%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : SEU337+2 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Oct 7 10:44:01 AM UTC 2026
% Result : Theorem 13.76s 2.66s
% Output : CNFRefutation 13.76s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 60
% Syntax : Number of formulae : 448 ( 270 unt; 0 typ; 0 def)
% Number of atoms : 2251 ( 734 equ; 0 cnn)
% Maximal formula atoms : 5 ( 5 avg)
% Number of connectives : 4170 ( 686 ~; 469 |; 39 &;2865 @)
% ( 13 <=>; 98 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 124 usr; 7 con; 0-6 aty)
% Number of variables : 709 ( 0 ^; 697 !; 12 ?; 709 :)
% Comments :
%------------------------------------------------------------------------------
thf(tp_antisymmetric,type,
antisymmetric: $i > $o ).
thf(tp_apply,type,
apply: $i > $i > $i ).
thf(tp_apply_binary,type,
apply_binary: $i > $i > $i > $i ).
thf(tp_apply_binary_as_element,type,
apply_binary_as_element: $i > $i > $i > $i > $i > $i > $i ).
thf(tp_are_equipotent,type,
are_equipotent: $i > $i > $o ).
thf(tp_being_limit_ordinal,type,
being_limit_ordinal: $i > $o ).
thf(tp_below,type,
below: $i > $i > $i > $o ).
thf(tp_cartesian_product2,type,
cartesian_product2: $i > $i > $i ).
thf(tp_cast_as_carrier_subset,type,
cast_as_carrier_subset: $i > $i ).
thf(tp_cast_to_subset,type,
cast_to_subset: $i > $i ).
thf(tp_closed_subset,type,
closed_subset: $i > $i > $o ).
thf(tp_closed_subsets,type,
closed_subsets: $i > $i > $o ).
thf(tp_complements_of_subsets,type,
complements_of_subsets: $i > $i > $i ).
thf(tp_connected,type,
connected: $i > $o ).
thf(tp_cup_closed,type,
cup_closed: $i > $o ).
thf(tp_diff_closed,type,
diff_closed: $i > $o ).
thf(tp_disjoint,type,
disjoint: $i > $i > $o ).
thf(tp_element,type,
element: $i > $i > $o ).
thf(tp_empty,type,
empty: $i > $o ).
thf(tp_empty_carrier,type,
empty_carrier: $i > $o ).
thf(tp_empty_carrier_subset,type,
empty_carrier_subset: $i > $i ).
thf(tp_empty_set,type,
empty_set: $i ).
thf(tp_epsilon_connected,type,
epsilon_connected: $i > $o ).
thf(tp_epsilon_transitive,type,
epsilon_transitive: $i > $o ).
thf(tp_equipotent,type,
equipotent: $i > $i > $o ).
thf(tp_fiber,type,
fiber: $i > $i > $i ).
thf(tp_finite,type,
finite: $i > $o ).
thf(tp_function,type,
function: $i > $o ).
thf(tp_function_inverse,type,
function_inverse: $i > $i ).
thf(tp_identity_relation,type,
identity_relation: $i > $i ).
thf(tp_in,type,
in: $i > $i > $o ).
thf(tp_inclusion_relation,type,
inclusion_relation: $i > $i ).
thf(tp_interior,type,
interior: $i > $i > $i ).
thf(tp_is_a_cover_of_carrier,type,
is_a_cover_of_carrier: $i > $i > $o ).
thf(tp_is_antisymmetric_in,type,
is_antisymmetric_in: $i > $i > $o ).
thf(tp_is_connected_in,type,
is_connected_in: $i > $i > $o ).
thf(tp_is_reflexive_in,type,
is_reflexive_in: $i > $i > $o ).
thf(tp_is_transitive_in,type,
is_transitive_in: $i > $i > $o ).
thf(tp_is_well_founded_in,type,
is_well_founded_in: $i > $i > $o ).
thf(tp_join,type,
join: $i > $i > $i > $i ).
thf(tp_join_commut,type,
join_commut: $i > $i > $i > $i ).
thf(tp_join_commutative,type,
join_commutative: $i > $o ).
thf(tp_join_semilatt_str,type,
join_semilatt_str: $i > $o ).
thf(tp_latt_str,type,
latt_str: $i > $o ).
thf(tp_meet,type,
meet: $i > $i > $i > $i ).
thf(tp_meet_absorbing,type,
meet_absorbing: $i > $o ).
thf(tp_meet_commut,type,
meet_commut: $i > $i > $i > $i ).
thf(tp_meet_commutative,type,
meet_commutative: $i > $o ).
thf(tp_meet_of_subsets,type,
meet_of_subsets: $i > $i > $i ).
thf(tp_meet_semilatt_str,type,
meet_semilatt_str: $i > $o ).
thf(tp_natural,type,
natural: $i > $o ).
thf(tp_omega,type,
omega: $i ).
thf(tp_one_sorted_str,type,
one_sorted_str: $i > $o ).
thf(tp_one_to_one,type,
one_to_one: $i > $o ).
thf(tp_open_subset,type,
open_subset: $i > $i > $o ).
thf(tp_open_subsets,type,
open_subsets: $i > $i > $o ).
thf(tp_ordered_pair,type,
ordered_pair: $i > $i > $i ).
thf(tp_ordinal,type,
ordinal: $i > $o ).
thf(tp_ordinal_subset,type,
ordinal_subset: $i > $i > $o ).
thf(tp_pair_first,type,
pair_first: $i > $i ).
thf(tp_pair_second,type,
pair_second: $i > $i ).
thf(tp_powerset,type,
powerset: $i > $i ).
thf(tp_preboolean,type,
preboolean: $i > $o ).
thf(tp_proper_subset,type,
proper_subset: $i > $i > $o ).
thf(tp_quasi_total,type,
quasi_total: $i > $i > $i > $o ).
thf(tp_reflexive,type,
reflexive: $i > $o ).
thf(tp_relation,type,
relation: $i > $o ).
thf(tp_relation_composition,type,
relation_composition: $i > $i > $i ).
thf(tp_relation_dom,type,
relation_dom: $i > $i ).
thf(tp_relation_dom_as_subset,type,
relation_dom_as_subset: $i > $i > $i > $i ).
thf(tp_relation_dom_restriction,type,
relation_dom_restriction: $i > $i > $i ).
thf(tp_relation_empty_yielding,type,
relation_empty_yielding: $i > $o ).
thf(tp_relation_field,type,
relation_field: $i > $i ).
thf(tp_relation_image,type,
relation_image: $i > $i > $i ).
thf(tp_relation_inverse,type,
relation_inverse: $i > $i ).
thf(tp_relation_inverse_image,type,
relation_inverse_image: $i > $i > $i ).
thf(tp_relation_isomorphism,type,
relation_isomorphism: $i > $i > $i > $o ).
thf(tp_relation_of2,type,
relation_of2: $i > $i > $i > $o ).
thf(tp_relation_of2_as_subset,type,
relation_of2_as_subset: $i > $i > $i > $o ).
thf(tp_relation_restriction,type,
relation_restriction: $i > $i > $i ).
thf(tp_relation_rng,type,
relation_rng: $i > $i ).
thf(tp_relation_rng_as_subset,type,
relation_rng_as_subset: $i > $i > $i > $i ).
thf(tp_relation_rng_restriction,type,
relation_rng_restriction: $i > $i > $i ).
thf(tp_sK1_A,type,
sK1_A: $i ).
thf(tp_sK2_SX0,type,
sK2_SX0: $i ).
thf(tp_sK3_SX0,type,
sK3_SX0: $i ).
thf(tp_sK4_SY75,type,
sK4_SY75: $i > $i ).
thf(tp_sK5_SY91,type,
sK5_SY91: $i > $i ).
thf(tp_sK6_SY102,type,
sK6_SY102: $i > $i > $i ).
thf(tp_sK7_SY73,type,
sK7_SY73: $i > $i ).
thf(tp_set_difference,type,
set_difference: $i > $i > $i ).
thf(tp_set_intersection2,type,
set_intersection2: $i > $i > $i ).
thf(tp_set_meet,type,
set_meet: $i > $i ).
thf(tp_set_union2,type,
set_union2: $i > $i > $i ).
thf(tp_singleton,type,
singleton: $i > $i ).
thf(tp_subset,type,
subset: $i > $i > $o ).
thf(tp_subset_complement,type,
subset_complement: $i > $i > $i ).
thf(tp_subset_difference,type,
subset_difference: $i > $i > $i > $i ).
thf(tp_subset_intersection2,type,
subset_intersection2: $i > $i > $i > $i ).
thf(tp_succ,type,
succ: $i > $i ).
thf(tp_the_L_join,type,
the_L_join: $i > $i ).
thf(tp_the_L_meet,type,
the_L_meet: $i > $i ).
thf(tp_the_carrier,type,
the_carrier: $i > $i ).
thf(tp_the_topology,type,
the_topology: $i > $i ).
thf(tp_top_str,type,
top_str: $i > $o ).
thf(tp_topological_space,type,
topological_space: $i > $o ).
thf(tp_topstr_closure,type,
topstr_closure: $i > $i > $i ).
thf(tp_transitive,type,
transitive: $i > $o ).
thf(tp_union,type,
union: $i > $i ).
thf(tp_union_of_subsets,type,
union_of_subsets: $i > $i > $i ).
thf(tp_unordered_pair,type,
unordered_pair: $i > $i > $i ).
thf(tp_unordered_triple,type,
unordered_triple: $i > $i > $i > $i ).
thf(tp_v1_int_1,type,
v1_int_1: $i > $o ).
thf(tp_v1_membered,type,
v1_membered: $i > $o ).
thf(tp_v1_rat_1,type,
v1_rat_1: $i > $o ).
thf(tp_v1_xcmplx_0,type,
v1_xcmplx_0: $i > $o ).
thf(tp_v1_xreal_0,type,
v1_xreal_0: $i > $o ).
thf(tp_v2_membered,type,
v2_membered: $i > $o ).
thf(tp_v3_membered,type,
v3_membered: $i > $o ).
thf(tp_v4_membered,type,
v4_membered: $i > $o ).
thf(tp_v5_membered,type,
v5_membered: $i > $o ).
thf(tp_well_founded_relation,type,
well_founded_relation: $i > $o ).
thf(tp_well_ordering,type,
well_ordering: $i > $o ).
thf(tp_well_orders,type,
well_orders: $i > $i > $o ).
thf(7,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_subset) ).
thf(9,axiom,
! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(59,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(67,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(78,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k7_setfam_1) ).
thf(80,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k3_subset_1) ).
thf(81,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k5_subset_1) ).
thf(130,axiom,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(148,axiom,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(153,axiom,
? [A: $i] : ( top_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_pre_topc) ).
thf(156,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u1_struct_0) ).
thf(157,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( element @ ( the_topology @ A ) @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u1_pre_topc) ).
thf(160,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_m1_subset_1) ).
thf(161,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_m1_relset_1) ).
thf(164,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_struct_0) ).
thf(165,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_pre_topc) ).
thf(167,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k9_relat_1) ).
thf(169,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k7_setfam_1) ).
thf(171,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_subset_1) ).
thf(172,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_setfam_1) ).
thf(174,axiom,
! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( topstr_closure @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_pre_topc) ).
thf(175,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_intersection2 @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_subset_1) ).
thf(176,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_setfam_1) ).
thf(179,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_ordinal2) ).
thf(180,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_xboole_0) ).
thf(181,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_tarski) ).
thf(185,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_xboole_0) ).
thf(186,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_tarski) ).
