## SET007 Axioms: SET007+188.ax

```%------------------------------------------------------------------------------
% File     : SET007+188 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Certain Facts about Families of Subsets of Many Sorted Sets
% 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    : mssubfam [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   98 (   1 unit)
%            Number of atoms       :  536 (  24 equality)
%            Maximal formula depth :   20 (   8 average)
%            Number of connectives :  463 (  25 ~  ;   1  |; 166  &)
%                                         (   8 <=>; 263 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   31 (   1 propositional; 0-4 arity)
%            Number of functors    :   37 (   4 constant; 0-4 arity)
%            Number of variables   :  336 (   0 singleton; 330 !;   6 ?)
%            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(cc1_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> v1_pre_circ(B,A) ) ) )).

fof(fc1_mssubfam,axiom,(
! [A] :
( v1_relat_1(k1_pboole(A))
& v3_relat_1(k1_pboole(A))
& v1_funct_1(k1_pboole(A))
& v1_pre_circ(k1_pboole(A),A) ) )).

fof(rc1_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m4_pboole(C,A,B)
& v1_relat_1(C)
& v3_relat_1(C)
& v1_funct_1(C)
& v1_pre_circ(C,A) ) ) )).

fof(cc2_mssubfam,axiom,(
! [A,B] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( m4_pboole(C,A,B)
=> v1_pre_circ(C,A) ) ) )).

fof(fc2_mssubfam,axiom,(
! [A,B,C] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k2_pboole(A,B,C))
& v1_funct_1(k2_pboole(A,B,C))
& v1_pre_circ(k2_pboole(A,B,C),A) ) ) )).

fof(fc3_mssubfam,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k3_pboole(A,B,C))
& v1_funct_1(k3_pboole(A,B,C))
& v1_pre_circ(k3_pboole(A,B,C),A) ) ) )).

fof(fc4_mssubfam,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k3_pboole(A,C,B))
& v1_funct_1(k3_pboole(A,C,B))
& v1_pre_circ(k3_pboole(A,C,B),A) ) ) )).

fof(fc5_mssubfam,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k4_pboole(A,C,B))
& v1_funct_1(k4_pboole(A,C,B))
& v1_pre_circ(k4_pboole(A,C,B),A) ) ) )).

fof(fc6_mssubfam,axiom,(
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k14_pboole(A,C,B))
& v1_funct_1(k14_pboole(A,C,B))
& v1_pre_circ(k14_pboole(A,C,B),A) ) ) )).

fof(fc7_mssubfam,axiom,(
! [A,B,C] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k11_pboole(A,B,C))
& v1_funct_1(k11_pboole(A,B,C))
& v1_pre_circ(k11_pboole(A,B,C),A) ) ) )).

fof(fc8_mssubfam,axiom,(
! [A,B] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ( v1_relat_1(k1_mboolean(A,B))
& v2_relat_1(k1_mboolean(A,B))
& v1_funct_1(k1_mboolean(A,B))
& v1_pre_circ(k1_mboolean(A,B),A) ) ) )).

fof(fc9_mssubfam,axiom,(
! [A,B,C] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k12_pboole(A,B,C))
& v1_funct_1(k12_pboole(A,B,C))
& v1_pre_circ(k12_pboole(A,B,C),A) ) ) )).

fof(fc10_mssubfam,axiom,(
! [A,B,C] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A)
& v1_pre_circ(C,A)
& m1_pboole(C,A) )
=> ( v1_relat_1(k5_pboole(A,B,C))
& v1_funct_1(k5_pboole(A,B,C))
& v1_pre_circ(k5_pboole(A,B,C),A) ) ) )).

fof(rc2_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C) ) ) )).

fof(rc3_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
& v1_relat_1(C)
& v3_relat_1(C)
& v1_funct_1(C)
& v1_pre_circ(C,A) ) ) )).

fof(rc4_mssubfam,axiom,(
! [A,B] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ? [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_pre_circ(C,A) ) ) )).

fof(rc5_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_mssubfam(C,A,B)
& v2_mssubfam(C,A,B)
& v3_mssubfam(C,A,B)
& v4_mssubfam(C,A,B)
& v5_mssubfam(C,A,B)
& v6_mssubfam(C,A,B) ) ) )).

fof(cc3_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v2_mssubfam(C,A,B)
=> v1_mssubfam(C,A,B) ) ) ) )).

fof(cc4_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v4_mssubfam(C,A,B)
=> v3_mssubfam(C,A,B) ) ) ) )).

fof(cc5_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v4_mssubfam(C,A,B)
=> v5_mssubfam(C,A,B) ) ) ) )).

