TPTP Problem File: SEU398+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU398+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t31_yellow19
% Version : [Urb07] axioms : Especial.
% English :
% Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source : [Urb07]
% Names : bushy-t31_yellow19 [Urb07]
% Status : Theorem
% Rating : 1.00 v3.3.0
% Syntax : Number of formulae : 141 ( 22 unt; 0 def)
% Number of atoms : 628 ( 37 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 580 ( 93 ~; 2 |; 310 &)
% ( 20 <=>; 155 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 42 ( 40 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 1 con; 0-4 aty)
% Number of variables : 304 ( 266 !; 38 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(abstractness_v6_waybel_0,axiom,
! [A,B] :
( ( one_sorted_str(A)
& net_str(B,A) )
=> ( strict_net_str(B,A)
=> B = net_str_of(A,the_carrier(B),the_InternalRel(B),the_mapping(A,B)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(cc1_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( empty(B)
=> ( open_subset(B,A)
& closed_subset(B,A) ) ) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc2_tops_1,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( empty(B)
=> boundary_set(B,A) ) ) ) ).
fof(cc3_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( empty(B)
=> nowhere_dense(B,A) ) ) ) ).
fof(cc4_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( nowhere_dense(B,A)
=> boundary_set(B,A) ) ) ) ).
fof(cc5_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( closed_subset(B,A)
& boundary_set(B,A) )
=> ( boundary_set(B,A)
& nowhere_dense(B,A) ) ) ) ) ).
fof(cc6_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( open_subset(B,A)
& nowhere_dense(B,A) )
=> ( empty(B)
& open_subset(B,A)
& closed_subset(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& boundary_set(B,A)
& nowhere_dense(B,A) ) ) ) ) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
fof(commutativity_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,C) = subset_intersection2(A,C,B) ) ).
fof(d12_waybel_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( is_often_in(A,B,C)
<=> ! [D] :
( element(D,the_carrier(B))
=> ? [E] :
( element(E,the_carrier(B))
& related(B,D,E)
& in(apply_netmap(A,B,E),C) ) ) ) ) ) ).
fof(d13_pre_topc,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( C = topstr_closure(A,B)
<=> ! [D] :
( in(D,the_carrier(A))
=> ( in(D,C)
<=> ! [E] :
( element(E,powerset(the_carrier(A)))
=> ~ ( open_subset(E,A)
& in(D,E)
& disjoint(B,E) ) ) ) ) ) ) ) ) ).
fof(d1_connsp_2,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( point_neighbourhood(C,A,B)
<=> in(B,interior(A,C)) ) ) ) ) ).
fof(d1_funct_2,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> ( ( ( B = empty_set
=> A = empty_set )
=> ( quasi_total(C,A,B)
<=> A = relation_dom_as_subset(A,B,C) ) )
& ( B = empty_set
=> ( A = empty_set
| ( quasi_total(C,A,B)
<=> C = empty_set ) ) ) ) ) ).
fof(d2_yellow19,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( netstr_induced_subset(C,A,B)
<=> ? [D] :
( element(D,the_carrier(B))
& C = relation_rng_as_subset(the_carrier(netstr_restr_to_element(A,B,D)),the_carrier(A),the_mapping(A,netstr_restr_to_element(A,B,D))) ) ) ) ) ) ).
fof(d3_xboole_0,axiom,
! [A,B,C] :
( C = set_intersection2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
& in(D,B) ) ) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(D,relation_dom(A))
& C = apply(A,D) ) ) ) ) ).
fof(d7_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
<=> set_intersection2(A,B) = empty_set ) ).
fof(d8_waybel_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(B))
=> apply_netmap(A,B,C) = apply_on_structs(B,A,the_mapping(A,B),C) ) ) ) ).
fof(d9_waybel_9,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(A))
=> ( is_a_cluster_point_of_netstr(A,B,C)
<=> ! [D] :
( point_neighbourhood(D,A,C)
=> is_often_in(A,B,D) ) ) ) ) ) ).