thf(187,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_subset_1) ).
thf(188,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_relat_1) ).
thf(190,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_zfmisc_1) ).
thf(191,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_xboole_0) ).
thf(193,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_tarski) ).
thf(194,axiom,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(195,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_relat_1) ).
thf(197,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_mcart_1) ).
thf(201,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_zfmisc_1) ).
thf(202,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_xboole_0) ).
thf(204,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_wellord1) ).
thf(205,axiom,
! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( interior @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_tops_1) ).
thf(206,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_tarski) ).
thf(207,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_setfam_1) ).
thf(208,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_relat_1) ).
thf(210,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_ordinal1) ).
thf(211,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_mcart_1) ).
thf(213,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_funct_1) ).
thf(214,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_enumset1) ).
thf(215,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_binop_1) ).
thf(216,axiom,
$true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k10_relat_1) ).
thf(301,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( subset_intersection2 @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k5_subset_1) ).
thf(315,axiom,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_finset_1) ).
thf(329,axiom,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( v1_membered @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc16_membered) ).
thf(340,axiom,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(394,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_setfam_1) ).
thf(474,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( closed_subsets @ B @ A )
<=> ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t16_tops_2) ).
thf(506,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) )
& ~ ( ( ( complements_of_subsets @ A @ B )
!= empty_set )
& ( B = empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_tops_2) ).
thf(537,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l71_subset_1) ).
thf(545,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3_subset_1) ).
thf(556,conjecture,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( open_subsets @ B @ A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_tops_2) ).
thf(557,negated_conjecture,
( ( ! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( open_subsets @ B @ A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ) )
= $false ),
inference(negate_conjecture,[status(cth)],[556]) ).
thf(558,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( open_subsets @ B @ A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ) )
= $false ),
inference(unfold_def,[status(thm)],[557]) ).
thf(559,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ) )
= $true ),
inference(unfold_def,[status(thm)],[7]) ).
thf(560,plain,
( ( ! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ) )
= $true ),
inference(unfold_def,[status(thm)],[9]) ).
thf(561,plain,
( ( ! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ) )
= $true ),
inference(unfold_def,[status(thm)],[14]) ).
thf(562,plain,
( ( ! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ) )
= $true ),
inference(unfold_def,[status(thm)],[59]) ).
thf(563,plain,
( ( ! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[67]) ).
thf(564,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ) )
= $true ),
inference(unfold_def,[status(thm)],[78]) ).
thf(565,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ) )
= $true ),
inference(unfold_def,[status(thm)],[80]) ).
thf(566,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ B )
= B ) ) )
= $true ),
inference(unfold_def,[status(thm)],[81]) ).
thf(567,plain,
( ( ! [A: $i] :
~ ( empty @ ( powerset @ A ) ) )
= $true ),
inference(unfold_def,[status(thm)],[130]) ).
thf(568,plain,
( ( ! [A: $i] :
? [B: $i] : ( element @ B @ A ) )
= $true ),
inference(unfold_def,[status(thm)],[148]) ).
thf(569,plain,
( ( ? [A: $i] : ( top_str @ A ) )
= $true ),
inference(unfold_def,[status(thm)],[153]) ).
thf(570,plain,
$true = $true,
inference(unfold_def,[status(thm)],[156]) ).
thf(571,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ( element @ ( the_topology @ A ) @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[157]) ).
thf(572,plain,
$true = $true,
inference(unfold_def,[status(thm)],[160]) ).
thf(573,plain,
$true = $true,
inference(unfold_def,[status(thm)],[161]) ).
thf(574,plain,
$true = $true,
inference(unfold_def,[status(thm)],[164]) ).
thf(575,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ) )
= $true ),
inference(unfold_def,[status(thm)],[165]) ).
thf(576,plain,
$true = $true,
inference(unfold_def,[status(thm)],[167]) ).
thf(577,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[169]) ).
thf(578,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[171]) ).
thf(579,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[172]) ).
thf(580,plain,
( ( ! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( topstr_closure @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[174]) ).
thf(581,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_intersection2 @ A @ B @ C ) @ ( powerset @ A ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[175]) ).
thf(582,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[176]) ).
thf(583,plain,
$true = $true,
inference(unfold_def,[status(thm)],[179]) ).
thf(584,plain,
$true = $true,
inference(unfold_def,[status(thm)],[180]) ).
thf(585,plain,
$true = $true,
inference(unfold_def,[status(thm)],[181]) ).
thf(586,plain,
$true = $true,
inference(unfold_def,[status(thm)],[185]) ).
thf(587,plain,
$true = $true,
inference(unfold_def,[status(thm)],[186]) ).
thf(588,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[187]) ).
thf(589,plain,
$true = $true,
inference(unfold_def,[status(thm)],[188]) ).
thf(590,plain,
$true = $true,
inference(unfold_def,[status(thm)],[190]) ).
thf(591,plain,
$true = $true,
inference(unfold_def,[status(thm)],[191]) ).
thf(592,plain,
$true = $true,
inference(unfold_def,[status(thm)],[193]) ).
thf(593,plain,
( ( ! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ) )
= $true ),
inference(unfold_def,[status(thm)],[194]) ).
thf(594,plain,
$true = $true,
inference(unfold_def,[status(thm)],[195]) ).
thf(595,plain,
$true = $true,
inference(unfold_def,[status(thm)],[197]) ).
thf(596,plain,
$true = $true,
inference(unfold_def,[status(thm)],[201]) ).
thf(597,plain,
$true = $true,
inference(unfold_def,[status(thm)],[202]) ).
thf(598,plain,
$true = $true,
inference(unfold_def,[status(thm)],[204]) ).
thf(599,plain,
( ( ! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( interior @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[205]) ).
thf(600,plain,
$true = $true,
inference(unfold_def,[status(thm)],[206]) ).
thf(601,plain,
$true = $true,
inference(unfold_def,[status(thm)],[207]) ).
thf(602,plain,
$true = $true,
inference(unfold_def,[status(thm)],[208]) ).
thf(603,plain,
$true = $true,
inference(unfold_def,[status(thm)],[210]) ).
thf(604,plain,
$true = $true,
inference(unfold_def,[status(thm)],[211]) ).
thf(605,plain,
$true = $true,
inference(unfold_def,[status(thm)],[213]) ).
thf(606,plain,
$true = $true,
inference(unfold_def,[status(thm)],[214]) ).
thf(607,plain,
$true = $true,
inference(unfold_def,[status(thm)],[215]) ).
thf(608,plain,
$true = $true,
inference(unfold_def,[status(thm)],[216]) ).
thf(609,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( subset_intersection2 @ A @ C @ B ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[301]) ).
thf(610,plain,
( ( ! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[315]) ).
thf(611,plain,
( ( ! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( v1_membered @ B ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[329]) ).
thf(612,plain,
( ( ! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ) )
= $true ),
inference(unfold_def,[status(thm)],[340]) ).
thf(613,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[394]) ).
thf(614,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( closed_subsets @ B @ A )
<=> ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[474]) ).
thf(615,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) )
& ~ ( ( ( complements_of_subsets @ A @ B )
!= empty_set )
& ( B = empty_set ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[506]) ).
thf(616,plain,
( ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[537]) ).
thf(617,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[545]) ).
thf(618,plain,
( ( ( top_str @ sK1_A )
=> ! [SY65: $i] :
( ( element @ SY65 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
=> ( ( open_subsets @ SY65 @ sK1_A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SY65 ) @ sK1_A ) ) ) )
= $false ),
inference(extcnf_forall_neg,[status(esa)],[558]) ).
thf(619,plain,
( ( top_str @ sK1_A )
= $true ),
inference(standard_cnf,[status(thm)],[618]) ).
thf(620,plain,
( ( ! [SY65: $i] :
( ( element @ SY65 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
=> ( ( open_subsets @ SY65 @ sK1_A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SY65 ) @ sK1_A ) ) ) )
= $false ),
inference(standard_cnf,[status(thm)],[618]) ).
thf(621,plain,
( ( ~ ! [SY65: $i] :
( ( element @ SY65 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
=> ( ( open_subsets @ SY65 @ sK1_A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SY65 ) @ sK1_A ) ) ) )
= $true ),
inference(polarity_switch,[status(thm)],[620]) ).
thf(622,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ) )
= $true ),
inference(copy,[status(thm)],[617]) ).
thf(623,plain,
( ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ) )
= $true ),
inference(copy,[status(thm)],[616]) ).
thf(624,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) )
& ~ ( ( ( complements_of_subsets @ A @ B )
!= empty_set )
& ( B = empty_set ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[615]) ).
thf(625,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( closed_subsets @ B @ A )
<=> ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[614]) ).
thf(626,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ) )
= $true ),
inference(copy,[status(thm)],[613]) ).
thf(627,plain,
( ( ! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ) )
= $true ),
inference(copy,[status(thm)],[612]) ).
thf(628,plain,
( ( ! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( v1_membered @ B ) ) ) )
= $true ),
inference(copy,[status(thm)],[611]) ).
thf(629,plain,
( ( ! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ) )
= $true ),
inference(copy,[status(thm)],[610]) ).
thf(630,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( subset_intersection2 @ A @ C @ B ) ) ) )
= $true ),
inference(copy,[status(thm)],[609]) ).
thf(631,plain,
$true = $true,
inference(copy,[status(thm)],[608]) ).
thf(632,plain,
$true = $true,
inference(copy,[status(thm)],[607]) ).
thf(633,plain,
$true = $true,
inference(copy,[status(thm)],[606]) ).
thf(634,plain,
$true = $true,
inference(copy,[status(thm)],[605]) ).
thf(635,plain,
$true = $true,
inference(copy,[status(thm)],[604]) ).
thf(636,plain,
$true = $true,
inference(copy,[status(thm)],[603]) ).
thf(637,plain,
$true = $true,
inference(copy,[status(thm)],[602]) ).
thf(638,plain,
$true = $true,
inference(copy,[status(thm)],[601]) ).
thf(639,plain,
$true = $true,
inference(copy,[status(thm)],[600]) ).
thf(640,plain,
( ( ! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( interior @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[599]) ).
thf(641,plain,
$true = $true,
inference(copy,[status(thm)],[598]) ).
thf(642,plain,
$true = $true,
inference(copy,[status(thm)],[597]) ).
thf(643,plain,
$true = $true,
inference(copy,[status(thm)],[596]) ).
thf(644,plain,
$true = $true,
inference(copy,[status(thm)],[595]) ).
thf(645,plain,
$true = $true,
inference(copy,[status(thm)],[594]) ).
thf(646,plain,
( ( ! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ) )
= $true ),
inference(copy,[status(thm)],[593]) ).
thf(647,plain,
$true = $true,
inference(copy,[status(thm)],[592]) ).
thf(648,plain,
$true = $true,
inference(copy,[status(thm)],[591]) ).
thf(649,plain,
$true = $true,
inference(copy,[status(thm)],[590]) ).
thf(650,plain,
$true = $true,
inference(copy,[status(thm)],[589]) ).
thf(651,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ) )
= $true ),
inference(copy,[status(thm)],[588]) ).
thf(652,plain,
$true = $true,
inference(copy,[status(thm)],[587]) ).
thf(653,plain,
$true = $true,
inference(copy,[status(thm)],[586]) ).
thf(654,plain,
$true = $true,
inference(copy,[status(thm)],[585]) ).
thf(655,plain,
$true = $true,
inference(copy,[status(thm)],[584]) ).
thf(656,plain,
$true = $true,
inference(copy,[status(thm)],[583]) ).
thf(657,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ) )
= $true ),
inference(copy,[status(thm)],[582]) ).
