TPTP Problem File: SLH0971^1.p
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%------------------------------------------------------------------------------
% 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 : Safe_Range_RC/0021_Relational_Calculus/prob_01681_063773__17503272_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1555 ( 680 unt; 277 typ; 0 def)
% Number of atoms : 3551 (1469 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9572 ( 448 ~; 52 |; 213 &;7377 @)
% ( 0 <=>;1482 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 1085 (1085 >; 0 *; 0 +; 0 <<)
% Number of symbols : 249 ( 246 usr; 20 con; 0-5 aty)
% Number of variables : 3183 ( 175 ^;2941 !; 67 ?;3183 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:21.891
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__Set__Oset_It__Nat__Onat_J_Mt__List__Olist_Itf__a_J_J,type,
relati3319051887937143740list_a: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
relati3346113412121145042list_a: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
relati1431216687008021250et_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
relati5436652119368751390list_a: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
relati2966265381950730542at_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
relati7098373631445621294et_nat: $tType ).
thf(ty_n_t__List__Olist_It__Relational____Calculus__Oterm_It__List__Olist_Itf__a_J_J_J,type,
list_R4687676760361139541list_a: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
relati6521690034043994866list_a: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
relati8428538799459208780_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__Relational____Calculus__Oterm_It__Nat__Onat_J_J,type,
list_R114826772386431851rm_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_It__Nat__Onat_Mt__Nat__Onat_J,type,
relati7126052417554554232at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Relational____Calculus__Oterm_Itf__a_J_J,type,
list_R6823256787227418703term_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
set_list_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relational_fmla_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
product_prod_b_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $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__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (246)
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
boolea4281390645585673690list_a: ( set_list_a > set_list_a > set_list_a ) > ( set_list_a > set_list_a > set_list_a ) > ( set_list_a > set_list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Nat__Onat_J,type,
boolea778851993438741648et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat ) > set_nat > set_nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
comple1750779994657059634st_a_o: set_list_a_o > list_a > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
comple903356909981783403list_a: set_set_list_a > set_list_a ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__List__Olist_Itf__a_J,type,
condit7729410879213921563list_a: ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Nat__Onat,type,
condit7935552474144124665dd_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__List__Olist_Itf__a_J,type,
condit5051389248180226297list_a: ( list_a > list_a > $o ) > set_list_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Nat__Onat,type,
condit4013746787832047771dd_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Filter_Ocofinite_001t__Nat__Onat,type,
cofinite_nat: filter_nat ).
thf(sy_c_Finite__Set_OFpow_001t__List__Olist_Itf__a_J,type,
finite_Fpow_list_a: set_list_a > set_set_list_a ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_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__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
inj_on_list_a_nat: ( list_a > nat ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
inj_on_nat_list_a: ( nat > list_a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
the_in2841908037993899827list_a: set_list_a > ( list_a > list_a ) > list_a > list_a ).
thf(sy_c_Fun_Othe__inv__into_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
the_in5654461283803998945_a_nat: set_list_a > ( list_a > nat ) > nat > list_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
the_in324026561121812671list_a: set_nat > ( nat > list_a ) > list_a > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Groups_Ocomm__monoid_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
comm_m7296721492685993625list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > $o ).
thf(sy_c_Groups_Ocomm__monoid_001t__Set__Oset_It__Nat__Onat_J,type,
comm_monoid_set_nat: ( set_nat > set_nat > set_nat ) > set_nat > $o ).
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__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
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_It__List__Olist_Itf__a_J_J_J,type,
minus_4782336368215558443list_a: set_set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
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_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Omonoid_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
monoid_set_list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > $o ).
thf(sy_c_Groups_Omonoid_001t__Set__Oset_It__Nat__Onat_J,type,
monoid_set_nat: ( set_nat > set_nat > set_nat ) > set_nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
uminus7925729386456332763list_a: set_list_a > set_list_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
inf_inf_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
inf_in4657809108759609906list_a: set_set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
semila4471139486997153862list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > $o ).
thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Nat__Onat_J,type,
semila1241773964035338532et_nat: ( set_nat > set_nat > set_nat ) > set_nat > $o ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
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__List__Olist_Itf__a_J_J,type,
sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).
