TPTP Problem File: SLH0026^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0015_Design_Extras/prob_00159_005410__27895072_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1414 ( 475 unt; 145 typ; 0 def)
% Number of atoms : 3464 (1081 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10593 ( 503 ~; 44 |; 204 &;8213 @)
% ( 0 <=>;1629 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 284 ( 284 >; 0 *; 0 +; 0 <<)
% Number of symbols : 133 ( 132 usr; 15 con; 0-3 aty)
% Number of variables : 3045 ( 108 ^;2865 !; 72 ?;3045 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:45:25.287
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J_J,type,
set_multiset_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
multiset_set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
multiset_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (132)
thf(sy_c_Design__Basics_Odesign_001t__Nat__Onat,type,
design_design_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Odesign_001t__Set__Oset_Itf__a_J,type,
design_design_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Odesign_001tf__a,type,
design_design_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Nat__Onat,type,
design5426232790142929158em_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Set__Oset_Itf__a_J,type,
design1749870844763721896_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001tf__a,type,
design9187838744727572296stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001t__Nat__Onat,type,
design3753904077504641269em_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001t__Set__Oset_Itf__a_J,type,
design9013482484999600761_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001tf__a,type,
design1863209521793301785stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001t__Nat__Onat,type,
design2875492832550762736nt_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001t__Set__Oset_Itf__a_J,type,
design4243878040612417342_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001tf__a,type,
design6447616907850319326ment_a: set_a > set_a > set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Ocomplement__blocks_001t__Nat__Onat,type,
design5569578106646884273ks_nat: set_nat > multiset_set_nat > multiset_set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Ocomplement__blocks_001t__Set__Oset_Itf__a_J,type,
design7413023778852989629_set_a: set_set_a > multiset_set_set_a > multiset_set_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Ocomplement__blocks_001tf__a,type,
design8640656491286871389ocks_a: set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Odesign__support_001tf__a,type,
design5397942185814921632port_a: multiset_set_a > set_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Oincident_001t__Nat__Onat,type,
design8502206366797944887nt_nat: multiset_set_nat > nat > set_nat > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Oincident_001t__Set__Oset_Itf__a_J,type,
design6773327923283668919_set_a: multiset_set_set_a > set_a > set_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Oincident_001tf__a,type,
design3210447939978979927dent_a: multiset_set_a > a > set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Opoint__indices_001tf__a,type,
design328527185268214962ices_a: set_a > multiset_set_a > nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001t__Nat__Onat,type,
design3853898657598026467rs_nat: set_nat > multiset_set_nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001t__Set__Oset_Itf__a_J,type,
design5240413817448814603_set_a: set_set_a > multiset_set_set_a > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001tf__a,type,
design8835372594653258411bers_a: set_a > multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001t__Nat__Onat,type,
design8152002643121538447es_nat: multiset_set_nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001tf__a,type,
design1769254222028858111izes_a: multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_On__intersect__number_001tf__a,type,
design735257067508376852mber_a: set_a > nat > set_a > nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001t__Nat__Onat,type,
design3571518413069006949er_nat: multiset_set_nat > nat > nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001t__Set__Oset_Itf__a_J,type,
design5008467512594872073_set_a: multiset_set_set_a > set_a > nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001tf__a,type,
design6637022207325878697mber_a: multiset_set_a > a > nat ).
thf(sy_c_Design__Basics_Opoints__index_001tf__a,type,
design254580327166089565ndex_a: multiset_set_a > set_a > nat ).
thf(sy_c_Design__Basics_Oproper__design_001t__Nat__Onat,type,
design435815215503836206gn_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Oproper__design_001tf__a,type,
design7287791228148780576sign_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Osimple__incidence__system_001tf__a,type,
design1338723777345758283stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001t__Nat__Onat,type,
design4725324266511619850ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001t__Set__Oset_Itf__a_J,type,
design7860908649167014820_set_a: multiset_set_set_a > set_set_a > multiset_set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001tf__a,type,
design4001997691126659652lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001t__Nat__Onat,type,
design8239173135376323853nt_nat: set_nat > nat > set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001t__Set__Oset_Itf__a_J,type,
design4648949625254728801_set_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001tf__a,type,
design2964366272795260673oint_a: set_a > a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001t__Nat__Onat,type,
design5698312687278145166ks_nat: multiset_set_nat > nat > set_set_nat > multiset_set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001t__Set__Oset_Itf__a_J,type,
design9181828390639750432_set_a: multiset_set_set_a > set_a > set_set_set_a > multiset_set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001tf__a,type,
design2935547469388721088ocks_a: multiset_set_a > a > set_set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__block_001tf__a,type,
design1146539425385464078lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001t__Nat__Onat,type,
design4269233978287968195nt_nat: set_nat > nat > set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001t__Set__Oset_Itf__a_J,type,
design7586725432863044395_set_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001tf__a,type,
design108908007054065099oint_a: set_a > a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001t__Nat__Onat,type,
design4832208198062110345ks_nat: multiset_set_nat > nat > multiset_set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001t__Set__Oset_Itf__a_J,type,
design5868748293471848933_set_a: multiset_set_set_a > set_a > multiset_set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001tf__a,type,
design6411949732824333445ocks_a: multiset_set_a > a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__block_001t__Nat__Onat,type,
design3550126062406151447ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__block_001tf__a,type,
design4241783006516448631lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001t__Nat__Onat,type,
design3278834155446248416ks_nat: multiset_set_nat > nat > multiset_set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001t__Set__Oset_Itf__a_J,type,
design9174845397011619918_set_a: multiset_set_set_a > set_a > multiset_set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001tf__a,type,
design5657747894866638574ocks_a: multiset_set_a > a > multiset_set_a ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
finite2815193924343055693_set_a: set_multiset_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
zero_z3157962936165190495et_nat: multiset_set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
zero_z6396401802697562811_set_a: multiset_set_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
zero_z5079479921072680283_set_a: multiset_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Nat__Onat,type,
count_nat: multiset_nat > nat > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Nat__Onat,type,
repeat_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_It__Nat__Onat_J,type,
repeat_mset_set_nat: nat > multiset_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
repeat3222187171979612824_set_a: nat > multiset_set_set_a > multiset_set_set_a ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_Itf__a_J,type,
repeat_mset_set_a: nat > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Orepeat__mset_001tf__a,type,
repeat_mset_a: nat > multiset_a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Nat__Onat_J,type,
set_mset_set_nat: multiset_set_nat > set_set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_mset_set_set_a: multiset_set_set_a > set_set_set_a ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Set__Oset_Itf__a_J,type,
subseteq_mset_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s7462436076474991978et_nat: multiset_set_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
size_s5830485025544567152_set_a: multiset_set_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
size_s6566526139600085008_set_a: multiset_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo9088538438451294192_set_a: set_multiset_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
member2747690772047059533_set_a: multiset_set_a > set_multiset_set_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_bl,type,
bl: set_a ).
thf(sy_v_block__collection,type,
block_collection: multiset_set_a ).
thf(sy_v_point__set,type,
point_set: set_a ).
% Relevant facts (1268)
thf(fact_0_delete__block__design,axiom,
! [Bl: set_a] : ( design_design_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ Bl ) ) ).
% delete_block_design
thf(fact_1_wf__design,axiom,
design_design_a @ point_set @ block_collection ).
% wf_design
thf(fact_2_proper__design__axioms,axiom,
design7287791228148780576sign_a @ point_set @ block_collection ).
% proper_design_axioms
thf(fact_3_add__point__existing__blocks,axiom,
! [Bs: set_set_a,P: a] :
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ Bs )
=> ( member_a @ P @ Bl2 ) )
=> ( ( design2935547469388721088ocks_a @ block_collection @ P @ Bs )
= block_collection ) ) ).
% add_point_existing_blocks
thf(fact_4_add__point__design,axiom,
! [P: a] : ( design_design_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection ) ).
% add_point_design
thf(fact_5_del__invalid__point__blocks,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( ( design6411949732824333445ocks_a @ block_collection @ P )
= block_collection ) ) ).
% del_invalid_point_blocks
thf(fact_6_delete__invalid__pt__strong__eq,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( block_collection
= ( design5657747894866638574ocks_a @ block_collection @ P ) ) ) ).
% delete_invalid_pt_strong_eq
thf(fact_7_delete__block__fin__incidence__sys,axiom,
! [B: set_a] : ( design9187838744727572296stem_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ B ) ) ).
% delete_block_fin_incidence_sys
thf(fact_8_delete__block__wf,axiom,
! [B: set_a] : ( design1863209521793301785stem_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ B ) ) ).
% delete_block_wf
thf(fact_9_multiple__is__design,axiom,
! [N: nat] : ( design_design_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ).
% multiple_is_design
thf(fact_10_del__invalid__point,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( ( design108908007054065099oint_a @ point_set @ P )
= point_set ) ) ).
% del_invalid_point
thf(fact_11_finite__incidence__system__axioms,axiom,
design9187838744727572296stem_a @ point_set @ block_collection ).
% finite_incidence_system_axioms
thf(fact_12_wf__invalid__point,axiom,
! [X: a,B: set_a] :
( ~ ( member_a @ X @ point_set )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ~ ( member_a @ X @ B ) ) ) ).
% wf_invalid_point
thf(fact_13_design_Odelete__block__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ ( design1146539425385464078lock_a @ Block_collection @ Bl ) ) ) ).
% design.delete_block_design
thf(fact_14_delete__block__subset,axiom,
! [B: set_a] : ( subseteq_mset_set_a @ ( design1146539425385464078lock_a @ block_collection @ B ) @ block_collection ) ).
% delete_block_subset
thf(fact_15_multiple__block__in__original,axiom,
! [B: set_a,N: nat] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) ) ) ).
% multiple_block_in_original
thf(fact_16_delete__invalid__block__eq,axiom,
! [B: set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ( design1146539425385464078lock_a @ block_collection @ B )
= block_collection ) ) ).
% delete_invalid_block_eq
thf(fact_17_delete__point__strong__block__not__in,axiom,
! [P: a,Bl: set_a] :
( ( member_a @ P @ Bl )
=> ~ ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ) ) ).
% delete_point_strong_block_not_in
thf(fact_18_delete__point__strong__block__in__orig,axiom,
! [Bl: set_a,P: a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) ) ).
% delete_point_strong_block_in_orig
thf(fact_19_delete__point__strong__block__in__iff,axiom,
! [Bl: set_a,P: a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) )
= ( ~ ( member_a @ P @ Bl ) ) ) ) ).
% delete_point_strong_block_in_iff
thf(fact_20_delete__point__strong__block__in,axiom,
! [P: a,Bl: set_a] :
( ~ ( member_a @ P @ Bl )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ) ) ) ).
% delete_point_strong_block_in
thf(fact_21_delete__point__p__not__in__bl__blocks,axiom,
! [P: a] :
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ~ ( member_a @ P @ Bl2 ) )
=> ( ( design6411949732824333445ocks_a @ block_collection @ P )
= block_collection ) ) ).
% delete_point_p_not_in_bl_blocks
thf(fact_22_add__delete__point__inv,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ P )
= point_set ) ) ).
% add_delete_point_inv
thf(fact_23_incidence__system__axioms,axiom,
design1863209521793301785stem_a @ point_set @ block_collection ).
% incidence_system_axioms
thf(fact_24_delete__point__strong__block__subset,axiom,
! [P: a] : ( subseteq_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) @ block_collection ) ).
% delete_point_strong_block_subset
thf(fact_25_repeat__mset__block__point__rel,axiom,
! [B: set_a,N: nat,X: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) )
=> ( ( member_a @ X @ B )
=> ( member_a @ X @ point_set ) ) ) ).
% repeat_mset_block_point_rel
thf(fact_26_multiple__is__finite,axiom,
! [N: nat] : ( design9187838744727572296stem_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ).
% multiple_is_finite
thf(fact_27_multiple__is__wellformed,axiom,
! [N: nat] : ( design1863209521793301785stem_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ).
% multiple_is_wellformed
thf(fact_28_add__point__finite,axiom,
! [P: a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection ) ).
% add_point_finite
thf(fact_29_add__point__wf,axiom,
! [P: a] : ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection ) ).
% add_point_wf
thf(fact_30_strong__del__point__finite,axiom,
! [P: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ).
% strong_del_point_finite
thf(fact_31_delete__point__finite,axiom,
! [P: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) ).
% delete_point_finite
thf(fact_32_strong__del__point__design,axiom,
! [P: a] : ( design_design_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ).
% strong_del_point_design
thf(fact_33_strong__del__point__incidence__wf,axiom,
! [P: a] : ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ).
% strong_del_point_incidence_wf
thf(fact_34_delete__point__wf,axiom,
! [P: a] : ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) ).
% delete_point_wf
thf(fact_35_add__point__to__blocks__finite,axiom,
! [P: a,Bs: set_set_a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ ( design2935547469388721088ocks_a @ block_collection @ P @ Bs ) ) ).
% add_point_to_blocks_finite
thf(fact_36_add__point__blocks__wf,axiom,
! [P: a,Bs: set_set_a] : ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ ( design2935547469388721088ocks_a @ block_collection @ P @ Bs ) ) ).
% add_point_blocks_wf
thf(fact_37_add__existing__point,axiom,
! [P: a] :
( ( member_a @ P @ point_set )
=> ( ( design2964366272795260673oint_a @ point_set @ P )
= point_set ) ) ).
% add_existing_point
thf(fact_38_incidence__alt__def,axiom,
! [P: a,B: set_a] :
( ( member_a @ P @ point_set )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ( design3210447939978979927dent_a @ block_collection @ P @ B )
= ( member_a @ P @ B ) ) ) ) ).
% incidence_alt_def
thf(fact_39_design__support__def,axiom,
( ( design5397942185814921632port_a @ block_collection )
= ( set_mset_set_a @ block_collection ) ) ).
% design_support_def
thf(fact_40_incidence__system_Oadd__point__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ Block_collection ) ) ).
% incidence_system.add_point_wf
thf(fact_41_incidence__system_Oadd__point_Ocong,axiom,
design2964366272795260673oint_a = design2964366272795260673oint_a ).
% incidence_system.add_point.cong
thf(fact_42_incidence__system_Odel__point_Ocong,axiom,
design108908007054065099oint_a = design108908007054065099oint_a ).
% incidence_system.del_point.cong
thf(fact_43_incidence__system_Odelete__point__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design6411949732824333445ocks_a @ Block_collection @ P ) ) ) ).
% incidence_system.delete_point_wf
thf(fact_44_incidence__system_Odel__invalid__point,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ P @ Point_set )
=> ( ( design7586725432863044395_set_a @ Point_set @ P )
= Point_set ) ) ) ).
% incidence_system.del_invalid_point
thf(fact_45_incidence__system_Odel__invalid__point,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ P @ Point_set )
=> ( ( design4269233978287968195nt_nat @ Point_set @ P )
= Point_set ) ) ) ).
% incidence_system.del_invalid_point
thf(fact_46_incidence__system_Odel__invalid__point,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ P @ Point_set )
=> ( ( design108908007054065099oint_a @ Point_set @ P )
= Point_set ) ) ) ).
% incidence_system.del_invalid_point
thf(fact_47_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_incidence__system_Oadd__existing__point,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ P @ Point_set )
=> ( ( design4648949625254728801_set_a @ Point_set @ P )
= Point_set ) ) ) ).
% incidence_system.add_existing_point
thf(fact_54_incidence__system_Oadd__existing__point,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ P @ Point_set )
=> ( ( design8239173135376323853nt_nat @ Point_set @ P )
= Point_set ) ) ) ).
% incidence_system.add_existing_point
thf(fact_55_incidence__system_Oadd__existing__point,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ P @ Point_set )
=> ( ( design2964366272795260673oint_a @ Point_set @ P )
= Point_set ) ) ) ).
% incidence_system.add_existing_point
thf(fact_56_incidence__system_Oadd__point__blocks__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,Bs: set_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ ( design2935547469388721088ocks_a @ Block_collection @ P @ Bs ) ) ) ).
% incidence_system.add_point_blocks_wf
thf(fact_57_incidence__system_Odelete__block__subset,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( subseteq_mset_set_a @ ( design1146539425385464078lock_a @ Block_collection @ B ) @ Block_collection ) ) ).
% incidence_system.delete_block_subset
thf(fact_58_incidence__system_Oadd__delete__point__inv,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ P @ Point_set )
=> ( ( design7586725432863044395_set_a @ ( design4648949625254728801_set_a @ Point_set @ P ) @ P )
= Point_set ) ) ) ).
