TPTP Problem File: ALG234+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ALG234+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Algebraic Operation on Subsets of Many Sorted Sets T16
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Mar97] Marasik (1997), Algebraic Operation on Subsets of Many
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t16_closure3 [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 154 ( 33 unt; 0 def)
% Number of atoms : 540 ( 26 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 451 ( 65 ~; 1 |; 222 &)
% ( 15 <=>; 148 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 41 ( 39 usr; 1 prp; 0-4 aty)
% Number of functors : 30 ( 30 usr; 1 con; 0-4 aty)
% Number of variables : 317 ( 274 !; 43 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t16_closure3,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v4_closure2(k6_msualg_2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(k6_msualg_2(A,B),k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) ) ) ) ).
fof(abstractness_v4_msualg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& l3_msualg_1(B,A) )
=> ( v4_msualg_1(B,A)
=> B = g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_closure2,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_fraenkel(A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_finset_1(A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(A) ) ).
fof(cc1_funct_2,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_partfun1(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,A,B) ) ) ) ).
fof(cc1_mssubfam,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> v1_pre_circ(B,A) ) ) ).
fof(cc1_msualg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k2_relat_1(u4_msualg_1(A,B)))
=> ~ v1_xboole_0(C) ) ) ).
fof(cc1_pboole,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v2_relat_1(B)
=> ~ v3_relat_1(B) ) ) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(cc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v1_closure2(C,A,B) ) ) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_finset_1(B) ) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(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(cc2_msualg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k2_relat_1(k6_pboole(u1_struct_0(A),u4_msualg_1(A,B))))
=> ~ v1_xboole_0(C) ) ) ).
fof(cc2_pboole,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> ~ v2_relat_1(B) ) ) ) ).
fof(cc3_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v3_closure2(C,A,B) ) ) ) ) ).
fof(cc3_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> v1_funcop_1(D) ) ) ).
fof(cc4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v5_closure2(C,A,B) ) ) ) ) ).
fof(cc5_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v5_closure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ) ) ).
fof(cc5_funct_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(cc6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v6_closure2(C,A,B) ) ) ) ) ).
fof(cc6_funct_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& ~ v1_xboole_0(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(cc7_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v6_closure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ) ) ).
fof(d12_msualg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( C = k6_msualg_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,k1_funct_2(u1_struct_0(A),k1_zfmisc_1(k3_card_3(u4_msualg_1(A,B)))))
& m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
& ! [E] :
( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,B))
=> ( E = D
=> v3_msualg_2(E,A,B) ) ) ) ) ) ) ) ).
fof(d1_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( C = k1_closure2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m4_pboole(D,A,B) ) ) ) ).
fof(d2_funct_2,axiom,
! [A,B,C] :
( C = k1_funct_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& D = E
& k1_relat_1(E) = A
& r1_tarski(k2_relat_1(E),B) ) ) ) ).
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(d3_pboole,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( m1_pboole(B,A)
<=> k1_relat_1(B) = A ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( r1_tarski(A,B)
<=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ).
fof(d4_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k3_card_3(A) = k3_tarski(k2_relat_1(A)) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( B = k2_relat_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( r2_hidden(D,k1_relat_1(A))
& C = k1_funct_1(A,D) ) ) ) ) ).
fof(d8_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r1_tarski(D,C)
=> r2_hidden(k6_mssubfam(A,B,k5_closure2(A,B,D)),C) ) ) ) ) ) ).
fof(dt_g3_msualg_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& m1_pboole(B,u1_struct_0(A))
& m3_pboole(C,u1_msualg_1(A),k7_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),B)),k7_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),B)) )
=> ( v4_msualg_1(g3_msualg_1(A,B,C),A)
& l3_msualg_1(g3_msualg_1(A,B,C),A) ) ) ).
fof(dt_k13_finseq_1,axiom,
$true ).
fof(dt_k1_closure2,axiom,
$true ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_funct_2,axiom,
$true ).
fof(dt_k1_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_pboole(k1_mboolean(A,B),A) ) ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_card_3,axiom,
$true ).
fof(dt_k3_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(k3_finseq_2(A))
& m1_finseq_2(k3_finseq_2(A),A) ) ).
fof(dt_k3_tarski,axiom,
$true ).
fof(dt_k4_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> m1_pboole(k4_closure2(A,B,C),A) ) ).
fof(dt_k5_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> m4_pboole(k5_closure2(A,B,C),A,k1_mboolean(A,B)) ) ).