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__Big_Oord__class_Oarg__min__on_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
lattic5043722365632780795_a_nat: ( list_a > nat ) > set_list_a > list_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
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_It__Set__Oset_It__List__Olist_Itf__a_J_J_M_Eo_J,type,
bot_bot_set_list_a_o: set_list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_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_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
bot_bot_filter_nat: filter_nat ).
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__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_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_It__List__Olist_Itf__a_J_J_J,type,
bot_bo3186585308812441520list_a: set_set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
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__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $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__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_eq_list_a_o: ( list_a > $o ) > ( list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $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__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $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_It__List__Olist_Itf__a_J_J_J,type,
ord_le8877086941679407844list_a: set_set_list_a > set_set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
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thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
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bind_b_nat: set_b > ( b > set_nat ) > set_nat ).
thf(sy_c_Set_Obind_001tf__b_001tf__a,type,
bind_b_a: set_b > ( b > set_a ) > set_a ).
thf(sy_c_Set_Obind_001tf__b_001tf__b,type,
bind_b_b: set_b > ( b > set_b ) > set_b ).
thf(sy_c_Set_Oimage_001_062_It__List__Olist_Itf__a_J_M_Eo_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5820879363088598756list_a: ( ( list_a > $o ) > set_list_a ) > set_list_a_o > set_set_list_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5749939591322298757list_a: ( set_list_a > set_list_a ) > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_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_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_Itf__a_J,type,
is_empty_list_a: set_list_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__b,type,
is_empty_b: set_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_Itf__a_J,type,
is_singleton_list_a: set_list_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
is_sin8525870043004244056list_a: set_set_list_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Opairwise_001t__List__Olist_Itf__a_J,type,
pairwise_list_a: ( list_a > list_a > $o ) > set_list_a > $o ).
thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
pairwise_set_list_a: ( set_list_a > set_list_a > $o ) > set_set_list_a > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_It__Nat__Onat_J,type,
pairwise_set_nat: ( set_nat > set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Set_Opairwise_001tf__a,type,
pairwise_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Set_Opairwise_001tf__b,type,
pairwise_b: ( b > b > $o ) > set_b > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_Itf__a_J,type,
remove_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_Itf__a_J,type,
the_elem_list_a: set_list_a > list_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
the_elem_set_list_a: set_set_list_a > set_list_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_or7033417953538090159list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1731685050470061051et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_or6279072120763780779list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_or4827425070552584583list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
% Relevant facts (1277)
thf(fact_0_fmla_Oinject_I2_J,axiom,
! [X2: $o,Y2: $o] :
( ( ( relational_Bool_a_b @ X2 )
= ( relational_Bool_a_b @ Y2 ) )
= ( X2 = Y2 ) ) ).
% fmla.inject(2)
thf(fact_1_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_2_empty__iff,axiom,
! [C: set_list_a] :
~ ( member_set_list_a @ C @ bot_bo3186585308812441520list_a ) ).
% empty_iff
thf(fact_3_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_4_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_5_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_6_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_7_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_8_all__not__in__conv,axiom,
! [A: set_set_list_a] :
( ( ! [X: set_list_a] :
~ ( member_set_list_a @ X @ A ) )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% all_not_in_conv
thf(fact_9_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_10_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_11_all__not__in__conv,axiom,
! [A: set_list_a] :
( ( ! [X: list_a] :
~ ( member_list_a @ X @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_12_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_13_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_14_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_15_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_16_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_17_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_18_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_21_bot__apply,axiom,
( bot_bot_list_a_o
= ( ^ [X: list_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_22_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_23_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_24_emptyE,axiom,
! [A2: set_list_a] :
~ ( member_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) ).
% emptyE
thf(fact_25_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_26_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_27_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_28_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_29_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_30_equals0D,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( A = bot_bo3186585308812441520list_a )
=> ~ ( member_set_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_31_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_32_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_33_equals0D,axiom,
! [A: set_list_a,A2: list_a] :
( ( A = bot_bot_set_list_a )
=> ~ ( member_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_34_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_35_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_36_equals0I,axiom,
! [A: set_set_list_a] :
( ! [Y: set_list_a] :
~ ( member_set_list_a @ Y @ A )
=> ( A = bot_bo3186585308812441520list_a ) ) ).
