,k1_zf_lang)
         => ! [C] :
              ( ( v1_zf_lang(C)
                & m2_finseq_1(C,k5_numbers) )
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k1_zf_lang,D)
                        & m2_relset_1(E,k1_zf_lang,D) )
                     => ( ( r1_xboole_0(k2_tarski(A,B),k2_zf_model(C))
                          & r1_zf_model(D,E,C) )
                       => r1_zf_model(D,E,k14_zf_lang(A,B,C)) ) ) ) ) ) ) )).

fof(t12_zfmodel1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k5_numbers,k1_zf_lang)
     => ! [B] :
          ( m2_subset_1(B,k5_numbers,k1_zf_lang)
         => ! [C] :
              ( m2_subset_1(C,k5_numbers,k1_zf_lang)
             => ! [D] :
                  ( ( v1_zf_lang(D)
                    & m2_finseq_1(D,k5_numbers) )
                 => ! [E] :
                      ( ~ v1_xboole_0(E)
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k1_zf_lang,E)
                            & m2_relset_1(F,k1_zf_lang,E) )
                         => ( ( r1_xboole_0(k1_enumset1(A,B,C),k2_zf_model(D))
                              & r1_zf_model(E,F,D) )
                           => r1_zf_model(E,F,k16_zf_lang(A,B,C,D)) ) ) ) ) ) ) ) )).

fof(d1_zfmodel1,axiom,(
    ! [A] :
      ( ( v1_zf_lang(A)
        & m2_finseq_1(A,k5_numbers) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k1_zf_lang,B)
                & m2_relset_1(C,k1_zf_lang,B) )
             => ( r1_zf_model(B,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(A,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0)))))))
               => ( r2_hidden(k2_zf_lang(np__0),k2_zf_model(A))
                  | ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,B,B)
                        & m2_relset_1(D,B,B) )
                     => ( D = k1_zfmodel1(A,B,C)
                      <=> ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,k1_zf_lang,B)
                              & m2_relset_1(E,k1_zf_lang,B) )
                           => ( ! [F] :
                                  ( m2_subset_1(F,k5_numbers,k1_zf_lang)
                                 => ~ ( k8_funct_2(k1_zf_lang,B,E,F) != k8_funct_2(k1_zf_lang,B,C,F)
                                      & k2_zf_lang(np__0) != F
                                      & k2_zf_lang(np__3) != F
                                      & k2_zf_lang(np__4) != F ) )
                             => ( r1_zf_model(B,E,A)
                              <=> k8_funct_2(B,B,D,k8_funct_2(k1_zf_lang,B,E,k2_zf_lang(np__3))) = k8_funct_2(k1_zf_lang,B,E,k2_zf_lang(np__4)) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_zfmodel1,axiom,(
    $true )).

fof(t14_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_zf_lang(B)
            & m2_finseq_1(B,k5_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k1_zf_lang,A)
                & m2_relset_1(C,k1_zf_lang,A) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k1_zf_lang,A)
                    & m2_relset_1(D,k1_zf_lang,A) )
                 => ( ( ! [E] :
                          ( m2_subset_1(E,k5_numbers,k1_zf_lang)
                         => ~ ( k8_funct_2(k1_zf_lang,A,C,E) != k8_funct_2(k1_zf_lang,A,D,E)
                              & r2_hidden(E,k2_zf_model(B)) ) )
                      & r1_zf_model(A,C,B) )
                   => r1_zf_model(A,D,B) ) ) ) ) ) )).

fof(d2_zfmodel1,axiom,(
    ! [A] :
      ( ( v1_zf_lang(A)
        & m2_finseq_1(A,k5_numbers) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( ( r1_tarski(k2_zf_model(A),k2_tarski(k2_zf_lang(np__3),k2_zf_lang(np__4)))
              & r2_zf_model(B,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(A,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,B,B)
                  & m2_relset_1(C,B,B) )
               => ( C = k2_zfmodel1(A,B)
                <=> ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k1_zf_lang,B)
                        & m2_relset_1(D,k1_zf_lang,B) )
                     => ( r1_zf_model(B,D,A)
                      <=> k8_funct_2(B,B,C,k8_funct_2(k1_zf_lang,B,D,k2_zf_lang(np__3))) = k8_funct_2(k1_zf_lang,B,D,k2_zf_lang(np__4)) ) ) ) ) ) ) ) )).

fof(d3_zfmodel1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( r1_zfmodel1(A,B)
          <=> ? [C] :
                ( v1_zf_lang(C)
                & m2_finseq_1(C,k5_numbers)
                & r1_tarski(k2_zf_model(C),k2_tarski(k2_zf_lang(np__3),k2_zf_lang(np__4)))
                & r2_zf_model(B,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(C,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0)))))))
                & A = k2_zfmodel1(C,B) ) ) ) ) )).

fof(d4_zfmodel1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( r2_zfmodel1(A,B)
          <=> ? [C] :
                ( v1_zf_lang(C)
                & m2_finseq_1(C,k5_numbers)
                & ? [D] :
                    ( v1_funct_1(D)
                    & v1_funct_2(D,k1_zf_lang,B)
                    & m2_relset_1(D,k1_zf_lang,B)
                    & r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(C))
                    & r1_zf_model(B,D,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(C,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0)))))))
                    & A = k1_zfmodel1(C,B,D) ) ) ) ) ) )).

