TPTP Problem File: SEU399+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU399+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_xboole_0__e2_25_1_2__wellord2
% 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-s1_xboole_0__e2_25_1_2__wellord2 [Urb07]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.75 v8.1.0, 0.67 v7.5.0, 0.78 v7.4.0, 0.60 v7.3.0, 0.62 v7.1.0, 0.61 v7.0.0, 0.70 v6.4.0, 0.69 v6.3.0, 0.67 v6.2.0, 0.68 v6.1.0, 0.73 v6.0.0, 0.74 v5.5.0, 0.78 v5.4.0, 0.75 v5.3.0, 0.81 v5.2.0, 0.75 v5.1.0, 0.76 v5.0.0, 0.79 v4.1.0, 0.78 v4.0.1, 0.87 v4.0.0, 0.88 v3.7.0, 0.85 v3.5.0, 0.84 v3.3.0
% Syntax : Number of formulae : 69 ( 8 unt; 0 def)
% Number of atoms : 338 ( 24 equ)
% Maximal formula atoms : 28 ( 4 avg)
% Number of connectives : 324 ( 55 ~; 0 |; 192 &)
% ( 3 <=>; 74 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 0 con; 1-4 aty)
% Number of variables : 154 ( 123 !; 31 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_xboole_0__e6_39_3__yellow19__1,conjecture,
! [A,B,C] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [D] :
? [E] :
! [F] :
( in(F,E)
<=> ( in(F,the_carrier(B))
& ? [G] :
( netstr_induced_subset(G,A,B)
& ? [H] :
( element(H,the_carrier(B))
& D = topstr_closure(A,G)
& F = H
& G = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,H)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,H))) ) ) ) ) ) ).
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(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(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(cartesian_product2(A,B)) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(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(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(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(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(rc2_waybel_7,axiom,
! [A] :
? [B] :
( element(B,powerset(powerset(A)))
& ~ empty(B)
& finite(B) ) ).
fof(rc3_waybel_7,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( element(B,powerset(powerset(the_carrier(A))))
& ~ empty(B)
& finite(B) ) ) ).
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(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_k2_zfmisc_1,axiom,
$true ).
fof(dt_u1_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
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(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_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(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
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(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_rng(A))
& relation(relation_rng(A)) ) ) ).
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(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
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_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
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(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
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(rc4_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( net_str(B,A)
& strict_net_str(B,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(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(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(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
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_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(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_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_pre_topc,axiom,
! [A] :
( top_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( net_str(B,A)
=> rel_str(B) ) ) ).
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_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(s1_tarski__e6_39_3__yellow19__1,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [D] :
( ! [E,F,G] :
( ( E = F
& ? [H] :
( netstr_induced_subset(H,A,B)
& ? [I] :
( element(I,the_carrier(B))
& D = topstr_closure(A,H)
& F = I
& H = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,I)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,I))) ) )
& E = G
& ? [J] :
( netstr_induced_subset(J,A,B)
& ? [K] :
( element(K,the_carrier(B))
& D = topstr_closure(A,J)
& G = K
& J = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,K)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,K))) ) ) )
=> F = G )
=> ? [E] :
! [F] :
( in(F,E)
<=> ? [G] :
( in(G,the_carrier(B))
& G = F
& ? [L] :
( netstr_induced_subset(L,A,B)
& ? [M] :
( element(M,the_carrier(B))
& D = topstr_closure(A,L)
& F = M
& L = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,M)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,M))) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------