% equals0I
thf(fact_37_equals0I,axiom,
! [A: set_b] :
( ! [Y: b] :
~ ( member_b @ Y @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_38_equals0I,axiom,
! [A: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_39_equals0I,axiom,
! [A: set_list_a] :
( ! [Y: list_a] :
~ ( member_list_a @ Y @ A )
=> ( A = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_40_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_41_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_42_ex__in__conv,axiom,
! [A: set_set_list_a] :
( ( ? [X: set_list_a] : ( member_set_list_a @ X @ A ) )
= ( A != bot_bo3186585308812441520list_a ) ) ).
% ex_in_conv
thf(fact_43_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X: b] : ( member_b @ X @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_44_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_45_ex__in__conv,axiom,
! [A: set_list_a] :
( ( ? [X: list_a] : ( member_list_a @ X @ A ) )
= ( A != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_46_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_47_bot__fun__def,axiom,
( bot_bot_list_a_o
= ( ^ [X: list_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_48_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_49_cpropagated__simps_I1_J,axiom,
! [B: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B ) ) ).
% cpropagated_simps(1)
thf(fact_50_csts_Osimps_I1_J,axiom,
! [B: $o] :
( ( relational_csts_a_b @ ( relational_Bool_a_b @ B ) )
= bot_bot_set_a ) ).
% csts.simps(1)
thf(fact_51_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_52_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_53_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_54_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_55_Set_Ois__empty__def,axiom,
( is_empty_b
= ( ^ [A3: set_b] : ( A3 = bot_bot_set_b ) ) ) ).
% Set.is_empty_def
thf(fact_56_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_57_Set_Ois__empty__def,axiom,
( is_empty_list_a
= ( ^ [A3: set_list_a] : ( A3 = bot_bot_set_list_a ) ) ) ).
% Set.is_empty_def
thf(fact_58_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_59_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_60_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_61_Collect__empty__eq__bot,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( P = bot_bot_list_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_62_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_63_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_64_bot__empty__eq,axiom,
( bot_bot_set_list_a_o
= ( ^ [X: set_list_a] : ( member_set_list_a @ X @ bot_bo3186585308812441520list_a ) ) ) ).
% bot_empty_eq
thf(fact_65_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X: b] : ( member_b @ X @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_66_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_67_bot__empty__eq,axiom,
( bot_bot_list_a_o
= ( ^ [X: list_a] : ( member_list_a @ X @ bot_bot_set_list_a ) ) ) ).
% bot_empty_eq
thf(fact_68_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_69_is__singletonI_H,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( member_set_nat @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_set_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_70_is__singletonI_H,axiom,
! [A: set_set_list_a] :
( ( A != bot_bo3186585308812441520list_a )
=> ( ! [X3: set_list_a,Y: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ( ( member_set_list_a @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin8525870043004244056list_a @ A ) ) ) ).
% is_singletonI'
thf(fact_71_is__singletonI_H,axiom,
! [A: set_b] :
( ( A != bot_bot_set_b )
=> ( ! [X3: b,Y: b] :
( ( member_b @ X3 @ A )
=> ( ( member_b @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_b @ A ) ) ) ).
% is_singletonI'
thf(fact_72_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_73_is__singletonI_H,axiom,
! [A: set_list_a] :
( ( A != bot_bot_set_list_a )
=> ( ! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ( member_list_a @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_list_a @ A ) ) ) ).
% is_singletonI'
thf(fact_74_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_75_fmla_Osimps_I128_J,axiom,
! [X2: $o] :
( ( relati3206271568683928597list_a @ ( relati5295267155521767763list_a @ X2 ) )
= bot_bot_set_list_a ) ).
% fmla.simps(128)
thf(fact_76_fmla_Osimps_I128_J,axiom,
! [X2: $o] :
( ( relati8645012329147373695_a_nat @ ( relati2772089302019829953_a_nat @ X2 ) )
= bot_bot_set_nat ) ).
% fmla.simps(128)
thf(fact_77_fmla_Osimps_I128_J,axiom,
! [X2: $o] :
( ( relati3314577606465187421list_a @ ( relati6665026616192419487list_a @ X2 ) )
= bot_bot_set_list_a ) ).
