SET007 Axioms: SET007+182.ax


%------------------------------------------------------------------------------
% File     : SET007+182 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Preliminaries to Circuits, I
% 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    : pre_circ [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   50 (   7 unit)
%            Number of atoms       :  331 (  45 equality)
%            Maximal formula depth :   23 (   7 average)
%            Number of connectives :  313 (  32 ~  ;   2  |; 167  &)
%                                         (   6 <=>; 106 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   37 (   1 propositional; 0-4 arity)
%            Number of functors    :   61 (  14 constant; 0-4 arity)
%            Number of variables   :  131 (   0 singleton; 121 !;  10 ?)
%            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_pre_circ,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A)
      & v1_membered(A)
      & v2_membered(A)
      & v3_membered(A)
      & v4_membered(A)
      & v5_membered(A) ) )).

fof(fc1_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v5_membered(A) )
     => ( v4_ordinal2(k2_seq_4(A))
        & v1_xcmplx_0(k2_seq_4(A))
        & v1_xreal_0(k2_seq_4(A)) ) ) )).

fof(rc2_pre_circ,axiom,(
    ! [A] :
    ? [B] :
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v2_relat_1(B)
      & v1_funct_1(B)
      & v1_pre_circ(B,A) ) )).

fof(fc2_pre_circ,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v2_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) )).

fof(fc3_pre_circ,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k5_relat_1(B,A))
        & v2_relat_1(k5_relat_1(B,A))
        & v1_funct_1(k5_relat_1(B,A)) ) ) )).

fof(cc1_pre_circ,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k5_trees_3(A))
         => v1_finset_1(B) ) ) )).

fof(fc4_pre_circ,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A)
        & m1_subset_1(B,k1_relat_1(A)) )
     => ( v1_relat_1(k5_trees_2(A,B))
        & v1_funct_1(k5_trees_2(A,B))
        & v1_finset_1(k5_trees_2(A,B))
        & v3_trees_2(k5_trees_2(A,B)) ) ) )).

fof(fc5_pre_circ,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A)
        & m1_subset_1(B,k1_relat_1(A))
        & v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finset_1(C)
        & v3_trees_2(C) )
     => ( v1_relat_1(k8_trees_2(A,B,C))
        & v1_funct_1(k8_trees_2(A,B,C))
        & v1_finset_1(k8_trees_2(A,B,C))
        & v3_trees_2(k8_trees_2(A,B,C)) ) ) )).

fof(fc6_pre_circ,axiom,(
    ! [A] :
      ( v1_relat_1(k1_trees_4(A))
      & v1_funct_1(k1_trees_4(A))
      & v1_finset_1(k1_trees_4(A))
      & v3_trees_2(k1_trees_4(A)) ) )).

fof(t1_pre_circ,axiom,(
    $true )).

fof(t2_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( ( k1_relat_1(A) = k1_tarski(B)
            & k2_relat_1(A) = k1_tarski(C) )
         => A = k1_tarski(k4_tarski(B,C)) ) ) )).

fof(t3_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( r1_tarski(A,B)
               => r1_tarski(k1_funct_4(A,C),k1_funct_4(B,C)) ) ) ) ) )).

fof(t4_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( r1_tarski(A,B)
                  & r1_xboole_0(k1_relat_1(A),k1_relat_1(C)) )
               => r1_tarski(A,k1_funct_4(B,C)) ) ) ) ) )).

fof(d1_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v2_membered(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( B = k1_pre_circ(A)
          <=> ( r2_hidden(B,A)
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ( r2_hidden(C,A)
                   => r1_xreal_0(C,B) ) ) ) ) ) ) )).

fof(t5_pre_circ,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => k1_funct_1(k6_pboole(A,B),k6_finseq_1(A)) = k1_tarski(k1_xboole_0) ) )).

fof(d2_pre_circ,axiom,(
    $true )).

fof(d3_pre_circ,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_pre_circ(B,A)
      <=> ! [C] :
            ( r2_hidden(C,A)
           => v1_finset_1(k1_funct_1(B,C)) ) ) ) )).

fof(t6_pre_circ,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,A) )
         => ~ v1_xboole_0(k3_tarski(k2_relat_1(B))) ) ) )).

fof(t7_pre_circ,axiom,(
    ! [A] : k4_funct_5(k2_pre_circ(A,k1_xboole_0)) = k1_xboole_0 )).

fof(t8_pre_circ,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( v2_relat_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( m3_pboole(D,A,k2_pre_circ(A,B),C)
             => k1_relat_1(k10_funct_6(D)) = B ) ) ) )).

fof(t9_pre_circ,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( m1_subset_1(D,k4_card_3(B))
             => ( ( r1_tarski(k1_relat_1(C),k1_relat_1(B))
                  & ! [E] :
                      ( r2_hidden(E,k1_relat_1(C))
                     => r2_hidden(k1_funct_1(C,E),k1_funct_1(B,E)) ) )
               => m1_subset_1(k1_funct_4(D,C),k4_card_3(B)) ) ) ) ) )).