thf(658,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_intersection2 @ A @ B @ C ) @ ( powerset @ A ) ) ) )
= $true ),
inference(copy,[status(thm)],[581]) ).
thf(659,plain,
( ( ! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( topstr_closure @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[580]) ).
thf(660,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ) )
= $true ),
inference(copy,[status(thm)],[579]) ).
thf(661,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ) )
= $true ),
inference(copy,[status(thm)],[578]) ).
thf(662,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[577]) ).
thf(663,plain,
$true = $true,
inference(copy,[status(thm)],[576]) ).
thf(664,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ) )
= $true ),
inference(copy,[status(thm)],[575]) ).
thf(665,plain,
$true = $true,
inference(copy,[status(thm)],[574]) ).
thf(666,plain,
$true = $true,
inference(copy,[status(thm)],[573]) ).
thf(667,plain,
$true = $true,
inference(copy,[status(thm)],[572]) ).
thf(668,plain,
( ( ! [A: $i] :
( ( top_str @ A )
=> ( element @ ( the_topology @ A ) @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[571]) ).
thf(669,plain,
$true = $true,
inference(copy,[status(thm)],[570]) ).
thf(670,plain,
( ( ? [A: $i] : ( top_str @ A ) )
= $true ),
inference(copy,[status(thm)],[569]) ).
thf(671,plain,
( ( ! [A: $i] :
? [B: $i] : ( element @ B @ A ) )
= $true ),
inference(copy,[status(thm)],[568]) ).
thf(672,plain,
( ( ! [A: $i] :
~ ( empty @ ( powerset @ A ) ) )
= $true ),
inference(copy,[status(thm)],[567]) ).
thf(673,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ B )
= B ) ) )
= $true ),
inference(copy,[status(thm)],[566]) ).
thf(674,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ) )
= $true ),
inference(copy,[status(thm)],[565]) ).
thf(675,plain,
( ( ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ) )
= $true ),
inference(copy,[status(thm)],[564]) ).
thf(676,plain,
( ( ! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ) )
= $true ),
inference(copy,[status(thm)],[563]) ).
thf(677,plain,
( ( ! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ) )
= $true ),
inference(copy,[status(thm)],[562]) ).
thf(678,plain,
( ( ! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ) )
= $true ),
inference(copy,[status(thm)],[561]) ).
thf(679,plain,
( ( ! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ) )
= $true ),
inference(copy,[status(thm)],[560]) ).
thf(680,plain,
( ( ! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ) )
= $true ),
inference(copy,[status(thm)],[559]) ).
thf(681,plain,
( ( top_str @ sK1_A )
= $true ),
inference(copy,[status(thm)],[619]) ).
thf(682,plain,
( ( ~ ! [SY65: $i] :
( ( element @ SY65 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
=> ( ( open_subsets @ SY65 @ sK1_A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SY65 ) @ sK1_A ) ) ) )
= $true ),
inference(copy,[status(thm)],[621]) ).
thf(683,plain,
( ( ! [SX0: $i,SX1: $i] :
~ ( ~ ( ~ ( element @ SX0 @ ( powerset @ SX1 ) )
| ( subset @ SX0 @ SX1 ) )
| ~ ( ~ ( subset @ SX0 @ SX1 )
| ( element @ SX0 @ ( powerset @ SX1 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[679]) ).
thf(684,plain,
( ( ~ ! [SX0: $i] :
~ ( top_str @ SX0 ) )
= $true ),
inference(unfold_def,[status(thm)],[670]) ).
thf(685,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ~ ( ~ ( top_str @ SX0 )
| ~ ( element @ SX1 @ ( powerset @ ( the_carrier @ SX0 ) ) ) )
| ( element @ ( topstr_closure @ SX0 @ SX1 ) @ ( powerset @ ( the_carrier @ SX0 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[659]) ).
thf(686,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ~ ( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ~ ( element @ SX2 @ ( powerset @ SX0 ) ) )
| ( ( subset_intersection2 @ SX0 @ SX1 @ SX2 )
= ( subset_intersection2 @ SX0 @ SX2 @ SX1 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[630]) ).
thf(687,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ( ( subset_complement @ SX0 @ ( subset_complement @ SX0 @ SX1 ) )
= SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[674]) ).
thf(688,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ~ ( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ~ ( element @ SX2 @ ( powerset @ SX0 ) ) )
| ( element @ ( subset_difference @ SX0 @ SX1 @ SX2 ) @ ( powerset @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[661]) ).
thf(689,plain,
( ( ! [SX0: $i] :
( ~ ~ ( empty @ SX0 )
| ~ ! [SX1: $i] :
~ ~ ( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ~ ~ ( empty @ SX1 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[676]) ).
thf(690,plain,
( ( ~ ! [SX0: $i] :
( ~ ( element @ SX0 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
| ~ ( ~ ( ~ ( open_subsets @ SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SX0 ) @ sK1_A ) )
| ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SX0 ) @ sK1_A )
| ( open_subsets @ SX0 @ sK1_A ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[682]) ).
thf(691,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( in @ SX0 @ SX1 )
| ( element @ SX0 @ SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[678]) ).
thf(692,plain,
( ( ! [SX0: $i] :
~ ! [SX1: $i] :
~ ~ ( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ~ ( empty @ SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[677]) ).
thf(693,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ SX0 ) ) )
| ( element @ ( meet_of_subsets @ SX0 @ SX1 ) @ ( powerset @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[660]) ).
thf(694,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ SX0 ) ) )
| ( ( complements_of_subsets @ SX0 @ ( complements_of_subsets @ SX0 @ SX1 ) )
= SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[675]) ).
thf(695,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ SX0 ) ) )
| ~ ~ ( ~ ( ( SX1 != empty_set ) )
| ( ( complements_of_subsets @ SX0 @ SX1 )
!= empty_set ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[626]) ).
thf(696,plain,
( ( ! [SX0: $i] :
( ~ ( v1_membered @ SX0 )
| ! [SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ( v1_membered @ SX1 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[628]) ).
thf(697,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ SX0 ) ) )
| ~ ( ~ ~ ~ ( ~ ( ( SX1 != empty_set ) )
| ( ( complements_of_subsets @ SX0 @ SX1 )
!= empty_set ) )
| ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SX0 @ SX1 )
!= empty_set ) )
| ( SX1 != empty_set ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[624]) ).
thf(698,plain,
( ( ! [SX0: $i] :
( ~ ( top_str @ SX0 )
| ( one_sorted_str @ SX0 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[664]) ).
thf(699,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ~ ( ~ ( in @ SX0 @ SX1 )
| ~ ( element @ SX1 @ ( powerset @ SX2 ) ) )
| ( element @ SX0 @ SX2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[680]) ).
thf(700,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ SX0 ) ) )
| ( element @ ( union_of_subsets @ SX0 @ SX1 ) @ ( powerset @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[657]) ).
thf(701,plain,
( ( ! [SX0: $i] :
( ~ ( top_str @ SX0 )
| ! [SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ ( the_carrier @ SX0 ) ) ) )
| ~ ( ~ ( ~ ( closed_subsets @ SX1 @ SX0 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SX0 ) @ SX1 ) @ SX0 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SX0 ) @ SX1 ) @ SX0 )
| ( closed_subsets @ SX1 @ SX0 ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[625]) ).
thf(702,plain,
( ( ! [SX0: $i] :
( ~ ( finite @ SX0 )
| ! [SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ( finite @ SX1 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[629]) ).
thf(703,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ~ ( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ~ ( element @ SX2 @ ( powerset @ SX0 ) ) )
| ( element @ ( subset_intersection2 @ SX0 @ SX1 @ SX2 ) @ ( powerset @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[658]) ).
thf(704,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ ( powerset @ SX0 ) ) )
| ( element @ ( complements_of_subsets @ SX0 @ SX1 ) @ ( powerset @ ( powerset @ SX0 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[662]) ).
thf(705,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ! [SX2: $i] :
( ~ ( in @ SX2 @ SX0 )
| ( in @ SX2 @ SX1 ) )
| ( element @ SX0 @ ( powerset @ SX1 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[623]) ).
thf(706,plain,
( ( ! [SX0: $i] :
~ ! [SX1: $i] :
~ ( element @ SX1 @ SX0 ) )
= $true ),
inference(unfold_def,[status(thm)],[671]) ).
thf(707,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ( element @ ( subset_complement @ SX0 @ SX1 ) @ ( powerset @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[651]) ).
thf(708,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ~ ( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ~ ( element @ SX2 @ ( powerset @ SX0 ) ) )
| ( ( subset_intersection2 @ SX0 @ SX1 @ SX1 )
= SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[673]) ).
thf(709,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ~ ( ~ ( top_str @ SX0 )
| ~ ( element @ SX1 @ ( powerset @ ( the_carrier @ SX0 ) ) ) )
| ( element @ ( interior @ SX0 @ SX1 ) @ ( powerset @ ( the_carrier @ SX0 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[640]) ).
thf(710,plain,
( ( ! [SX0: $i] :
( ~ ( top_str @ SX0 )
| ( element @ ( the_topology @ SX0 ) @ ( powerset @ ( powerset @ ( the_carrier @ SX0 ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[668]) ).
thf(711,plain,
( ( ! [SX0: $i,SX1: $i] :
( ~ ( element @ SX1 @ ( powerset @ SX0 ) )
| ! [SX2: $i] :
( ~ ( in @ SX2 @ SX1 )
| ( in @ SX2 @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[622]) ).
thf(712,plain,
! [SV1: $i] :
( ( ( union @ ( powerset @ SV1 ) )
= SV1 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[627]) ).
thf(713,plain,
! [SV2: $i] :
( ( element @ ( cast_to_subset @ SV2 ) @ ( powerset @ SV2 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[646]) ).
thf(714,plain,
! [SV3: $i] :
( ( ~ ( empty @ ( powerset @ SV3 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[672]) ).
thf(715,plain,
! [SV4: $i] :
( ( ! [SY66: $i] :
~ ( ~ ( ~ ( element @ SV4 @ ( powerset @ SY66 ) )
| ( subset @ SV4 @ SY66 ) )
| ~ ( ~ ( subset @ SV4 @ SY66 )
| ( element @ SV4 @ ( powerset @ SY66 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[683]) ).
thf(716,plain,
( ( ! [SX0: $i] :
~ ( top_str @ SX0 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[684]) ).
thf(717,plain,
! [SV5: $i] :
( ( ! [SY67: $i] :
( ~ ~ ( ~ ( top_str @ SV5 )
| ~ ( element @ SY67 @ ( powerset @ ( the_carrier @ SV5 ) ) ) )
| ( element @ ( topstr_closure @ SV5 @ SY67 ) @ ( powerset @ ( the_carrier @ SV5 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[685]) ).
thf(718,plain,
! [SV6: $i] :
( ( ! [SY68: $i,SY69: $i] :
( ~ ~ ( ~ ( element @ SY68 @ ( powerset @ SV6 ) )
| ~ ( element @ SY69 @ ( powerset @ SV6 ) ) )
| ( ( subset_intersection2 @ SV6 @ SY68 @ SY69 )
= ( subset_intersection2 @ SV6 @ SY69 @ SY68 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[686]) ).
thf(719,plain,
! [SV7: $i] :
( ( ! [SY70: $i] :
( ~ ( element @ SY70 @ ( powerset @ SV7 ) )
| ( ( subset_complement @ SV7 @ ( subset_complement @ SV7 @ SY70 ) )
= SY70 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[687]) ).