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_k5_relat_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> v1_relat_1(k5_relat_1(A,B)) ) ).
fof(dt_k6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_closure2(k6_closure2(A,B),A,B)
& v2_closure2(k6_closure2(A,B),A,B)
& v3_closure2(k6_closure2(A,B),A,B)
& v4_closure2(k6_closure2(A,B),A,B)
& v5_closure2(k6_closure2(A,B),A,B)
& v6_closure2(k6_closure2(A,B),A,B)
& m1_subset_1(k6_closure2(A,B),k1_zfmisc_1(k1_closure2(A,B))) ) ) ).
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(dt_k6_msualg_2,axiom,
$true ).
fof(dt_k6_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_pboole(k6_pboole(A,B),k3_finseq_2(A)) ) ).
fof(dt_k7_pboole,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_pboole(D,B) )
=> m1_pboole(k7_pboole(A,B,C,D),A) ) ).
fof(dt_k8_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k8_setfam_1(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_l1_msualg_1,axiom,
! [A] :
( l1_msualg_1(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_msualg_1,axiom,
$true ).
fof(dt_l3_msualg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> l2_msualg_1(B,A) ) ) ).
fof(dt_m1_finseq_2,axiom,
$true ).
fof(dt_m1_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_m3_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> m1_pboole(D,A) ) ) ).
fof(dt_m4_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,B)
=> m1_pboole(C,A) ) ) ).
fof(dt_u1_msualg_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u2_msualg_1,axiom,
! [A] :
( l1_msualg_1(A)
=> ( v1_funct_1(u2_msualg_1(A))
& v1_funct_2(u2_msualg_1(A),u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)))
& m2_relset_1(u2_msualg_1(A),u1_msualg_1(A),k3_finseq_2(u1_struct_0(A))) ) ) ).
fof(dt_u3_msualg_1,axiom,
! [A] :
( l1_msualg_1(A)
=> ( v1_funct_1(u3_msualg_1(A))
& v1_funct_2(u3_msualg_1(A),u1_msualg_1(A),u1_struct_0(A))
& m2_relset_1(u3_msualg_1(A),u1_msualg_1(A),u1_struct_0(A)) ) ) ).
fof(dt_u4_msualg_1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A) )
=> m1_pboole(u4_msualg_1(A,B),u1_struct_0(A)) ) ).
fof(dt_u5_msualg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& l3_msualg_1(B,A) )
=> m3_pboole(u5_msualg_1(A,B),u1_msualg_1(A),k7_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),u4_msualg_1(A,B))),k7_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),u4_msualg_1(A,B))) ) ).
fof(existence_l1_msualg_1,axiom,
? [A] : l1_msualg_1(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_l2_msualg_1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] : l2_msualg_1(B,A) ) ).
fof(existence_l3_msualg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ? [B] : l3_msualg_1(B,A) ) ).
fof(existence_m1_finseq_2,axiom,
! [A] :
? [B] : m1_finseq_2(B,A) ).
fof(existence_m1_pboole,axiom,
! [A] :
? [B] : m1_pboole(B,A) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(existence_m3_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ? [D] : m3_pboole(D,A,B,C) ) ).
fof(existence_m4_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] : m4_pboole(C,A,B) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_zfmisc_1(A,B)) ) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k5_relat_1(A,B))
& v1_funct_1(k5_relat_1(A,B)) ) ) ).
fof(fc1_funct_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ~ v1_xboole_0(k1_funct_2(A,B)) ) ).
fof(fc1_msualg_1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v5_msualg_1(B,A)
& l2_msualg_1(B,A) )
=> ( v1_relat_1(u4_msualg_1(A,B))
& v2_relat_1(u4_msualg_1(A,B))
& v1_funct_1(u4_msualg_1(A,B)) ) ) ).
