SET007 Axioms: SET007+864.ax


%------------------------------------------------------------------------------
% File     : SET007+864 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Equivalences of Inconsistency and Henkin 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    : henmodel [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   32 (   6 unit)
%            Number of atoms       :  174 (  12 equality)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :  169 (  27 ~  ;   4  |;  50  &)
%                                         (  12 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   0 propositional; 1-4 arity)
%            Number of functors    :   33 (  10 constant; 0-4 arity)
%            Number of variables   :   74 (   0 singleton;  70 !;   4 ?)
%            Maximal term depth    :    4 (   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(rc1_henmodel,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
      & v1_henmodel(A) ) )).

fof(t1_henmodel,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k5_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( ( v2_funct_1(C)
                  & k5_relset_1(A,B,C) = B
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(E,k4_relset_1(A,B,C))
                              & r2_hidden(D,k4_relset_1(A,B,C)) )
                           => ( r1_xreal_0(E,D)
                              | r2_hidden(k1_funct_1(C,D),k1_funct_1(C,E)) ) ) ) ) )
               => ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k1_ordinal1(D) != A )
                  | k1_funct_1(C,k3_tarski(A)) = k3_tarski(k5_relset_1(A,B,C)) ) ) ) ) ) )).

fof(t2_henmodel,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ( r2_hidden(k3_tarski(A),A)
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ~ r2_hidden(B,k3_tarski(A))
              & B != k3_tarski(A) ) ) ) )).

fof(d1_henmodel,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( ( ( ~ v1_xboole_0(A)
            & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
         => ( B = k1_henmodel(A)
          <=> ( r2_hidden(B,A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,A)
                   => r1_xreal_0(B,C) ) ) ) ) )
        & ( ~ ( ~ v1_xboole_0(A)
              & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
         => ( B = k1_henmodel(A)
          <=> B = np__0 ) ) ) ) )).

fof(t3_henmodel,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,A)
            & m2_relset_1(B,k5_numbers,A) )
         => ! [C] :
              ( v1_finset_1(C)
             => ~ ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(E,k4_relset_1(k5_numbers,A,B))
                              & r2_hidden(D,k4_relset_1(k5_numbers,A,B)) )
                           => ( r1_xreal_0(E,D)
                              | r1_tarski(k8_funct_2(k5_numbers,A,B,D),k8_funct_2(k5_numbers,A,B,E)) ) ) ) )
                  & r1_tarski(C,k3_tarski(k5_relset_1(k5_numbers,A,B)))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ r1_tarski(C,k8_funct_2(k5_numbers,A,B,D)) ) ) ) ) ) )).

fof(d2_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ! [B] :
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ( r1_henmodel(A,B)
          <=> ? [C] :
                ( m2_finseq_1(C,k7_cqc_lang)
                & r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,C),A)
                & r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) )).

fof(d3_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ( v1_henmodel(A)
      <=> ! [B] :
            ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
           => ~ ( r1_henmodel(A,B)
                & r1_henmodel(A,k10_cqc_lang(B)) ) ) ) ) )).

fof(d4_henmodel,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k7_cqc_lang)
     => ( v2_henmodel(A)
      <=> ! [B] :
            ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
           => ~ ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,B)))
                & r4_calcul_1(k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(B)))) ) ) ) ) )).

fof(t4_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ! [B] :
          ( m2_finseq_1(B,k7_cqc_lang)
         => ( ( v1_henmodel(A)
              & r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,B),A) )
           => v2_henmodel(B) ) ) ) )).

fof(t5_henmodel,axiom,(
    ! [A] :
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => ! [B] :
          ( m2_finseq_1(B,k7_cqc_lang)
         => ! [C] :
              ( m2_finseq_1(C,k7_cqc_lang)
             => ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A)))
               => r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,B,C),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) )).

fof(t6_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ( ~ v1_henmodel(A)
      <=> ! [B] :
            ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
           => r1_henmodel(A,B) ) ) ) )).

fof(t7_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ~ ( ~ v1_henmodel(A)
          & ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
             => ~ ( r1_tarski(B,A)
                  & v1_finset_1(B)
                  & ~ v1_henmodel(B) ) ) ) ) )).

fof(t8_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ! [B] :
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ! [C] :
              ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
             => ~ ( r1_henmodel(k4_subset_1(k7_cqc_lang,A,k6_domain_1(k7_cqc_lang,B)),C)
                  & ! [D] :
                      ( m2_finseq_1(D,k7_cqc_lang)
                     => ~ ( r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,D),A)
                          & r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,B)),k12_finseq_1(k7_cqc_lang,C))) ) ) ) ) ) ) )).

fof(t9_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ! [B] :
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ( r1_henmodel(A,B)
          <=> ~ v1_henmodel(k4_subset_1(k7_cqc_lang,A,k6_domain_1(k7_cqc_lang,k10_cqc_lang(B)))) ) ) ) )).

fof(t10_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ! [B] :
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ( r1_henmodel(A,k10_cqc_lang(B))
          <=> ~ v1_henmodel(k4_subset_1(k7_cqc_lang,A,k6_domain_1(k7_cqc_lang,B))) ) ) ) )).