% incidence_system.add_delete_point_inv
thf(fact_59_incidence__system_Oadd__delete__point__inv,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ P @ Point_set )
=> ( ( design4269233978287968195nt_nat @ ( design8239173135376323853nt_nat @ Point_set @ P ) @ P )
= Point_set ) ) ) ).
% incidence_system.add_delete_point_inv
thf(fact_60_incidence__system_Oadd__delete__point__inv,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ P @ Point_set )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ P )
= Point_set ) ) ) ).
% incidence_system.add_delete_point_inv
thf(fact_61_incidence__system_Odel__point__blocks_Ocong,axiom,
design6411949732824333445ocks_a = design6411949732824333445ocks_a ).
% incidence_system.del_point_blocks.cong
thf(fact_62_finite__incidence__system_Oadd__point__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ Block_collection ) ) ).
% finite_incidence_system.add_point_finite
thf(fact_63_incidence__system_Odelete__invalid__block__eq,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design1146539425385464078lock_a @ Block_collection @ B )
= Block_collection ) ) ) ).
% incidence_system.delete_invalid_block_eq
thf(fact_64_incidence__system_Oadd__point__to__blocks_Ocong,axiom,
design2935547469388721088ocks_a = design2935547469388721088ocks_a ).
% incidence_system.add_point_to_blocks.cong
thf(fact_65_incidence__system_Odel__invalid__point__blocks,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ P @ Point_set )
=> ( ( design5868748293471848933_set_a @ Block_collection @ P )
= Block_collection ) ) ) ).
% incidence_system.del_invalid_point_blocks
thf(fact_66_incidence__system_Odel__invalid__point__blocks,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ P @ Point_set )
=> ( ( design4832208198062110345ks_nat @ Block_collection @ P )
= Block_collection ) ) ) ).
% incidence_system.del_invalid_point_blocks
thf(fact_67_incidence__system_Odel__invalid__point__blocks,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ P @ Point_set )
=> ( ( design6411949732824333445ocks_a @ Block_collection @ P )
= Block_collection ) ) ) ).
% incidence_system.del_invalid_point_blocks
thf(fact_68_incidence__system_Oadd__point__existing__blocks,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bs: set_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_set_a] :
( ( member_set_set_a @ Bl2 @ Bs )
=> ( member_set_a @ P @ Bl2 ) )
=> ( ( design9181828390639750432_set_a @ Block_collection @ P @ Bs )
= Block_collection ) ) ) ).
% incidence_system.add_point_existing_blocks
thf(fact_69_incidence__system_Oadd__point__existing__blocks,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bs: set_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ Bs )
=> ( member_nat @ P @ Bl2 ) )
=> ( ( design5698312687278145166ks_nat @ Block_collection @ P @ Bs )
= Block_collection ) ) ) ).
% incidence_system.add_point_existing_blocks
thf(fact_70_incidence__system_Oadd__point__existing__blocks,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bs: set_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ Bs )
=> ( member_a @ P @ Bl2 ) )
=> ( ( design2935547469388721088ocks_a @ Block_collection @ P @ Bs )
= Block_collection ) ) ) ).
% incidence_system.add_point_existing_blocks
thf(fact_71_incidence__system_Ostr__del__point__blocks_Ocong,axiom,
design5657747894866638574ocks_a = design5657747894866638574ocks_a ).
% incidence_system.str_del_point_blocks.cong
thf(fact_72_finite__incidence__system_Odelete__point__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design6411949732824333445ocks_a @ Block_collection @ P ) ) ) ).
% finite_incidence_system.delete_point_finite
thf(fact_73_incidence__system_Odelete__invalid__pt__strong__eq,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ P @ Point_set )
=> ( Block_collection
= ( design9174845397011619918_set_a @ Block_collection @ P ) ) ) ) ).
% incidence_system.delete_invalid_pt_strong_eq
thf(fact_74_incidence__system_Odelete__invalid__pt__strong__eq,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ P @ Point_set )
=> ( Block_collection
= ( design3278834155446248416ks_nat @ Block_collection @ P ) ) ) ) ).
% incidence_system.delete_invalid_pt_strong_eq
thf(fact_75_incidence__system_Odelete__invalid__pt__strong__eq,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ P @ Point_set )
=> ( Block_collection
= ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ) ).
% incidence_system.delete_invalid_pt_strong_eq
thf(fact_76_incidence__system_Odelete__point__strong__block__in,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a,Bl: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ P @ Bl )
=> ( ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ Block_collection ) )
=> ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ ( design9174845397011619918_set_a @ Block_collection @ P ) ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_in
thf(fact_77_incidence__system_Odelete__point__strong__block__in,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ P @ Bl )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ Block_collection @ P ) ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_in
thf(fact_78_incidence__system_Odelete__point__strong__block__in,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ P @ Bl )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_in
thf(fact_79_incidence__system_Ostrong__del__point__incidence__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ).
% incidence_system.strong_del_point_incidence_wf
thf(fact_80_finite__incidence__system_Ostrong__del__point__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ).
% finite_incidence_system.strong_del_point_finite
thf(fact_81_incidence__system_Odelete__point__p__not__in__bl__blocks,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_set_a] :
( ( member_set_set_a @ Bl2 @ ( set_mset_set_set_a @ Block_collection ) )
=> ~ ( member_set_a @ P @ Bl2 ) )
=> ( ( design5868748293471848933_set_a @ Block_collection @ P )
= Block_collection ) ) ) ).
% incidence_system.delete_point_p_not_in_bl_blocks
thf(fact_82_incidence__system_Odelete__point__p__not__in__bl__blocks,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ~ ( member_nat @ P @ Bl2 ) )
=> ( ( design4832208198062110345ks_nat @ Block_collection @ P )
= Block_collection ) ) ) ).
% incidence_system.delete_point_p_not_in_bl_blocks
thf(fact_83_incidence__system_Odelete__point__p__not__in__bl__blocks,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ~ ( member_a @ P @ Bl2 ) )
=> ( ( design6411949732824333445ocks_a @ Block_collection @ P )
= Block_collection ) ) ) ).
% incidence_system.delete_point_p_not_in_bl_blocks
thf(fact_84_incidence__system_Odelete__point__strong__block__in__iff,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bl: set_set_a,P: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ Block_collection ) )
=> ( ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ ( design9174845397011619918_set_a @ Block_collection @ P ) ) )
= ( ~ ( member_set_a @ P @ Bl ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_in_iff
thf(fact_85_incidence__system_Odelete__point__strong__block__in__iff,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ Block_collection @ P ) ) )
= ( ~ ( member_nat @ P @ Bl ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_in_iff
thf(fact_86_incidence__system_Odelete__point__strong__block__in__iff,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) )
= ( ~ ( member_a @ P @ Bl ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_in_iff
thf(fact_87_incidence__system_Odelete__point__strong__block__not__in,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a,Bl: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ P @ Bl )
=> ~ ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ ( design9174845397011619918_set_a @ Block_collection @ P ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_not_in
thf(fact_88_incidence__system_Odelete__point__strong__block__not__in,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ P @ Bl )
=> ~ ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ Block_collection @ P ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_not_in
thf(fact_89_incidence__system_Odelete__point__strong__block__not__in,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ P @ Bl )
=> ~ ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ) ) ).
% incidence_system.delete_point_strong_block_not_in
thf(fact_90_incidence__system_Odelete__point__strong__block__subset,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( subseteq_mset_set_a @ ( design5657747894866638574ocks_a @ Block_collection @ P ) @ Block_collection ) ) ).
% incidence_system.delete_point_strong_block_subset
thf(fact_91_finite__incidence__system_Oadd__point__to__blocks__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,Bs: set_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ ( design2935547469388721088ocks_a @ Block_collection @ P @ Bs ) ) ) ).
% finite_incidence_system.add_point_to_blocks_finite
thf(fact_92_incidence__system_Odelete__point__strong__block__in__orig,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) ) ) ).
% incidence_system.delete_point_strong_block_in_orig
thf(fact_93_design_Ostrong__del__point__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ).
% design.strong_del_point_design
thf(fact_94_incidence__system_Odelete__block__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ Point_set @ ( design1146539425385464078lock_a @ Block_collection @ B ) ) ) ).
% incidence_system.delete_block_wf
thf(fact_95_finite__incidence__system_Odelete__block__fin__incidence__sys,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ ( design1146539425385464078lock_a @ Block_collection @ B ) ) ) ).
% finite_incidence_system.delete_block_fin_incidence_sys
thf(fact_96_design_Oadd__point__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ Block_collection ) ) ).
% design.add_point_design
thf(fact_97_incidence__system_Odel__block_Ocong,axiom,
design1146539425385464078lock_a = design1146539425385464078lock_a ).
% incidence_system.del_block.cong
thf(fact_98_complement__finite,axiom,
design9187838744727572296stem_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ).
% complement_finite
thf(fact_99_del__invalid__add__block__eq,axiom,
! [Bl: set_a] :
( ~ ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ block_collection @ Bl ) @ Bl )
= ( design4001997691126659652lock_a @ block_collection @ Bl ) ) ) ).
% del_invalid_add_block_eq
thf(fact_100_del__add__block__inv,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ block_collection @ Bl ) @ Bl )
= block_collection ) ) ).
% del_add_block_inv
thf(fact_101_block__complement__inv,axiom,
! [Bl: set_a,Bl22: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( ( design6447616907850319326ment_a @ point_set @ Bl )
= Bl22 )
=> ( ( design6447616907850319326ment_a @ point_set @ Bl22 )
= Bl ) ) ) ).
% block_complement_inv
thf(fact_102_wellformed,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_eq_set_a @ B @ point_set ) ) ).
% wellformed
thf(fact_103_delete__point__blocks__sub,axiom,
! [B: set_a,P: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) )
=> ~ ! [Bl2: set_a] :
~ ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
& ( ord_less_eq_set_a @ B @ Bl2 ) ) ) ).
% delete_point_blocks_sub
thf(fact_104_incidence__system_Omultiple__block__in__original,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a,N: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ Block_collection ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) ) ) ) ).
% incidence_system.multiple_block_in_original
thf(fact_105_incidence__system_Orepeat__mset__block__point__rel,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,B: set_set_a,N: nat,X: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ ( repeat3222187171979612824_set_a @ N @ Block_collection ) ) )
=> ( ( member_set_a @ X @ B )
=> ( member_set_a @ X @ Point_set ) ) ) ) ).
% incidence_system.repeat_mset_block_point_rel
thf(fact_106_incidence__system_Orepeat__mset__block__point__rel,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat,N: nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ Block_collection ) ) )
=> ( ( member_nat @ X @ B )
=> ( member_nat @ X @ Point_set ) ) ) ) ).
% incidence_system.repeat_mset_block_point_rel
thf(fact_107_incidence__system_Orepeat__mset__block__point__rel,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a,N: nat,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ Block_collection ) ) )
=> ( ( member_a @ X @ B )
=> ( member_a @ X @ Point_set ) ) ) ) ).
% incidence_system.repeat_mset_block_point_rel
thf(fact_108_complete__block__size__eq__points,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( ( finite_card_a @ Bl )
= ( finite_card_a @ point_set ) )
=> ( Bl = point_set ) ) ) ).
% complete_block_size_eq_points
thf(fact_109_del__block__b_I2_J,axiom,
! [Bl: set_a] :
( ~ ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ block_collection @ Bl ) )
= ( size_s6566526139600085008_set_a @ block_collection ) ) ) ).
% del_block_b(2)
thf(fact_110_multiple__1__same,axiom,
( ( repeat_mset_set_a @ one_one_nat @ block_collection )
= block_collection ) ).
% multiple_1_same
thf(fact_111_block__set__nempty__imp__block__ex,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ? [Bl2: set_a] : ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) ) ).
% block_set_nempty_imp_block_ex
thf(fact_112_design__blocks__nempty,axiom,
block_collection != zero_z5079479921072680283_set_a ).
% design_blocks_nempty
thf(fact_113_del__point__block__count,axiom,
! [P: a] :
( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) )
= ( size_s6566526139600085008_set_a @ block_collection ) ) ).
% del_point_block_count
thf(fact_114_block__comp__elem__alt__left,axiom,
! [X: a,Bl: set_a,Ps: set_a] :
( ( member_a @ X @ Bl )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl ) )
=> ~ ( member_a @ X @ Ps ) ) ) ).
% block_comp_elem_alt_left
thf(fact_115_block__comp__elem__alt__right,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ! [X3: a] :
( ( member_a @ X3 @ Ps )
=> ~ ( member_a @ X3 @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl ) ) ) ) ).
% block_comp_elem_alt_right
thf(fact_116_block__complement__elem__iff,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl ) ) ) ) ) ).
% block_complement_elem_iff
thf(fact_117_block__complement__subset__points,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ point_set ) ) ).
% block_complement_subset_points
thf(fact_118_complete__block__all__subsets,axiom,
! [Bl: set_a,Ps: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( ( finite_card_a @ Bl )
= ( finite_card_a @ point_set ) )
=> ( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ord_less_eq_set_a @ Ps @ Bl ) ) ) ) ).
% complete_block_all_subsets
thf(fact_119_complement__blocks__wf,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) )
=> ( ord_less_eq_set_a @ Bl @ point_set ) ) ).
% complement_blocks_wf
thf(fact_120_obtain__comp__block__orig,axiom,
! [Bl1: set_a] :
( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) )
=> ~ ! [Bl23: set_a] :
( ( member_set_a @ Bl23 @ ( set_mset_set_a @ block_collection ) )
=> ( Bl1
!= ( design6447616907850319326ment_a @ point_set @ Bl23 ) ) ) ) ).
% obtain_comp_block_orig
thf(fact_121_add__del__block__inv,axiom,
! [Bl: set_a] :
( ( ord_less_eq_set_a @ Bl @ point_set )
=> ( ( design1146539425385464078lock_a @ ( design4001997691126659652lock_a @ block_collection @ Bl ) @ Bl )
= block_collection ) ) ).
% add_del_block_inv
thf(fact_122_add__block__fin__cond,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ point_set )
=> ( design9187838744727572296stem_a @ point_set @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ) ).
% add_block_fin_cond
thf(fact_123_add__block__wf__cond,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ point_set )
=> ( design1863209521793301785stem_a @ point_set @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ) ).
% add_block_wf_cond
thf(fact_124_assms,axiom,
ord_less_nat @ one_one_nat @ ( size_s6566526139600085008_set_a @ block_collection ) ).
% assms
thf(fact_125_proper__designII,axiom,
! [V: set_a,B2: multiset_set_a] :
( ( design_design_a @ V @ B2 )
=> ( ( B2 != zero_z5079479921072680283_set_a )
=> ( design7287791228148780576sign_a @ V @ B2 ) ) ) ).
% proper_designII
thf(fact_126_complement__same__b,axiom,
( ( size_s6566526139600085008_set_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) )
= ( size_s6566526139600085008_set_a @ block_collection ) ) ).
% complement_same_b
thf(fact_127_complement__wf,axiom,
design1863209521793301785stem_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ).
% complement_wf
thf(fact_128_incidence__system_Ocomplement__same__b,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( size_s6566526139600085008_set_a @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) )
= ( size_s6566526139600085008_set_a @ Block_collection ) ) ) ).
% incidence_system.complement_same_b
thf(fact_129_incidence__system_Oadd__block_Ocong,axiom,
design4001997691126659652lock_a = design4001997691126659652lock_a ).
% incidence_system.add_block.cong
thf(fact_130_incidence__system_Oblock__comp__elem__alt__left,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a,Bl: set_set_a,Ps: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ X @ Bl )
=> ( ( ord_le3724670747650509150_set_a @ Ps @ ( design4243878040612417342_set_a @ Point_set @ Bl ) )
=> ~ ( member_set_a @ X @ Ps ) ) ) ) ).
% incidence_system.block_comp_elem_alt_left
thf(fact_131_incidence__system_Oblock__comp__elem__alt__left,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat,Bl: set_nat,Ps: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ Bl )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ Point_set @ Bl ) )
=> ~ ( member_nat @ X @ Ps ) ) ) ) ).
% incidence_system.block_comp_elem_alt_left
thf(fact_132_incidence__system_Oblock__comp__elem__alt__left,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a,Bl: set_a,Ps: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ X @ Bl )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ Point_set @ Bl ) )
=> ~ ( member_a @ X @ Ps ) ) ) ) ).