% fmla.simps(128)
thf(fact_78_fmla_Osimps_I128_J,axiom,
! [X2: $o] :
( ( relati2567137625159299127at_nat @ ( relati4833799250026832501at_nat @ X2 ) )
= bot_bot_set_nat ) ).
% fmla.simps(128)
thf(fact_79_fmla_Osimps_I128_J,axiom,
! [X2: $o] :
( ( relati8924981150291758614la_a_b @ ( relational_Bool_a_b @ X2 ) )
= bot_bot_set_b ) ).
% fmla.simps(128)
thf(fact_80_empty__bind,axiom,
! [F: nat > set_nat] :
( ( bind_nat_nat @ bot_bot_set_nat @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_81_empty__bind,axiom,
! [F: list_a > set_nat] :
( ( bind_list_a_nat @ bot_bot_set_list_a @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_82_empty__bind,axiom,
! [F: nat > set_list_a] :
( ( bind_nat_list_a @ bot_bot_set_nat @ F )
= bot_bot_set_list_a ) ).
% empty_bind
thf(fact_83_empty__bind,axiom,
! [F: list_a > set_list_a] :
( ( bind_list_a_list_a @ bot_bot_set_list_a @ F )
= bot_bot_set_list_a ) ).
% empty_bind
thf(fact_84_empty__bind,axiom,
! [F: nat > set_b] :
( ( bind_nat_b @ bot_bot_set_nat @ F )
= bot_bot_set_b ) ).
% empty_bind
thf(fact_85_empty__bind,axiom,
! [F: nat > set_a] :
( ( bind_nat_a @ bot_bot_set_nat @ F )
= bot_bot_set_a ) ).
% empty_bind
thf(fact_86_empty__bind,axiom,
! [F: b > set_nat] :
( ( bind_b_nat @ bot_bot_set_b @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_87_empty__bind,axiom,
! [F: b > set_b] :
( ( bind_b_b @ bot_bot_set_b @ F )
= bot_bot_set_b ) ).
% empty_bind
thf(fact_88_empty__bind,axiom,
! [F: b > set_a] :
( ( bind_b_a @ bot_bot_set_b @ F )
= bot_bot_set_a ) ).
% empty_bind
thf(fact_89_empty__bind,axiom,
! [F: a > set_nat] :
( ( bind_a_nat @ bot_bot_set_a @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_90_fmla_Osimps_I121_J,axiom,
! [X2: $o] :
( ( relati4910493452150799124list_a @ ( relati5295267155521767763list_a @ X2 ) )
= bot_bot_set_list_a ) ).
% fmla.simps(121)
thf(fact_91_fmla_Osimps_I121_J,axiom,
! [X2: $o] :
( ( relati6972889302305856_a_nat @ ( relati2772089302019829953_a_nat @ X2 ) )
= bot_bot_set_list_a ) ).
% fmla.simps(121)
thf(fact_92_fmla_Osimps_I121_J,axiom,
! [X2: $o] :
( ( relati3899910203474895390list_a @ ( relati6665026616192419487list_a @ X2 ) )
= bot_bot_set_nat ) ).
% fmla.simps(121)
thf(fact_93_fmla_Osimps_I121_J,axiom,
! [X2: $o] :
( ( relati6321887899146193334at_nat @ ( relati4833799250026832501at_nat @ X2 ) )
= bot_bot_set_nat ) ).
% fmla.simps(121)
thf(fact_94_fmla_Osimps_I121_J,axiom,
! [X2: $o] :
( ( relati3071123380395136021la_a_b @ ( relational_Bool_a_b @ X2 ) )
= bot_bot_set_a ) ).
% fmla.simps(121)
thf(fact_95_qp__impl_Osimps_I4_J,axiom,
! [V: $o] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Bool_a_b @ V ) ) ).
% qp_impl.simps(4)
thf(fact_96_rrb__simps_I1_J,axiom,
! [B: $o] : ( relational_rrb_a_b @ ( relational_Bool_a_b @ B ) ) ).
% rrb_simps(1)
thf(fact_97_sr__False,axiom,
relational_sr_a_b @ ( relational_Bool_a_b @ $false ) ).
