SET007 Axioms: SET007+29.ax


%------------------------------------------------------------------------------
% File     : SET007+29 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Non-Negative Real Numbers. Part 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    : arytm_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   51 (  11 unit)
%            Number of atoms       :  248 (  55 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :  233 (  36 ~  ;   3  |;  77  &)
%                                         (  17 <=>; 100 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   10 (   1 propositional; 0-3 arity)
%            Number of functors    :   24 (   8 constant; 0-2 arity)
%            Number of variables   :  113 (   0 singleton;  96 !;  17 ?)
%            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(fc1_arytm_2,axiom,(
    ~ v1_xboole_0(k1_arytm_2) )).

fof(fc2_arytm_2,axiom,(
    ~ v1_xboole_0(k2_arytm_2) )).

fof(t1_arytm_2,axiom,(
    r1_tarski(k6_arytm_3,k2_arytm_2) )).

fof(t2_arytm_2,axiom,(
    r1_tarski(k5_ordinal2,k2_arytm_2) )).

fof(t3_arytm_2,axiom,(
    ! [A] : ~ r2_hidden(k4_tarski(k12_arytm_3,A),k2_arytm_2) )).

fof(d4_arytm_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( ? [C] :
                  ( m1_subset_1(C,k6_arytm_3)
                  & ! [D] :
                      ( m1_subset_1(D,k6_arytm_3)
                     => ( r2_hidden(D,A)
                      <=> ~ r3_arytm_3(C,D) ) ) )
             => ( B = k4_arytm_2(A)
              <=> ? [C] :
                    ( m1_subset_1(C,k6_arytm_3)
                    & B = C
                    & ! [D] :
                        ( m1_subset_1(D,k6_arytm_3)
                       => ( r2_hidden(D,A)
                        <=> ~ r3_arytm_3(C,D) ) ) ) ) )
            & ( ! [C] :
                  ( m1_subset_1(C,k6_arytm_3)
                 => ~ ! [D] :
                        ( m1_subset_1(D,k6_arytm_3)
                       => ( r2_hidden(D,A)
                        <=> ~ r3_arytm_3(C,D) ) ) )
             => ( B = k4_arytm_2(A)
              <=> B = A ) ) ) ) ) )).

fof(d5_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( ( r2_hidden(A,k6_arytm_3)
                & r2_hidden(B,k6_arytm_3) )
             => ( r1_arytm_2(A,B)
              <=> ? [C] :
                    ( m1_subset_1(C,k6_arytm_3)
                    & ? [D] :
                        ( m1_subset_1(D,k6_arytm_3)
                        & A = C
                        & B = D
                        & r3_arytm_3(C,D) ) ) ) )
            & ( r2_hidden(A,k6_arytm_3)
             => ( r2_hidden(B,k6_arytm_3)
                | ( r1_arytm_2(A,B)
                <=> r2_hidden(A,B) ) ) )
            & ( r2_hidden(B,k6_arytm_3)
             => ( r2_hidden(A,k6_arytm_3)
                | ( r1_arytm_2(A,B)
                <=> ~ r2_hidden(B,A) ) ) )
            & ~ ( ~ ( r2_hidden(A,k6_arytm_3)
                    & r2_hidden(B,k6_arytm_3) )
                & ~ ( r2_hidden(A,k6_arytm_3)
                    & ~ r2_hidden(B,k6_arytm_3) )
                & ~ ( ~ r2_hidden(A,k6_arytm_3)
                    & r2_hidden(B,k6_arytm_3) )
                & ~ ( r1_arytm_2(A,B)
                  <=> r1_tarski(A,B) ) ) ) ) ) )).

fof(d8_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( B = k12_arytm_3
             => k7_arytm_2(A,B) = A )
            & ( A = k12_arytm_3
             => k7_arytm_2(A,B) = B )
            & ~ ( B != k12_arytm_3
                & A != k12_arytm_3
                & k7_arytm_2(A,B) != k4_arytm_2(k5_arytm_2(k3_arytm_2(A),k3_arytm_2(B))) ) ) ) ) )).

fof(d9_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => k8_arytm_2(A,B) = k4_arytm_2(k6_arytm_2(k3_arytm_2(A),k3_arytm_2(B))) ) ) )).

fof(t4_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k12_arytm_3
           => k8_arytm_2(A,B) = k12_arytm_3 ) ) ) )).

fof(t5_arytm_2,axiom,(
    $true )).

fof(t6_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( k7_arytm_2(A,B) = k12_arytm_3
           => A = k12_arytm_3 ) ) ) )).

fof(t7_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] :
              ( m1_subset_1(C,k2_arytm_2)
             => k7_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k7_arytm_2(A,B),C) ) ) ) )).