fof(dt_g1_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ( strict_net_str(net_str_of(A,B,C,D),A)
& net_str(net_str_of(A,B,C,D),A) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> element(empty_carrier_subset(A),powerset(the_carrier(A))) ) ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tops_1,axiom,
! [A,B] :
( ( top_str(A)
& element(B,powerset(the_carrier(A))) )
=> element(interior(A,B),powerset(the_carrier(A))) ) ).
fof(dt_k1_waybel_0,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& one_sorted_str(B)
& function(C)
& quasi_total(C,the_carrier(A),the_carrier(B))
& relation_of2(C,the_carrier(A),the_carrier(B))
& element(D,the_carrier(A)) )
=> element(apply_on_structs(A,B,C,D),the_carrier(B)) ) ).
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_waybel_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> element(apply_netmap(A,B,C),the_carrier(A)) ) ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_dom_as_subset(A,B,C),powerset(A)) ) ).
fof(dt_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_rng_as_subset(A,B,C),powerset(B)) ) ).
fof(dt_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> element(subset_intersection2(A,B,C),powerset(A)) ) ).
fof(dt_k5_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( strict_net_str(netstr_restr_to_element(A,B,C),A)
& net_str(netstr_restr_to_element(A,B,C),A) ) ) ).
fof(dt_k6_pre_topc,axiom,
! [A,B] :
( ( top_str(A)
& element(B,powerset(the_carrier(A))) )
=> element(topstr_closure(A,B),powerset(the_carrier(A))) ) ).
fof(dt_k6_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( strict_net_str(subnetstr_of_element(A,B,C),A)
& subnet(subnetstr_of_element(A,B,C),A,B) ) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( top_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l1_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( net_str(B,A)
=> rel_str(B) ) ) ).
fof(dt_m1_connsp_2,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& element(B,the_carrier(A)) )
=> ! [C] :
( point_neighbourhood(C,A,B)
=> element(C,powerset(the_carrier(A))) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m1_yellow19,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( netstr_induced_subset(C,A,B)
=> element(C,powerset(the_carrier(A))) ) ) ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> element(C,powerset(cartesian_product2(A,B))) ) ).
fof(dt_m2_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( subnet(C,A,B)
=> ( ~ empty_carrier(C)
& transitive_relstr(C)
& directed_relstr(C)
& net_str(C,A) ) ) ) ).
fof(dt_u1_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u1_waybel_0,axiom,
! [A,B] :
( ( one_sorted_str(A)
& net_str(B,A) )
=> ( function(the_mapping(A,B))
& quasi_total(the_mapping(A,B),the_carrier(B),the_carrier(A))
& relation_of2_as_subset(the_mapping(A,B),the_carrier(B),the_carrier(A)) ) ) ).
fof(existence_l1_orders_2,axiom,
? [A] : rel_str(A) ).
fof(existence_l1_pre_topc,axiom,
? [A] : top_str(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_l1_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] : net_str(B,A) ) ).
fof(existence_m1_connsp_2,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& element(B,the_carrier(A)) )
=> ? [C] : point_neighbourhood(C,A,B) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : relation_of2(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(existence_m1_yellow19,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A) )
=> ? [C] : netstr_induced_subset(C,A,B) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : relation_of2_as_subset(C,A,B) ).
fof(existence_m2_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ? [C] : subnet(C,A,B) ) ).
fof(fc10_finset_1,axiom,
! [A,B] :
( finite(B)
=> finite(set_intersection2(A,B)) ) ).
fof(fc10_tops_1,axiom,
! [A,B] :
( ( top_str(A)
& boundary_set(B,A)
& element(B,powerset(the_carrier(A))) )
=> ( empty(interior(A,B))
& v1_membered(interior(A,B))
& v2_membered(interior(A,B))
& v3_membered(interior(A,B))
& v4_membered(interior(A,B))
& v5_membered(interior(A,B))
& boundary_set(interior(A,B),A) ) ) ).