% sr_False
thf(fact_98_sr__def,axiom,
( relational_sr_a_b
= ( ^ [Q: relational_fmla_a_b] :
( ( ( relati62690040636126068ns_a_b @ Q )
= bot_bot_set_nat )
& ( relational_rrb_a_b @ Q ) ) ) ) ).
% sr_def
thf(fact_99_is__singletonI,axiom,
! [X4: set_nat] : ( is_singleton_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ).
% is_singletonI
thf(fact_100_is__singletonI,axiom,
! [X4: set_list_a] : ( is_sin8525870043004244056list_a @ ( insert_set_list_a @ X4 @ bot_bo3186585308812441520list_a ) ) ).
% is_singletonI
thf(fact_101_is__singletonI,axiom,
! [X4: b] : ( is_singleton_b @ ( insert_b @ X4 @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_102_is__singletonI,axiom,
! [X4: a] : ( is_singleton_a @ ( insert_a @ X4 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_103_is__singletonI,axiom,
! [X4: list_a] : ( is_singleton_list_a @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ).
% is_singletonI
thf(fact_104_is__singletonI,axiom,
! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_105_fmla_Opred__mono__strong,axiom,
! [P1: nat > $o,P2: nat > $o,X4: relati7126052417554554232at_nat,P1a: nat > $o,P2a: nat > $o] :
( ( relati6725209092851823240at_nat @ P1 @ P2 @ X4 )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati6321887899146193334at_nat @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati2567137625159299127at_nat @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati6725209092851823240at_nat @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_106_fmla_Opred__mono__strong,axiom,
! [P1: nat > $o,P2: list_a > $o,X4: relati6521690034043994866list_a,P1a: nat > $o,P2a: list_a > $o] :
( ( relati1651932441426877132list_a @ P1 @ P2 @ X4 )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati3899910203474895390list_a @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3314577606465187421list_a @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati1651932441426877132list_a @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_107_fmla_Opred__mono__strong,axiom,
! [P1: list_a > $o,P2: nat > $o,X4: relati8428538799459208780_a_nat,P1a: list_a > $o,P2a: nat > $o] :
( ( relati6982367164109063406_a_nat @ P1 @ P2 @ X4 )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati6972889302305856_a_nat @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati8645012329147373695_a_nat @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati6982367164109063406_a_nat @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_108_fmla_Opred__mono__strong,axiom,
! [P1: list_a > $o,P2: list_a > $o,X4: relati5436652119368751390list_a,P1a: list_a > $o,P2a: list_a > $o] :
( ( relati4741328337081057510list_a @ P1 @ P2 @ X4 )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati4910493452150799124list_a @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3206271568683928597list_a @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati4741328337081057510list_a @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_109_fmla_Opred__mono__strong,axiom,
! [P1: a > $o,P2: b > $o,X4: relational_fmla_a_b,P1a: a > $o,P2a: b > $o] :
( ( relati3660035184769383399la_a_b @ P1 @ P2 @ X4 )
=> ( ! [Z1: a] :
( ( member_a @ Z1 @ ( relati3071123380395136021la_a_b @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: b] :
( ( member_b @ Z2 @ ( relati8924981150291758614la_a_b @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati3660035184769383399la_a_b @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_110_fmla_Opred__mono__strong,axiom,
! [P1: nat > $o,P2: set_nat > $o,X4: relati7098373631445621294et_nat,P1a: nat > $o,P2a: set_nat > $o] :
( ( relati6450431918034959934et_nat @ P1 @ P2 @ X4 )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati7069291191333231980et_nat @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( relati1785867741103292397et_nat @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati6450431918034959934et_nat @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_111_fmla_Opred__mono__strong,axiom,
! [P1: set_nat > $o,P2: nat > $o,X4: relati2966265381950730542at_nat,P1a: set_nat > $o,P2a: nat > $o] :
( ( relati2884797013379590718at_nat @ P1 @ P2 @ X4 )
=> ( ! [Z1: set_nat] :
( ( member_set_nat @ Z1 @ ( relati3503656286677862764at_nat @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati7443604873302698989at_nat @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati2884797013379590718at_nat @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_112_fmla_Opred__mono__strong,axiom,