thf(720,plain,
! [SV8: $i] :
( ( ! [SY71: $i,SY72: $i] :
( ~ ~ ( ~ ( element @ SY71 @ ( powerset @ SV8 ) )
| ~ ( element @ SY72 @ ( powerset @ SV8 ) ) )
| ( element @ ( subset_difference @ SV8 @ SY71 @ SY72 ) @ ( powerset @ SV8 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[688]) ).
thf(721,plain,
! [SV9: $i] :
( ( ~ ~ ( empty @ SV9 )
| ~ ! [SY73: $i] :
~ ~ ( ~ ( element @ SY73 @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ SY73 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[689]) ).
thf(722,plain,
( ( ! [SX0: $i] :
( ~ ( element @ SX0 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
| ~ ( ~ ( ~ ( open_subsets @ SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SX0 ) @ sK1_A ) )
| ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ SX0 ) @ sK1_A )
| ( open_subsets @ SX0 @ sK1_A ) ) ) ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[690]) ).
thf(723,plain,
! [SV10: $i] :
( ( ! [SY74: $i] :
( ~ ( in @ SV10 @ SY74 )
| ( element @ SV10 @ SY74 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[691]) ).
thf(724,plain,
! [SV11: $i] :
( ( ~ ! [SY75: $i] :
~ ~ ( ~ ( element @ SY75 @ ( powerset @ SV11 ) )
| ~ ( empty @ SY75 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[692]) ).
thf(725,plain,
! [SV12: $i] :
( ( ! [SY76: $i] :
( ~ ( element @ SY76 @ ( powerset @ ( powerset @ SV12 ) ) )
| ( element @ ( meet_of_subsets @ SV12 @ SY76 ) @ ( powerset @ SV12 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[693]) ).
thf(726,plain,
! [SV13: $i] :
( ( ! [SY77: $i] :
( ~ ( element @ SY77 @ ( powerset @ ( powerset @ SV13 ) ) )
| ( ( complements_of_subsets @ SV13 @ ( complements_of_subsets @ SV13 @ SY77 ) )
= SY77 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[694]) ).
thf(727,plain,
! [SV14: $i] :
( ( ! [SY78: $i] :
( ~ ( element @ SY78 @ ( powerset @ ( powerset @ SV14 ) ) )
| ~ ~ ( ~ ( ( SY78 != empty_set ) )
| ( ( complements_of_subsets @ SV14 @ SY78 )
!= empty_set ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[695]) ).
thf(728,plain,
! [SV15: $i] :
( ( ~ ( v1_membered @ SV15 )
| ! [SY79: $i] :
( ~ ( element @ SY79 @ ( powerset @ SV15 ) )
| ( v1_membered @ SY79 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[696]) ).
thf(729,plain,
! [SV16: $i] :
( ( ! [SY80: $i] :
( ~ ( element @ SY80 @ ( powerset @ ( powerset @ SV16 ) ) )
| ~ ( ~ ~ ~ ( ~ ( ( SY80 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SY80 )
!= empty_set ) )
| ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SY80 )
!= empty_set ) )
| ( SY80 != empty_set ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[697]) ).
thf(730,plain,
! [SV17: $i] :
( ( ~ ( top_str @ SV17 )
| ( one_sorted_str @ SV17 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[698]) ).
thf(731,plain,
! [SV18: $i] :
( ( ! [SY81: $i,SY82: $i] :
( ~ ~ ( ~ ( in @ SV18 @ SY81 )
| ~ ( element @ SY81 @ ( powerset @ SY82 ) ) )
| ( element @ SV18 @ SY82 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[699]) ).
thf(732,plain,
! [SV19: $i] :
( ( ! [SY83: $i] :
( ~ ( element @ SY83 @ ( powerset @ ( powerset @ SV19 ) ) )
| ( element @ ( union_of_subsets @ SV19 @ SY83 ) @ ( powerset @ SV19 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[700]) ).
thf(733,plain,
! [SV20: $i] :
( ( ~ ( top_str @ SV20 )
| ! [SY84: $i] :
( ~ ( element @ SY84 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
| ~ ( ~ ( ~ ( closed_subsets @ SY84 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SY84 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SY84 ) @ SV20 )
| ( closed_subsets @ SY84 @ SV20 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[701]) ).
thf(734,plain,
! [SV21: $i] :
( ( ~ ( finite @ SV21 )
| ! [SY85: $i] :
( ~ ( element @ SY85 @ ( powerset @ SV21 ) )
| ( finite @ SY85 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[702]) ).
thf(735,plain,
! [SV22: $i] :
( ( ! [SY86: $i,SY87: $i] :
( ~ ~ ( ~ ( element @ SY86 @ ( powerset @ SV22 ) )
| ~ ( element @ SY87 @ ( powerset @ SV22 ) ) )
| ( element @ ( subset_intersection2 @ SV22 @ SY86 @ SY87 ) @ ( powerset @ SV22 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[703]) ).
thf(736,plain,
! [SV23: $i] :
( ( ! [SY88: $i] :
( ~ ( element @ SY88 @ ( powerset @ ( powerset @ SV23 ) ) )
| ( element @ ( complements_of_subsets @ SV23 @ SY88 ) @ ( powerset @ ( powerset @ SV23 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[704]) ).
thf(737,plain,
! [SV24: $i] :
( ( ! [SY89: $i] :
( ~ ! [SY90: $i] :
( ~ ( in @ SY90 @ SV24 )
| ( in @ SY90 @ SY89 ) )
| ( element @ SV24 @ ( powerset @ SY89 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[705]) ).
thf(738,plain,
! [SV25: $i] :
( ( ~ ! [SY91: $i] :
~ ( element @ SY91 @ SV25 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[706]) ).
thf(739,plain,
! [SV26: $i] :
( ( ! [SY92: $i] :
( ~ ( element @ SY92 @ ( powerset @ SV26 ) )
| ( element @ ( subset_complement @ SV26 @ SY92 ) @ ( powerset @ SV26 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[707]) ).
thf(740,plain,
! [SV27: $i] :
( ( ! [SY93: $i,SY94: $i] :
( ~ ~ ( ~ ( element @ SY93 @ ( powerset @ SV27 ) )
| ~ ( element @ SY94 @ ( powerset @ SV27 ) ) )
| ( ( subset_intersection2 @ SV27 @ SY93 @ SY93 )
= SY93 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[708]) ).
thf(741,plain,
! [SV28: $i] :
( ( ! [SY95: $i] :
( ~ ~ ( ~ ( top_str @ SV28 )
| ~ ( element @ SY95 @ ( powerset @ ( the_carrier @ SV28 ) ) ) )
| ( element @ ( interior @ SV28 @ SY95 ) @ ( powerset @ ( the_carrier @ SV28 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[709]) ).
thf(742,plain,
! [SV29: $i] :
( ( ~ ( top_str @ SV29 )
| ( element @ ( the_topology @ SV29 ) @ ( powerset @ ( powerset @ ( the_carrier @ SV29 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[710]) ).
thf(743,plain,
! [SV30: $i] :
( ( ! [SY96: $i] :
( ~ ( element @ SY96 @ ( powerset @ SV30 ) )
| ! [SY97: $i] :
( ~ ( in @ SY97 @ SY96 )
| ( in @ SY97 @ SV30 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[711]) ).
thf(744,plain,
! [SV3: $i] :
( ( empty @ ( powerset @ SV3 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[714]) ).
thf(745,plain,
! [SV31: $i,SV4: $i] :
( ( ~ ( ~ ( ~ ( element @ SV4 @ ( powerset @ SV31 ) )
| ( subset @ SV4 @ SV31 ) )
| ~ ( ~ ( subset @ SV4 @ SV31 )
| ( element @ SV4 @ ( powerset @ SV31 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[715]) ).
thf(746,plain,
( ( ~ ( top_str @ sK2_SX0 ) )
= $false ),
inference(extcnf_forall_neg,[status(esa)],[716]) ).
thf(747,plain,
! [SV32: $i,SV5: $i] :
( ( ~ ~ ( ~ ( top_str @ SV5 )
| ~ ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) ) )
| ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[717]) ).
thf(748,plain,
! [SV6: $i,SV33: $i] :
( ( ! [SY98: $i] :
( ~ ~ ( ~ ( element @ SV33 @ ( powerset @ SV6 ) )
| ~ ( element @ SY98 @ ( powerset @ SV6 ) ) )
| ( ( subset_intersection2 @ SV6 @ SV33 @ SY98 )
= ( subset_intersection2 @ SV6 @ SY98 @ SV33 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[718]) ).
thf(749,plain,
! [SV7: $i,SV34: $i] :
( ( ~ ( element @ SV34 @ ( powerset @ SV7 ) )
| ( ( subset_complement @ SV7 @ ( subset_complement @ SV7 @ SV34 ) )
= SV34 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[719]) ).
thf(750,plain,
! [SV8: $i,SV35: $i] :
( ( ! [SY99: $i] :
( ~ ~ ( ~ ( element @ SV35 @ ( powerset @ SV8 ) )
| ~ ( element @ SY99 @ ( powerset @ SV8 ) ) )
| ( element @ ( subset_difference @ SV8 @ SV35 @ SY99 ) @ ( powerset @ SV8 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[720]) ).
thf(751,plain,
! [SV9: $i] :
( ( ( ~ ~ ( empty @ SV9 ) )
= $true )
| ( ( ~ ! [SY73: $i] :
~ ~ ( ~ ( element @ SY73 @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ SY73 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[721]) ).
thf(752,plain,
( ( ~ ( element @ sK3_SX0 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
| ~ ( ~ ( ~ ( open_subsets @ sK3_SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) )
| ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) ) )
= $false ),
inference(extcnf_forall_neg,[status(esa)],[722]) ).
thf(753,plain,
! [SV36: $i,SV10: $i] :
( ( ~ ( in @ SV10 @ SV36 )
| ( element @ SV10 @ SV36 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[723]) ).
thf(754,plain,
! [SV11: $i] :
( ( ! [SY75: $i] :
~ ~ ( ~ ( element @ SY75 @ ( powerset @ SV11 ) )
| ~ ( empty @ SY75 ) ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[724]) ).
thf(755,plain,
! [SV12: $i,SV37: $i] :
( ( ~ ( element @ SV37 @ ( powerset @ ( powerset @ SV12 ) ) )
| ( element @ ( meet_of_subsets @ SV12 @ SV37 ) @ ( powerset @ SV12 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[725]) ).
thf(756,plain,
! [SV13: $i,SV38: $i] :
( ( ~ ( element @ SV38 @ ( powerset @ ( powerset @ SV13 ) ) )
| ( ( complements_of_subsets @ SV13 @ ( complements_of_subsets @ SV13 @ SV38 ) )
= SV38 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[726]) ).
thf(757,plain,
! [SV14: $i,SV39: $i] :
( ( ~ ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
| ~ ~ ( ~ ( ( SV39 != empty_set ) )
| ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[727]) ).
thf(758,plain,
! [SV15: $i] :
( ( ( ~ ( v1_membered @ SV15 ) )
= $true )
| ( ( ! [SY79: $i] :
( ~ ( element @ SY79 @ ( powerset @ SV15 ) )
| ( v1_membered @ SY79 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[728]) ).
thf(759,plain,
! [SV16: $i,SV40: $i] :
( ( ~ ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
| ~ ( ~ ~ ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[729]) ).
thf(760,plain,
! [SV17: $i] :
( ( ( ~ ( top_str @ SV17 ) )
= $true )
| ( ( one_sorted_str @ SV17 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[730]) ).
thf(761,plain,
! [SV41: $i,SV18: $i] :
( ( ! [SY100: $i] :
( ~ ~ ( ~ ( in @ SV18 @ SV41 )
| ~ ( element @ SV41 @ ( powerset @ SY100 ) ) )
| ( element @ SV18 @ SY100 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[731]) ).