% incidence_system.block_comp_elem_alt_left
thf(fact_133_incidence__system_Oblock__comp__elem__alt__right,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Ps: set_set_a,Bl: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( ord_le3724670747650509150_set_a @ Ps @ Point_set )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ Ps )
=> ~ ( member_set_a @ X3 @ Bl ) )
=> ( ord_le3724670747650509150_set_a @ Ps @ ( design4243878040612417342_set_a @ Point_set @ Bl ) ) ) ) ) ).
% incidence_system.block_comp_elem_alt_right
thf(fact_134_incidence__system_Oblock__comp__elem__alt__right,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Ps: set_nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ Ps @ Point_set )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ Ps )
=> ~ ( member_nat @ X3 @ Bl ) )
=> ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ Point_set @ Bl ) ) ) ) ) ).
% incidence_system.block_comp_elem_alt_right
thf(fact_135_incidence__system_Oblock__comp__elem__alt__right,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Ps: set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Ps @ Point_set )
=> ( ! [X3: a] :
( ( member_a @ X3 @ Ps )
=> ~ ( member_a @ X3 @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ Point_set @ Bl ) ) ) ) ) ).
% incidence_system.block_comp_elem_alt_right
thf(fact_136_incidence__system_Oblock__complement__elem__iff,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Ps: set_set_a,Bl: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( ord_le3724670747650509150_set_a @ Ps @ Point_set )
=> ( ( ord_le3724670747650509150_set_a @ Ps @ ( design4243878040612417342_set_a @ Point_set @ Bl ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ Ps )
=> ~ ( member_set_a @ X2 @ Bl ) ) ) ) ) ) ).
% incidence_system.block_complement_elem_iff
thf(fact_137_incidence__system_Oblock__complement__elem__iff,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Ps: set_nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ Ps @ Point_set )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ Point_set @ Bl ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ Ps )
=> ~ ( member_nat @ X2 @ Bl ) ) ) ) ) ) ).
% incidence_system.block_complement_elem_iff
thf(fact_138_incidence__system_Oblock__complement__elem__iff,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Ps: set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Ps @ Point_set )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ Point_set @ Bl ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl ) ) ) ) ) ) ).
% incidence_system.block_complement_elem_iff
thf(fact_139_incidence__system_Oblock__complement__subset__points,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Ps: set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ Point_set @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ Point_set ) ) ) ).
% incidence_system.block_complement_subset_points
thf(fact_140_incidence__system_Ocomplement__blocks_Ocong,axiom,
design8640656491286871389ocks_a = design8640656491286871389ocks_a ).
% incidence_system.complement_blocks.cong
thf(fact_141_incidence__system_Oblock__complement_Ocong,axiom,
design6447616907850319326ment_a = design6447616907850319326ment_a ).
% incidence_system.block_complement.cong
thf(fact_142_incidence__system_Odesign__support_Ocong,axiom,
design5397942185814921632port_a = design5397942185814921632port_a ).
% incidence_system.design_support.cong
thf(fact_143_incidence__system_Oincident_Ocong,axiom,
design3210447939978979927dent_a = design3210447939978979927dent_a ).
% incidence_system.incident.cong
thf(fact_144_incidence__system_Oadd__block__wf__cond,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ B @ Point_set )
=> ( design1863209521793301785stem_a @ Point_set @ ( design4001997691126659652lock_a @ Block_collection @ B ) ) ) ) ).
% incidence_system.add_block_wf_cond
thf(fact_145_finite__incidence__system_Oadd__block__fin__cond,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ B @ Point_set )
=> ( design9187838744727572296stem_a @ Point_set @ ( design4001997691126659652lock_a @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin_cond
thf(fact_146_proper__design_Odesign__blocks__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( Block_collection != zero_z5079479921072680283_set_a ) ) ).
% proper_design.design_blocks_nempty
thf(fact_147_incidence__system_Ocomplement__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ).
% incidence_system.complement_wf
thf(fact_148_incidence__system_Ocomplement__blocks__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) )
=> ( ord_less_eq_set_a @ Bl @ Point_set ) ) ) ).
% incidence_system.complement_blocks_wf
thf(fact_149_finite__incidence__system_Ocomplement__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ).
% finite_incidence_system.complement_finite
thf(fact_150_incidence__system_Oobtain__comp__block__orig,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl1: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) )
=> ~ ! [Bl23: set_a] :
( ( member_set_a @ Bl23 @ ( set_mset_set_a @ Block_collection ) )
=> ( Bl1
!= ( design6447616907850319326ment_a @ Point_set @ Bl23 ) ) ) ) ) ).
% incidence_system.obtain_comp_block_orig
thf(fact_151_finite__incidence__system_Ocomplete__block__all__subsets,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat,Ps: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ( finite_card_nat @ Bl )
= ( finite_card_nat @ Point_set ) )
=> ( ( ord_less_eq_set_nat @ Ps @ Point_set )
=> ( ord_less_eq_set_nat @ Ps @ Bl ) ) ) ) ) ).
% finite_incidence_system.complete_block_all_subsets
thf(fact_152_finite__incidence__system_Ocomplete__block__all__subsets,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a,Ps: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ( finite_card_a @ Bl )
= ( finite_card_a @ Point_set ) )
=> ( ( ord_less_eq_set_a @ Ps @ Point_set )
=> ( ord_less_eq_set_a @ Ps @ Bl ) ) ) ) ) ).
% finite_incidence_system.complete_block_all_subsets
thf(fact_153_incidence__system_Oadd__del__block__inv,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Bl @ Point_set )
=> ( ( design1146539425385464078lock_a @ ( design4001997691126659652lock_a @ Block_collection @ Bl ) @ Bl )
= Block_collection ) ) ) ).
% incidence_system.add_del_block_inv
thf(fact_154_incidence__system__def,axiom,
( design1863209521793301785stem_a
= ( ^ [Point_set2: set_a,Block_collection2: multiset_set_a] :
! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ Block_collection2 ) )
=> ( ord_less_eq_set_a @ B3 @ Point_set2 ) ) ) ) ).
% incidence_system_def
thf(fact_155_incidence__system_Owellformed,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_set_a @ B @ Point_set ) ) ) ).
% incidence_system.wellformed
thf(fact_156_incidence__system_Ointro,axiom,
! [Block_collection: multiset_set_a,Point_set: set_a] :
( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_set_a @ B4 @ Point_set ) )
=> ( design1863209521793301785stem_a @ Point_set @ Block_collection ) ) ).
% incidence_system.intro
thf(fact_157_incidence__system_Oblock__set__nempty__imp__block__ex,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ? [Bl2: set_a] : ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) ) ) ) ).
% incidence_system.block_set_nempty_imp_block_ex
thf(fact_158_incidence__system_Omultiple__1__same,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( repeat_mset_set_a @ one_one_nat @ Block_collection )
= Block_collection ) ) ).
% incidence_system.multiple_1_same
thf(fact_159_incidence__system_Oblock__complement__inv,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a,Bl22: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ( design6447616907850319326ment_a @ Point_set @ Bl )
= Bl22 )
=> ( ( design6447616907850319326ment_a @ Point_set @ Bl22 )
= Bl ) ) ) ) ).
% incidence_system.block_complement_inv
thf(fact_160_finite__incidence__system_Ocomplete__block__size__eq__points,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ( finite_card_nat @ Bl )
= ( finite_card_nat @ Point_set ) )
=> ( Bl = Point_set ) ) ) ) ).
% finite_incidence_system.complete_block_size_eq_points
thf(fact_161_finite__incidence__system_Ocomplete__block__size__eq__points,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ( finite_card_a @ Bl )
= ( finite_card_a @ Point_set ) )
=> ( Bl = Point_set ) ) ) ) ).
% finite_incidence_system.complete_block_size_eq_points
thf(fact_162_incidence__system_Odel__point__block__count,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ Block_collection @ P ) )
= ( size_s6566526139600085008_set_a @ Block_collection ) ) ) ).
% incidence_system.del_point_block_count
thf(fact_163_incidence__system_Odesign__support__def,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( design5397942185814921632port_a @ Block_collection )
= ( set_mset_set_a @ Block_collection ) ) ) ).
% incidence_system.design_support_def
thf(fact_164_incidence__system_Oincidence__alt__def,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a,B: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ P @ Point_set )
=> ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ Block_collection ) )
=> ( ( design6773327923283668919_set_a @ Block_collection @ P @ B )
= ( member_set_a @ P @ B ) ) ) ) ) ).
% incidence_system.incidence_alt_def
thf(fact_165_incidence__system_Oincidence__alt__def,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat,B: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ P @ Point_set )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( design8502206366797944887nt_nat @ Block_collection @ P @ B )
= ( member_nat @ P @ B ) ) ) ) ) ).
% incidence_system.incidence_alt_def
thf(fact_166_incidence__system_Oincidence__alt__def,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ P @ Point_set )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design3210447939978979927dent_a @ Block_collection @ P @ B )
= ( member_a @ P @ B ) ) ) ) ) ).
% incidence_system.incidence_alt_def
thf(fact_167_incidence__system_Odel__block__b_I2_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ Block_collection @ Bl ) )
= ( size_s6566526139600085008_set_a @ Block_collection ) ) ) ) ).
% incidence_system.del_block_b(2)
thf(fact_168_incidence__system_Odelete__point__blocks__sub,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ Block_collection @ P ) ) )
=> ~ ! [Bl2: set_a] :
~ ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
& ( ord_less_eq_set_a @ B @ Bl2 ) ) ) ) ).
% incidence_system.delete_point_blocks_sub
thf(fact_169_design_Owf__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ Block_collection ) ) ).
% design.wf_design
thf(fact_170_proper__design_Ois__proper,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( design7287791228148780576sign_a @ Point_set @ Block_collection ) ) ).
% proper_design.is_proper
thf(fact_171_incidence__system_Odel__invalid__add__block__eq,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ Block_collection @ Bl ) @ Bl )
= ( design4001997691126659652lock_a @ Block_collection @ Bl ) ) ) ) ).
% incidence_system.del_invalid_add_block_eq
thf(fact_172_incidence__system_Odel__add__block__inv,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ Block_collection @ Bl ) @ Bl )
= Block_collection ) ) ) ).
% incidence_system.del_add_block_inv
thf(fact_173_incidence__system_Owf__invalid__point,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a,B: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ X @ Point_set )
=> ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ Block_collection ) )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% incidence_system.wf_invalid_point
thf(fact_174_incidence__system_Owf__invalid__point,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat,B: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ X @ Point_set )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ Block_collection ) )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% incidence_system.wf_invalid_point
thf(fact_175_incidence__system_Owf__invalid__point,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ X @ Point_set )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% incidence_system.wf_invalid_point
thf(fact_176_finite__incidence__system_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ Point_set @ Block_collection ) ) ).
% finite_incidence_system.axioms(1)
thf(fact_177_incidence__system_Omultiple__is__wellformed,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ).
% incidence_system.multiple_is_wellformed
thf(fact_178_design_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ Block_collection ) ) ).
% design.axioms(1)
thf(fact_179_proper__design_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ Block_collection ) ) ).
% proper_design.axioms(1)
thf(fact_180_design_Omultiple__is__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ).
% design.multiple_is_design
thf(fact_181_finite__incidence__system_Omultiple__is__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ).
% finite_incidence_system.multiple_is_finite
thf(fact_182_sys__block__sizes__in,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( member_nat @ ( finite_card_a @ Bl ) @ ( design1769254222028858111izes_a @ block_collection ) ) ) ).
% sys_block_sizes_in
thf(fact_183_sys__block__sizes__obtain__bl,axiom,
! [X: nat] :
( ( member_nat @ X @ ( design1769254222028858111izes_a @ block_collection ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_mset_set_a @ block_collection ) )
& ( ( finite_card_a @ X3 )
= X ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_184_complement__proper__design,axiom,
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) ) )
=> ( design7287791228148780576sign_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) ) ).
% complement_proper_design
thf(fact_185_complement__design,axiom,
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) ) )
=> ( design_design_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) ) ).
% complement_design
thf(fact_186_add__block__design__cond,axiom,
! [Bl: set_a] :
( ( ord_less_eq_set_a @ Bl @ point_set )
=> ( ( Bl != bot_bot_set_a )
=> ( design_design_a @ point_set @ ( design4001997691126659652lock_a @ block_collection @ Bl ) ) ) ) ).
% add_block_design_cond
thf(fact_187_v__eq0__imp__b__eq__0,axiom,
( ( ( finite_card_a @ point_set )
= zero_zero_nat )
=> ( ( size_s6566526139600085008_set_a @ block_collection )
= zero_zero_nat ) ) ).
% v_eq0_imp_b_eq_0
thf(fact_188_repeat__mset__empty,axiom,
! [N: nat] :
( ( repeat_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_empty
thf(fact_189_subset__mset_Oextremum__unique,axiom,
! [A: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( A = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_unique
thf(fact_190_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_set_a] :
( ( subseteq_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= ( N = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_191_del__block__b_I1_J,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ block_collection @ Bl ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ one_one_nat ) ) ) ).
% del_block_b(1)
thf(fact_192_design__points__nempty,axiom,
point_set != bot_bot_set_a ).
% design_points_nempty
thf(fact_193_b__non__zero,axiom,
( ( size_s6566526139600085008_set_a @ block_collection )
!= zero_zero_nat ) ).
% b_non_zero
thf(fact_194_blocks__nempty__alt,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ block_collection ) )
=> ( X4 != bot_bot_set_a ) ) ).
% blocks_nempty_alt
thf(fact_195_blocks__nempty,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( Bl != bot_bot_set_a ) ) ).
% blocks_nempty
thf(fact_196_subset__mset_Oorder__refl,axiom,
! [X: multiset_set_a] : ( subseteq_mset_set_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_197_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_198_v__non__zero,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ point_set ) ).
% v_non_zero
thf(fact_199_b__positive,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ block_collection ) ).
% b_positive
thf(fact_200_block__set__nempty__imp__points,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ( point_set != bot_bot_set_a ) ) ).
% block_set_nempty_imp_points
thf(fact_201_block__size__gt__0,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl ) ) ) ).
% block_size_gt_0
thf(fact_202_multiple__block__in,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ) ).
% multiple_block_in
thf(fact_203_del__point__order,axiom,
! [P: a] :
( ( member_a @ P @ point_set )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_a @ point_set ) @ one_one_nat ) ) ) ).
% del_point_order
thf(fact_204_block__complement__size,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ point_set )
=> ( ( finite_card_a @ ( design6447616907850319326ment_a @ point_set @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ point_set ) @ ( finite_card_a @ B ) ) ) ) ).
% block_complement_size
thf(fact_205_multiple__block__sizes__same,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design1769254222028858111izes_a @ block_collection )
= ( design1769254222028858111izes_a @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ).
% multiple_block_sizes_same
thf(fact_206_block__sizes__non__empty,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ block_collection ) ) ) ) ).
% block_sizes_non_empty
thf(fact_207_incomplete__alt__size,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) )
=> ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) ) ) ).
% incomplete_alt_size
thf(fact_208_incomplete__alt__in,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) ) ).
% incomplete_alt_in
thf(fact_209_multiple__blocks__sub,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( subseteq_mset_set_a @ block_collection @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ).
% multiple_blocks_sub
thf(fact_210_b__non__zero__imp__v__non__zero,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ point_set ) ) ) ).
% b_non_zero_imp_v_non_zero
thf(fact_211_multiple__proper__design,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( design7287791228148780576sign_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ).
% multiple_proper_design
thf(fact_212_block__comp__incomplete,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ point_set @ Bl ) ) ) ) ).
% block_comp_incomplete
thf(fact_213_block__comp__incomplete__nempty,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) )
=> ( ( design6447616907850319326ment_a @ point_set @ Bl )
!= bot_bot_set_a ) ) ).
% block_comp_incomplete_nempty
thf(fact_214_set__mset__eq__empty__iff,axiom,
! [M: multiset_set_a] :
( ( ( set_mset_set_a @ M )
= bot_bot_set_set_a )
= ( M = zero_z5079479921072680283_set_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_215_set__mset__eq__empty__iff,axiom,
! [M: multiset_a] :
( ( ( set_mset_a @ M )
= bot_bot_set_a )
= ( M = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_216_set__mset__eq__empty__iff,axiom,
! [M: multiset_nat] :
( ( ( set_mset_nat @ M )
= bot_bot_set_nat )
= ( M = zero_z7348594199698428585et_nat ) ) ).
% set_mset_eq_empty_iff
thf(fact_217_set__mset__empty,axiom,
( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
= bot_bot_set_set_a ) ).