fof(cc6_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v5_mssubfam(C,A,B)
=> v2_relat_1(C) ) ) ) )).

fof(cc7_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v2_mssubfam(C,A,B)
=> v6_mssubfam(C,A,B) ) ) ) )).

fof(cc8_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v6_mssubfam(C,A,B)
=> v2_relat_1(C) ) ) ) )).

fof(t1_mssubfam,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B != k1_xboole_0
=> r1_tarski(k8_setfam_1(A,B),k5_setfam_1(A,B)) ) ) )).

fof(t2_mssubfam,axiom,(
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(B,C)
=> r1_tarski(k8_setfam_1(A,C),B) ) ) )).

fof(t3_mssubfam,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(k1_xboole_0,B)
=> k8_setfam_1(A,B) = k1_xboole_0 ) ) )).

fof(t4_mssubfam,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( r2_hidden(D,B)
=> r1_tarski(C,D) )
=> r1_tarski(C,k8_setfam_1(A,B)) ) ) ) )).

fof(t5_mssubfam,axiom,(
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ! [D] :
( r2_hidden(D,C)
=> r1_tarski(B,D) )
=> ( C = k1_xboole_0
| r1_tarski(B,k8_setfam_1(A,C)) ) ) ) )).

fof(t6_mssubfam,axiom,(
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( r2_hidden(B,D)
& r1_tarski(B,C) )
=> r1_tarski(k8_setfam_1(A,D),C) ) ) )).

fof(t7_mssubfam,axiom,(
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( r2_hidden(B,D)
& r1_xboole_0(B,C) )
=> r1_xboole_0(k8_setfam_1(A,D),C) ) ) )).

fof(t8_mssubfam,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B = k4_subset_1(k1_zfmisc_1(A),C,D)
=> k8_setfam_1(A,B) = k5_subset_1(A,k8_setfam_1(A,C),k8_setfam_1(A,D)) ) ) ) ) )).

fof(t9_mssubfam,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( B = k1_tarski(C)
=> k8_setfam_1(A,B) = C ) ) ) )).

fof(t10_mssubfam,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( B = k2_tarski(C,D)
=> k8_setfam_1(A,B) = k5_subset_1(A,C,D) ) ) ) ) )).

fof(t11_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pboole(A,B,C)
=> m2_pboole(B,A,C) ) ) ) )).

fof(t12_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ( m2_pboole(B,A,C)
=> r1_pboole(A,B,C) ) ) ) )).

fof(t13_mssubfam,axiom,(
! [A,B,C] :
( ( v1_funcop_1(C)
& m1_pboole(C,B) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,B)
& D = k1_funct_1(C,A) )
=> k1_funct_1(k2_mssubfam(B,C),A) = k2_relat_1(D) ) ) ) )).

fof(t14_mssubfam,axiom,(
! [A,B,C] :
( ( v1_funcop_1(C)
& m1_pboole(C,B) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,B)
& D = k1_funct_1(C,A) )
=> k1_funct_1(k1_mssubfam(B,C),A) = k1_relat_1(D) ) ) ) )).

fof(t15_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,A) )
=> ( v1_funcop_1(k13_pboole(B,C))
& m1_pboole(k13_pboole(B,C),A) ) ) ) )).

fof(t16_mssubfam,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m3_pboole(C,A,B,k1_pboole(A))
=> r6_pboole(A,C,k1_pboole(A)) ) ) )).

fof(t17_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( ( v1_funcop_1(D)
& m1_pboole(D,A) )
=> ( ( r1_pzfmisc1(A,B,C)
& m3_pboole(D,A,B,C) )
=> ( r6_pboole(A,k1_mssubfam(A,D),B)
& r2_pboole(A,k2_mssubfam(A,D),C) ) ) ) ) ) )).

fof(t18_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( ( r2_pboole(A,B,C)
& v1_pre_circ(C,A) )
=> v1_pre_circ(B,A) ) ) ) )).

fof(t19_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( ( v2_relat_1(B)
& v1_pre_circ(k11_pboole(A,C,B),A) )
=> v1_pre_circ(C,A) ) ) ) )).

fof(t20_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( ( v2_relat_1(B)
& v1_pre_circ(k11_pboole(A,B,C),A) )
=> v1_pre_circ(C,A) ) ) ) )).

fof(t21_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v1_pre_circ(B,A)
<=> v1_pre_circ(k1_mboolean(A,B),A) ) ) )).