thf(762,plain,
! [SV19: $i,SV42: $i] :
( ( ~ ( element @ SV42 @ ( powerset @ ( powerset @ SV19 ) ) )
| ( element @ ( union_of_subsets @ SV19 @ SV42 ) @ ( powerset @ SV19 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[732]) ).
thf(763,plain,
! [SV20: $i] :
( ( ( ~ ( top_str @ SV20 ) )
= $true )
| ( ( ! [SY84: $i] :
( ~ ( element @ SY84 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
| ~ ( ~ ( ~ ( closed_subsets @ SY84 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SY84 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SY84 ) @ SV20 )
| ( closed_subsets @ SY84 @ SV20 ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[733]) ).
thf(764,plain,
! [SV21: $i] :
( ( ( ~ ( finite @ SV21 ) )
= $true )
| ( ( ! [SY85: $i] :
( ~ ( element @ SY85 @ ( powerset @ SV21 ) )
| ( finite @ SY85 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[734]) ).
thf(765,plain,
! [SV22: $i,SV43: $i] :
( ( ! [SY101: $i] :
( ~ ~ ( ~ ( element @ SV43 @ ( powerset @ SV22 ) )
| ~ ( element @ SY101 @ ( powerset @ SV22 ) ) )
| ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SY101 ) @ ( powerset @ SV22 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[735]) ).
thf(766,plain,
! [SV23: $i,SV44: $i] :
( ( ~ ( element @ SV44 @ ( powerset @ ( powerset @ SV23 ) ) )
| ( element @ ( complements_of_subsets @ SV23 @ SV44 ) @ ( powerset @ ( powerset @ SV23 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[736]) ).
thf(767,plain,
! [SV45: $i,SV24: $i] :
( ( ~ ! [SY102: $i] :
( ~ ( in @ SY102 @ SV24 )
| ( in @ SY102 @ SV45 ) )
| ( element @ SV24 @ ( powerset @ SV45 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[737]) ).
thf(768,plain,
! [SV25: $i] :
( ( ! [SY91: $i] :
~ ( element @ SY91 @ SV25 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[738]) ).
thf(769,plain,
! [SV26: $i,SV46: $i] :
( ( ~ ( element @ SV46 @ ( powerset @ SV26 ) )
| ( element @ ( subset_complement @ SV26 @ SV46 ) @ ( powerset @ SV26 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[739]) ).
thf(770,plain,
! [SV27: $i,SV47: $i] :
( ( ! [SY103: $i] :
( ~ ~ ( ~ ( element @ SV47 @ ( powerset @ SV27 ) )
| ~ ( element @ SY103 @ ( powerset @ SV27 ) ) )
| ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[740]) ).
thf(771,plain,
! [SV48: $i,SV28: $i] :
( ( ~ ~ ( ~ ( top_str @ SV28 )
| ~ ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) ) )
| ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[741]) ).
thf(772,plain,
! [SV29: $i] :
( ( ( ~ ( top_str @ SV29 ) )
= $true )
| ( ( element @ ( the_topology @ SV29 ) @ ( powerset @ ( powerset @ ( the_carrier @ SV29 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[742]) ).
thf(773,plain,
! [SV30: $i,SV49: $i] :
( ( ~ ( element @ SV49 @ ( powerset @ SV30 ) )
| ! [SY104: $i] :
( ~ ( in @ SY104 @ SV49 )
| ( in @ SY104 @ SV30 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[743]) ).
thf(774,plain,
! [SV31: $i,SV4: $i] :
( ( ~ ( ~ ( element @ SV4 @ ( powerset @ SV31 ) )
| ( subset @ SV4 @ SV31 ) )
| ~ ( ~ ( subset @ SV4 @ SV31 )
| ( element @ SV4 @ ( powerset @ SV31 ) ) ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[745]) ).
thf(775,plain,
( ( top_str @ sK2_SX0 )
= $true ),
inference(extcnf_not_neg,[status(thm)],[746]) ).
thf(776,plain,
! [SV32: $i,SV5: $i] :
( ( ( ~ ~ ( ~ ( top_str @ SV5 )
| ~ ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) ) ) )
= $true )
| ( ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[747]) ).
thf(777,plain,
! [SV50: $i,SV6: $i,SV33: $i] :
( ( ~ ~ ( ~ ( element @ SV33 @ ( powerset @ SV6 ) )
| ~ ( element @ SV50 @ ( powerset @ SV6 ) ) )
| ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[748]) ).
thf(778,plain,
! [SV7: $i,SV34: $i] :
( ( ( ~ ( element @ SV34 @ ( powerset @ SV7 ) ) )
= $true )
| ( ( ( subset_complement @ SV7 @ ( subset_complement @ SV7 @ SV34 ) )
= SV34 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[749]) ).
thf(779,plain,
! [SV51: $i,SV8: $i,SV35: $i] :
( ( ~ ~ ( ~ ( element @ SV35 @ ( powerset @ SV8 ) )
| ~ ( element @ SV51 @ ( powerset @ SV8 ) ) )
| ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[750]) ).
thf(780,plain,
! [SV9: $i] :
( ( ( ~ ( empty @ SV9 ) )
= $false )
| ( ( ~ ! [SY73: $i] :
~ ~ ( ~ ( element @ SY73 @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ SY73 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[751]) ).
thf(781,plain,
( ( ~ ( element @ sK3_SX0 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) ) )
= $false ),
inference(extcnf_or_neg,[status(thm)],[752]) ).
thf(782,plain,
( ( ~ ( ~ ( ~ ( open_subsets @ sK3_SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) )
| ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) ) )
= $false ),
inference(extcnf_or_neg,[status(thm)],[752]) ).
thf(783,plain,
! [SV36: $i,SV10: $i] :
( ( ( ~ ( in @ SV10 @ SV36 ) )
= $true )
| ( ( element @ SV10 @ SV36 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[753]) ).
thf(784,plain,
! [SV11: $i] :
( ( ~ ~ ( ~ ( element @ ( sK4_SY75 @ SV11 ) @ ( powerset @ SV11 ) )
| ~ ( empty @ ( sK4_SY75 @ SV11 ) ) ) )
= $false ),
inference(extcnf_forall_neg,[status(esa)],[754]) ).
thf(785,plain,
! [SV12: $i,SV37: $i] :
( ( ( ~ ( element @ SV37 @ ( powerset @ ( powerset @ SV12 ) ) ) )
= $true )
| ( ( element @ ( meet_of_subsets @ SV12 @ SV37 ) @ ( powerset @ SV12 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[755]) ).
thf(786,plain,
! [SV13: $i,SV38: $i] :
( ( ( ~ ( element @ SV38 @ ( powerset @ ( powerset @ SV13 ) ) ) )
= $true )
| ( ( ( complements_of_subsets @ SV13 @ ( complements_of_subsets @ SV13 @ SV38 ) )
= SV38 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[756]) ).
thf(787,plain,
! [SV14: $i,SV39: $i] :
( ( ( ~ ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) ) )
= $true )
| ( ( ~ ~ ( ~ ( ( SV39 != empty_set ) )
| ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[757]) ).
thf(788,plain,
! [SV15: $i] :
( ( ( v1_membered @ SV15 )
= $false )
| ( ( ! [SY79: $i] :
( ~ ( element @ SY79 @ ( powerset @ SV15 ) )
| ( v1_membered @ SY79 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[758]) ).
thf(789,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) ) )
= $true )
| ( ( ~ ( ~ ~ ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[759]) ).
thf(790,plain,
! [SV17: $i] :
( ( ( top_str @ SV17 )
= $false )
| ( ( one_sorted_str @ SV17 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[760]) ).
thf(791,plain,
! [SV52: $i,SV41: $i,SV18: $i] :
( ( ~ ~ ( ~ ( in @ SV18 @ SV41 )
| ~ ( element @ SV41 @ ( powerset @ SV52 ) ) )
| ( element @ SV18 @ SV52 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[761]) ).
thf(792,plain,
! [SV19: $i,SV42: $i] :
( ( ( ~ ( element @ SV42 @ ( powerset @ ( powerset @ SV19 ) ) ) )
= $true )
| ( ( element @ ( union_of_subsets @ SV19 @ SV42 ) @ ( powerset @ SV19 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[762]) ).
thf(793,plain,
! [SV20: $i] :
( ( ( top_str @ SV20 )
= $false )
| ( ( ! [SY84: $i] :
( ~ ( element @ SY84 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
| ~ ( ~ ( ~ ( closed_subsets @ SY84 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SY84 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SY84 ) @ SV20 )
| ( closed_subsets @ SY84 @ SV20 ) ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[763]) ).
thf(794,plain,
! [SV21: $i] :
( ( ( finite @ SV21 )
= $false )
| ( ( ! [SY85: $i] :
( ~ ( element @ SY85 @ ( powerset @ SV21 ) )
| ( finite @ SY85 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[764]) ).
thf(795,plain,
! [SV53: $i,SV22: $i,SV43: $i] :
( ( ~ ~ ( ~ ( element @ SV43 @ ( powerset @ SV22 ) )
| ~ ( element @ SV53 @ ( powerset @ SV22 ) ) )
| ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[765]) ).
thf(796,plain,
! [SV23: $i,SV44: $i] :
( ( ( ~ ( element @ SV44 @ ( powerset @ ( powerset @ SV23 ) ) ) )
= $true )
| ( ( element @ ( complements_of_subsets @ SV23 @ SV44 ) @ ( powerset @ ( powerset @ SV23 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[766]) ).
thf(797,plain,
! [SV45: $i,SV24: $i] :
( ( ( ~ ! [SY102: $i] :
( ~ ( in @ SY102 @ SV24 )
| ( in @ SY102 @ SV45 ) ) )
= $true )
| ( ( element @ SV24 @ ( powerset @ SV45 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[767]) ).
thf(798,plain,
! [SV25: $i] :
( ( ~ ( element @ ( sK5_SY91 @ SV25 ) @ SV25 ) )
= $false ),
inference(extcnf_forall_neg,[status(esa)],[768]) ).
thf(799,plain,
! [SV26: $i,SV46: $i] :
( ( ( ~ ( element @ SV46 @ ( powerset @ SV26 ) ) )
= $true )
| ( ( element @ ( subset_complement @ SV26 @ SV46 ) @ ( powerset @ SV26 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[769]) ).
thf(800,plain,
! [SV54: $i,SV27: $i,SV47: $i] :
( ( ~ ~ ( ~ ( element @ SV47 @ ( powerset @ SV27 ) )
| ~ ( element @ SV54 @ ( powerset @ SV27 ) ) )
| ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[770]) ).
thf(801,plain,
! [SV48: $i,SV28: $i] :
( ( ( ~ ~ ( ~ ( top_str @ SV28 )
| ~ ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) ) ) )
= $true )
| ( ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[771]) ).
thf(802,plain,
! [SV29: $i] :
( ( ( top_str @ SV29 )
= $false )
| ( ( element @ ( the_topology @ SV29 ) @ ( powerset @ ( powerset @ ( the_carrier @ SV29 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[772]) ).
thf(803,plain,
! [SV30: $i,SV49: $i] :
( ( ( ~ ( element @ SV49 @ ( powerset @ SV30 ) ) )
= $true )
| ( ( ! [SY104: $i] :
( ~ ( in @ SY104 @ SV49 )
| ( in @ SY104 @ SV30 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[773]) ).