fof(t10_pre_circ,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v2_relat_1(C)
                & m1_pboole(C,A) )
             => ! [D] :
                  ( m3_pboole(D,A,k2_pre_circ(A,B),C)
                 => ! [E] :
                      ( m1_subset_1(E,B)
                     => ? [F] :
                          ( m1_pboole(F,A)
                          & F = k1_funct_1(k10_funct_6(D),E)
                          & r3_pboole(A,F,C) ) ) ) ) ) ) )).

fof(t11_pre_circ,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ( r2_hidden(C,k4_card_3(B))
               => r2_hidden(k5_relat_1(D,C),k4_card_3(k5_relat_1(D,B))) ) ) ) ) )).

fof(t12_pre_circ,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : k4_card_3(k2_finseq_2(A,k1_tarski(B))) = k1_tarski(k2_finseq_2(A,B)) ) )).

fof(t15_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & v1_trees_1(B) )
         => ! [C] :
              ( m1_trees_1(C,A)
             => ! [D] :
                  ( m2_finseq_1(D,k5_numbers)
                 => ~ ( r2_hidden(D,k5_trees_1(A,C,B))
                      & r1_tarski(C,D)
                      & ! [E] :
                          ( m1_trees_1(E,B)
                         => D != k8_finseq_1(k5_numbers,C,E) ) ) ) ) ) ) )).

fof(t16_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v4_trees_3(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A))
           => k4_trees_1(k13_trees_3(A),k12_finseq_1(k5_numbers,B)) = k1_funct_1(A,k1_nat_1(B,np__1)) ) ) ) )).

fof(t17_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v6_trees_3(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A))
           => r2_hidden(k12_finseq_1(k5_numbers,B),k13_trees_3(k2_funct_6(A))) ) ) ) )).

fof(t18_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v4_trees_3(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B)
            & v4_trees_3(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(A))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(A))
                       => ( D = k1_nat_1(C,np__1)
                          | k1_funct_1(A,D) = k1_funct_1(B,D) ) ) ) )
               => ! [D] :
                    ( ( ~ v1_xboole_0(D)
                      & v1_trees_1(D) )
                   => ( k1_funct_1(B,k1_nat_1(C,np__1)) = D
                     => k13_trees_3(B) = k5_trees_1(k13_trees_3(A),k12_finseq_1(k5_numbers,C),D) ) ) ) ) ) ) )).

fof(t19_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finset_1(B)
            & v3_trees_2(B) )
         => ! [C,D] :
              ( m2_subset_1(D,k1_numbers,k5_numbers)
             => ! [E] :
                  ( ( v1_relat_1(E)
                    & v1_funct_1(E)
                    & v1_finseq_1(E)
                    & v6_trees_3(E) )
                 => ~ ( r2_hidden(k12_finseq_1(k5_numbers,D),k1_relat_1(A))
                      & A = k4_trees_4(C,E)
                      & ! [F] :
                          ( ( v1_relat_1(F)
                            & v1_funct_1(F)
                            & v1_finseq_1(F)
                            & v6_trees_3(F) )
                         => ~ ( k8_trees_2(A,k12_finseq_1(k5_numbers,D),B) = k4_trees_4(C,F)
                              & k3_finseq_1(F) = k3_finseq_1(E)
                              & k1_funct_1(F,k1_nat_1(D,np__1)) = B
                              & ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ( r2_hidden(G,k4_finseq_1(E))
                                   => ( G = k1_nat_1(D,np__1)
                                      | k1_funct_1(F,G) = k1_funct_1(E,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t20_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ~ ( B != k1_xboole_0
              & r1_xreal_0(k4_card_1(A),k4_card_1(k4_trees_1(A,B))) ) ) ) )).

fof(t21_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_card_1(A) = k1_card_1(k1_relat_1(A)) ) )).

fof(t22_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & v1_trees_1(B) )
         => ! [C] :
              ( m1_trees_1(C,A)
             => k1_nat_1(k4_card_1(k5_trees_1(A,C,B)),k4_card_1(k4_trees_1(A,C))) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) )).

fof(t23_pre_circ,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finset_1(B)
            & v3_trees_2(B) )
         => ! [C] :
              ( m1_trees_1(C,k1_relat_1(A))
             => k1_nat_1(k4_card_1(k8_trees_2(A,C,B)),k4_card_1(k5_trees_2(A,C))) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) )).

fof(t24_pre_circ,axiom,(
    ! [A] : k4_card_1(k1_trees_4(A)) = np__1 )).

fof(s2_pre_circ,axiom,(
    ? [A] :
      ( m1_pboole(A,f1_s2_pre_circ)
      & ! [B] :
          ( m1_subset_1(B,f1_s2_pre_circ)
         => ( r2_hidden(B,f1_s2_pre_circ)
           => ( ( p1_s2_pre_circ(B)
               => k1_funct_1(A,B) = f2_s2_pre_circ(B) )
              & ( ~ p1_s2_pre_circ(B)
               => k1_funct_1(A,B) = f3_s2_pre_circ(B) ) ) ) ) ) )).