! [P1: nat > $o,P2: set_list_a > $o,X4: relati3346113412121145042list_a,P1a: nat > $o,P2a: set_list_a > $o] :
( ( relati1595616085582258860list_a @ P1 @ P2 @ X4 )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati5613704384950755838list_a @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: set_list_a] :
( ( member_set_list_a @ Z2 @ ( relati6609543971511857725list_a @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati1595616085582258860list_a @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_113_fmla_Opred__mono__strong,axiom,
! [P1: list_a > $o,P2: set_nat > $o,X4: relati1431216687008021250et_nat,P1a: list_a > $o,P2a: set_nat > $o] :
( ( relati3360304113585600676et_nat @ P1 @ P2 @ X4 )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati7378392412954097654et_nat @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( relati8374231999515199541et_nat @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati3360304113585600676et_nat @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_114_fmla_Opred__mono__strong,axiom,
! [P1: set_nat > $o,P2: list_a > $o,X4: relati3319051887937143740list_a,P1a: set_nat > $o,P2a: list_a > $o] :
( ( relati2391026702087272342list_a @ P1 @ P2 @ X4 )
=> ( ! [Z1: set_nat] :
( ( member_set_nat @ Z1 @ ( relati6409115001455769320list_a @ X4 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati7404954588016871207list_a @ X4 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( relati2391026702087272342list_a @ P1a @ P2a @ X4 ) ) ) ) ).
% fmla.pred_mono_strong
thf(fact_115_fmla_Opred__cong,axiom,
! [X4: relati7126052417554554232at_nat,Ya: relati7126052417554554232at_nat,P1: nat > $o,P1a: nat > $o,P2: nat > $o,P2a: nat > $o] :
( ( X4 = Ya )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati6321887899146193334at_nat @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati2567137625159299127at_nat @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati6725209092851823240at_nat @ P1 @ P2 @ X4 )
= ( relati6725209092851823240at_nat @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_116_fmla_Opred__cong,axiom,
! [X4: relati6521690034043994866list_a,Ya: relati6521690034043994866list_a,P1: nat > $o,P1a: nat > $o,P2: list_a > $o,P2a: list_a > $o] :
( ( X4 = Ya )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati3899910203474895390list_a @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3314577606465187421list_a @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati1651932441426877132list_a @ P1 @ P2 @ X4 )
= ( relati1651932441426877132list_a @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_117_fmla_Opred__cong,axiom,
! [X4: relati8428538799459208780_a_nat,Ya: relati8428538799459208780_a_nat,P1: list_a > $o,P1a: list_a > $o,P2: nat > $o,P2a: nat > $o] :
( ( X4 = Ya )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati6972889302305856_a_nat @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati8645012329147373695_a_nat @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati6982367164109063406_a_nat @ P1 @ P2 @ X4 )
= ( relati6982367164109063406_a_nat @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_118_fmla_Opred__cong,axiom,
! [X4: relati5436652119368751390list_a,Ya: relati5436652119368751390list_a,P1: list_a > $o,P1a: list_a > $o,P2: list_a > $o,P2a: list_a > $o] :
( ( X4 = Ya )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati4910493452150799124list_a @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3206271568683928597list_a @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati4741328337081057510list_a @ P1 @ P2 @ X4 )
= ( relati4741328337081057510list_a @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_119_fmla_Opred__cong,axiom,
! [X4: relational_fmla_a_b,Ya: relational_fmla_a_b,P1: a > $o,P1a: a > $o,P2: b > $o,P2a: b > $o] :
( ( X4 = Ya )
=> ( ! [Z1: a] :
( ( member_a @ Z1 @ ( relati3071123380395136021la_a_b @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: b] :
( ( member_b @ Z2 @ ( relati8924981150291758614la_a_b @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati3660035184769383399la_a_b @ P1 @ P2 @ X4 )
= ( relati3660035184769383399la_a_b @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_120_fmla_Opred__cong,axiom,