thf(804,plain,
! [SV31: $i,SV4: $i] :
( ( ~ ( ~ ( element @ SV4 @ ( powerset @ SV31 ) )
| ( subset @ SV4 @ SV31 ) ) )
= $false ),
inference(extcnf_or_neg,[status(thm)],[774]) ).
thf(805,plain,
! [SV31: $i,SV4: $i] :
( ( ~ ( ~ ( subset @ SV4 @ SV31 )
| ( element @ SV4 @ ( powerset @ SV31 ) ) ) )
= $false ),
inference(extcnf_or_neg,[status(thm)],[774]) ).
thf(806,plain,
! [SV32: $i,SV5: $i] :
( ( ( ~ ( ~ ( top_str @ SV5 )
| ~ ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) ) ) )
= $false )
| ( ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[776]) ).
thf(807,plain,
! [SV50: $i,SV6: $i,SV33: $i] :
( ( ( ~ ~ ( ~ ( element @ SV33 @ ( powerset @ SV6 ) )
| ~ ( element @ SV50 @ ( powerset @ SV6 ) ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[777]) ).
thf(808,plain,
! [SV7: $i,SV34: $i] :
( ( ( element @ SV34 @ ( powerset @ SV7 ) )
= $false )
| ( ( ( subset_complement @ SV7 @ ( subset_complement @ SV7 @ SV34 ) )
= SV34 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[778]) ).
thf(809,plain,
! [SV51: $i,SV8: $i,SV35: $i] :
( ( ( ~ ~ ( ~ ( element @ SV35 @ ( powerset @ SV8 ) )
| ~ ( element @ SV51 @ ( powerset @ SV8 ) ) ) )
= $true )
| ( ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[779]) ).
thf(810,plain,
! [SV9: $i] :
( ( ( empty @ SV9 )
= $true )
| ( ( ~ ! [SY73: $i] :
~ ~ ( ~ ( element @ SY73 @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ SY73 ) ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[780]) ).
thf(811,plain,
( ( element @ sK3_SX0 @ ( powerset @ ( powerset @ ( the_carrier @ sK1_A ) ) ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[781]) ).
thf(812,plain,
( ( ~ ( ~ ( open_subsets @ sK3_SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) )
| ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[782]) ).
thf(813,plain,
! [SV36: $i,SV10: $i] :
( ( ( in @ SV10 @ SV36 )
= $false )
| ( ( element @ SV10 @ SV36 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[783]) ).
thf(814,plain,
! [SV11: $i] :
( ( ~ ( ~ ( element @ ( sK4_SY75 @ SV11 ) @ ( powerset @ SV11 ) )
| ~ ( empty @ ( sK4_SY75 @ SV11 ) ) ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[784]) ).
thf(815,plain,
! [SV12: $i,SV37: $i] :
( ( ( element @ SV37 @ ( powerset @ ( powerset @ SV12 ) ) )
= $false )
| ( ( element @ ( meet_of_subsets @ SV12 @ SV37 ) @ ( powerset @ SV12 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[785]) ).
thf(816,plain,
! [SV13: $i,SV38: $i] :
( ( ( element @ SV38 @ ( powerset @ ( powerset @ SV13 ) ) )
= $false )
| ( ( ( complements_of_subsets @ SV13 @ ( complements_of_subsets @ SV13 @ SV38 ) )
= SV38 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[786]) ).
thf(817,plain,
! [SV14: $i,SV39: $i] :
( ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false )
| ( ( ~ ~ ( ~ ( ( SV39 != empty_set ) )
| ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[787]) ).
thf(818,plain,
! [SV15: $i,SV55: $i] :
( ( ( ~ ( element @ SV55 @ ( powerset @ SV15 ) )
| ( v1_membered @ SV55 ) )
= $true )
| ( ( v1_membered @ SV15 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[788]) ).
thf(819,plain,
! [SV16: $i,SV40: $i] :
( ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false )
| ( ( ~ ( ~ ~ ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[789]) ).
thf(820,plain,
! [SV52: $i,SV41: $i,SV18: $i] :
( ( ( ~ ~ ( ~ ( in @ SV18 @ SV41 )
| ~ ( element @ SV41 @ ( powerset @ SV52 ) ) ) )
= $true )
| ( ( element @ SV18 @ SV52 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[791]) ).
thf(821,plain,
! [SV19: $i,SV42: $i] :
( ( ( element @ SV42 @ ( powerset @ ( powerset @ SV19 ) ) )
= $false )
| ( ( element @ ( union_of_subsets @ SV19 @ SV42 ) @ ( powerset @ SV19 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[792]) ).
thf(822,plain,
! [SV20: $i,SV56: $i] :
( ( ( ~ ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
| ~ ( ~ ( ~ ( closed_subsets @ SV56 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
| ( closed_subsets @ SV56 @ SV20 ) ) ) )
= $true )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[793]) ).
thf(823,plain,
! [SV21: $i,SV57: $i] :
( ( ( ~ ( element @ SV57 @ ( powerset @ SV21 ) )
| ( finite @ SV57 ) )
= $true )
| ( ( finite @ SV21 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[794]) ).
thf(824,plain,
! [SV53: $i,SV22: $i,SV43: $i] :
( ( ( ~ ~ ( ~ ( element @ SV43 @ ( powerset @ SV22 ) )
| ~ ( element @ SV53 @ ( powerset @ SV22 ) ) ) )
= $true )
| ( ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[795]) ).
thf(825,plain,
! [SV23: $i,SV44: $i] :
( ( ( element @ SV44 @ ( powerset @ ( powerset @ SV23 ) ) )
= $false )
| ( ( element @ ( complements_of_subsets @ SV23 @ SV44 ) @ ( powerset @ ( powerset @ SV23 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[796]) ).
thf(826,plain,
! [SV45: $i,SV24: $i] :
( ( ( ! [SY102: $i] :
( ~ ( in @ SY102 @ SV24 )
| ( in @ SY102 @ SV45 ) ) )
= $false )
| ( ( element @ SV24 @ ( powerset @ SV45 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[797]) ).
thf(827,plain,
! [SV25: $i] :
( ( element @ ( sK5_SY91 @ SV25 ) @ SV25 )
= $true ),
inference(extcnf_not_neg,[status(thm)],[798]) ).
thf(828,plain,
! [SV26: $i,SV46: $i] :
( ( ( element @ SV46 @ ( powerset @ SV26 ) )
= $false )
| ( ( element @ ( subset_complement @ SV26 @ SV46 ) @ ( powerset @ SV26 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[799]) ).
thf(829,plain,
! [SV54: $i,SV27: $i,SV47: $i] :
( ( ( ~ ~ ( ~ ( element @ SV47 @ ( powerset @ SV27 ) )
| ~ ( element @ SV54 @ ( powerset @ SV27 ) ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[800]) ).
thf(830,plain,
! [SV48: $i,SV28: $i] :
( ( ( ~ ( ~ ( top_str @ SV28 )
| ~ ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) ) ) )
= $false )
| ( ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[801]) ).
thf(831,plain,
! [SV30: $i,SV49: $i] :
( ( ( element @ SV49 @ ( powerset @ SV30 ) )
= $false )
| ( ( ! [SY104: $i] :
( ~ ( in @ SY104 @ SV49 )
| ( in @ SY104 @ SV30 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[803]) ).
thf(832,plain,
! [SV31: $i,SV4: $i] :
( ( ~ ( element @ SV4 @ ( powerset @ SV31 ) )
| ( subset @ SV4 @ SV31 ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[804]) ).
thf(833,plain,
! [SV31: $i,SV4: $i] :
( ( ~ ( subset @ SV4 @ SV31 )
| ( element @ SV4 @ ( powerset @ SV31 ) ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[805]) ).
thf(834,plain,
! [SV32: $i,SV5: $i] :
( ( ( ~ ( top_str @ SV5 )
| ~ ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) ) )
= $true )
| ( ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[806]) ).
thf(835,plain,
! [SV50: $i,SV6: $i,SV33: $i] :
( ( ( ~ ( ~ ( element @ SV33 @ ( powerset @ SV6 ) )
| ~ ( element @ SV50 @ ( powerset @ SV6 ) ) ) )
= $false )
| ( ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[807]) ).
thf(836,plain,
! [SV51: $i,SV8: $i,SV35: $i] :
( ( ( ~ ( ~ ( element @ SV35 @ ( powerset @ SV8 ) )
| ~ ( element @ SV51 @ ( powerset @ SV8 ) ) ) )
= $false )
| ( ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[809]) ).
thf(837,plain,
! [SV9: $i] :
( ( ( ! [SY73: $i] :
~ ~ ( ~ ( element @ SY73 @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ SY73 ) ) )
= $false )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[810]) ).
thf(838,plain,
( ( ( ~ ( ~ ( open_subsets @ sK3_SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) ) )
= $true )
| ( ( ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[812]) ).
thf(839,plain,
! [SV11: $i] :
( ( ~ ( element @ ( sK4_SY75 @ SV11 ) @ ( powerset @ SV11 ) )
| ~ ( empty @ ( sK4_SY75 @ SV11 ) ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[814]) ).
thf(840,plain,
! [SV14: $i,SV39: $i] :
( ( ( ~ ( ~ ( ( SV39 != empty_set ) )
| ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set ) ) )
= $false )
| ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[817]) ).
thf(841,plain,
! [SV15: $i,SV55: $i] :
( ( ( ~ ( element @ SV55 @ ( powerset @ SV15 ) ) )
= $true )
| ( ( v1_membered @ SV55 )
= $true )
| ( ( v1_membered @ SV15 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[818]) ).
thf(842,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ~ ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) )
= $false )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[819]) ).
thf(843,plain,
! [SV52: $i,SV41: $i,SV18: $i] :
( ( ( ~ ( ~ ( in @ SV18 @ SV41 )
| ~ ( element @ SV41 @ ( powerset @ SV52 ) ) ) )
= $false )
| ( ( element @ SV18 @ SV52 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[820]) ).
thf(844,plain,
! [SV20: $i,SV56: $i] :
( ( ( ~ ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) ) )
= $true )
| ( ( ~ ( ~ ( ~ ( closed_subsets @ SV56 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
| ( closed_subsets @ SV56 @ SV20 ) ) ) )
= $true )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[822]) ).
thf(845,plain,
! [SV21: $i,SV57: $i] :
( ( ( ~ ( element @ SV57 @ ( powerset @ SV21 ) ) )
= $true )
| ( ( finite @ SV57 )
= $true )
| ( ( finite @ SV21 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[823]) ).
thf(846,plain,
! [SV53: $i,SV22: $i,SV43: $i] :
( ( ( ~ ( ~ ( element @ SV43 @ ( powerset @ SV22 ) )
| ~ ( element @ SV53 @ ( powerset @ SV22 ) ) ) )
= $false )
| ( ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[824]) ).
thf(847,plain,
! [SV24: $i,SV45: $i] :
( ( ( ~ ( in @ ( sK6_SY102 @ SV45 @ SV24 ) @ SV24 )
| ( in @ ( sK6_SY102 @ SV45 @ SV24 ) @ SV45 ) )
= $false )
| ( ( element @ SV24 @ ( powerset @ SV45 ) )
= $true ) ),
inference(extcnf_forall_neg,[status(esa)],[826]) ).
thf(848,plain,
! [SV54: $i,SV27: $i,SV47: $i] :
( ( ( ~ ( ~ ( element @ SV47 @ ( powerset @ SV27 ) )
| ~ ( element @ SV54 @ ( powerset @ SV27 ) ) ) )
= $false )
| ( ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[829]) ).