fof(fc1_pboole,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ( v1_relat_1(k6_pboole(A,B))
& v2_relat_1(k6_pboole(A,B))
& ~ v3_relat_1(k6_pboole(A,B))
& v1_funct_1(k6_pboole(A,B)) ) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( ~ v1_xboole_0(k1_closure2(A,B))
& v1_fraenkel(k1_closure2(A,B))
& v1_pralg_2(k1_closure2(A,B)) ) ) ).
fof(fc2_funct_2,axiom,
! [A] : ~ v1_xboole_0(k1_funct_2(A,A)) ).
fof(fc2_pboole,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> ~ v1_xboole_0(k1_funct_1(B,C)) ) ).
fof(fc4_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ( v1_relat_1(k4_closure2(A,B,C))
& v3_relat_1(k4_closure2(A,B,C))
& v1_funct_1(k4_closure2(A,B,C))
& v1_pre_circ(k4_closure2(A,B,C),A) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fc5_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ( v1_relat_1(k4_closure2(A,B,C))
& v2_relat_1(k4_closure2(A,B,C))
& v1_funct_1(k4_closure2(A,B,C)) ) ) ).
fof(fc6_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) )
=> v1_setfam_1(k2_relat_1(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(free_g3_msualg_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& m1_pboole(B,u1_struct_0(A))
& m3_pboole(C,u1_msualg_1(A),k7_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),B)),k7_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),B)) )
=> ! [D,E,F] :
( g3_msualg_1(A,B,C) = g3_msualg_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(rc1_closure2,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_finset_1(A)
& v1_fraenkel(A) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc1_funct_2,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B) ) ).
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(rc1_pboole,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& ~ v1_xboole_0(C)
& v1_fraenkel(C)
& v1_pralg_2(C) ) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(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(rc2_pboole,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v3_relat_1(B)
& v1_funct_1(B) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v2_funct_1(C)
& v1_finset_1(C)
& v1_fraenkel(C) ) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
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(rc3_pboole,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& ~ v1_xboole_0(C)
& v1_fraenkel(C)
& v1_pralg_2(C)
& v1_closure2(C,A,B)
& v2_closure2(C,A,B)
& v3_closure2(C,A,B)
& v4_closure2(C,A,B)
& v5_closure2(C,A,B)
& v6_closure2(C,A,B) ) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A)
& v1_funct_1(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(rc4_pboole,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funcop_1(B) ) ).
fof(rc5_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc5_msualg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v4_msualg_1(B,A) ) ) ).
fof(rc5_pboole,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ? [C] :
( m4_pboole(C,A,B)
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C) ) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(rc6_msualg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v4_msualg_1(B,A)
& v5_msualg_1(B,A) ) ) ).
fof(redefinition_k3_finseq_2,axiom,
! [A] : k3_finseq_2(A) = k13_finseq_1(A) ).
fof(redefinition_k5_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> k5_closure2(A,B,C) = k4_closure2(A,B,C) ) ).
fof(redefinition_k6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> k6_closure2(A,B) = k1_closure2(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(redefinition_k7_pboole,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_pboole(D,B) )
=> k7_pboole(A,B,C,D) = k5_relat_1(C,D) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(redefinition_r6_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ( r6_pboole(A,B,C)
<=> B = C ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(reflexivity_r6_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> r6_pboole(A,B,B) ) ).
fof(symmetry_r6_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ( r6_pboole(A,B,C)
=> r6_pboole(A,C,B) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t1_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(B,C) )
=> r1_tarski(A,C) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t3_closure3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B))))
=> ( r1_tarski(C,k6_msualg_2(A,B))
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ( r6_pboole(u1_struct_0(A),D,k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),k5_closure2(u1_struct_0(A),u4_msualg_1(A,B),C)))
=> v3_msualg_2(D,A,B) ) ) ) ) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(t92_zfmisc_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> r1_tarski(A,k3_tarski(B)) ) ).
%------------------------------------------------------------------------------