fof(t22_mssubfam,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ( ( v1_pre_circ(B,A)
& ! [C] :
( m1_pboole(C,A)
=> ( r1_pboole(A,C,B)
=> v1_pre_circ(C,A) ) ) )
=> v1_pre_circ(k2_mboolean(A,B),A) ) ) )).

fof(t23_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v1_pre_circ(k2_mboolean(A,B),A)
=> ( v1_pre_circ(B,A)
& ! [C] :
( m1_pboole(C,A)
=> ( r1_pboole(A,C,B)
=> v1_pre_circ(C,A) ) ) ) ) ) )).

fof(t24_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ( v1_pre_circ(k1_mssubfam(A,B),A)
=> v1_pre_circ(k2_mssubfam(A,B),A) ) ) )).

fof(t25_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,A) )
=> ( ( r2_pboole(A,B,k2_mssubfam(A,C))
& ! [D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(D,A)
& E = k1_funct_1(C,D) )
=> v1_finset_1(k10_relat_1(E,k1_funct_1(B,D))) ) ) )
=> v1_pre_circ(B,A) ) ) ) )).

fof(t26_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ~ ( v1_pre_circ(B,A)
& r2_pboole(A,B,k11_pboole(A,C,D))
& ! [E] :
( m1_pboole(E,A)
=> ! [F] :
( m1_pboole(F,A)
=> ~ ( v1_pre_circ(E,A)
& r2_pboole(A,E,C)
& v1_pre_circ(F,A)
& r2_pboole(A,F,D)
& r2_pboole(A,B,k11_pboole(A,E,F)) ) ) ) ) ) ) ) )).

fof(t27_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ~ ( v1_pre_circ(B,A)
& v1_pre_circ(C,A)
& r2_pboole(A,B,k11_pboole(A,D,C))
& ! [E] :
( m1_pboole(E,A)
=> ~ ( v1_pre_circ(E,A)
& r2_pboole(A,E,D)
& r2_pboole(A,B,k11_pboole(A,E,C)) ) ) ) ) ) ) )).

fof(t28_mssubfam,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ~ ( ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ~ ( r1_pboole(A,C,B)
& r1_pboole(A,D,B)
& ~ r2_pboole(A,C,D)
& ~ r2_pboole(A,D,C) ) ) )
& ! [C] :
( m1_pboole(C,A)
=> ~ ( r1_pboole(A,C,B)
& ! [D] :
( m1_pboole(D,A)
=> ( r1_pboole(A,D,B)
=> r2_pboole(A,C,D) ) ) ) ) ) ) )).

fof(t29_mssubfam,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ~ ( ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ~ ( r1_pboole(A,C,B)
& r1_pboole(A,D,B)
& ~ r2_pboole(A,C,D)
& ~ r2_pboole(A,D,C) ) ) )
& ! [C] :
( m1_pboole(C,A)
=> ~ ( r1_pboole(A,C,B)
& ! [D] :
( m1_pboole(D,A)
=> ( r1_pboole(A,D,B)
=> r2_pboole(A,D,C) ) ) ) ) ) ) )).

fof(t30_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> ~ ( v1_pre_circ(C,A)
& r2_pboole(A,C,k2_mssubfam(A,B))
& ! [D] :
( m1_pboole(D,A)
=> ~ ( r2_pboole(A,D,k1_mssubfam(A,B))
& v1_pre_circ(D,A)
& r6_pboole(A,k14_pboole(A,D,B),C) ) ) ) ) ) )).

fof(t31_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v3_relat_1(k1_pboole(A))
& v1_pre_circ(k1_pboole(A),A)
& m4_pboole(k1_pboole(A),A,k1_mboolean(A,B)) ) ) )).

fof(t32_mssubfam,axiom,(
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( m4_pboole(D,B,k1_mboolean(B,C))
=> ( r2_hidden(A,B)
=> m1_subset_1(k1_funct_1(D,A),k1_zfmisc_1(k1_zfmisc_1(k1_funct_1(C,A)))) ) ) ) )).

fof(t33_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,C))
=> ( r1_pboole(A,B,D)
=> m4_pboole(B,A,C) ) ) ) ) )).

fof(t34_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> m4_pboole(k2_pboole(A,C,D),A,k1_mboolean(A,B)) ) ) ) )).

fof(t35_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> m4_pboole(k3_pboole(A,C,D),A,k1_mboolean(A,B)) ) ) ) )).

fof(t36_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,C))
=> m4_pboole(k4_pboole(A,D,B),A,k1_mboolean(A,C)) ) ) ) )).

fof(t37_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> m4_pboole(k5_pboole(A,C,D),A,k1_mboolean(A,B)) ) ) ) )).