fof(s3_pre_circ,axiom,
    ( ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,f1_s3_pre_circ)
        & m2_relset_1(A,k5_numbers,f1_s3_pre_circ)
        & k8_funct_2(k5_numbers,f1_s3_pre_circ,A,np__0) = f2_s3_pre_circ
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k8_funct_2(k5_numbers,f1_s3_pre_circ,A,k1_nat_1(B,np__1)) = f3_s3_pre_circ(B,k8_funct_2(k5_numbers,f1_s3_pre_circ,A,B)) ) )
    & ! [A] :
        ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,f1_s3_pre_circ)
          & m2_relset_1(A,k5_numbers,f1_s3_pre_circ) )
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,f1_s3_pre_circ)
              & m2_relset_1(B,k5_numbers,f1_s3_pre_circ) )
           => ( ( k8_funct_2(k5_numbers,f1_s3_pre_circ,A,np__0) = f2_s3_pre_circ
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,f1_s3_pre_circ,A,k1_nat_1(C,np__1)) = f3_s3_pre_circ(C,k8_funct_2(k5_numbers,f1_s3_pre_circ,A,C)) )
                & k8_funct_2(k5_numbers,f1_s3_pre_circ,B,np__0) = f2_s3_pre_circ
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,f1_s3_pre_circ,B,k1_nat_1(C,np__1)) = f3_s3_pre_circ(C,k8_funct_2(k5_numbers,f1_s3_pre_circ,B,C)) ) )
             => A = B ) ) ) )).

fof(s4_pre_circ,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s4_pre_circ)
       => ( r2_hidden(A,f2_s4_pre_circ)
         => ( v1_funct_1(f5_s4_pre_circ(A))
            & v1_funct_2(f5_s4_pre_circ(A),k1_funct_1(f3_s4_pre_circ,A),k1_funct_1(f4_s4_pre_circ,A))
            & m2_relset_1(f5_s4_pre_circ(A),k1_funct_1(f3_s4_pre_circ,A),k1_funct_1(f4_s4_pre_circ,A)) ) ) )
   => ? [A] :
        ( m3_pboole(A,f2_s4_pre_circ,f3_s4_pre_circ,f4_s4_pre_circ)
        & ! [B] :
            ( m1_subset_1(B,f1_s4_pre_circ)
           => ( r2_hidden(B,f2_s4_pre_circ)
             => k1_funct_1(A,B) = f5_s4_pre_circ(B) ) ) ) )).

fof(dt_k1_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v2_membered(A) )
     => v1_xreal_0(k1_pre_circ(A)) ) )).

fof(redefinition_k1_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v2_membered(A) )
     => k1_pre_circ(A) = k2_seq_4(A) ) )).

fof(dt_k2_pre_circ,axiom,(
    ! [A,B] : m1_pboole(k2_pre_circ(A,B),A) )).

fof(redefinition_k2_pre_circ,axiom,(
    ! [A,B] : k2_pre_circ(A,B) = k2_funcop_1(A,B) )).

fof(dt_k3_pre_circ,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_pboole(k3_pre_circ(A,B,C),C) ) )).

fof(redefinition_k3_pre_circ,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k3_pre_circ(A,B,C) = k7_relat_1(B,C) ) )).

fof(t13_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => r2_tarski(k4_trees_1(A,B),a_2_0_pre_circ(A,B)) ) ) )).

fof(t14_pre_circ,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & v1_trees_1(B) )
         => ! [C] :
              ( m1_trees_1(C,A)
             => k5_trees_1(A,C,B) = k2_xboole_0(a_2_1_pre_circ(A,C),a_3_0_pre_circ(A,B,C)) ) ) ) )).

fof(s1_pre_circ,axiom,(
    v1_finset_1(a_0_0_pre_circ) )).

fof(fraenkel_a_2_0_pre_circ,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & v1_trees_1(B)
        & m1_trees_1(C,B) )
     => ( r2_hidden(A,a_2_0_pre_circ(B,C))
      <=> ? [D] :
            ( m1_trees_1(D,B)
            & A = D
            & r1_tarski(C,D) ) ) ) )).

fof(fraenkel_a_2_1_pre_circ,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & v1_trees_1(B)
        & m1_trees_1(C,B) )
     => ( r2_hidden(A,a_2_1_pre_circ(B,C))
      <=> ? [D] :
            ( m1_trees_1(D,B)
            & A = D
            & ~ r1_tarski(C,D) ) ) ) )).

fof(fraenkel_a_3_0_pre_circ,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & v1_trees_1(B)
        & ~ v1_xboole_0(C)
        & v1_finset_1(C)
        & v1_trees_1(C)
        & m1_trees_1(D,B) )
     => ( r2_hidden(A,a_3_0_pre_circ(B,C,D))
      <=> ? [E] :
            ( m1_trees_1(E,C)
            & A = k8_finseq_1(k5_numbers,D,E) ) ) ) )).

fof(fraenkel_a_0_0_pre_circ,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_pre_circ)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_pre_circ)
          & A = f2_s1_pre_circ(B)
          & p1_s1_pre_circ(B) ) ) )).
%------------------------------------------------------------------------------