SET007 Axioms: SET007+196.ax


%------------------------------------------------------------------------------
% File     : SET007+196 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Abian's Fixed Point Theorem
% 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    : abian [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (   0 unit)
%            Number of atoms       :  277 (  14 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  281 (  47 ~  ;   0  |; 169  &)
%                                         (  10 <=>;  55 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   31 (   0 propositional; 1-3 arity)
%            Number of functors    :   23 (   5 constant; 0-4 arity)
%            Number of variables   :   98 (   0 singleton;  86 !;  12 ?)
%            Maximal term depth    :    3 (   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_abian,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_int_1(A)
      & v1_abian(A) ) )).

fof(rc2_abian,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_int_1(A)
      & ~ v1_abian(A) ) )).

fof(rc3_abian,axiom,(
    ? [A] :
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v1_int_1(A)
      & v1_abian(A) ) )).

fof(rc4_abian,axiom,(
    ? [A] :
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v1_int_1(A)
      & ~ v1_abian(A) ) )).

fof(fc1_abian,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_xcmplx_0(k3_xcmplx_0(np__2,A))
        & v1_xreal_0(k3_xcmplx_0(np__2,A))
        & v1_int_1(k3_xcmplx_0(np__2,A))
        & v1_abian(k3_xcmplx_0(np__2,A)) ) ) )).

fof(fc2_abian,axiom,(
    ! [A] :
      ( ( v1_int_1(A)
        & v1_abian(A) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,np__1))
        & v1_xreal_0(k2_xcmplx_0(A,np__1))
        & v1_int_1(k2_xcmplx_0(A,np__1))
        & ~ v1_abian(k2_xcmplx_0(A,np__1)) ) ) )).

fof(fc3_abian,axiom,(
    ! [A] :
      ( ( v1_int_1(A)
        & ~ v1_abian(A) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,np__1))
        & v1_xreal_0(k2_xcmplx_0(A,np__1))
        & v1_int_1(k2_xcmplx_0(A,np__1))
        & v1_abian(k2_xcmplx_0(A,np__1)) ) ) )).

fof(fc4_abian,axiom,(
    ! [A] :
      ( ( v1_int_1(A)
        & v1_abian(A) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,np__1))
        & v1_xreal_0(k6_xcmplx_0(A,np__1))
        & v1_int_1(k6_xcmplx_0(A,np__1))
        & ~ v1_abian(k6_xcmplx_0(A,np__1)) ) ) )).

fof(fc5_abian,axiom,(
    ! [A] :
      ( ( v1_int_1(A)
        & ~ v1_abian(A) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,np__1))
        & v1_xreal_0(k6_xcmplx_0(A,np__1))
        & v1_int_1(k6_xcmplx_0(A,np__1))
        & v1_abian(k6_xcmplx_0(A,np__1)) ) ) )).

fof(fc6_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_int_1(k3_xcmplx_0(A,B))
        & v1_abian(k3_xcmplx_0(A,B)) ) ) )).

fof(fc7_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(B,A))
        & v1_xreal_0(k3_xcmplx_0(B,A))
        & v1_int_1(k3_xcmplx_0(B,A))
        & v1_abian(k3_xcmplx_0(B,A)) ) ) )).

fof(fc8_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & ~ v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_int_1(k3_xcmplx_0(A,B))
        & ~ v1_abian(k3_xcmplx_0(A,B)) ) ) )).

fof(fc9_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B)
        & v1_abian(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_int_1(k2_xcmplx_0(A,B))
        & v1_abian(k2_xcmplx_0(A,B)) ) ) )).

fof(fc10_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_int_1(k2_xcmplx_0(A,B))
        & ~ v1_abian(k2_xcmplx_0(A,B)) ) ) )).

fof(fc11_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(B,A))
        & v1_xreal_0(k2_xcmplx_0(B,A))
        & v1_int_1(k2_xcmplx_0(B,A))
        & ~ v1_abian(k2_xcmplx_0(B,A)) ) ) )).

fof(fc12_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & ~ v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_int_1(k2_xcmplx_0(A,B))
        & v1_abian(k2_xcmplx_0(A,B)) ) ) )).

fof(fc13_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_int_1(k6_xcmplx_0(A,B))
        & ~ v1_abian(k6_xcmplx_0(A,B)) ) ) )).

fof(fc14_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(B,A))
        & v1_xreal_0(k6_xcmplx_0(B,A))
        & v1_int_1(k6_xcmplx_0(B,A))
        & ~ v1_abian(k6_xcmplx_0(B,A)) ) ) )).

fof(fc15_abian,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & ~ v1_abian(A)
        & v1_int_1(B)
        & ~ v1_abian(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_int_1(k6_xcmplx_0(A,B))
        & v1_abian(k6_xcmplx_0(A,B)) ) ) )).

fof(rc5_abian,axiom,(
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      & ~ v1_xboole_0(B)
      & v1_finset_1(B)
      & v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(cc1_abian,axiom,(
    ! [A] :
      ( v2_setfam_1(A)
     => v1_realset1(A) ) )).

fof(rc6_abian,axiom,(
    ! [A,B] :
      ( ~ v2_setfam_1(B)
     => ? [C] :
          ( m1_relset_1(C,A,B)
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,A,B) ) ) )).