thf(849,plain,
! [SV48: $i,SV28: $i] :
( ( ( ~ ( top_str @ SV28 )
| ~ ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) ) )
= $true )
| ( ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[830]) ).
thf(850,plain,
! [SV30: $i,SV49: $i,SV58: $i] :
( ( ( ~ ( in @ SV58 @ SV49 )
| ( in @ SV58 @ SV30 ) )
= $true )
| ( ( element @ SV49 @ ( powerset @ SV30 ) )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[831]) ).
thf(851,plain,
! [SV31: $i,SV4: $i] :
( ( ( ~ ( element @ SV4 @ ( powerset @ SV31 ) ) )
= $true )
| ( ( subset @ SV4 @ SV31 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[832]) ).
thf(852,plain,
! [SV31: $i,SV4: $i] :
( ( ( ~ ( subset @ SV4 @ SV31 ) )
= $true )
| ( ( element @ SV4 @ ( powerset @ SV31 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[833]) ).
thf(853,plain,
! [SV32: $i,SV5: $i] :
( ( ( ~ ( top_str @ SV5 ) )
= $true )
| ( ( ~ ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) ) )
= $true )
| ( ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[834]) ).
thf(854,plain,
! [SV50: $i,SV6: $i,SV33: $i] :
( ( ( ~ ( element @ SV33 @ ( powerset @ SV6 ) )
| ~ ( element @ SV50 @ ( powerset @ SV6 ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[835]) ).
thf(855,plain,
! [SV51: $i,SV8: $i,SV35: $i] :
( ( ( ~ ( element @ SV35 @ ( powerset @ SV8 ) )
| ~ ( element @ SV51 @ ( powerset @ SV8 ) ) )
= $true )
| ( ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[836]) ).
thf(856,plain,
! [SV9: $i] :
( ( ( ~ ~ ( ~ ( element @ ( sK7_SY73 @ SV9 ) @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ ( sK7_SY73 @ SV9 ) ) ) )
= $false )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_forall_neg,[status(esa)],[837]) ).
thf(857,plain,
( ( ( ~ ( open_subsets @ sK3_SX0 @ sK1_A )
| ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) )
= $false )
| ( ( ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[838]) ).
thf(858,plain,
! [SV11: $i] :
( ( ~ ( element @ ( sK4_SY75 @ SV11 ) @ ( powerset @ SV11 ) ) )
= $false ),
inference(extcnf_or_neg,[status(thm)],[839]) ).
thf(859,plain,
! [SV11: $i] :
( ( ~ ( empty @ ( sK4_SY75 @ SV11 ) ) )
= $false ),
inference(extcnf_or_neg,[status(thm)],[839]) ).
thf(860,plain,
! [SV14: $i,SV39: $i] :
( ( ( ~ ( ( SV39 != empty_set ) )
| ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set ) )
= $true )
| ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[840]) ).
thf(861,plain,
! [SV15: $i,SV55: $i] :
( ( ( element @ SV55 @ ( powerset @ SV15 ) )
= $false )
| ( ( v1_membered @ SV55 )
= $true )
| ( ( v1_membered @ SV15 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[841]) ).
thf(862,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ~ ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) ) )
= $false )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_or_neg,[status(thm)],[842]) ).
thf(863,plain,
! [SV40: $i,SV16: $i] :
( ( ( ~ ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) )
= $false )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_or_neg,[status(thm)],[842]) ).
thf(864,plain,
! [SV52: $i,SV41: $i,SV18: $i] :
( ( ( ~ ( in @ SV18 @ SV41 )
| ~ ( element @ SV41 @ ( powerset @ SV52 ) ) )
= $true )
| ( ( element @ SV18 @ SV52 )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[843]) ).
thf(865,plain,
! [SV20: $i,SV56: $i] :
( ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( ~ ( ~ ( ~ ( closed_subsets @ SV56 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
| ( closed_subsets @ SV56 @ SV20 ) ) ) )
= $true )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[844]) ).
thf(866,plain,
! [SV21: $i,SV57: $i] :
( ( ( element @ SV57 @ ( powerset @ SV21 ) )
= $false )
| ( ( finite @ SV57 )
= $true )
| ( ( finite @ SV21 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[845]) ).
thf(867,plain,
! [SV53: $i,SV22: $i,SV43: $i] :
( ( ( ~ ( element @ SV43 @ ( powerset @ SV22 ) )
| ~ ( element @ SV53 @ ( powerset @ SV22 ) ) )
= $true )
| ( ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[846]) ).
thf(868,plain,
! [SV24: $i,SV45: $i] :
( ( ( ~ ( in @ ( sK6_SY102 @ SV45 @ SV24 ) @ SV24 ) )
= $false )
| ( ( element @ SV24 @ ( powerset @ SV45 ) )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[847]) ).
thf(869,plain,
! [SV24: $i,SV45: $i] :
( ( ( in @ ( sK6_SY102 @ SV45 @ SV24 ) @ SV45 )
= $false )
| ( ( element @ SV24 @ ( powerset @ SV45 ) )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[847]) ).
thf(870,plain,
! [SV54: $i,SV27: $i,SV47: $i] :
( ( ( ~ ( element @ SV47 @ ( powerset @ SV27 ) )
| ~ ( element @ SV54 @ ( powerset @ SV27 ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[848]) ).
thf(871,plain,
! [SV48: $i,SV28: $i] :
( ( ( ~ ( top_str @ SV28 ) )
= $true )
| ( ( ~ ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) ) )
= $true )
| ( ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[849]) ).
thf(872,plain,
! [SV30: $i,SV49: $i,SV58: $i] :
( ( ( ~ ( in @ SV58 @ SV49 ) )
= $true )
| ( ( in @ SV58 @ SV30 )
= $true )
| ( ( element @ SV49 @ ( powerset @ SV30 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[850]) ).
thf(873,plain,
! [SV31: $i,SV4: $i] :
( ( ( element @ SV4 @ ( powerset @ SV31 ) )
= $false )
| ( ( subset @ SV4 @ SV31 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[851]) ).
thf(874,plain,
! [SV31: $i,SV4: $i] :
( ( ( subset @ SV4 @ SV31 )
= $false )
| ( ( element @ SV4 @ ( powerset @ SV31 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[852]) ).
thf(875,plain,
! [SV32: $i,SV5: $i] :
( ( ( top_str @ SV5 )
= $false )
| ( ( ~ ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) ) )
= $true )
| ( ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[853]) ).
thf(876,plain,
! [SV50: $i,SV6: $i,SV33: $i] :
( ( ( ~ ( element @ SV33 @ ( powerset @ SV6 ) ) )
= $true )
| ( ( ~ ( element @ SV50 @ ( powerset @ SV6 ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[854]) ).
thf(877,plain,
! [SV51: $i,SV8: $i,SV35: $i] :
( ( ( ~ ( element @ SV35 @ ( powerset @ SV8 ) ) )
= $true )
| ( ( ~ ( element @ SV51 @ ( powerset @ SV8 ) ) )
= $true )
| ( ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[855]) ).
thf(878,plain,
! [SV9: $i] :
( ( ( ~ ( ~ ( element @ ( sK7_SY73 @ SV9 ) @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ ( sK7_SY73 @ SV9 ) ) ) )
= $true )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[856]) ).
thf(879,plain,
( ( ( ~ ( open_subsets @ sK3_SX0 @ sK1_A ) )
= $false )
| ( ( ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[857]) ).
thf(880,plain,
( ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $false )
| ( ( ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[857]) ).
thf(881,plain,
! [SV11: $i] :
( ( element @ ( sK4_SY75 @ SV11 ) @ ( powerset @ SV11 ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[858]) ).
thf(882,plain,
! [SV11: $i] :
( ( empty @ ( sK4_SY75 @ SV11 ) )
= $true ),
inference(extcnf_not_neg,[status(thm)],[859]) ).
thf(883,plain,
! [SV14: $i,SV39: $i] :
( ( ( ~ ( ( SV39 != empty_set ) ) )
= $true )
| ( ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set )
= $true )
| ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[860]) ).
thf(884,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) ) )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[862]) ).
thf(885,plain,
! [SV40: $i,SV16: $i] :
( ( ( ~ ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[863]) ).
thf(886,plain,
! [SV52: $i,SV41: $i,SV18: $i] :
( ( ( ~ ( in @ SV18 @ SV41 ) )
= $true )
| ( ( ~ ( element @ SV41 @ ( powerset @ SV52 ) ) )
= $true )
| ( ( element @ SV18 @ SV52 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[864]) ).
thf(887,plain,
! [SV20: $i,SV56: $i] :
( ( ( ~ ( ~ ( closed_subsets @ SV56 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) )
| ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
| ( closed_subsets @ SV56 @ SV20 ) ) )
= $false )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[865]) ).
thf(888,plain,
! [SV53: $i,SV22: $i,SV43: $i] :
( ( ( ~ ( element @ SV43 @ ( powerset @ SV22 ) ) )
= $true )
| ( ( ~ ( element @ SV53 @ ( powerset @ SV22 ) ) )
= $true )
| ( ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[867]) ).
thf(889,plain,
! [SV24: $i,SV45: $i] :
( ( ( in @ ( sK6_SY102 @ SV45 @ SV24 ) @ SV24 )
= $true )
| ( ( element @ SV24 @ ( powerset @ SV45 ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[868]) ).
thf(890,plain,
! [SV54: $i,SV27: $i,SV47: $i] :
( ( ( ~ ( element @ SV47 @ ( powerset @ SV27 ) ) )
= $true )
| ( ( ~ ( element @ SV54 @ ( powerset @ SV27 ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[870]) ).
thf(891,plain,
! [SV48: $i,SV28: $i] :
( ( ( top_str @ SV28 )
= $false )
| ( ( ~ ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) ) )
= $true )
| ( ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[871]) ).
thf(892,plain,
! [SV30: $i,SV49: $i,SV58: $i] :
( ( ( in @ SV58 @ SV49 )
= $false )
| ( ( in @ SV58 @ SV30 )
= $true )
| ( ( element @ SV49 @ ( powerset @ SV30 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[872]) ).
thf(893,plain,
! [SV5: $i,SV32: $i] :
( ( ( element @ SV32 @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $false )
| ( ( top_str @ SV5 )
= $false )
| ( ( element @ ( topstr_closure @ SV5 @ SV32 ) @ ( powerset @ ( the_carrier @ SV5 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[875]) ).
thf(894,plain,
! [SV50: $i,SV6: $i,SV33: $i] :
( ( ( element @ SV33 @ ( powerset @ SV6 ) )
= $false )
| ( ( ~ ( element @ SV50 @ ( powerset @ SV6 ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[876]) ).
thf(895,plain,
! [SV51: $i,SV8: $i,SV35: $i] :
( ( ( element @ SV35 @ ( powerset @ SV8 ) )
= $false )
| ( ( ~ ( element @ SV51 @ ( powerset @ SV8 ) ) )
= $true )
| ( ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[877]) ).
thf(896,plain,
! [SV9: $i] :
( ( ( ~ ( element @ ( sK7_SY73 @ SV9 ) @ ( powerset @ SV9 ) )
| ~ ~ ( empty @ ( sK7_SY73 @ SV9 ) ) )
= $false )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[878]) ).
thf(897,plain,
( ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $true )
| ( ( ~ ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) ) )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[879]) ).
thf(898,plain,
( ( ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) )
= $false )
| ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[880]) ).
thf(899,plain,
! [SV14: $i,SV39: $i] :
( ( ( SV39 != empty_set )
= $false )
| ( ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set )
= $true )
| ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[883]) ).