% set_mset_empty
thf(fact_218_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_219_set__mset__empty,axiom,
( ( set_mset_nat @ zero_z7348594199698428585et_nat )
= bot_bot_set_nat ) ).
% set_mset_empty
thf(fact_220_size__empty,axiom,
( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_221_size__eq__0__iff__empty,axiom,
! [M: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ M )
= zero_zero_nat )
= ( M = zero_z5079479921072680283_set_a ) ) ).
% size_eq_0_iff_empty
thf(fact_222_repeat__mset__0,axiom,
! [M: multiset_set_a] :
( ( repeat_mset_set_a @ zero_zero_nat @ M )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_0
thf(fact_223_incomplete__alt__imp,axiom,
! [Bl: set_a] :
( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) ) ) ) ).
% incomplete_alt_imp
thf(fact_224_proper__designI,axiom,
( ( ( size_s6566526139600085008_set_a @ block_collection )
!= zero_zero_nat )
=> ( design7287791228148780576sign_a @ point_set @ block_collection ) ) ).
% proper_designI
thf(fact_225_designI,axiom,
( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( set_mset_set_a @ block_collection ) )
=> ( B4 != bot_bot_set_a ) )
=> ( ( block_collection != zero_z5079479921072680283_set_a )
=> ( ( point_set != bot_bot_set_a )
=> ( design_design_a @ point_set @ block_collection ) ) ) ) ).
% designI
thf(fact_226_n__inter__num__zero,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ block_collection ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ block_collection ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ).
% n_inter_num_zero
thf(fact_227_incidence__system_Osys__block__sizes_Ocong,axiom,
design1769254222028858111izes_a = design1769254222028858111izes_a ).
% incidence_system.sys_block_sizes.cong
thf(fact_228_incidence__system_Oblock__sizes__non__empty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ Block_collection ) ) ) ) ) ).
% incidence_system.block_sizes_non_empty
thf(fact_229_repeat__mset__cancel1,axiom,
! [A: nat,A2: multiset_set_a,B2: multiset_set_a] :
( ( ( repeat_mset_set_a @ A @ A2 )
= ( repeat_mset_set_a @ A @ B2 ) )
= ( ( A2 = B2 )
| ( A = zero_zero_nat ) ) ) ).
% repeat_mset_cancel1
thf(fact_230_nonempty__has__size,axiom,
! [S: multiset_set_a] :
( ( S != zero_z5079479921072680283_set_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ S ) ) ) ).
% nonempty_has_size
thf(fact_231_incidence__system_Omultiple__block__sizes__same,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design1769254222028858111izes_a @ Block_collection )
= ( design1769254222028858111izes_a @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ) ).
% incidence_system.multiple_block_sizes_same
thf(fact_232_proper__design_Ov__non__zero,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design435815215503836206gn_nat @ Point_set @ Block_collection )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ Point_set ) ) ) ).
% proper_design.v_non_zero
thf(fact_233_proper__design_Ov__non__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Point_set ) ) ) ).
% proper_design.v_non_zero
thf(fact_234_proper__design_Ob__positive,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) ) ) ).
% proper_design.b_positive
thf(fact_235_proper__design_Omultiple__proper__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( design7287791228148780576sign_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ).
% proper_design.multiple_proper_design
thf(fact_236_proper__design_Odesign__points__nempty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design435815215503836206gn_nat @ Point_set @ Block_collection )
=> ( Point_set != bot_bot_set_nat ) ) ).
% proper_design.design_points_nempty
thf(fact_237_proper__design_Odesign__points__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( Point_set != bot_bot_set_a ) ) ).
% proper_design.design_points_nempty
thf(fact_238_repeat__mset__eq__empty__iff,axiom,
! [N: nat,A2: multiset_set_a] :
( ( ( repeat_mset_set_a @ N @ A2 )
= zero_z5079479921072680283_set_a )
= ( ( N = zero_zero_nat )
| ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_239_design_Oblock__size__gt__0,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ Bl ) ) ) ) ).
% design.block_size_gt_0
thf(fact_240_design_Oblock__size__gt__0,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl ) ) ) ) ).
% design.block_size_gt_0
thf(fact_241_design_Ob__non__zero__imp__v__non__zero,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7462436076474991978et_nat @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ Point_set ) ) ) ) ).
% design.b_non_zero_imp_v_non_zero
thf(fact_242_design_Ob__non__zero__imp__v__non__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Point_set ) ) ) ) ).
% design.b_non_zero_imp_v_non_zero
thf(fact_243_incidence__system_Omultiple__block__in,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ) ) ).
% incidence_system.multiple_block_in
thf(fact_244_incidence__system_Omultiple__blocks__sub,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( subseteq_mset_set_a @ Block_collection @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ).
% incidence_system.multiple_blocks_sub
thf(fact_245_finite__incidence__system_Oblock__comp__incomplete__nempty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ( design2875492832550762736nt_nat @ Point_set @ Bl )
!= bot_bot_set_nat ) ) ) ).
% finite_incidence_system.block_comp_incomplete_nempty
thf(fact_246_finite__incidence__system_Oblock__comp__incomplete__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ( design6447616907850319326ment_a @ Point_set @ Bl )
!= bot_bot_set_a ) ) ) ).
% finite_incidence_system.block_comp_incomplete_nempty
thf(fact_247_finite__incidence__system_Oblock__comp__incomplete,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design2875492832550762736nt_nat @ Point_set @ Bl ) ) ) ) ) ).
% finite_incidence_system.block_comp_incomplete
thf(fact_248_finite__incidence__system_Oblock__comp__incomplete,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ Point_set @ Bl ) ) ) ) ) ).
% finite_incidence_system.block_comp_incomplete
thf(fact_249_design_Oblocks__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( Bl != bot_bot_set_a ) ) ) ).
% design.blocks_nempty
thf(fact_250_design_Oblocks__nempty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( Bl != bot_bot_set_nat ) ) ) ).
% design.blocks_nempty
thf(fact_251_design_Oblocks__nempty__alt,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ Block_collection ) )
=> ( X4 != bot_bot_set_a ) ) ) ).
% design.blocks_nempty_alt
thf(fact_252_design_Oblocks__nempty__alt,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_mset_set_nat @ Block_collection ) )
=> ( X4 != bot_bot_set_nat ) ) ) ).
% design.blocks_nempty_alt
thf(fact_253_design_Oblock__set__nempty__imp__points,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ( Point_set != bot_bot_set_a ) ) ) ).
% design.block_set_nempty_imp_points
thf(fact_254_design_Oblock__set__nempty__imp__points,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z3157962936165190495et_nat )
=> ( Point_set != bot_bot_set_nat ) ) ) ).
% design.block_set_nempty_imp_points
thf(fact_255_proper__design_Ob__non__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ( size_s6566526139600085008_set_a @ Block_collection )
!= zero_zero_nat ) ) ).
% proper_design.b_non_zero
thf(fact_256_subset__mset_Otrans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_257_subset__mset_Oeq__iff,axiom,
( ( ^ [Y: multiset_set_a,Z: multiset_set_a] : ( Y = Z ) )
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ A3 @ B3 )
& ( subseteq_mset_set_a @ B3 @ A3 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_258_subset__mset_Oantisym,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_259_subset__mset_Oeq__refl,axiom,
! [X: multiset_set_a,Y2: multiset_set_a] :
( ( X = Y2 )
=> ( subseteq_mset_set_a @ X @ Y2 ) ) ).
% subset_mset.eq_refl
thf(fact_260_subset__mset_Oorder__trans,axiom,
! [X: multiset_set_a,Y2: multiset_set_a,Z2: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y2 )
=> ( ( subseteq_mset_set_a @ Y2 @ Z2 )
=> ( subseteq_mset_set_a @ X @ Z2 ) ) ) ).
% subset_mset.order_trans
thf(fact_261_subset__mset_Oantisym__conv,axiom,
! [Y2: multiset_set_a,X: multiset_set_a] :
( ( subseteq_mset_set_a @ Y2 @ X )
=> ( ( subseteq_mset_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% subset_mset.antisym_conv
thf(fact_262_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y: multiset_set_a,Z: multiset_set_a] : ( Y = Z ) )
= ( ^ [X2: multiset_set_a,Y3: multiset_set_a] :
( ( subseteq_mset_set_a @ X2 @ Y3 )
& ( subseteq_mset_set_a @ Y3 @ X2 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_263_subset__mset_Oorder__antisym,axiom,
! [X: multiset_set_a,Y2: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y2 )
=> ( ( subseteq_mset_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% subset_mset.order_antisym
thf(fact_264_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( A = B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_265_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_266_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_267_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y: multiset_set_a,Z: multiset_set_a] : ( Y = Z ) )
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ B3 @ A3 )
& ( subseteq_mset_set_a @ A3 @ B3 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_268_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A )
=> ( ( subseteq_mset_set_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_269_incidence__system_Oincomplete__alt__in,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) ) ) ).
% incidence_system.incomplete_alt_in
thf(fact_270_incidence__system_Oincomplete__alt__in,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) ) ) ).
% incidence_system.incomplete_alt_in
thf(fact_271_incidence__system_Oincomplete__alt__imp,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) ) ) ) ) ).
% incidence_system.incomplete_alt_imp
thf(fact_272_incidence__system_Oincomplete__alt__imp,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) ) ) ) ) ).
% incidence_system.incomplete_alt_imp
thf(fact_273_incidence__system_Oincomplete__alt__size,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) ) ) ) ).
% incidence_system.incomplete_alt_size
thf(fact_274_incidence__system_Oincomplete__alt__size,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) ) ) ) ).
% incidence_system.incomplete_alt_size
thf(fact_275_design_Oadd__block__design__cond,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ Bl @ Point_set )
=> ( ( Bl != bot_bot_set_nat )
=> ( design_design_nat @ Point_set @ ( design4725324266511619850ck_nat @ Block_collection @ Bl ) ) ) ) ) ).
% design.add_block_design_cond
thf(fact_276_design_Oadd__block__design__cond,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Bl @ Point_set )
=> ( ( Bl != bot_bot_set_a )
=> ( design_design_a @ Point_set @ ( design4001997691126659652lock_a @ Block_collection @ Bl ) ) ) ) ) ).
% design.add_block_design_cond
thf(fact_277_design_Ov__eq0__imp__b__eq__0,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( ( finite_card_nat @ Point_set )
= zero_zero_nat )
=> ( ( size_s7462436076474991978et_nat @ Block_collection )
= zero_zero_nat ) ) ) ).
% design.v_eq0_imp_b_eq_0
thf(fact_278_design_Ov__eq0__imp__b__eq__0,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ( finite_card_a @ Point_set )
= zero_zero_nat )
=> ( ( size_s6566526139600085008_set_a @ Block_collection )
= zero_zero_nat ) ) ) ).
% design.v_eq0_imp_b_eq_0
thf(fact_279_design_Oproper__designI,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ( size_s6566526139600085008_set_a @ Block_collection )
!= zero_zero_nat )
=> ( design7287791228148780576sign_a @ Point_set @ Block_collection ) ) ) ).
% design.proper_designI
thf(fact_280_finite__incidence__system_Oblock__complement__size,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ B @ Point_set )
=> ( ( finite_card_nat @ ( design2875492832550762736nt_nat @ Point_set @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ Point_set ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% finite_incidence_system.block_complement_size
thf(fact_281_finite__incidence__system_Oblock__complement__size,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ B @ Point_set )
=> ( ( finite_card_a @ ( design6447616907850319326ment_a @ Point_set @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ Point_set ) @ ( finite_card_a @ B ) ) ) ) ) ).
% finite_incidence_system.block_complement_size
thf(fact_282_finite__incidence__system_Odel__point__order,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ P @ Point_set )
=> ( ( finite_card_set_a @ ( design7586725432863044395_set_a @ Point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_set_a @ Point_set ) @ one_one_nat ) ) ) ) ).
% finite_incidence_system.del_point_order
thf(fact_283_finite__incidence__system_Odel__point__order,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ P @ Point_set )
=> ( ( finite_card_nat @ ( design4269233978287968195nt_nat @ Point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_nat @ Point_set ) @ one_one_nat ) ) ) ) ).
% finite_incidence_system.del_point_order
thf(fact_284_finite__incidence__system_Odel__point__order,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ P @ Point_set )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ Point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_a @ Point_set ) @ one_one_nat ) ) ) ) ).
% finite_incidence_system.del_point_order
thf(fact_285_design_Ocomplement__design,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) ) ) )
=> ( design_design_nat @ Point_set @ ( design5569578106646884273ks_nat @ Point_set @ Block_collection ) ) ) ) ).
% design.complement_design
thf(fact_286_design_Ocomplement__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) ) ) )
=> ( design_design_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ) ).
% design.complement_design
thf(fact_287_proper__design_Ocomplement__proper__design,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design435815215503836206gn_nat @ Point_set @ Block_collection )
=> ( ! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) ) ) )
=> ( design435815215503836206gn_nat @ Point_set @ ( design5569578106646884273ks_nat @ Point_set @ Block_collection ) ) ) ) ).
% proper_design.complement_proper_design
thf(fact_288_proper__design_Ocomplement__proper__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) ) ) )
=> ( design7287791228148780576sign_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ) ).
% proper_design.complement_proper_design
thf(fact_289_finite__incidence__system_OdesignI,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ! [B4: set_nat] :
( ( member_set_nat @ B4 @ ( set_mset_set_nat @ Block_collection ) )
=> ( B4 != bot_bot_set_nat ) )
=> ( ( Block_collection != zero_z3157962936165190495et_nat )
=> ( ( Point_set != bot_bot_set_nat )
=> ( design_design_nat @ Point_set @ Block_collection ) ) ) ) ) ).
% finite_incidence_system.designI
thf(fact_290_finite__incidence__system_OdesignI,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( set_mset_set_a @ Block_collection ) )
=> ( B4 != bot_bot_set_a ) )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ( ( Point_set != bot_bot_set_a )
=> ( design_design_a @ Point_set @ Block_collection ) ) ) ) ) ).
% finite_incidence_system.designI
thf(fact_291_incidence__system_Osys__block__sizes__in,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( member_nat @ ( finite_card_nat @ Bl ) @ ( design8152002643121538447es_nat @ Block_collection ) ) ) ) ).
% incidence_system.sys_block_sizes_in
thf(fact_292_incidence__system_Osys__block__sizes__in,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( member_nat @ ( finite_card_a @ Bl ) @ ( design1769254222028858111izes_a @ Block_collection ) ) ) ) ).
% incidence_system.sys_block_sizes_in
thf(fact_293_incidence__system_Osys__block__sizes__obtain__bl,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ ( design8152002643121538447es_nat @ Block_collection ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_mset_set_nat @ Block_collection ) )
& ( ( finite_card_nat @ X3 )
= X ) ) ) ) ).
% incidence_system.sys_block_sizes_obtain_bl
thf(fact_294_incidence__system_Osys__block__sizes__obtain__bl,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ ( design1769254222028858111izes_a @ Block_collection ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_mset_set_a @ Block_collection ) )
& ( ( finite_card_a @ X3 )
= X ) ) ) ) ).
% incidence_system.sys_block_sizes_obtain_bl
thf(fact_295_incidence__system_Odel__block__b_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ Block_collection @ Bl ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) @ one_one_nat ) ) ) ) ).
% incidence_system.del_block_b(1)
thf(fact_296_mset__subset__eqD,axiom,
! [A2: multiset_a,B2: multiset_a,X: a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( member_a @ X @ ( set_mset_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_297_mset__subset__eqD,axiom,
! [A2: multiset_nat,B2: multiset_nat,X: nat] :
( ( subseteq_mset_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ ( set_mset_nat @ A2 ) )
=> ( member_nat @ X @ ( set_mset_nat @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_298_mset__subset__eqD,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,X: set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ ( set_mset_set_a @ A2 ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_299_subset__mset_Ozero__le,axiom,
! [X: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X ) ).
% subset_mset.zero_le
thf(fact_300_subset__mset_Obot__least,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A ) ).
% subset_mset.bot_least
thf(fact_301_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( A = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_302_empty__le,axiom,
! [A2: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A2 ) ).
% empty_le
thf(fact_303_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_304_multiset__nonemptyE,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ ( set_mset_nat @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_305_multiset__nonemptyE,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
=> ~ ! [X3: set_a] :
~ ( member_set_a @ X3 @ ( set_mset_set_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_306_repeat__mset__cancel2,axiom,
! [A: nat,A2: multiset_set_a,B: nat] :
( ( ( repeat_mset_set_a @ A @ A2 )
= ( repeat_mset_set_a @ B @ A2 ) )
= ( ( A = B )
| ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_cancel2
thf(fact_307_set__mset__mono,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( set_mset_set_a @ A2 ) @ ( set_mset_set_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_308_set__mset__mono,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_309_multiple__not__simple,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( block_collection != zero_z5079479921072680283_set_a )
=> ~ ( design1338723777345758283stem_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ).