fof(fc16_abian,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_setfam_1(B)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => ~ v1_xboole_0(k1_funct_1(C,D)) ) )).

fof(fc17_abian,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v1_xboole_0(k1_zfmisc_1(A))
        & ~ v2_setfam_1(k1_zfmisc_1(A)) ) ) )).

fof(d1_abian,axiom,(
    ! [A] :
      ( v1_abian(A)
    <=> ? [B] :
          ( v1_int_1(B)
          & A = k3_xcmplx_0(np__2,B) ) ) )).

fof(d2_abian,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_abian(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & A = k2_nat_1(np__2,B) ) ) ) )).

fof(t1_abian,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( ~ ( ~ v1_abian(A)
            & ! [B] :
                ( v1_int_1(B)
               => A != k2_xcmplx_0(k3_xcmplx_0(np__2,B),np__1) ) )
        & ~ ( ? [B] :
                ( v1_int_1(B)
                & A = k2_xcmplx_0(k3_xcmplx_0(np__2,B),np__1) )
            & v1_abian(A) ) ) ) )).

fof(t2_abian,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ( r2_hidden(np__0,A)
       => k10_cqc_sim1(A) = np__0 ) ) )).

fof(t3_abian,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => k8_funct_2(A,A,k1_abian(A,B,np__0),C) = C ) ) ) )).

fof(d3_abian,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_abian(A,B)
      <=> ( r2_hidden(A,k1_relat_1(B))
          & A = k1_funct_1(B,A) ) ) ) )).

fof(d4_abian,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,A)
                & m2_relset_1(C,A,A) )
             => ( r2_abian(A,B,C)
              <=> B = k8_funct_2(A,A,C,B) ) ) ) ) )).

fof(d5_abian,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( r3_abian(A)
      <=> ? [B] : r1_abian(B,A) ) ) )).

fof(d6_abian,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,A)
     => ( v2_abian(B,A)
      <=> k3_tarski(B) = k3_tarski(k3_tarski(A)) ) ) )).

fof(t4_abian,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      <=> k5_setfam_1(A,B) = A ) ) )).

fof(t5_abian,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( ~ v1_xboole_0(C)
            & v2_abian(C,k1_zfmisc_1(k1_zfmisc_1(A)))
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ~ ( ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(A),C)
                 => r1_xboole_0(D,k2_funct_2(A,A,B,D)) )
              & r3_abian(B) ) ) ) )).

fof(d7_abian,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( C = k2_abian(A,B)
          <=> ! [D,E] :
                ( ( r2_hidden(D,A)
                  & r2_hidden(E,A) )
               => ( r2_hidden(k4_tarski(D,E),C)
                <=> ? [F] :
                      ( m2_subset_1(F,k1_numbers,k5_numbers)
                      & ? [G] :
                          ( m2_subset_1(G,k1_numbers,k5_numbers)
                          & k1_funct_1(k1_abian(A,B,F),D) = k1_funct_1(k1_abian(A,B,G),E) ) ) ) ) ) ) ) )).

fof(t6_abian,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( m2_subset_1(C,k1_zfmisc_1(A),k8_eqrel_1(A,k2_abian(A,B)))
             => ! [D] :
                  ( m2_subset_1(D,A,C)
                 => r2_hidden(k8_funct_2(A,A,B,D),C) ) ) ) ) )).

fof(t7_abian,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( m2_subset_1(C,k1_zfmisc_1(A),k8_eqrel_1(A,k2_abian(A,B)))
             => ! [D] :
                  ( m2_subset_1(D,A,C)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => r2_hidden(k8_funct_2(A,A,k1_abian(A,B,E),D),C) ) ) ) ) ) )).

fof(t8_abian,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,A)
            & m2_relset_1(B,A,A) )
         => ~ ( ~ r3_abian(B)
              & ! [C,D,E] :
                  ~ ( k2_xboole_0(k2_xboole_0(C,D),E) = A
                    & r1_xboole_0(k2_funct_2(A,A,B,C),C)
                    & r1_xboole_0(k2_funct_2(A,A,B,D),D)
                    & r1_xboole_0(k2_funct_2(A,A,B,E),E) ) ) ) ) )).

fof(redefinition_r2_abian,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,A)
        & m1_relset_1(C,A,A) )
     => ( r2_abian(A,B,C)
      <=> r1_abian(B,C) ) ) )).

fof(dt_k1_abian,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m1_relset_1(B,A,A)
        & m1_subset_1(C,k5_numbers) )
     => ( v1_funct_1(k1_abian(A,B,C))
        & v1_funct_2(k1_abian(A,B,C),A,A)
        & m2_relset_1(k1_abian(A,B,C),A,A) ) ) )).

fof(redefinition_k1_abian,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m1_relset_1(B,A,A)
        & m1_subset_1(C,k5_numbers) )
     => k1_abian(A,B,C) = k9_funct_7(B,C) ) )).

fof(dt_k2_abian,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m1_relset_1(B,A,A) )
     => ( v3_relat_2(k2_abian(A,B))
        & v8_relat_2(k2_abian(A,B))
        & v1_partfun1(k2_abian(A,B),A,A)
        & m2_relset_1(k2_abian(A,B),A,A) ) ) )).
%------------------------------------------------------------------------------