thf(900,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) ) )
= $false )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[884]) ).
thf(901,plain,
! [SV40: $i,SV16: $i] :
( ( ( ~ ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) ) )
= $false )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[885]) ).
thf(902,plain,
! [SV52: $i,SV41: $i,SV18: $i] :
( ( ( in @ SV18 @ SV41 )
= $false )
| ( ( ~ ( element @ SV41 @ ( powerset @ SV52 ) ) )
= $true )
| ( ( element @ SV18 @ SV52 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[886]) ).
thf(903,plain,
! [SV20: $i,SV56: $i] :
( ( ( ~ ( ~ ( closed_subsets @ SV56 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) ) )
= $false )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_or_neg,[status(thm)],[887]) ).
thf(904,plain,
! [SV56: $i,SV20: $i] :
( ( ( ~ ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
| ( closed_subsets @ SV56 @ SV20 ) ) )
= $false )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_or_neg,[status(thm)],[887]) ).
thf(905,plain,
! [SV53: $i,SV22: $i,SV43: $i] :
( ( ( element @ SV43 @ ( powerset @ SV22 ) )
= $false )
| ( ( ~ ( element @ SV53 @ ( powerset @ SV22 ) ) )
= $true )
| ( ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[888]) ).
thf(906,plain,
! [SV54: $i,SV27: $i,SV47: $i] :
( ( ( element @ SV47 @ ( powerset @ SV27 ) )
= $false )
| ( ( ~ ( element @ SV54 @ ( powerset @ SV27 ) ) )
= $true )
| ( ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[890]) ).
thf(907,plain,
! [SV28: $i,SV48: $i] :
( ( ( element @ SV48 @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $false )
| ( ( top_str @ SV28 )
= $false )
| ( ( element @ ( interior @ SV28 @ SV48 ) @ ( powerset @ ( the_carrier @ SV28 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[891]) ).
thf(908,plain,
! [SV33: $i,SV6: $i,SV50: $i] :
( ( ( element @ SV50 @ ( powerset @ SV6 ) )
= $false )
| ( ( element @ SV33 @ ( powerset @ SV6 ) )
= $false )
| ( ( ( subset_intersection2 @ SV6 @ SV33 @ SV50 )
= ( subset_intersection2 @ SV6 @ SV50 @ SV33 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[894]) ).
thf(909,plain,
! [SV35: $i,SV8: $i,SV51: $i] :
( ( ( element @ SV51 @ ( powerset @ SV8 ) )
= $false )
| ( ( element @ SV35 @ ( powerset @ SV8 ) )
= $false )
| ( ( element @ ( subset_difference @ SV8 @ SV35 @ SV51 ) @ ( powerset @ SV8 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[895]) ).
thf(910,plain,
! [SV9: $i] :
( ( ( ~ ( element @ ( sK7_SY73 @ SV9 ) @ ( powerset @ SV9 ) ) )
= $false )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[896]) ).
thf(911,plain,
! [SV9: $i] :
( ( ( ~ ~ ( empty @ ( sK7_SY73 @ SV9 ) ) )
= $false )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[896]) ).
thf(912,plain,
( ( ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
| ( open_subsets @ sK3_SX0 @ sK1_A ) )
= $false )
| ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[897]) ).
thf(913,plain,
( ( ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) )
= $false )
| ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $false ) ),
inference(extcnf_or_neg,[status(thm)],[898]) ).
thf(914,plain,
( ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $false )
| ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $false ) ),
inference(extcnf_or_neg,[status(thm)],[898]) ).
thf(915,plain,
! [SV14: $i,SV39: $i] :
( ( ( SV39 = empty_set )
= $true )
| ( ( ( complements_of_subsets @ SV14 @ SV39 )
!= empty_set )
= $true )
| ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[899]) ).
thf(916,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ( ( SV40 != empty_set ) )
| ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[900]) ).
thf(917,plain,
! [SV40: $i,SV16: $i] :
( ( ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) )
| ( SV40 != empty_set ) )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[901]) ).
thf(918,plain,
! [SV18: $i,SV52: $i,SV41: $i] :
( ( ( element @ SV41 @ ( powerset @ SV52 ) )
= $false )
| ( ( in @ SV18 @ SV41 )
= $false )
| ( ( element @ SV18 @ SV52 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[902]) ).
thf(919,plain,
! [SV20: $i,SV56: $i] :
( ( ( ~ ( closed_subsets @ SV56 @ SV20 )
| ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) )
= $true )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[903]) ).
thf(920,plain,
! [SV56: $i,SV20: $i] :
( ( ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
| ( closed_subsets @ SV56 @ SV20 ) )
= $true )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[904]) ).
thf(921,plain,
! [SV43: $i,SV22: $i,SV53: $i] :
( ( ( element @ SV53 @ ( powerset @ SV22 ) )
= $false )
| ( ( element @ SV43 @ ( powerset @ SV22 ) )
= $false )
| ( ( element @ ( subset_intersection2 @ SV22 @ SV43 @ SV53 ) @ ( powerset @ SV22 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[905]) ).
thf(922,plain,
! [SV47: $i,SV27: $i,SV54: $i] :
( ( ( element @ SV54 @ ( powerset @ SV27 ) )
= $false )
| ( ( element @ SV47 @ ( powerset @ SV27 ) )
= $false )
| ( ( ( subset_intersection2 @ SV27 @ SV47 @ SV47 )
= SV47 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[906]) ).
thf(923,plain,
! [SV9: $i] :
( ( ( element @ ( sK7_SY73 @ SV9 ) @ ( powerset @ SV9 ) )
= $true )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[910]) ).
thf(924,plain,
! [SV9: $i] :
( ( ( ~ ( empty @ ( sK7_SY73 @ SV9 ) ) )
= $true )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[911]) ).
thf(925,plain,
( ( ( ~ ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A ) )
= $false )
| ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[912]) ).
thf(926,plain,
( ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $false )
| ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $true ) ),
inference(extcnf_or_neg,[status(thm)],[912]) ).
thf(927,plain,
( ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $true )
| ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[913]) ).
thf(928,plain,
! [SV39: $i,SV14: $i] :
( ( ( ( complements_of_subsets @ SV14 @ SV39 )
= empty_set )
= $false )
| ( ( SV39 = empty_set )
= $true )
| ( ( element @ SV39 @ ( powerset @ ( powerset @ SV14 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[915]) ).
thf(929,plain,
! [SV16: $i,SV40: $i] :
( ( ( ~ ( ( SV40 != empty_set ) ) )
= $true )
| ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[916]) ).
thf(930,plain,
! [SV40: $i,SV16: $i] :
( ( ( ~ ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set ) ) )
= $true )
| ( ( SV40 != empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[917]) ).
thf(931,plain,
! [SV20: $i,SV56: $i] :
( ( ( ~ ( closed_subsets @ SV56 @ SV20 ) )
= $true )
| ( ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
= $true )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[919]) ).
thf(932,plain,
! [SV56: $i,SV20: $i] :
( ( ( ~ ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 ) )
= $true )
| ( ( closed_subsets @ SV56 @ SV20 )
= $true )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[920]) ).
thf(933,plain,
! [SV9: $i] :
( ( ( empty @ ( sK7_SY73 @ SV9 ) )
= $false )
| ( ( empty @ SV9 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[924]) ).
thf(934,plain,
( ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ sK1_A ) @ sK3_SX0 ) @ sK1_A )
= $true )
| ( ( open_subsets @ sK3_SX0 @ sK1_A )
= $true ) ),
inference(extcnf_not_neg,[status(thm)],[925]) ).
thf(935,plain,
! [SV16: $i,SV40: $i] :
( ( ( SV40 != empty_set )
= $false )
| ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[929]) ).
thf(936,plain,
! [SV40: $i,SV16: $i] :
( ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set )
= $false )
| ( ( SV40 != empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[930]) ).
thf(937,plain,
! [SV20: $i,SV56: $i] :
( ( ( closed_subsets @ SV56 @ SV20 )
= $false )
| ( ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
= $true )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[931]) ).
thf(938,plain,
! [SV56: $i,SV20: $i] :
( ( ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ SV20 ) @ SV56 ) @ SV20 )
= $false )
| ( ( closed_subsets @ SV56 @ SV20 )
= $true )
| ( ( element @ SV56 @ ( powerset @ ( powerset @ ( the_carrier @ SV20 ) ) ) )
= $false )
| ( ( top_str @ SV20 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[932]) ).
thf(939,plain,
! [SV16: $i,SV40: $i] :
( ( ( SV40 = empty_set )
= $true )
| ( ( ( complements_of_subsets @ SV16 @ SV40 )
!= empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[935]) ).
thf(940,plain,
! [SV40: $i,SV16: $i] :
( ( ( ( complements_of_subsets @ SV16 @ SV40 )
= empty_set )
= $true )
| ( ( SV40 != empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_neg,[status(thm)],[936]) ).
thf(941,plain,
! [SV40: $i,SV16: $i] :
( ( ( ( complements_of_subsets @ SV16 @ SV40 )
= empty_set )
= $false )
| ( ( SV40 = empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[939]) ).
thf(942,plain,
! [SV16: $i,SV40: $i] :
( ( ( SV40 = empty_set )
= $false )
| ( ( ( complements_of_subsets @ SV16 @ SV40 )
= empty_set )
= $true )
| ( ( element @ SV40 @ ( powerset @ ( powerset @ SV16 ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[940]) ).
thf(943,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[631,942,941,938,937,934,933,928,927,926,923,922,921,918,914,909,908,907,893,892,889,882,881,874,873,869,866,861,828,827,825,821,816,815,813,811,808,802,790,775,744,713,712,681,669,667,666,665,663,656,655,654,653,652,650,649,648,647,645,644,643,642,641,639,638,637,636,635,634,633,632]) ).
thf(944,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[943]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU337+2 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.04 % Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.22 % Computer : n018.cluster.edu
% 0.09/0.22 % Model : x86_64 x86_64
% 0.09/0.22 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.22 % Memory : 8046.5625MB
% 0.09/0.22 % OS : Linux 6.8.0-71-generic
% 0.09/0.22 % CPULimit : 300
% 0.09/0.22 % WCLimit : 300
% 0.09/0.22 % DateTime : Tue Oct 6 15:16:37 UTC 2026
% 0.09/0.22 % CPUTime :
% 0.09/0.22 Running leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 13.76/2.66 .....
% 13.76/2.66
% 13.76/2.66 No.of.Axioms: 59
% 13.76/2.66
% 13.76/2.66 Length.of.Defs: 0
% 13.76/2.66
% 13.76/2.66 Contains.Choice.Funs: false
% 13.76/2.66 ................
% 13.76/2.66
% 13.76/2.66 ********************************
% 13.76/2.66 * All subproblems solved! *
% 13.76/2.66 ********************************
% 13.76/2.66 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:1,axioms:60,ps:0,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:false,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:943,loop_count:0,foatp_calls:1,translation:fof_full)
% 13.76/2.66
% 13.76/2.66 %**** Beginning of derivation protocol ****
% 13.76/2.66 % SZS output start CNFRefutation
% See solution above
% 13.76/2.66
% 13.76/2.66 %**** End of derivation protocol ****
% 13.76/2.66 %**** no. of clauses in derivation: 448 ****
% 13.76/2.66 %**** clause counter: 943 ****
% 13.76/2.66
% 13.76/2.66 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:1,axioms:60,ps:0,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:false,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:943,loop_count:0,foatp_calls:1,translation:fof_full)
%------------------------------------------------------------------------------