fof(t38_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_pboole(A,B,C)
=> m4_pboole(k1_pzfmisc1(A,B),A,k1_mboolean(A,C)) ) ) ) )).

fof(t39_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r2_pboole(A,B,C)
& r2_pboole(A,D,C) )
=> m4_pboole(k2_pzfmisc1(A,B,D),A,k1_mboolean(A,C)) ) ) ) ) )).

fof(t40_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> r2_pboole(A,k2_mboolean(A,C),B) ) ) )).

fof(d1_mssubfam,axiom,(
\$true )).

fof(d2_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k5_mssubfam(A,B,C)
<=> ! [E] :
~ ( r2_hidden(E,A)
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(k1_funct_1(B,E))))
=> ~ ( F = k1_funct_1(C,E)
& k1_funct_1(D,E) = k8_setfam_1(k1_funct_1(B,E),F) ) ) ) ) ) ) ) )).

fof(t41_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( r6_pboole(A,C,k1_pboole(A))
=> r6_pboole(A,k6_mssubfam(A,B,C),B) ) ) ) )).

fof(t42_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> r2_pboole(A,k6_mssubfam(A,B,C),k2_mboolean(A,C)) ) ) )).

fof(t43_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> ( r1_pboole(A,C,D)
=> r2_pboole(A,k6_mssubfam(A,B,D),C) ) ) ) ) )).

fof(t44_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( r1_pboole(A,k1_pboole(A),C)
=> r6_pboole(A,k6_mssubfam(A,B,C),k1_pboole(A)) ) ) ) )).

fof(t45_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( ( v2_relat_1(D)
& m4_pboole(D,A,k1_mboolean(A,C)) )
=> ( ! [E] :
( m1_pboole(E,A)
=> ( r1_pboole(A,E,D)
=> r2_pboole(A,B,E) ) )
=> r2_pboole(A,B,k6_mssubfam(A,C,D)) ) ) ) ) )).

fof(t46_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> ( r2_pboole(A,C,D)
=> r2_pboole(A,k6_mssubfam(A,B,D),k6_mssubfam(A,B,C)) ) ) ) ) )).

fof(t47_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m4_pboole(E,A,k1_mboolean(A,B))
=> ( ( r1_pboole(A,C,E)
& r2_pboole(A,C,D) )
=> r2_pboole(A,k6_mssubfam(A,B,E),D) ) ) ) ) ) )).

fof(t48_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m4_pboole(E,A,k1_mboolean(A,B))
=> ( ( r1_pboole(A,C,E)
& r6_pboole(A,k3_pboole(A,C,D),k1_pboole(A)) )
=> r6_pboole(A,k3_pboole(A,k6_mssubfam(A,B,E),D),k1_pboole(A)) ) ) ) ) ) )).

fof(t49_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> ! [E] :
( m4_pboole(E,A,k1_mboolean(A,B))
=> ( r6_pboole(A,C,k2_pboole(A,D,E))
=> r6_pboole(A,k6_mssubfam(A,B,C),k3_pboole(A,k6_mssubfam(A,B,D),k6_mssubfam(A,B,E))) ) ) ) ) ) )).

fof(t50_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,B)
=> ( r6_pboole(A,C,k1_pzfmisc1(A,D))
=> r6_pboole(A,k6_mssubfam(A,B,C),D) ) ) ) ) )).

fof(t51_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ! [D] :
( m4_pboole(D,A,B)
=> ! [E] :
( m4_pboole(E,A,B)
=> ( r6_pboole(A,C,k2_pzfmisc1(A,D,E))
=> r6_pboole(A,k6_mssubfam(A,B,C),k3_pboole(A,D,E)) ) ) ) ) ) )).

fof(t52_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> ( r1_pboole(A,C,k6_mssubfam(A,B,D))
=> ! [E] :
( m1_pboole(E,A)
=> ( r1_pboole(A,E,D)
=> r1_pboole(A,C,E) ) ) ) ) ) ) )).

fof(t53_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( ( v2_relat_1(D)
& m4_pboole(D,A,k1_mboolean(A,C)) )
=> ( ( r1_pboole(A,B,C)
& ! [E] :
( m1_pboole(E,A)
=> ( r1_pboole(A,E,D)
=> r1_pboole(A,B,E) ) ) )
=> r1_pboole(A,B,k6_mssubfam(A,C,D)) ) ) ) ) )).

fof(d3_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v1_mssubfam(C,A,B)
<=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_pboole(E,A)
=> ( ( r1_pboole(A,D,C)
& r1_pboole(A,E,C) )
=> r1_pboole(A,k2_pboole(A,D,E),C) ) ) ) ) ) ) )).