% multiple_not_simple
thf(fact_310_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_311_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_312_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_313_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_314_wf__design__iff,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( design_design_a @ point_set @ block_collection )
= ( ( ord_less_eq_set_a @ Bl @ point_set )
& ( finite_finite_a @ point_set )
& ( Bl != bot_bot_set_a ) ) ) ) ).
% wf_design_iff
thf(fact_315_incomplete__block__proper__subset,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) )
=> ( ord_less_set_a @ Bl @ point_set ) ) ).
% incomplete_block_proper_subset
thf(fact_316_block__complement__def,axiom,
! [B: set_a] :
( ( design6447616907850319326ment_a @ point_set @ B )
= ( minus_minus_set_a @ point_set @ B ) ) ).
% block_complement_def
thf(fact_317_finite__sets,axiom,
finite_finite_a @ point_set ).
% finite_sets
thf(fact_318_finite__blocks,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( finite_finite_a @ B ) ) ).
% finite_blocks
thf(fact_319_block__sizes__non__empty__set,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ( ( design1769254222028858111izes_a @ block_collection )
!= bot_bot_set_nat ) ) ).
% block_sizes_non_empty_set
thf(fact_320_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_321_finite__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_322_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_323_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_324_finite__Diff2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_325_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_326_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_327_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_328_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_329_block__size__lt__v,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) ) ) ).
% block_size_lt_v
thf(fact_330_multiple__blocks__gt,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ).
% multiple_blocks_gt
thf(fact_331_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_332_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_333_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ A )
= zero_z5079479921072680283_set_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_334_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_335_diff__zero,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% diff_zero
thf(fact_336_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_337_zero__diff,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_z5079479921072680283_set_a ) ).
% zero_diff
thf(fact_338_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_339_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_340_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_341_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_342_finite__set__mset,axiom,
! [M: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M ) ) ).
% finite_set_mset
thf(fact_343_finite__set__mset,axiom,
! [M: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M ) ) ).
% finite_set_mset
thf(fact_344_finite__set__mset,axiom,
! [M: multiset_set_a] : ( finite_finite_set_a @ ( set_mset_set_a @ M ) ) ).
% finite_set_mset
thf(fact_345_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_346_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_347_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_348_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_349_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_350_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_351_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_352_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_353_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_354_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_355_finite__sysI,axiom,
( ( finite_finite_a @ point_set )
=> ( design9187838744727572296stem_a @ point_set @ block_collection ) ) ).
% finite_sysI
thf(fact_356_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_357_diff__card__le__card__Diff,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_358_diff__card__le__card__Diff,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_359_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_360_card__le__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_361_card__le__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_362_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_363_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_364_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_365_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_366_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_367_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_368_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_369_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_370_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_371_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_372_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_373_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_374_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_375_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_376_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_377_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_378_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_379_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_380_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_381_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_382_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_383_Diff__infinite__finite,axiom,
! [T: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_384_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_385_finite__psubset__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_386_finite__psubset__induct,axiom,
! [A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ! [B5: set_set_a] :
( ( ord_less_set_set_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_387_finite__psubset__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ! [B5: set_a] :
( ( ord_less_set_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_388_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_389_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_390_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_391_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_392_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_393_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_394_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_395_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_396_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_397_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_398_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_399_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_400_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_401_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_402_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_403_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_404_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_405_card__less__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_406_card__less__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_407_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_408_psubset__card__mono,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_409_psubset__card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_410_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_411_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C2: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_412_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_413_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_414_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T2 @ S )
=> ( ( ( finite_card_set_a @ T2 )
= N )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_415_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T2: set_a] :
( ( ord_less_eq_set_a @ T2 @ S )
=> ( ( ( finite_card_a @ T2 )
= N )
=> ~ ( finite_finite_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_416_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_417_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ C2 )
& ( ( finite_card_set_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_418_exists__subset__between,axiom,
! [A2: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A2 @ B6 )
& ( ord_less_eq_set_a @ B6 @ C2 )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_419_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_420_card__seteq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B2 ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_421_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_422_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_423_card__mono,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_424_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_425_diff__size__le__size__Diff,axiom,
! [M: multiset_set_a,M4: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ M4 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ M4 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_426_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_427_card__psubset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_428_card__psubset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_429_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_430_card__Diff__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_431_card__Diff__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_432_diff__subset__eq__self,axiom,
! [M: multiset_set_a,N3: multiset_set_a] : ( subseteq_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N3 ) @ M ) ).
% diff_subset_eq_self
thf(fact_433_in__diffD,axiom,
! [A: a,M: multiset_a,N3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N3 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_434_in__diffD,axiom,
! [A: nat,M: multiset_nat,N3: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N3 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ M ) ) ) ).
% in_diffD
thf(fact_435_in__diffD,axiom,
! [A: set_a,M: multiset_set_a,N3: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N3 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ M ) ) ) ).
% in_diffD
thf(fact_436_Multiset_Odiff__cancel,axiom,
! [A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A2 @ A2 )
= zero_z5079479921072680283_set_a ) ).
% Multiset.diff_cancel
thf(fact_437_diff__empty,axiom,
! [M: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ M @ zero_z5079479921072680283_set_a )
= M )
& ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ M )
= zero_z5079479921072680283_set_a ) ) ).
% diff_empty
thf(fact_438_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_439_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_440_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_441_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_442_finite__incidence__system_Ofinite__sets,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( finite_finite_nat @ Point_set ) ) ).
% finite_incidence_system.finite_sets
thf(fact_443_finite__incidence__system_Ofinite__sets,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( finite_finite_set_a @ Point_set ) ) ).
% finite_incidence_system.finite_sets
thf(fact_444_finite__incidence__system_Ofinite__sets,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( finite_finite_a @ Point_set ) ) ).
% finite_incidence_system.finite_sets
thf(fact_445_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_446_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ( finite_card_set_a @ B6 )
= N )
& ( ord_le3724670747650509150_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_447_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_448_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_449_card__subset__eq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_450_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_451_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_452_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_453_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_454_simple__incidence__system_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ Point_set @ Block_collection ) ) ).
% simple_incidence_system.axioms(1)
thf(fact_455_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_456_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_457_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_458_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_459_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_460_card__ge__0__finite,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_461_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ A2 @ B2 )
= zero_z5079479921072680283_set_a )
= ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_462_left__diff__repeat__mset__distrib_H,axiom,
! [I: nat,J: nat,U: multiset_set_a] :
( ( repeat_mset_set_a @ ( minus_minus_nat @ I @ J ) @ U )
= ( minus_706656509937749387_set_a @ ( repeat_mset_set_a @ I @ U ) @ ( repeat_mset_set_a @ J @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_463_size__mset__mono,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ A2 ) @ ( size_s6566526139600085008_set_a @ B2 ) ) ) ).
% size_mset_mono
thf(fact_464_finite__incidence__system_Ofinite__blocks,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ Block_collection ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_incidence_system.finite_blocks
thf(fact_465_finite__incidence__system_Ofinite__blocks,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,B: set_set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ Block_collection ) )
=> ( finite_finite_set_a @ B ) ) ) ).
% finite_incidence_system.finite_blocks
thf(fact_466_finite__incidence__system_Ofinite__blocks,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_incidence_system.finite_blocks
thf(fact_467_incidence__system_Ofinite__sysI,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( finite_finite_nat @ Point_set )
=> ( design5426232790142929158em_nat @ Point_set @ Block_collection ) ) ) ).
% incidence_system.finite_sysI
thf(fact_468_incidence__system_Ofinite__sysI,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( finite_finite_set_a @ Point_set )
=> ( design1749870844763721896_set_a @ Point_set @ Block_collection ) ) ) ).
% incidence_system.finite_sysI
thf(fact_469_incidence__system_Ofinite__sysI,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( finite_finite_a @ Point_set )
=> ( design9187838744727572296stem_a @ Point_set @ Block_collection ) ) ) ).
% incidence_system.finite_sysI
thf(fact_470_incidence__system_Oblock__complement__def,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( design6447616907850319326ment_a @ Point_set @ B )
= ( minus_minus_set_a @ Point_set @ B ) ) ) ).
% incidence_system.block_complement_def
thf(fact_471_card__gt__0__iff,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
= ( ( A2 != bot_bot_set_set_a )
& ( finite_finite_set_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_472_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_473_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_474_zero__reorient,axiom,
! [X: multiset_set_a] :
( ( zero_z5079479921072680283_set_a = X )
= ( X = zero_z5079479921072680283_set_a ) ) ).
% zero_reorient
thf(fact_475_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_476_size__Diff__submset,axiom,
! [M: multiset_set_a,M4: multiset_set_a] :
( ( subseteq_mset_set_a @ M @ M4 )
=> ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M4 @ M ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M4 ) @ ( size_s6566526139600085008_set_a @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_477_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_478_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_479_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_480_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P2 @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P2 @ M5 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_481_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P2 @ M5 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_482_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_483_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_484_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_485_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_486_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_487_size__neq__size__imp__neq,axiom,
! [X: multiset_set_a,Y2: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ X )
!= ( size_s6566526139600085008_set_a @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_488_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_489_design_Oblock__size__lt__v,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) ) ) ) ).
% design.block_size_lt_v
thf(fact_490_design_Oblock__size__lt__v,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) ) ) ) ).
% design.block_size_lt_v
thf(fact_491_finite__incidence__system_Oblock__size__lt__order,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) ) ) ) ).
% finite_incidence_system.block_size_lt_order
thf(fact_492_finite__incidence__system_Oblock__size__lt__order,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) ) ) ) ).
% finite_incidence_system.block_size_lt_order
thf(fact_493_incidence__system_Oblock__sizes__non__empty__set,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ( ( design1769254222028858111izes_a @ Block_collection )
!= bot_bot_set_nat ) ) ) ).
% incidence_system.block_sizes_non_empty_set
thf(fact_494_simple__incidence__system_Osimple__block__size__eq__card,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ( ( size_s6566526139600085008_set_a @ Block_collection )
= ( finite_card_set_a @ ( design5397942185814921632port_a @ Block_collection ) ) ) ) ).
% simple_incidence_system.simple_block_size_eq_card
thf(fact_495_design_Owf__design__iff,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bl: set_set_a] :
( ( design_design_set_a @ Point_set @ Block_collection )
=> ( ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ Block_collection ) )
=> ( ( design_design_set_a @ Point_set @ Block_collection )
= ( ( ord_le3724670747650509150_set_a @ Bl @ Point_set )
& ( finite_finite_set_a @ Point_set )
& ( Bl != bot_bot_set_set_a ) ) ) ) ) ).
% design.wf_design_iff
thf(fact_496_design_Owf__design__iff,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( design_design_nat @ Point_set @ Block_collection )
= ( ( ord_less_eq_set_nat @ Bl @ Point_set )
& ( finite_finite_nat @ Point_set )
& ( Bl != bot_bot_set_nat ) ) ) ) ) ).
% design.wf_design_iff
thf(fact_497_design_Owf__design__iff,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design_design_a @ Point_set @ Block_collection )
= ( ( ord_less_eq_set_a @ Bl @ Point_set )
& ( finite_finite_a @ Point_set )
& ( Bl != bot_bot_set_a ) ) ) ) ) ).
% design.wf_design_iff
thf(fact_498_finite__incidence__system_Oincomplete__block__proper__subset,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ord_less_set_nat @ Bl @ Point_set ) ) ) ).
% finite_incidence_system.incomplete_block_proper_subset
thf(fact_499_finite__incidence__system_Oincomplete__block__proper__subset,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ord_less_set_a @ Bl @ Point_set ) ) ) ).
% finite_incidence_system.incomplete_block_proper_subset
thf(fact_500_wf__design__implies,axiom,
! [B7: multiset_set_set_a,V: set_set_a] :
( ! [B4: set_set_a] :
( ( member_set_set_a @ B4 @ ( set_mset_set_set_a @ B7 ) )
=> ( ord_le3724670747650509150_set_a @ B4 @ V ) )
=> ( ! [B4: set_set_a] :
( ( member_set_set_a @ B4 @ ( set_mset_set_set_a @ B7 ) )
=> ( B4 != bot_bot_set_set_a ) )
=> ( ( finite_finite_set_a @ V )
=> ( ( B7 != zero_z6396401802697562811_set_a )
=> ( ( V != bot_bot_set_set_a )
=> ( design_design_set_a @ V @ B7 ) ) ) ) ) ) ).
% wf_design_implies
thf(fact_501_wf__design__implies,axiom,
! [B7: multiset_set_nat,V: set_nat] :
( ! [B4: set_nat] :
( ( member_set_nat @ B4 @ ( set_mset_set_nat @ B7 ) )
=> ( ord_less_eq_set_nat @ B4 @ V ) )
=> ( ! [B4: set_nat] :
( ( member_set_nat @ B4 @ ( set_mset_set_nat @ B7 ) )
=> ( B4 != bot_bot_set_nat ) )
=> ( ( finite_finite_nat @ V )
=> ( ( B7 != zero_z3157962936165190495et_nat )
=> ( ( V != bot_bot_set_nat )
=> ( design_design_nat @ V @ B7 ) ) ) ) ) ) ).
% wf_design_implies
thf(fact_502_wf__design__implies,axiom,
! [B7: multiset_set_a,V: set_a] :
( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( set_mset_set_a @ B7 ) )
=> ( ord_less_eq_set_a @ B4 @ V ) )
=> ( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( set_mset_set_a @ B7 ) )
=> ( B4 != bot_bot_set_a ) )
=> ( ( finite_finite_a @ V )
=> ( ( B7 != zero_z5079479921072680283_set_a )
=> ( ( V != bot_bot_set_a )
=> ( design_design_a @ V @ B7 ) ) ) ) ) ) ).
% wf_design_implies
thf(fact_503_incidence__system_Omultiple__blocks__gt,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) @ ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ) ).
% incidence_system.multiple_blocks_gt
thf(fact_504_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_505_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_506_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_507_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_508_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_509_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_510_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_511_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_512_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_513_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_514_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_515_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_516_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_517_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_518_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_519_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P2 @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P2 @ M5 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_520_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_521_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_522_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_523_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_524_incidence__system_Omultiple__not__simple,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ~ ( design1338723777345758283stem_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ) ).
% incidence_system.multiple_not_simple
thf(fact_525_design_On__inter__num__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B1: set_a,B22: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B1 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ) ).
% design.n_inter_num_zero
thf(fact_526_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_527_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_528_strong__del__block__des,axiom,
! [B: set_a] :
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ~ ( ord_less_set_a @ Bl2 @ B ) )
=> ( design_design_a @ ( minus_minus_set_a @ point_set @ B ) @ ( design4241783006516448631lock_a @ block_collection @ B ) ) ) ).
% strong_del_block_des
thf(fact_529_strong__del__block__wf,axiom,
! [B: set_a] : ( design1863209521793301785stem_a @ ( minus_minus_set_a @ point_set @ B ) @ ( design4241783006516448631lock_a @ block_collection @ B ) ) ).
% strong_del_block_wf
thf(fact_530_strong__del__block__fin,axiom,
! [B: set_a] : ( design9187838744727572296stem_a @ ( minus_minus_set_a @ point_set @ B ) @ ( design4241783006516448631lock_a @ block_collection @ B ) ) ).
% strong_del_block_fin
thf(fact_531_replication__numbers__non__empty,axiom,
( ( point_set != bot_bot_set_a )
=> ( ( design8835372594653258411bers_a @ point_set @ block_collection )
!= bot_bot_set_nat ) ) ).
% replication_numbers_non_empty
thf(fact_532_add__block__design,axiom,
! [Bl: set_a] :
( ( finite_finite_a @ Bl )
=> ( ( Bl != bot_bot_set_a )
=> ( design_design_a @ ( sup_sup_set_a @ point_set @ Bl ) @ ( design4001997691126659652lock_a @ block_collection @ Bl ) ) ) ) ).
% add_block_design
thf(fact_533_delete__point__blocks__wf,axiom,
! [B: set_a,P: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) )
=> ( ord_less_eq_set_a @ B @ ( minus_minus_set_a @ point_set @ ( insert_a @ P @ bot_bot_set_a ) ) ) ) ).