fof(t9_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k2_arytm_2))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k2_arytm_2))
         => ~ ( ? [C] :
                  ( m1_subset_1(C,k2_arytm_2)
                  & r2_hidden(C,A) )
              & ? [C] :
                  ( m1_subset_1(C,k2_arytm_2)
                  & r2_hidden(C,B) )
              & ! [C] :
                  ( m1_subset_1(C,k2_arytm_2)
                 => ! [D] :
                      ( m1_subset_1(D,k2_arytm_2)
                     => ( ( r2_hidden(C,A)
                          & r2_hidden(D,B) )
                       => r1_arytm_2(C,D) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,k2_arytm_2)
                 => ? [D] :
                      ( m1_subset_1(D,k2_arytm_2)
                      & ? [E] :
                          ( m1_subset_1(E,k2_arytm_2)
                          & r2_hidden(D,A)
                          & r2_hidden(E,B)
                          & ~ ( r1_arytm_2(D,C)
                              & r1_arytm_2(C,E) ) ) ) ) ) ) ) )).

fof(t10_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ( r1_arytm_2(A,B)
              & ! [C] :
                  ( m1_subset_1(C,k2_arytm_2)
                 => k7_arytm_2(A,C) != B ) ) ) ) )).

fof(t11_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ! [C] :
                ( m1_subset_1(C,k2_arytm_2)
               => ( k7_arytm_2(A,C) != B
                  & k7_arytm_2(B,C) != A ) ) ) ) )).

fof(t12_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] :
              ( m1_subset_1(C,k2_arytm_2)
             => ( k7_arytm_2(A,B) = k7_arytm_2(A,C)
               => B = C ) ) ) ) )).

fof(t13_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] :
              ( m1_subset_1(C,k2_arytm_2)
             => k8_arytm_2(A,k8_arytm_2(B,C)) = k8_arytm_2(k8_arytm_2(A,B),C) ) ) ) )).

fof(t14_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] :
              ( m1_subset_1(C,k2_arytm_2)
             => k8_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k8_arytm_2(A,B),k8_arytm_2(A,C)) ) ) ) )).

fof(t15_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ~ ( A != k12_arytm_3
          & ! [B] :
              ( m1_subset_1(B,k2_arytm_2)
             => k8_arytm_2(A,B) != k13_arytm_3 ) ) ) )).

fof(t16_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k13_arytm_3
           => k8_arytm_2(A,B) = B ) ) ) )).

fof(t17_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( r2_hidden(A,k5_ordinal2)
              & r2_hidden(B,k5_ordinal2) )
           => r2_hidden(k7_arytm_2(B,A),k5_ordinal2) ) ) ) )).

fof(t18_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k2_arytm_2))
     => ( ( r2_hidden(k12_arytm_3,A)
          & ! [B] :
              ( m1_subset_1(B,k2_arytm_2)
             => ! [C] :
                  ( m1_subset_1(C,k2_arytm_2)
                 => ( ( r2_hidden(B,A)
                      & C = k13_arytm_3 )
                   => r2_hidden(k7_arytm_2(B,C),A) ) ) ) )
       => r1_tarski(k5_ordinal2,A) ) ) )).

fof(t19_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ( r2_hidden(A,k5_ordinal2)
       => ! [B] :
            ( m1_subset_1(B,k2_arytm_2)
           => ( r2_hidden(B,A)
            <=> ( r2_hidden(B,k5_ordinal2)
                & B != A
                & r1_arytm_2(B,A) ) ) ) ) ) )).

fof(t20_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] :
              ( m1_subset_1(C,k2_arytm_2)
             => ( A = k7_arytm_2(B,C)
               => r1_arytm_2(C,A) ) ) ) ) )).

fof(t21_arytm_2,axiom,
    ( r2_hidden(k12_arytm_3,k2_arytm_2)
    & r2_hidden(k13_arytm_3,k2_arytm_2) )).

fof(t22_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ( r2_hidden(A,k6_arytm_3)
              & r2_hidden(B,k6_arytm_3)
              & ! [C] :
                  ( m1_subset_1(C,k6_arytm_3)
                 => ! [D] :
                      ( m1_subset_1(D,k6_arytm_3)
                     => ~ ( A = C
                          & B = D
                          & k8_arytm_2(A,B) = k11_arytm_3(C,D) ) ) ) ) ) ) )).

fof(connectedness_r1_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => ( r1_arytm_2(A,B)
        | r1_arytm_2(B,A) ) ) )).

fof(dt_k1_arytm_2,axiom,(
    m1_subset_1(k1_arytm_2,k1_zfmisc_1(k1_zfmisc_1(k6_arytm_3))) )).

fof(dt_k2_arytm_2,axiom,(
    $true )).

fof(dt_k3_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => m2_subset_1(k3_arytm_2(A),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) )).