fof(fc11_finset_1,axiom,
! [A,B] :
( finite(A)
=> finite(set_intersection2(A,B)) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(cartesian_product2(A,B)) ) ).
fof(fc15_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A) )
=> ( ~ empty(the_mapping(A,B))
& relation(the_mapping(A,B))
& function(the_mapping(A,B))
& quasi_total(the_mapping(A,B),the_carrier(B),the_carrier(A)) ) ) ).
fof(fc1_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> ( empty(empty_carrier_subset(A))
& v1_membered(empty_carrier_subset(A))
& v2_membered(empty_carrier_subset(A))
& v3_membered(empty_carrier_subset(A))
& v4_membered(empty_carrier_subset(A))
& v5_membered(empty_carrier_subset(A)) ) ) ).
fof(fc1_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(set_intersection2(A,B)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ( empty(empty_carrier_subset(A))
& closed_subset(empty_carrier_subset(A),A)
& v1_membered(empty_carrier_subset(A))
& v2_membered(empty_carrier_subset(A))
& v3_membered(empty_carrier_subset(A))
& v4_membered(empty_carrier_subset(A))
& v5_membered(empty_carrier_subset(A)) ) ) ).
fof(fc1_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( empty(empty_carrier_subset(A))
& strongly_connected_rel_subset(empty_carrier_subset(A),A)
& finite(empty_carrier_subset(A))
& v1_membered(empty_carrier_subset(A))
& v2_membered(empty_carrier_subset(A))
& v3_membered(empty_carrier_subset(A))
& v4_membered(empty_carrier_subset(A))
& v5_membered(empty_carrier_subset(A))
& directed_subset(empty_carrier_subset(A),A)
& filtered_subset(empty_carrier_subset(A),A) ) ) ).
fof(fc22_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( ~ empty_carrier(netstr_restr_to_element(A,B,C))
& strict_net_str(netstr_restr_to_element(A,B,C),A) ) ) ).
fof(fc26_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( ~ empty_carrier(netstr_restr_to_element(A,B,C))
& transitive_relstr(netstr_restr_to_element(A,B,C))
& strict_net_str(netstr_restr_to_element(A,B,C),A)
& directed_relstr(netstr_restr_to_element(A,B,C)) ) ) ).
fof(fc2_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> closed_subset(topstr_closure(A,B),A) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc5_tops_1,axiom,
! [A] :
( top_str(A)
=> ( empty(interior(A,empty_carrier_subset(A)))
& v1_membered(interior(A,empty_carrier_subset(A)))
& v2_membered(interior(A,empty_carrier_subset(A)))
& v3_membered(interior(A,empty_carrier_subset(A)))
& v4_membered(interior(A,empty_carrier_subset(A)))
& v5_membered(interior(A,empty_carrier_subset(A))) ) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc6_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> open_subset(interior(A,B),A) ) ).
fof(fc6_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& ~ empty(B)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ( ~ empty_carrier(net_str_of(A,B,C,D))
& strict_net_str(net_str_of(A,B,C,D),A) ) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_dom(A))
& relation(relation_dom(A)) ) ) ).
fof(fc7_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ( empty(empty_carrier_subset(A))
& open_subset(empty_carrier_subset(A),A)
& closed_subset(empty_carrier_subset(A),A)
& v1_membered(empty_carrier_subset(A))
& v2_membered(empty_carrier_subset(A))
& v3_membered(empty_carrier_subset(A))
& v4_membered(empty_carrier_subset(A))
& v5_membered(empty_carrier_subset(A)) ) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_rng(A))
& relation(relation_rng(A)) ) ) ).
fof(fraenkel_a_3_0_waybel_9,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(B)
& one_sorted_str(B)
& ~ empty_carrier(C)
& net_str(C,B)
& element(D,the_carrier(C)) )
=> ( in(A,a_3_0_waybel_9(B,C,D))
<=> ? [E] :
( element(E,the_carrier(C))
& A = E
& related(C,D,E) ) ) ) ).