fof(t15_zfmodel1,axiom,(
    $true )).

fof(t16_zfmodel1,axiom,(
    $true )).

fof(t17_zfmodel1,axiom,(
    $true )).

fof(t18_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_zfmodel1(B,A)
           => r2_zfmodel1(B,A) ) ) ) )).

fof(t19_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( ! [B] :
              ( ( v1_zf_lang(B)
                & m2_finseq_1(B,k5_numbers) )
             => ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
               => r2_zf_model(A,k11_zf_model(B)) ) )
        <=> ! [B] :
              ( ( v1_zf_lang(B)
                & m2_finseq_1(B,k5_numbers) )
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k1_zf_lang,A)
                    & m2_relset_1(C,k1_zf_lang,A) )
                 => ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
                      & r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
                   => ! [D] :
                        ( m1_subset_1(D,A)
                       => r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) ) ) ) )).

fof(t20_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_ordinal1(A)
       => ( ! [B] :
              ( ( v1_zf_lang(B)
                & m2_finseq_1(B,k5_numbers) )
             => ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
               => r2_zf_model(A,k11_zf_model(B)) ) )
        <=> ! [B] :
              ( ( v1_relat_1(B)
                & v1_funct_1(B) )
             => ( r2_zfmodel1(B,A)
               => ! [C] :
                    ( r2_hidden(C,A)
                   => r2_hidden(k9_relat_1(B,C),A) ) ) ) ) ) ) )).

fof(t21_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_zf_model(A)
       => ( v1_ordinal1(A)
          & ! [B] :
              ( m1_subset_1(B,A)
             => ! [C] :
                  ( m1_subset_1(C,A)
                 => ( ! [D] :
                        ( m1_subset_1(D,A)
                       => ( r2_hidden(D,B)
                        <=> r2_hidden(D,C) ) )
                   => B = C ) ) )
          & ! [B] :
              ( m1_subset_1(B,A)
             => ! [C] :
                  ( m1_subset_1(C,A)
                 => r2_hidden(k2_tarski(B,C),A) ) )
          & ! [B] :
              ( m1_subset_1(B,A)
             => r2_hidden(k3_tarski(B),A) )
          & ? [B] :
              ( m1_subset_1(B,A)
              & B != k1_xboole_0
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => ~ ( r2_hidden(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,A)
                         => ~ ( r2_xboole_0(C,D)
                              & r2_hidden(D,B) ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,A)
             => r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) )
          & ! [B] :
              ( ( v1_zf_lang(B)
                & m2_finseq_1(B,k5_numbers) )
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k1_zf_lang,A)
                    & m2_relset_1(C,k1_zf_lang,A) )
                 => ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
                      & r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
                   => ! [D] :
                        ( m1_subset_1(D,A)
                       => r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) ) ) ) )).

fof(t22_zfmodel1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ( v1_ordinal1(A)
          & ! [B] :
              ( m1_subset_1(B,A)
             => ! [C] :
                  ( m1_subset_1(C,A)
                 => r2_hidden(k2_tarski(B,C),A) ) )
          & ! [B] :
              ( m1_subset_1(B,A)
             => r2_hidden(k3_tarski(B),A) )
          & ! [B] :
              ( m1_subset_1(B,A)
             => r2_hidden(k3_xboole_0(A,k1_zfmisc_1(B)),A) )
          & ! [B] :
              ( ( v1_zf_lang(B)
                & m2_finseq_1(B,k5_numbers) )
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k1_zf_lang,A)
                    & m2_relset_1(C,k1_zf_lang,A) )
                 => ( ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
                      & r1_zf_model(A,C,k8_zf_lang(k2_zf_lang(np__3),k13_zf_lang(k2_zf_lang(np__0),k8_zf_lang(k2_zf_lang(np__4),k12_zf_lang(B,k4_zf_lang(k2_zf_lang(np__4),k2_zf_lang(np__0))))))) )
                   => ! [D] :
                        ( m1_subset_1(D,A)
                       => r2_hidden(k2_funct_2(A,A,k1_zfmodel1(B,A,C),D),A) ) ) ) ) )
       => ( ! [B] :
              ( m1_subset_1(B,A)
             => ~ ( B != k1_xboole_0
                  & ! [C] :
                      ( m1_subset_1(C,A)
                     => ~ ( r2_hidden(C,B)
                          & ! [D] :
                              ( m1_subset_1(D,A)
                             => ~ ( r2_xboole_0(C,D)
                                  & r2_hidden(D,B) ) ) ) ) ) )
          | v1_zf_model(A) ) ) ) )).

fof(dt_k1_zfmodel1,axiom,(
    ! [A,B,C] :
      ( ( v1_zf_lang(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k1_zf_lang,B)
        & m1_relset_1(C,k1_zf_lang,B) )
     => ( v1_funct_1(k1_zfmodel1(A,B,C))
        & v1_funct_2(k1_zfmodel1(A,B,C),B,B)
        & m2_relset_1(k1_zfmodel1(A,B,C),B,B) ) ) )).

fof(dt_k2_zfmodel1,axiom,(
    ! [A,B] :
      ( ( v1_zf_lang(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B) )
     => ( v1_funct_1(k2_zfmodel1(A,B))
        & v1_funct_2(k2_zfmodel1(A,B),B,B)
        & m2_relset_1(k2_zfmodel1(A,B),B,B) ) ) )).
%------------------------------------------------------------------------------