fof(d4_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v2_mssubfam(C,A,B)
<=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> ( r2_pboole(A,D,C)
=> r1_pboole(A,k2_mboolean(A,D),C) ) ) ) ) ) )).

fof(d5_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v3_mssubfam(C,A,B)
<=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_pboole(E,A)
=> ( ( r1_pboole(A,D,C)
& r1_pboole(A,E,C) )
=> r1_pboole(A,k3_pboole(A,D,E),C) ) ) ) ) ) ) )).

fof(d6_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v4_mssubfam(C,A,B)
<=> ! [D] :
( m4_pboole(D,A,k1_mboolean(A,B))
=> ( r2_pboole(A,D,C)
=> r1_pboole(A,k6_mssubfam(A,B,D),C) ) ) ) ) ) )).

fof(d7_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v5_mssubfam(C,A,B)
<=> r1_pboole(A,B,C) ) ) ) )).

fof(d8_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,k1_mboolean(A,B))
=> ( v6_mssubfam(C,A,B)
<=> r1_pboole(A,k1_pboole(A),C) ) ) ) )).

fof(s1_mssubfam,axiom,
( ! [A] :
( r2_hidden(A,f1_s1_mssubfam)
=> ! [B] :
~ ( r2_hidden(B,k1_funct_1(f2_s1_mssubfam,A))
& ! [C] :
~ ( r2_hidden(C,k1_funct_1(f3_s1_mssubfam,A))
& p1_s1_mssubfam(C,B,A) ) ) )
=> ? [A] :
( m3_pboole(A,f1_s1_mssubfam,f2_s1_mssubfam,f3_s1_mssubfam)
& ! [B] :
~ ( r2_hidden(B,f1_s1_mssubfam)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_funct_1(f2_s1_mssubfam,B),k1_funct_1(f3_s1_mssubfam,B))
& m2_relset_1(C,k1_funct_1(f2_s1_mssubfam,B),k1_funct_1(f3_s1_mssubfam,B)) )
=> ~ ( C = k1_funct_1(A,B)
& ! [D] :
( r2_hidden(D,k1_funct_1(f2_s1_mssubfam,B))
=> p1_s1_mssubfam(k1_funct_1(C,D),D,B) ) ) ) ) ) )).

fof(dt_k1_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> m1_pboole(k1_mssubfam(A,B),A) ) )).

fof(redefinition_k1_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> k1_mssubfam(A,B) = k2_funct_6(B) ) )).

fof(dt_k2_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> m1_pboole(k2_mssubfam(A,B),A) ) )).

fof(redefinition_k2_mssubfam,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> k2_mssubfam(A,B) = k3_funct_6(B) ) )).

fof(dt_k3_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> m4_pboole(k3_mssubfam(A,B),A,k1_mboolean(A,B)) ) )).

fof(redefinition_k3_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> k3_mssubfam(A,B) = k1_mboolean(A,B) ) )).

fof(dt_k4_mssubfam,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A)
& m4_pboole(C,A,k1_mboolean(A,B))
& m1_subset_1(D,A) )
=> m1_subset_1(k4_mssubfam(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(k1_funct_1(B,D)))) ) )).

fof(redefinition_k4_mssubfam,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A)
& m4_pboole(C,A,k1_mboolean(A,B))
& m1_subset_1(D,A) )
=> k4_mssubfam(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k5_mssubfam,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> m1_pboole(k5_mssubfam(A,B,C),A) ) )).

fof(dt_k6_mssubfam,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> m4_pboole(k6_mssubfam(A,B,C),A,B) ) )).

fof(redefinition_k6_mssubfam,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> k6_mssubfam(A,B,C) = k5_mssubfam(A,B,C) ) )).

fof(dt_k7_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v1_mssubfam(k7_mssubfam(A,B),A,B)
& v2_mssubfam(k7_mssubfam(A,B),A,B)
& v3_mssubfam(k7_mssubfam(A,B),A,B)
& v4_mssubfam(k7_mssubfam(A,B),A,B)
& v5_mssubfam(k7_mssubfam(A,B),A,B)
& v6_mssubfam(k7_mssubfam(A,B),A,B)
& m4_pboole(k7_mssubfam(A,B),A,k1_mboolean(A,B)) ) ) )).

fof(redefinition_k7_mssubfam,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> k7_mssubfam(A,B) = k1_mboolean(A,B) ) )).
%------------------------------------------------------------------------------
```