fof(t11_henmodel,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_zfmisc_1(k7_cqc_lang))
        & m2_relset_1(A,k5_numbers,k1_zfmisc_1(k7_cqc_lang)) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ( r2_hidden(C,k4_relset_1(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A))
                    & r2_hidden(B,k4_relset_1(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A)) )
                 => ( r1_xreal_0(C,B)
                    | ( v1_henmodel(k8_funct_2(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A,B))
                      & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A,B),k8_funct_2(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A,C)) ) ) ) ) )
       => v1_henmodel(k5_setfam_1(k7_cqc_lang,k5_relset_1(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A))) ) ) )).

fof(t12_henmodel,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( ~ v1_henmodel(A)
           => ! [C] :
                ( m1_valuat_1(C,B)
               => ! [D] :
                    ( m2_fraenkel(D,k2_qc_lang1,B,k2_valuat_1(B))
                   => ~ r6_calcul_1(A,B,C,D) ) ) ) ) ) )).

fof(t13_henmodel,axiom,(
    v1_henmodel(k6_domain_1(k7_cqc_lang,k9_cqc_lang)) )).

fof(d5_henmodel,axiom,(
    k2_henmodel = k2_qc_lang1 )).

fof(d6_henmodel,axiom,(
    ! [A] :
      ( ( v1_henmodel(A)
        & m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
     => ! [B] :
          ( m1_valuat_1(B,k2_henmodel)
         => ( m1_henmodel(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k5_qc_lang1)
               => ! [D] :
                    ( m1_subset_1(D,k3_margrel1(k2_henmodel))
                   => ( r1_margrel1(k2_henmodel,k8_funct_2(k5_qc_lang1,k3_margrel1(k2_henmodel),B,C),D)
                     => ! [E] :
                          ( r2_hidden(E,D)
                        <=> ? [F] :
                              ( v1_cqc_lang(F,k6_qc_lang1(C))
                              & m1_qc_lang1(F,k6_qc_lang1(C))
                              & E = F
                              & r1_henmodel(A,k3_henmodel(C,F)) ) ) ) ) ) ) ) ) )).

fof(d7_henmodel,axiom,(
    k4_henmodel = k13_cqc_sim1(k2_qc_lang1) )).

fof(t14_henmodel,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_cqc_lang(B,A)
            & m1_qc_lang1(B,A) )
         => k8_valuat_1(k2_henmodel,A,B,k4_henmodel) = B ) ) )).

fof(t15_henmodel,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k7_cqc_lang)
     => r4_calcul_1(k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,k9_cqc_lang))) ) )).

fof(t16_henmodel,axiom,(
    ! [A] :
      ( ( v1_henmodel(A)
        & m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
     => ! [B] :
          ( m1_henmodel(B,A)
         => ( r1_valuat_1(k2_henmodel,k9_cqc_lang,B,k4_henmodel)
          <=> r1_henmodel(A,k9_cqc_lang) ) ) ) )).

fof(t17_henmodel,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
         => ! [C] :
              ( ( v1_cqc_lang(C,A)
                & m1_qc_lang1(C,A) )
             => ! [D] :
                  ( ( v1_henmodel(D)
                    & m1_subset_1(D,k1_zfmisc_1(k7_cqc_lang)) )
                 => ! [E] :
                      ( m1_henmodel(E,D)
                     => ( r1_valuat_1(k2_henmodel,k8_cqc_lang(A,B,C),E,k4_henmodel)
                      <=> r1_henmodel(D,k8_cqc_lang(A,B,C)) ) ) ) ) ) ) )).

fof(dt_m1_henmodel,axiom,(
    ! [A] :
      ( ( v1_henmodel(A)
        & m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
     => ! [B] :
          ( m1_henmodel(B,A)
         => m1_valuat_1(B,k2_henmodel) ) ) )).

fof(existence_m1_henmodel,axiom,(
    ! [A] :
      ( ( v1_henmodel(A)
        & m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
     => ? [B] : m1_henmodel(B,A) ) )).

fof(dt_k1_henmodel,axiom,(
    ! [A] : m2_subset_1(k1_henmodel(A),k1_numbers,k5_numbers) )).

fof(dt_k2_henmodel,axiom,(
    ~ v1_xboole_0(k2_henmodel) )).

fof(dt_k3_henmodel,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_qc_lang1)
        & v1_cqc_lang(B,k6_qc_lang1(A))
        & m1_qc_lang1(B,k6_qc_lang1(A)) )
     => m2_subset_1(k3_henmodel(A,B),k8_qc_lang1,k7_cqc_lang) ) )).

fof(redefinition_k3_henmodel,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_qc_lang1)
        & v1_cqc_lang(B,k6_qc_lang1(A))
        & m1_qc_lang1(B,k6_qc_lang1(A)) )
     => k3_henmodel(A,B) = k9_qc_lang1(A,B) ) )).

fof(dt_k4_henmodel,axiom,(
    m2_fraenkel(k4_henmodel,k2_qc_lang1,k2_henmodel,k2_valuat_1(k2_henmodel)) )).
%------------------------------------------------------------------------------