% delete_point_blocks_wf
thf(fact_534_simple__not__multiplicity,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ one_one_nat @ ( count_set_a @ block_collection @ B ) )
=> ~ ( design1338723777345758283stem_a @ point_set @ block_collection ) ) ) ).
% simple_not_multiplicity
thf(fact_535_finite__block__sizes,axiom,
finite_finite_nat @ ( design1769254222028858111izes_a @ block_collection ) ).
% finite_block_sizes
thf(fact_536_finite__design__support,axiom,
finite_finite_set_a @ ( design5397942185814921632port_a @ block_collection ) ).
% finite_design_support
thf(fact_537_add__point__def,axiom,
! [P: a] :
( ( design2964366272795260673oint_a @ point_set @ P )
= ( insert_a @ P @ point_set ) ) ).
% add_point_def
thf(fact_538_replication__numbers__finite,axiom,
finite_finite_nat @ ( design8835372594653258411bers_a @ point_set @ block_collection ) ).
% replication_numbers_finite
thf(fact_539_del__point__def,axiom,
! [P: a] :
( ( design108908007054065099oint_a @ point_set @ P )
= ( minus_minus_set_a @ point_set @ ( insert_a @ P @ bot_bot_set_a ) ) ) ).
% del_point_def
thf(fact_540_block__original__count__le,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_a @ block_collection @ B ) @ ( count_set_a @ ( repeat_mset_set_a @ N @ block_collection ) @ B ) ) ) ).
% block_original_count_le
thf(fact_541_add__block__wf,axiom,
! [B: set_a] : ( design1863209521793301785stem_a @ ( sup_sup_set_a @ point_set @ B ) @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ).
% add_block_wf
thf(fact_542_str__del__block__del__point,axiom,
! [X: a] :
( ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( set_mset_set_a @ block_collection ) )
=> ( ( design4241783006516448631lock_a @ block_collection @ ( insert_a @ X @ bot_bot_set_a ) )
= ( design6411949732824333445ocks_a @ block_collection @ X ) ) ) ).
% str_del_block_del_point
thf(fact_543_remove__invalid__point__block,axiom,
! [P: a,Bl: set_a] :
( ~ ( member_a @ P @ point_set )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( minus_minus_set_a @ Bl @ ( insert_a @ P @ bot_bot_set_a ) )
= Bl ) ) ) ).
% remove_invalid_point_block
thf(fact_544_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_545_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_546_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_547_finite__Un,axiom,
! [F2: set_set_a,G2: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G2 ) )
= ( ( finite_finite_set_a @ F2 )
& ( finite_finite_set_a @ G2 ) ) ) ).
% finite_Un
thf(fact_548_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_549_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_550_add__block__fin,axiom,
! [B: set_a] :
( ( finite_finite_a @ B )
=> ( design9187838744727572296stem_a @ ( sup_sup_set_a @ point_set @ B ) @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ) ).
% add_block_fin
thf(fact_551_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_552_finite__Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B2: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_553_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_554_count__empty,axiom,
! [A: set_a] :
( ( count_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_555_count__diff,axiom,
! [M: multiset_set_a,N3: multiset_set_a,A: set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ M @ N3 ) @ A )
= ( minus_minus_nat @ ( count_set_a @ M @ A ) @ ( count_set_a @ N3 @ A ) ) ) ).
% count_diff
thf(fact_556_count__greater__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M @ X ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_557_count__greater__zero__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( count_nat @ M @ X ) )
= ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_558_count__greater__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_set_a @ M @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_559_count__greater__eq__one__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M @ X ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_560_count__greater__eq__one__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_nat @ M @ X ) )
= ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_561_count__greater__eq__one__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_set_a @ M @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_562_card__Diff__insert,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ A @ B2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_563_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_564_card__Diff__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_565_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_566_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_567_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_568_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_569_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_570_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_571_finite__incidence__system_Oreplication__numbers__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( finite_finite_nat @ ( design8835372594653258411bers_a @ Point_set @ Block_collection ) ) ) ).
% finite_incidence_system.replication_numbers_finite
thf(fact_572_incidence__system_Oreplication__numbers_Ocong,axiom,
design8835372594653258411bers_a = design8835372594653258411bers_a ).
% incidence_system.replication_numbers.cong
thf(fact_573_incidence__system_Ostr__del__block_Ocong,axiom,
design4241783006516448631lock_a = design4241783006516448631lock_a ).
% incidence_system.str_del_block.cong
thf(fact_574_multiset__eq__iff,axiom,
( ( ^ [Y: multiset_set_a,Z: multiset_set_a] : ( Y = Z ) )
= ( ^ [M6: multiset_set_a,N5: multiset_set_a] :
! [A3: set_a] :
( ( count_set_a @ M6 @ A3 )
= ( count_set_a @ N5 @ A3 ) ) ) ) ).
% multiset_eq_iff
thf(fact_575_multiset__eqI,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ! [X3: set_a] :
( ( count_set_a @ A2 @ X3 )
= ( count_set_a @ B2 @ X3 ) )
=> ( A2 = B2 ) ) ).
% multiset_eqI
thf(fact_576_count__inject,axiom,
! [X: multiset_set_a,Y2: multiset_set_a] :
( ( ( count_set_a @ X )
= ( count_set_a @ Y2 ) )
= ( X = Y2 ) ) ).
% count_inject
thf(fact_577_mset__subset__eq__count,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,A: set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( count_set_a @ A2 @ A ) @ ( count_set_a @ B2 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_578_subseteq__mset__def,axiom,
( subseteq_mset_set_a
= ( ^ [A5: multiset_set_a,B8: multiset_set_a] :
! [A3: set_a] : ( ord_less_eq_nat @ ( count_set_a @ A5 @ A3 ) @ ( count_set_a @ B8 @ A3 ) ) ) ) ).
% subseteq_mset_def
thf(fact_579_mset__subset__eqI,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ! [A6: set_a] : ( ord_less_eq_nat @ ( count_set_a @ A2 @ A6 ) @ ( count_set_a @ B2 @ A6 ) )
=> ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% mset_subset_eqI
thf(fact_580_minus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa2: multiset_set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ X @ Xa2 ) )
= ( ^ [A3: set_a] : ( minus_minus_nat @ ( count_set_a @ X @ A3 ) @ ( count_set_a @ Xa2 @ A3 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_581_finite__UnI,axiom,
! [F2: set_set_a,G2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( finite_finite_set_a @ G2 )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_582_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_583_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_584_Un__infinite,axiom,
! [S: set_set_a,T: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_585_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_586_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_587_infinite__Un,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_set_a @ S )
| ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_Un
thf(fact_588_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_589_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_590_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_591_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_592_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_593_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_594_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_595_subset__mset_Ofinite__has__maximal2,axiom,
! [A2: set_multiset_set_a,A: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( member2747690772047059533_set_a @ A @ A2 )
=> ? [X3: multiset_set_a] :
( ( member2747690772047059533_set_a @ X3 @ A2 )
& ( subseteq_mset_set_a @ A @ X3 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal2
thf(fact_596_subset__mset_Ofinite__has__minimal2,axiom,
! [A2: set_multiset_set_a,A: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( member2747690772047059533_set_a @ A @ A2 )
=> ? [X3: multiset_set_a] :
( ( member2747690772047059533_set_a @ X3 @ A2 )
& ( subseteq_mset_set_a @ X3 @ A )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal2
thf(fact_597_subset__mset_Ofinite__has__minimal,axiom,
! [A2: set_multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( A2 != bot_bo9088538438451294192_set_a )
=> ? [X3: multiset_set_a] :
( ( member2747690772047059533_set_a @ X3 @ A2 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_598_subset__mset_Ofinite__has__maximal,axiom,
! [A2: set_multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( A2 != bot_bo9088538438451294192_set_a )
=> ? [X3: multiset_set_a] :
( ( member2747690772047059533_set_a @ X3 @ A2 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_599_count__inI,axiom,
! [M: multiset_a,X: a] :
( ( ( count_a @ M @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_inI
thf(fact_600_count__inI,axiom,
! [M: multiset_nat,X: nat] :
( ( ( count_nat @ M @ X )
!= zero_zero_nat )
=> ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% count_inI
thf(fact_601_count__inI,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M @ X )
!= zero_zero_nat )
=> ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_inI
thf(fact_602_count__eq__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ( count_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_603_count__eq__zero__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ( count_nat @ M @ X )
= zero_zero_nat )
= ( ~ ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_604_count__eq__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_605_zero__multiset_Orep__eq,axiom,
( ( count_set_a @ zero_z5079479921072680283_set_a )
= ( ^ [A3: set_a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_606_in__diff__count,axiom,
! [A: a,M: multiset_a,N3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N3 ) ) )
= ( ord_less_nat @ ( count_a @ N3 @ A ) @ ( count_a @ M @ A ) ) ) ).
% in_diff_count
thf(fact_607_in__diff__count,axiom,
! [A: nat,M: multiset_nat,N3: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N3 ) ) )
= ( ord_less_nat @ ( count_nat @ N3 @ A ) @ ( count_nat @ M @ A ) ) ) ).
% in_diff_count
thf(fact_608_in__diff__count,axiom,
! [A: set_a,M: multiset_set_a,N3: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N3 ) ) )
= ( ord_less_nat @ ( count_set_a @ N3 @ A ) @ ( count_set_a @ M @ A ) ) ) ).
% in_diff_count
thf(fact_609_infinite__finite__induct,axiom,
! [P2: set_set_a > $o,A2: set_set_a] :
( ! [A4: set_set_a] :
( ~ ( finite_finite_set_a @ A4 )
=> ( P2 @ A4 ) )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_610_infinite__finite__induct,axiom,
! [P2: set_a > $o,A2: set_a] :
( ! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( P2 @ A4 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_611_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A2: set_nat] :
( ! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( P2 @ A4 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_612_finite__empty__induct,axiom,
! [A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A6: set_a,A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ A6 @ A4 )
=> ( ( P2 @ A4 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ A6 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_613_finite__empty__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A6: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A6 @ A4 )
=> ( ( P2 @ A4 )
=> ( P2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_614_finite__empty__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A6: a,A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A6 @ A4 )
=> ( ( P2 @ A4 )
=> ( P2 @ ( minus_minus_set_a @ A4 @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_615_infinite__coinduct,axiom,
! [X5: set_set_a > $o,A2: set_set_a] :
( ( X5 @ A2 )
=> ( ! [A4: set_set_a] :
( ( X5 @ A4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
& ( ( X5 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_616_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A4: set_nat] :
( ( X5 @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( X5 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_617_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A4: set_a] :
( ( X5 @ A4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A4 )
& ( ( X5 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_618_finite__ne__induct,axiom,
! [F2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P2 @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_619_finite__ne__induct,axiom,
! [F2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P2 @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_620_finite__ne__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_621_infinite__remove,axiom,
! [S: set_set_a,A: set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_622_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_623_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_624_finite__induct,axiom,
! [F2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_625_finite__induct,axiom,
! [F2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_626_finite__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_627_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A3: set_set_a] :
( ( A3 = bot_bot_set_set_a )
| ? [A5: set_set_a,B3: set_a] :
( ( A3
= ( insert_set_a @ B3 @ A5 ) )
& ( finite_finite_set_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_628_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A5: set_a,B3: a] :
( ( A3
= ( insert_a @ B3 @ A5 ) )
& ( finite_finite_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_629_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A5: set_nat,B3: nat] :
( ( A3
= ( insert_nat @ B3 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_630_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A4: set_set_a] :
( ? [A6: set_a] :
( A
= ( insert_set_a @ A6 @ A4 ) )
=> ~ ( finite_finite_set_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_631_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A4: set_a] :
( ? [A6: a] :
( A
= ( insert_a @ A6 @ A4 ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_632_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A4: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A4 ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_633_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_634_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_635_incidence__system_Oadd__block__wf,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( design3753904077504641269em_nat @ ( sup_sup_set_nat @ Point_set @ B ) @ ( design4725324266511619850ck_nat @ Block_collection @ B ) ) ) ).
% incidence_system.add_block_wf
thf(fact_636_incidence__system_Oadd__block__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ ( sup_sup_set_a @ Point_set @ B ) @ ( design4001997691126659652lock_a @ Block_collection @ B ) ) ) ).
% incidence_system.add_block_wf
thf(fact_637_incidence__system_Oadd__point__def,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( design8239173135376323853nt_nat @ Point_set @ P )
= ( insert_nat @ P @ Point_set ) ) ) ).
% incidence_system.add_point_def
thf(fact_638_incidence__system_Oadd__point__def,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( design2964366272795260673oint_a @ Point_set @ P )
= ( insert_a @ P @ Point_set ) ) ) ).
% incidence_system.add_point_def
thf(fact_639_incidence__system_Ofinite__block__sizes,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( finite_finite_nat @ ( design1769254222028858111izes_a @ Block_collection ) ) ) ).
% incidence_system.finite_block_sizes
thf(fact_640_incidence__system_Ostrong__del__block__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( design1863209521793301785stem_a @ ( minus_minus_set_a @ Point_set @ B ) @ ( design4241783006516448631lock_a @ Block_collection @ B ) ) ) ).
% incidence_system.strong_del_block_wf
thf(fact_641_finite__incidence__system_Ostrong__del__block__fin,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( minus_minus_set_a @ Point_set @ B ) @ ( design4241783006516448631lock_a @ Block_collection @ B ) ) ) ).
% finite_incidence_system.strong_del_block_fin
thf(fact_642_finite__incidence__system_Ofinite__design__support,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( finite_finite_set_a @ ( design5397942185814921632port_a @ Block_collection ) ) ) ).
% finite_incidence_system.finite_design_support
thf(fact_643_incidence__system_Ostr__del__block__del__point,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( design3550126062406151447ck_nat @ Block_collection @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( design4832208198062110345ks_nat @ Block_collection @ X ) ) ) ) ).
% incidence_system.str_del_block_del_point
thf(fact_644_incidence__system_Ostr__del__block__del__point,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design4241783006516448631lock_a @ Block_collection @ ( insert_a @ X @ bot_bot_set_a ) )
= ( design6411949732824333445ocks_a @ Block_collection @ X ) ) ) ) ).
% incidence_system.str_del_block_del_point
thf(fact_645_finite__subset__induct_H,axiom,
! [F2: set_set_a,A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_646_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_647_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_648_finite__subset__induct,axiom,
! [F2: set_set_a,A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_649_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_650_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A2 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_651_remove__induct,axiom,
! [P2: set_set_a > $o,B2: set_set_a] :
( ( P2 @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_652_remove__induct,axiom,
! [P2: set_nat > $o,B2: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_653_remove__induct,axiom,
! [P2: set_a > $o,B2: set_a] :
( ( P2 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_654_finite__remove__induct,axiom,
! [B2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_655_finite__remove__induct,axiom,
! [B2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_656_finite__remove__induct,axiom,
! [B2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_657_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_658_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_659_finite__induct__select,axiom,
! [S: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ S )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [T2: set_set_a] :
( ( ord_less_set_set_a @ T2 @ S )
=> ( ( P2 @ T2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ ( minus_5736297505244876581_set_a @ S @ T2 ) )
& ( P2 @ ( insert_set_a @ X4 @ T2 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_660_finite__induct__select,axiom,
! [S: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [T2: set_nat] :
( ( ord_less_set_nat @ T2 @ S )
=> ( ( P2 @ T2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ S @ T2 ) )
& ( P2 @ ( insert_nat @ X4 @ T2 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_661_finite__induct__select,axiom,
! [S: set_a,P2: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [T2: set_a] :
( ( ord_less_set_a @ T2 @ S )
=> ( ( P2 @ T2 )
=> ? [X4: a] :
( ( member_a @ X4 @ ( minus_minus_set_a @ S @ T2 ) )
& ( P2 @ ( insert_a @ X4 @ T2 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_662_incidence__system_Oremove__invalid__point__block,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a,Bl: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ P @ Point_set )
=> ( ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ Block_collection ) )
=> ( ( minus_5736297505244876581_set_a @ Bl @ ( insert_set_a @ P @ bot_bot_set_set_a ) )
= Bl ) ) ) ) ).
% incidence_system.remove_invalid_point_block
thf(fact_663_incidence__system_Oremove__invalid__point__block,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat,Bl: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ P @ Point_set )
=> ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( minus_minus_set_nat @ Bl @ ( insert_nat @ P @ bot_bot_set_nat ) )
= Bl ) ) ) ) ).
% incidence_system.remove_invalid_point_block
thf(fact_664_incidence__system_Oremove__invalid__point__block,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,Bl: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ P @ Point_set )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( minus_minus_set_a @ Bl @ ( insert_a @ P @ bot_bot_set_a ) )
= Bl ) ) ) ) ).