fof(dt_k4_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_arytm_2)
     => m1_subset_1(k4_arytm_2(A),k2_arytm_2) ) )).

fof(dt_k5_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_arytm_2)
        & m1_subset_1(B,k1_arytm_2) )
     => m2_subset_1(k5_arytm_2(A,B),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) )).

fof(commutativity_k5_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_arytm_2)
        & m1_subset_1(B,k1_arytm_2) )
     => k5_arytm_2(A,B) = k5_arytm_2(B,A) ) )).

fof(dt_k6_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_arytm_2)
        & m1_subset_1(B,k1_arytm_2) )
     => m2_subset_1(k6_arytm_2(A,B),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) )).

fof(commutativity_k6_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_arytm_2)
        & m1_subset_1(B,k1_arytm_2) )
     => k6_arytm_2(A,B) = k6_arytm_2(B,A) ) )).

fof(dt_k7_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => m1_subset_1(k7_arytm_2(A,B),k2_arytm_2) ) )).

fof(commutativity_k7_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => k7_arytm_2(A,B) = k7_arytm_2(B,A) ) )).

fof(dt_k8_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => m1_subset_1(k8_arytm_2(A,B),k2_arytm_2) ) )).

fof(commutativity_k8_arytm_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => k8_arytm_2(A,B) = k8_arytm_2(B,A) ) )).

fof(d1_arytm_2,axiom,(
    k1_arytm_2 = k4_xboole_0(a_0_0_arytm_2,k1_tarski(k6_arytm_3)) )).

fof(d2_arytm_2,axiom,(
    k2_arytm_2 = k4_xboole_0(k2_xboole_0(k6_arytm_3,k1_arytm_2),a_0_1_arytm_2) )).

fof(d3_arytm_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
         => ( ( r2_hidden(A,k6_arytm_3)
             => ( B = k3_arytm_2(A)
              <=> ? [C] :
                    ( m1_subset_1(C,k6_arytm_3)
                    & A = C
                    & B = a_1_0_arytm_2(C) ) ) )
            & ( ~ r2_hidden(A,k6_arytm_3)
             => ( B = k3_arytm_2(A)
              <=> B = A ) ) ) ) ) )).

fof(d6_arytm_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
     => ! [B] :
          ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
         => k5_arytm_2(A,B) = a_2_0_arytm_2(A,B) ) ) )).

fof(d7_arytm_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
     => ! [B] :
          ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
         => k6_arytm_2(A,B) = a_2_1_arytm_2(A,B) ) ) )).

fof(t8_arytm_2,axiom,(
    v6_ordinal1(a_0_0_arytm_2) )).

fof(fraenkel_a_0_0_arytm_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_arytm_2)
    <=> ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k6_arytm_3))
          & A = B
          & ! [C] :
              ( m1_subset_1(C,k6_arytm_3)
             => ( r2_hidden(C,B)
               => ( ! [D] :
                      ( m1_subset_1(D,k6_arytm_3)
                     => ( r3_arytm_3(D,C)
                       => r2_hidden(D,B) ) )
                  & ? [D] :
                      ( m1_subset_1(D,k6_arytm_3)
                      & r2_hidden(D,B)
                      & ~ r3_arytm_3(D,C) ) ) ) ) ) ) )).

fof(fraenkel_a_0_1_arytm_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_arytm_2)
    <=> ? [B] :
          ( m1_subset_1(B,k6_arytm_3)
          & A = a_1_0_arytm_2(B)
          & B != k12_arytm_3 ) ) )).

fof(fraenkel_a_1_0_arytm_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k6_arytm_3)
     => ( r2_hidden(A,a_1_0_arytm_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k6_arytm_3)
            & A = C
            & ~ r3_arytm_3(B,C) ) ) ) )).

fof(fraenkel_a_2_0_arytm_2,axiom,(
    ! [A,B,C] :
      ( ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
        & m2_subset_1(C,k1_zfmisc_1(k6_arytm_3),k1_arytm_2) )
     => ( r2_hidden(A,a_2_0_arytm_2(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,k6_arytm_3)
            & m1_subset_1(E,k6_arytm_3)
            & A = k10_arytm_3(D,E)
            & r2_hidden(D,B)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_2_1_arytm_2,axiom,(
    ! [A,B,C] :
      ( ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
        & m2_subset_1(C,k1_zfmisc_1(k6_arytm_3),k1_arytm_2) )
     => ( r2_hidden(A,a_2_1_arytm_2(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,k6_arytm_3)
            & m1_subset_1(E,k6_arytm_3)
            & A = k11_arytm_3(D,E)
            & r2_hidden(D,B)
            & r2_hidden(E,C) ) ) ) )).
%------------------------------------------------------------------------------