fof(fraenkel_a_3_3_yellow19,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(B)
& topological_space(B)
& top_str(B)
& ~ empty_carrier(C)
& transitive_relstr(C)
& directed_relstr(C)
& net_str(C,B)
& element(D,the_carrier(C)) )
=> ( in(A,a_3_3_yellow19(B,C,D))
<=> ? [E] :
( element(E,the_carrier(C))
& A = E
& related(C,D,E) ) ) ) ).
fof(free_g1_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ! [E,F,G,H] :
( net_str_of(A,B,C,D) = net_str_of(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(idempotence_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,B) = B ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& open_subset(B,A) ) ) ).
fof(rc1_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& directed_subset(B,A)
& filtered_subset(B,A) ) ) ).
fof(rc1_waybel_9,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ? [C] :
( subnet(C,A,B)
& ~ empty_carrier(C)
& transitive_relstr(C)
& strict_net_str(C,A)
& directed_relstr(C) ) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& open_subset(B,A)
& closed_subset(B,A) ) ) ).
fof(rc2_waybel_7,axiom,
! [A] :
? [B] :
( element(B,powerset(powerset(A)))
& ~ empty(B)
& finite(B) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc3_tops_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B)
& open_subset(B,A)
& closed_subset(B,A) ) ) ).
fof(rc3_waybel_7,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( element(B,powerset(powerset(the_carrier(A))))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_tops_1,axiom,
! [A] :
( top_str(A)
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& empty(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& boundary_set(B,A) ) ) ).
fof(rc4_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( net_str(B,A)
& strict_net_str(B,A) ) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(rc5_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& empty(B)
& open_subset(B,A)
& closed_subset(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& boundary_set(B,A)
& nowhere_dense(B,A) ) ) ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& closed_subset(B,A) ) ) ).
fof(rc7_pre_topc,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B)
& closed_subset(B,A) ) ) ).
fof(redefinition_k1_waybel_0,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& one_sorted_str(B)
& function(C)
& quasi_total(C,the_carrier(A),the_carrier(B))
& relation_of2(C,the_carrier(A),the_carrier(B))
& element(D,the_carrier(A)) )
=> apply_on_structs(A,B,C,D) = apply(C,D) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_dom_as_subset(A,B,C) = relation_dom(C) ) ).
fof(redefinition_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_rng_as_subset(A,B,C) = relation_rng(C) ) ).
fof(redefinition_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,C) = set_intersection2(B,C) ) ).
fof(redefinition_k6_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> subnetstr_of_element(A,B,C) = netstr_restr_to_element(A,B,C) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(symmetry_r1_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
=> disjoint(B,A) ) ).
fof(t12_waybel_9,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(B))
=> the_carrier(netstr_restr_to_element(A,B,C)) = a_3_0_waybel_9(A,B,C) ) ) ) ).
fof(t16_waybel_9,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(B))
=> ! [D] :
( element(D,the_carrier(B))
=> ! [E] :
( element(E,the_carrier(netstr_restr_to_element(A,B,C)))
=> ( D = E
=> apply_netmap(A,B,D) = apply_netmap(A,netstr_restr_to_element(A,B,C),E) ) ) ) ) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t23_yellow19,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( element(C,the_carrier(A))
=> ( in(C,topstr_closure(A,B))
<=> ? [D] :
( ~ empty_carrier(D)
& transitive_relstr(D)
& directed_relstr(D)
& net_str(D,A)
& is_eventually_in(A,D,B)
& is_a_cluster_point_of_netstr(A,D,C) ) ) ) ) ) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ) ).
fof(t31_yellow19,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(A))
=> ( is_a_cluster_point_of_netstr(A,B,C)
<=> ! [D] :
( netstr_induced_subset(D,A,B)
=> in(C,topstr_closure(A,D)) ) ) ) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t44_tops_1,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> subset(interior(A,B),B) ) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
fof(t9_yellow19,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( netstr_induced_subset(C,A,B)
=> is_eventually_in(A,B,C) ) ) ) ).
%------------------------------------------------------------------------------