% incidence_system.remove_invalid_point_block
thf(fact_665_simple__incidence__system_Osimple__alt__def__all,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( count_set_a @ Block_collection @ X4 )
= one_one_nat ) ) ) ).
% simple_incidence_system.simple_alt_def_all
thf(fact_666_simple__incidence__system_Osimple,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( count_set_a @ Block_collection @ Bl )
= one_one_nat ) ) ) ).
% simple_incidence_system.simple
thf(fact_667_incidence__system_Odel__point__def,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( design4269233978287968195nt_nat @ Point_set @ P )
= ( minus_minus_set_nat @ Point_set @ ( insert_nat @ P @ bot_bot_set_nat ) ) ) ) ).
% incidence_system.del_point_def
thf(fact_668_incidence__system_Odel__point__def,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( design108908007054065099oint_a @ Point_set @ P )
= ( minus_minus_set_a @ Point_set @ ( insert_a @ P @ bot_bot_set_a ) ) ) ) ).
% incidence_system.del_point_def
thf(fact_669_finite__incidence__system_Oadd__block__fin,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,B: set_set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( ( finite_finite_set_a @ B )
=> ( design1749870844763721896_set_a @ ( sup_sup_set_set_a @ Point_set @ B ) @ ( design7860908649167014820_set_a @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin
thf(fact_670_finite__incidence__system_Oadd__block__fin,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( finite_finite_nat @ B )
=> ( design5426232790142929158em_nat @ ( sup_sup_set_nat @ Point_set @ B ) @ ( design4725324266511619850ck_nat @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin
thf(fact_671_finite__incidence__system_Oadd__block__fin,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( finite_finite_a @ B )
=> ( design9187838744727572296stem_a @ ( sup_sup_set_a @ Point_set @ B ) @ ( design4001997691126659652lock_a @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin
thf(fact_672_incidence__system_Oreplication__numbers__non__empty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( Point_set != bot_bot_set_nat )
=> ( ( design3853898657598026467rs_nat @ Point_set @ Block_collection )
!= bot_bot_set_nat ) ) ) ).
% incidence_system.replication_numbers_non_empty
thf(fact_673_incidence__system_Oreplication__numbers__non__empty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( Point_set != bot_bot_set_a )
=> ( ( design8835372594653258411bers_a @ Point_set @ Block_collection )
!= bot_bot_set_nat ) ) ) ).
% incidence_system.replication_numbers_non_empty
thf(fact_674_card__Diff1__less__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) )
= ( ( finite_finite_set_a @ A2 )
& ( member_set_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_675_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_676_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_677_card__Diff2__less,axiom,
! [A2: set_set_a,X: set_a,Y2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( member_set_a @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_678_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_679_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y2: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_680_card__Diff1__less,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_681_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_682_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_683_incidence__system_Oblock__original__count__le,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_a @ Block_collection @ B ) @ ( count_set_a @ ( repeat_mset_set_a @ N @ Block_collection ) @ B ) ) ) ) ).
% incidence_system.block_original_count_le
thf(fact_684_card__Diff__singleton__if,axiom,
! [X: set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_685_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_686_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_687_card__Diff__singleton,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_688_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_689_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_690_design_Oadd__block__design,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bl: set_set_a] :
( ( design_design_set_a @ Point_set @ Block_collection )
=> ( ( finite_finite_set_a @ Bl )
=> ( ( Bl != bot_bot_set_set_a )
=> ( design_design_set_a @ ( sup_sup_set_set_a @ Point_set @ Bl ) @ ( design7860908649167014820_set_a @ Block_collection @ Bl ) ) ) ) ) ).
% design.add_block_design
thf(fact_691_design_Oadd__block__design,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( finite_finite_nat @ Bl )
=> ( ( Bl != bot_bot_set_nat )
=> ( design_design_nat @ ( sup_sup_set_nat @ Point_set @ Bl ) @ ( design4725324266511619850ck_nat @ Block_collection @ Bl ) ) ) ) ) ).
% design.add_block_design
thf(fact_692_design_Oadd__block__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( finite_finite_a @ Bl )
=> ( ( Bl != bot_bot_set_a )
=> ( design_design_a @ ( sup_sup_set_a @ Point_set @ Bl ) @ ( design4001997691126659652lock_a @ Block_collection @ Bl ) ) ) ) ) ).
% design.add_block_design
thf(fact_693_card__insert__le__m1,axiom,
! [N: nat,Y2: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y2 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y2 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_694_card__insert__le__m1,axiom,
! [N: nat,Y2: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y2 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y2 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_695_incidence__system_Osimple__not__multiplicity,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ one_one_nat @ ( count_set_a @ Block_collection @ B ) )
=> ~ ( design1338723777345758283stem_a @ Point_set @ Block_collection ) ) ) ) ).
% incidence_system.simple_not_multiplicity
thf(fact_696_incidence__system_Odelete__point__blocks__wf,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat,P: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( design4832208198062110345ks_nat @ Block_collection @ P ) ) )
=> ( ord_less_eq_set_nat @ B @ ( minus_minus_set_nat @ Point_set @ ( insert_nat @ P @ bot_bot_set_nat ) ) ) ) ) ).
% incidence_system.delete_point_blocks_wf
thf(fact_697_incidence__system_Odelete__point__blocks__wf,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ Block_collection @ P ) ) )
=> ( ord_less_eq_set_a @ B @ ( minus_minus_set_a @ Point_set @ ( insert_a @ P @ bot_bot_set_a ) ) ) ) ) ).
% incidence_system.delete_point_blocks_wf
thf(fact_698_design_Ostrong__del__block__des,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ~ ( ord_less_set_a @ Bl2 @ B ) )
=> ( design_design_a @ ( minus_minus_set_a @ Point_set @ B ) @ ( design4241783006516448631lock_a @ Block_collection @ B ) ) ) ) ).
% design.strong_del_block_des
thf(fact_699_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_700_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_701_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_702_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_703_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_704_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_705_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_706_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_707_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_708_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_709_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_710_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_711_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_712_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_713_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_714_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_715_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_716_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_717_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_718_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_719_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_720_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_721_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_722_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_723_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
= ( insert_nat @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_724_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_725_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_726_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_727_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_728_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_729_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_730_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_731_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_732_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_733_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_734_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_735_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_736_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_737_Un__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
| ( member_set_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_738_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_739_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_740_UnCI,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ A2 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_741_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_742_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_743_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_744_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_745_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_746_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_747_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_748_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_749_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_750_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_751_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_752_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_753_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_754_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_755_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_756_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_757_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_758_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_759_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_760_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_761_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_762_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_763_insert__Diff1,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_764_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_765_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_766_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_767_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_768_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_769_Un__insert__right,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_770_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_771_Un__insert__left,axiom,
! [A: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_772_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_773_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_774_Un__Diff__cancel2,axiom,
! [B2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_a @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_775_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_776_Un__Diff__cancel,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_777_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_778_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_779_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_780_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_781_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_782_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_783_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_784_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_785_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_786_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_787_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_788_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_789_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_790_set__eq__subset,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A5: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A5 @ B8 )
& ( ord_less_eq_set_a @ B8 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_791_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_792_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_793_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_794_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B8: set_set_a] :
! [T4: set_a] :
( ( member_set_a @ T4 @ A5 )
=> ( member_set_a @ T4 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_795_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B8: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A5 )
=> ( member_nat @ T4 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_796_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B8: set_a] :
! [T4: a] :
( ( member_a @ T4 @ A5 )
=> ( member_a @ T4 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_797_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_798_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_799_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B8: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_800_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B8: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_801_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B8: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_802_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_803_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_804_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_805_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_806_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_807_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_808_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_809_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B6: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_810_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_811_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_812_insert__commute,axiom,
! [X: a,Y2: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y2 @ A2 ) )
= ( insert_a @ Y2 @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_813_insert__commute,axiom,
! [X: nat,Y2: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y2 @ A2 ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_814_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C3 ) )
& ~ ( member_set_a @ B @ C3 )
& ( B2
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_815_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_816_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_817_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_818_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_819_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_820_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_821_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_822_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_823_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B6: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_824_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_825_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_826_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_827_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_828_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_829_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_830_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_831_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_832_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_833_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_834_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_835_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_836_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_837_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_838_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_839_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_840_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_841_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_842_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_843_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_844_Un__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_845_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_846_Un__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_847_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_848_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B8: set_a] : ( sup_sup_set_a @ B8 @ A5 ) ) ) ).
% Un_commute
thf(fact_849_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B8: set_nat] : ( sup_sup_set_nat @ B8 @ A5 ) ) ) ).
% Un_commute
thf(fact_850_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_851_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_852_Un__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_853_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_854_ball__Un,axiom,
! [A2: set_a,B2: set_a,P2: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_855_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_856_bex__Un,axiom,
! [A2: set_a,B2: set_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_857_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_858_UnI2,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_859_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_860_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_861_UnI1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_862_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_863_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_864_UnE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ~ ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_865_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_866_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_867_psubsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_868_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_869_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_870_psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_871_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_872_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_873_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_874_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_875_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_876_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_877_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_878_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_879_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_880_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_881_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_882_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_883_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_884_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_885_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_886_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_887_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_888_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_889_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_890_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_891_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_892_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_893_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_894_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_895_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_896_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_897_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_898_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_899_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B8: set_nat] :
( ( sup_sup_set_nat @ A5 @ B8 )
= B8 ) ) ) ).
% subset_Un_eq
thf(fact_900_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B8: set_a] :
( ( sup_sup_set_a @ A5 @ B8 )
= B8 ) ) ) ).
% subset_Un_eq
thf(fact_901_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A2 )
=> ! [B9: set_nat] :
( ( ord_less_eq_set_nat @ B9 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A7 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_902_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A2 )
=> ! [B9: set_a] :
( ( ord_less_eq_set_a @ B9 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_903_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_904_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_905_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_906_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_907_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_908_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_909_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_910_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_911_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_912_Un__least,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_913_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_914_Un__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_915_insert__Diff__if,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_916_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_917_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_918_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_919_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_920_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_921_Un__Diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C2 ) @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_922_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_923_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A5 @ B8 )
& ( A5 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_924_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_925_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_926_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A5 @ B8 )
& ~ ( ord_less_eq_set_a @ B8 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_927_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_928_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B8: set_a] :
( ( ord_less_set_a @ A5 @ B8 )
| ( A5 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_929_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ? [B4: set_a] : ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_930_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_931_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_932_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_933_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_934_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_935_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_936_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_937_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_938_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_939_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_940_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_941_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_942_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_943_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_944_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_945_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_946_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_947_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_948_Un__singleton__iff,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_949_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_950_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_951_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_952_subset__Diff__insert,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a @ X @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C2 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_953_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_954_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_955_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_956_Diff__partition,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_957_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_958_Diff__subset__conv,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_959_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_960_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_961_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_962_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_963_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_964_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_965_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_966_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_967_mset__size__ne0__set__card,axiom,
! [A2: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_mset_a @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_968_mset__size__ne0__set__card,axiom,
! [A2: multiset_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5917832649809541300et_nat @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( set_mset_nat @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_969_mset__size__ne0__set__card,axiom,
! [A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ ( set_mset_set_a @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_970_set__card__diff__ge__zero,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_971_set__card__diff__ge__zero,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_972_set__card__diff__ge__zero,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_973_add__point__sys__rep__numbers,axiom,
! [P: a] :
( ( design8835372594653258411bers_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection )
= ( sup_sup_set_nat @ ( design8835372594653258411bers_a @ point_set @ block_collection ) @ ( insert_nat @ ( design6637022207325878697mber_a @ block_collection @ P ) @ bot_bot_set_nat ) ) ) ).
% add_point_sys_rep_numbers
thf(fact_974_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_975_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_976_add__block__rep__number__not__in,axiom,
! [X: a,B: set_a] :
( ~ ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ X )
= ( design6637022207325878697mber_a @ block_collection @ X ) ) ) ).
% add_block_rep_number_not_in
thf(fact_977_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_978_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_979_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_980_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_981_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_982_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_983_sup__left__idem,axiom,
! [X: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) )
= ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_984_sup__left__idem,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_985_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_986_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_987_point__rep__number__in__set,axiom,
! [X: a] :
( ( member_a @ X @ point_set )
=> ( member_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) @ ( design8835372594653258411bers_a @ point_set @ block_collection ) ) ) ).
% point_rep_number_in_set
thf(fact_988_obtain__point__with__rep,axiom,
! [R: nat] :
( ( member_nat @ R @ ( design8835372594653258411bers_a @ point_set @ block_collection ) )
=> ? [X3: a] :
( ( member_a @ X3 @ point_set )
& ( ( design6637022207325878697mber_a @ block_collection @ X3 )
= R ) ) ) ).
% obtain_point_with_rep
thf(fact_989_rep__number__non__zero__system__point,axiom,
! [X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) )
=> ( member_a @ X @ point_set ) ) ).
% rep_number_non_zero_system_point
thf(fact_990_point__in__block__rep__min__iff,axiom,
! [X: a] :
( ( member_a @ X @ point_set )
=> ? [Bl2: set_a] :
( ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
& ( member_a @ X @ Bl2 ) )
= ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) ) ) ) ).
% point_in_block_rep_min_iff
thf(fact_991_complement__rep__number,axiom,
! [X: a,R: nat] :
( ( member_a @ X @ point_set )
=> ( ( ( design6637022207325878697mber_a @ block_collection @ X )
= R )
=> ( ( design6637022207325878697mber_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) @ X )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ R ) ) ) ) ).
% complement_rep_number
thf(fact_992_le__sup__iff,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_993_le__sup__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_994_le__sup__iff,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_995_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_996_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_997_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_998_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_999_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_1000_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1001_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1002_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1003_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1004_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1005_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1006_sup__eq__bot__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X @ Y2 )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1007_sup__eq__bot__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1008_bot__eq__sup__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y2 ) )
= ( ( X = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1009_bot__eq__sup__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1010_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_1011_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_1012_point__rep__non__existance,axiom,
! [X: a] :
( ~ ( member_a @ X @ point_set )
=> ( ( design6637022207325878697mber_a @ block_collection @ X )
= zero_zero_nat ) ) ).
% point_rep_non_existance
thf(fact_1013_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1014_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1015_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1016_incidence__system_Opoint__rep__non__existance,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ X @ Point_set )
=> ( ( design5008467512594872073_set_a @ Block_collection @ X )
= zero_zero_nat ) ) ) ).
% incidence_system.point_rep_non_existance
thf(fact_1017_incidence__system_Opoint__rep__non__existance,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ X @ Point_set )
=> ( ( design3571518413069006949er_nat @ Block_collection @ X )
= zero_zero_nat ) ) ) ).
% incidence_system.point_rep_non_existance
thf(fact_1018_incidence__system_Opoint__rep__non__existance,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ X @ Point_set )
=> ( ( design6637022207325878697mber_a @ Block_collection @ X )
= zero_zero_nat ) ) ) ).
% incidence_system.point_rep_non_existance
thf(fact_1019_max__point__rep,axiom,
! [B2: multiset_set_a,X: a] : ( ord_less_eq_nat @ ( design6637022207325878697mber_a @ B2 @ X ) @ ( size_s6566526139600085008_set_a @ B2 ) ) ).
% max_point_rep
thf(fact_1020_incidence__system_Oadd__block__rep__number__not__in,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a,B: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( design5008467512594872073_set_a @ ( design7860908649167014820_set_a @ Block_collection @ B ) @ X )
= ( design5008467512594872073_set_a @ Block_collection @ X ) ) ) ) ).
% incidence_system.add_block_rep_number_not_in
thf(fact_1021_incidence__system_Oadd__block__rep__number__not__in,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat,B: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ X @ B )
=> ( ( design3571518413069006949er_nat @ ( design4725324266511619850ck_nat @ Block_collection @ B ) @ X )
= ( design3571518413069006949er_nat @ Block_collection @ X ) ) ) ) ).
% incidence_system.add_block_rep_number_not_in
thf(fact_1022_incidence__system_Oadd__block__rep__number__not__in,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ Block_collection @ B ) @ X )
= ( design6637022207325878697mber_a @ Block_collection @ X ) ) ) ) ).
% incidence_system.add_block_rep_number_not_in
thf(fact_1023_incidence__system_Oobtain__point__with__rep,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,R: nat] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_nat @ R @ ( design5240413817448814603_set_a @ Point_set @ Block_collection ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ Point_set )
& ( ( design5008467512594872073_set_a @ Block_collection @ X3 )
= R ) ) ) ) ).
% incidence_system.obtain_point_with_rep
thf(fact_1024_incidence__system_Oobtain__point__with__rep,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,R: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ R @ ( design3853898657598026467rs_nat @ Point_set @ Block_collection ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ Point_set )
& ( ( design3571518413069006949er_nat @ Block_collection @ X3 )
= R ) ) ) ) ).
% incidence_system.obtain_point_with_rep
thf(fact_1025_incidence__system_Oobtain__point__with__rep,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,R: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_nat @ R @ ( design8835372594653258411bers_a @ Point_set @ Block_collection ) )
=> ? [X3: a] :
( ( member_a @ X3 @ Point_set )
& ( ( design6637022207325878697mber_a @ Block_collection @ X3 )
= R ) ) ) ) ).
% incidence_system.obtain_point_with_rep
thf(fact_1026_incidence__system_Opoint__rep__number__in__set,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ X @ Point_set )
=> ( member_nat @ ( design5008467512594872073_set_a @ Block_collection @ X ) @ ( design5240413817448814603_set_a @ Point_set @ Block_collection ) ) ) ) ).
% incidence_system.point_rep_number_in_set
thf(fact_1027_incidence__system_Opoint__rep__number__in__set,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ Point_set )
=> ( member_nat @ ( design3571518413069006949er_nat @ Block_collection @ X ) @ ( design3853898657598026467rs_nat @ Point_set @ Block_collection ) ) ) ) ).
% incidence_system.point_rep_number_in_set
thf(fact_1028_incidence__system_Opoint__rep__number__in__set,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ X @ Point_set )
=> ( member_nat @ ( design6637022207325878697mber_a @ Block_collection @ X ) @ ( design8835372594653258411bers_a @ Point_set @ Block_collection ) ) ) ) ).
% incidence_system.point_rep_number_in_set
thf(fact_1029_rep__number__g0__exists,axiom,
! [B2: multiset_set_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( design5008467512594872073_set_a @ B2 @ X ) )
=> ~ ! [B4: set_set_a] :
( ( member_set_set_a @ B4 @ ( set_mset_set_set_a @ B2 ) )
=> ~ ( member_set_a @ X @ B4 ) ) ) ).
% rep_number_g0_exists
thf(fact_1030_rep__number__g0__exists,axiom,
! [B2: multiset_set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ B2 @ X ) )
=> ~ ! [B4: set_nat] :
( ( member_set_nat @ B4 @ ( set_mset_set_nat @ B2 ) )
=> ~ ( member_nat @ X @ B4 ) ) ) ).
% rep_number_g0_exists
thf(fact_1031_rep__number__g0__exists,axiom,
! [B2: multiset_set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ B2 @ X ) )
=> ~ ! [B4: set_a] :
( ( member_set_a @ B4 @ ( set_mset_set_a @ B2 ) )
=> ~ ( member_a @ X @ B4 ) ) ) ).
% rep_number_g0_exists
thf(fact_1032_incidence__system_Orep__number__non__zero__system__point,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( design5008467512594872073_set_a @ Block_collection @ X ) )
=> ( member_set_a @ X @ Point_set ) ) ) ).
% incidence_system.rep_number_non_zero_system_point
thf(fact_1033_incidence__system_Orep__number__non__zero__system__point,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ Block_collection @ X ) )
=> ( member_nat @ X @ Point_set ) ) ) ).
% incidence_system.rep_number_non_zero_system_point
thf(fact_1034_incidence__system_Orep__number__non__zero__system__point,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ Block_collection @ X ) )
=> ( member_a @ X @ Point_set ) ) ) ).
% incidence_system.rep_number_non_zero_system_point
thf(fact_1035_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) )
= ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_1036_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_1037_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_1038_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_1039_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_1040_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_1041_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1042_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1043_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1044_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1045_sup__assoc,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_1046_sup__assoc,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_1047_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_1048_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_1049_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_1050_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_1051_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1052_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1053_sup__left__commute,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_1054_sup__left__commute,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_1055_incidence__system_Opoint__in__block__rep__min__iff,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ X @ Point_set )
=> ? [Bl2: set_set_a] :
( ( ( member_set_set_a @ Bl2 @ ( set_mset_set_set_a @ Block_collection ) )
& ( member_set_a @ X @ Bl2 ) )
= ( ord_less_nat @ zero_zero_nat @ ( design5008467512594872073_set_a @ Block_collection @ X ) ) ) ) ) ).
% incidence_system.point_in_block_rep_min_iff
thf(fact_1056_incidence__system_Opoint__in__block__rep__min__iff,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ Point_set )
=> ? [Bl2: set_nat] :
( ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
& ( member_nat @ X @ Bl2 ) )
= ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ Block_collection @ X ) ) ) ) ) ).
% incidence_system.point_in_block_rep_min_iff
thf(fact_1057_incidence__system_Opoint__in__block__rep__min__iff,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ X @ Point_set )
=> ? [Bl2: set_a] :
( ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
& ( member_a @ X @ Bl2 ) )
= ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ Block_collection @ X ) ) ) ) ) ).
% incidence_system.point_in_block_rep_min_iff
thf(fact_1058_incidence__system_Ocomplement__rep__number,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a,R: nat] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ X @ Point_set )
=> ( ( ( design5008467512594872073_set_a @ Block_collection @ X )
= R )
=> ( ( design5008467512594872073_set_a @ ( design7413023778852989629_set_a @ Point_set @ Block_collection ) @ X )
= ( minus_minus_nat @ ( size_s5830485025544567152_set_a @ Block_collection ) @ R ) ) ) ) ) ).
% incidence_system.complement_rep_number
thf(fact_1059_incidence__system_Ocomplement__rep__number,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat,R: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ Point_set )
=> ( ( ( design3571518413069006949er_nat @ Block_collection @ X )
= R )
=> ( ( design3571518413069006949er_nat @ ( design5569578106646884273ks_nat @ Point_set @ Block_collection ) @ X )
= ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ Block_collection ) @ R ) ) ) ) ) ).
% incidence_system.complement_rep_number
thf(fact_1060_incidence__system_Ocomplement__rep__number,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a,R: nat] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ X @ Point_set )
=> ( ( ( design6637022207325878697mber_a @ Block_collection @ X )
= R )
=> ( ( design6637022207325878697mber_a @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) @ X )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) @ R ) ) ) ) ) ).
% incidence_system.complement_rep_number
thf(fact_1061_incidence__system_Oadd__point__sys__rep__numbers,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( design8835372594653258411bers_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ Block_collection )
= ( sup_sup_set_nat @ ( design8835372594653258411bers_a @ Point_set @ Block_collection ) @ ( insert_nat @ ( design6637022207325878697mber_a @ Block_collection @ P ) @ bot_bot_set_nat ) ) ) ) ).
% incidence_system.add_point_sys_rep_numbers
thf(fact_1062_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_1063_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_1064_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_1065_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_1066_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_1067_inf__sup__ord_I3_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_1068_le__supE,axiom,
! [A: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1069_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_1070_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1071_le__supI,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1072_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1073_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1074_sup__ge1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_1075_sup__ge1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_1076_sup__ge1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_1077_sup__ge2,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_1078_sup__ge2,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_1079_sup__ge2,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_1080_le__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_1081_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_1082_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_1083_le__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_1084_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_1085_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_1086_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1087_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1088_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1089_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1090_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1091_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1092_sup__least,axiom,
! [Y2: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1093_sup__least,axiom,
! [Y2: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1094_sup__least,axiom,
! [Y2: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1095_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1096_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1097_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1098_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1099_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1100_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1101_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_1102_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_1103_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_1104_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_1105_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( ( ord_less_eq_set_a @ Z3 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_1106_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y2: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_1107_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1108_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1109_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1110_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1111_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1112_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1113_sup__absorb1,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_1114_sup__absorb1,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( sup_sup_set_a @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_1115_sup__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( sup_sup_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_1116_sup__absorb2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_1117_sup__absorb2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( sup_sup_set_a @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_1118_sup__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_1119_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1120_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1121_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1122_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1123_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1124_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1125_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1126_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1127_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1128_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1129_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1130_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1131_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1132_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1133_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1134_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1135_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1136_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1137_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1138_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1139_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1140_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1141_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1142_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1143_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1144_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1145_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1146_less__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X @ A )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1147_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1148_less__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_1149_less__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X @ B )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1150_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1151_less__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_1152_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1153_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1154_sup_Oabsorb3,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1155_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1156_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1157_sup_Oabsorb4,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1158_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1159_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1160_sup_Ostrict__boundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1161_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1162_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1163_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1164_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1165_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1166_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1167_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1168_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1169_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1170_elem__in__repeat__in__original,axiom,
! [A: a,N: nat,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( repeat_mset_a @ N @ A2 ) ) )
=> ( member_a @ A @ ( set_mset_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_1171_elem__in__repeat__in__original,axiom,
! [A: nat,N: nat,A2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( repeat_mset_nat @ N @ A2 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_1172_elem__in__repeat__in__original,axiom,
! [A: set_a,N: nat,A2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_1173_set__diff__non__empty__not__subset,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
=> ( ( C2 != bot_bot_set_nat )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( B2 != bot_bot_set_nat )
=> ~ ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_1174_set__diff__non__empty__not__subset,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
=> ( ( C2 != bot_bot_set_a )
=> ( ( A2 != bot_bot_set_a )
=> ( ( B2 != bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_1175_elem__exists__non__empty__set,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ~ ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_1176_elem__exists__non__empty__set,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_1177_elem__exists__non__empty__set,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_1178_mset__nempty__set__nempty,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
= ( ( set_mset_set_a @ A2 )
!= bot_bot_set_set_a ) ) ).
% mset_nempty_set_nempty
thf(fact_1179_mset__nempty__set__nempty,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
= ( ( set_mset_a @ A2 )
!= bot_bot_set_a ) ) ).
% mset_nempty_set_nempty
thf(fact_1180_mset__nempty__set__nempty,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
= ( ( set_mset_nat @ A2 )
!= bot_bot_set_nat ) ) ).
% mset_nempty_set_nempty
thf(fact_1181_set__count__size__min,axiom,
! [N: nat,A2: multiset_set_a,A: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ A2 @ A ) )
=> ( ord_less_eq_nat @ N @ ( size_s6566526139600085008_set_a @ A2 ) ) ) ).
% set_count_size_min
thf(fact_1182_repeat__mset__subset__in,axiom,
! [A2: multiset_set_set_a,B2: set_set_a,X5: set_set_a,N: nat,X: set_a] :
( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ ( set_mset_set_set_a @ A2 ) )
=> ( ord_le3724670747650509150_set_a @ A6 @ B2 ) )
=> ( ( member_set_set_a @ X5 @ ( set_mset_set_set_a @ ( repeat3222187171979612824_set_a @ N @ A2 ) ) )
=> ( ( member_set_a @ X @ X5 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_1183_repeat__mset__subset__in,axiom,
! [A2: multiset_set_nat,B2: set_nat,X5: set_nat,N: nat,X: nat] :
( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ ( set_mset_set_nat @ A2 ) )
=> ( ord_less_eq_set_nat @ A6 @ B2 ) )
=> ( ( member_set_nat @ X5 @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ A2 ) ) )
=> ( ( member_nat @ X @ X5 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_1184_repeat__mset__subset__in,axiom,
! [A2: multiset_set_a,B2: set_a,X5: set_a,N: nat,X: a] :
( ! [A6: set_a] :
( ( member_set_a @ A6 @ ( set_mset_set_a @ A2 ) )
=> ( ord_less_eq_set_a @ A6 @ B2 ) )
=> ( ( member_set_a @ X5 @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) )
=> ( ( member_a @ X @ X5 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_1185_card__subset__not__gt__card,axiom,
! [A2: set_nat,Ps: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ Ps ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_1186_card__subset__not__gt__card,axiom,
! [A2: set_set_a,Ps: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ Ps ) )
=> ~ ( ord_le3724670747650509150_set_a @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_1187_card__subset__not__gt__card,axiom,
! [A2: set_a,Ps: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ Ps ) )
=> ~ ( ord_less_eq_set_a @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_1188_elem__in__original__in__repeat,axiom,
! [N: nat,A: a,A2: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ( member_a @ A @ ( set_mset_a @ ( repeat_mset_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_1189_elem__in__original__in__repeat,axiom,
! [N: nat,A: nat,A2: multiset_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
=> ( member_nat @ A @ ( set_mset_nat @ ( repeat_mset_nat @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_1190_elem__in__original__in__repeat,axiom,
! [N: nat,A: set_a,A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_1191_repeat__mset__not__empty,axiom,
! [N: nat,A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( A2 != zero_z5079479921072680283_set_a )
=> ( ( repeat_mset_set_a @ N @ A2 )
!= zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_not_empty
thf(fact_1192_delete__point__index__eq,axiom,
! [Ps: set_a,P: a] :
( ( ord_less_eq_set_a @ Ps @ ( design108908007054065099oint_a @ point_set @ P ) )
=> ( ( design254580327166089565ndex_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) @ Ps )
= ( design254580327166089565ndex_a @ block_collection @ Ps ) ) ) ).
% delete_point_index_eq
thf(fact_1193_add__block__rep__number__in,axiom,
! [X: a,B: set_a] :
( ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ X )
= ( plus_plus_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) @ one_one_nat ) ) ) ).
% add_block_rep_number_in
thf(fact_1194_finite__linorder__max__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B4: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_nat @ X4 @ B4 ) )
=> ( ( P2 @ A4 )
=> ( P2 @ ( insert_nat @ B4 @ A4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1195_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1196_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1197_add__block__index__not__in,axiom,
! [Ps: set_a,B: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ Ps )
= ( design254580327166089565ndex_a @ block_collection @ Ps ) ) ) ).
% add_block_index_not_in
thf(fact_1198_points__index__count__min,axiom,
! [N: nat,Bl: set_a,Ps: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ block_collection @ Bl ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl )
=> ( ord_less_eq_nat @ N @ ( design254580327166089565ndex_a @ block_collection @ Ps ) ) ) ) ).
% points_index_count_min
thf(fact_1199_points__index__ps__nin,axiom,
! [Ps: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ( design254580327166089565ndex_a @ block_collection @ Ps )
= zero_zero_nat ) ) ).
% points_index_ps_nin
thf(fact_1200_add__block__index__in,axiom,
! [Ps: set_a,B: set_a] :
( ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ Ps )
= ( plus_plus_nat @ ( design254580327166089565ndex_a @ block_collection @ Ps ) @ one_one_nat ) ) ) ).
% add_block_index_in
thf(fact_1201_points__index__zero,axiom,
! [Ps: set_a] :
( ( ord_less_nat @ ( finite_card_a @ point_set ) @ ( finite_card_a @ Ps ) )
=> ( ( design254580327166089565ndex_a @ block_collection @ Ps )
= zero_zero_nat ) ) ).
% points_index_zero
thf(fact_1202_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1203_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1204_add_Oright__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.right_neutral
thf(fact_1205_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1206_add__cancel__left__left,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_left
thf(fact_1207_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1208_add__cancel__left__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_right
thf(fact_1209_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1210_add__cancel__right__left,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_left
thf(fact_1211_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1212_add__cancel__right__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_right
thf(fact_1213_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1214_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1215_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1216_add__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% add_0
thf(fact_1217_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1218_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1219_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1220_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1221_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1222_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1223_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1224_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1225_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1226_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1227_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1228_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1229_point__indices__elem__in,axiom,
! [Ps: set_a,T3: nat] :
( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ( ( finite_card_a @ Ps )
= T3 )
=> ( member_nat @ ( design254580327166089565ndex_a @ block_collection @ Ps ) @ ( design328527185268214962ices_a @ point_set @ block_collection @ T3 ) ) ) ) ).
% point_indices_elem_in
thf(fact_1230_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1231_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1232_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1233_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1234_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1235_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1236_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1237_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1238_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1239_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1240_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1241_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1242_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1243_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1244_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1245_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1246_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1247_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1248_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1249_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1250_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1251_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1252_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1253_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1254_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1255_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1256_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M7: nat,N4: nat] :
( ( ord_less_nat @ M7 @ N4 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1257_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1258_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1259_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1260_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1261_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1262_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1263_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1264_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1265_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1266_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1267_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
% Conjectures (1)
thf(conj_0,conjecture,
design_design_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ bl ) ).
%------------------------------------------------------------------------------