! [X4: relati7098373631445621294et_nat,Ya: relati7098373631445621294et_nat,P1: nat > $o,P1a: nat > $o,P2: set_nat > $o,P2a: set_nat > $o] :
( ( X4 = Ya )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati7069291191333231980et_nat @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( relati1785867741103292397et_nat @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati6450431918034959934et_nat @ P1 @ P2 @ X4 )
= ( relati6450431918034959934et_nat @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_121_fmla_Opred__cong,axiom,
! [X4: relati2966265381950730542at_nat,Ya: relati2966265381950730542at_nat,P1: set_nat > $o,P1a: set_nat > $o,P2: nat > $o,P2a: nat > $o] :
( ( X4 = Ya )
=> ( ! [Z1: set_nat] :
( ( member_set_nat @ Z1 @ ( relati3503656286677862764at_nat @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati7443604873302698989at_nat @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati2884797013379590718at_nat @ P1 @ P2 @ X4 )
= ( relati2884797013379590718at_nat @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_122_fmla_Opred__cong,axiom,
! [X4: relati3346113412121145042list_a,Ya: relati3346113412121145042list_a,P1: nat > $o,P1a: nat > $o,P2: set_list_a > $o,P2a: set_list_a > $o] :
( ( X4 = Ya )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati5613704384950755838list_a @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: set_list_a] :
( ( member_set_list_a @ Z2 @ ( relati6609543971511857725list_a @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati1595616085582258860list_a @ P1 @ P2 @ X4 )
= ( relati1595616085582258860list_a @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_123_fmla_Opred__cong,axiom,
! [X4: relati1431216687008021250et_nat,Ya: relati1431216687008021250et_nat,P1: list_a > $o,P1a: list_a > $o,P2: set_nat > $o,P2a: set_nat > $o] :
( ( X4 = Ya )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati7378392412954097654et_nat @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( relati8374231999515199541et_nat @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati3360304113585600676et_nat @ P1 @ P2 @ X4 )
= ( relati3360304113585600676et_nat @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_124_fmla_Opred__cong,axiom,
! [X4: relati3319051887937143740list_a,Ya: relati3319051887937143740list_a,P1: set_nat > $o,P1a: set_nat > $o,P2: list_a > $o,P2a: list_a > $o] :
( ( X4 = Ya )
=> ( ! [Z1: set_nat] :
( ( member_set_nat @ Z1 @ ( relati6409115001455769320list_a @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati7404954588016871207list_a @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( relati2391026702087272342list_a @ P1 @ P2 @ X4 )
= ( relati2391026702087272342list_a @ P1a @ P2a @ Ya ) ) ) ) ) ).
% fmla.pred_cong
thf(fact_125_is__singleton__def,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
? [X: set_nat] :
( A3
= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_126_is__singleton__def,axiom,
( is_sin8525870043004244056list_a
= ( ^ [A3: set_set_list_a] :
? [X: set_list_a] :
( A3
= ( insert_set_list_a @ X @ bot_bo3186585308812441520list_a ) ) ) ) ).
% is_singleton_def
thf(fact_127_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A3: set_b] :
? [X: b] :
( A3
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_128_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X: a] :
( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_129_is__singleton__def,axiom,
( is_singleton_list_a
= ( ^ [A3: set_list_a] :
? [X: list_a] :
( A3
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_def
thf(fact_130_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_131_is__singletonE,axiom,
! [A: set_set_nat] :
( ( is_singleton_set_nat @ A )
=> ~ ! [X3: set_nat] :
( A
!= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% is_singletonE
thf(fact_132_is__singletonE,axiom,
! [A: set_set_list_a] :
( ( is_sin8525870043004244056list_a @ A )
=> ~ ! [X3: set_list_a] :
( A
!= ( insert_set_list_a @ X3 @ bot_bo3186585308812441520list_a ) ) ) ).
% is_singletonE
thf(fact_133_is__singletonE,axiom,
! [A: set_b] :
( ( is_singleton_b @ A )
=> ~ ! [X3: b] :
( A
!= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_134_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_135_is__singletonE,axiom,
! [A: set_list_a] :
( ( is_singleton_list_a @ A )
=> ~ ! [X3: list_a] :
( A
!= ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ).
% is_singletonE
thf(fact_136_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_137_fmla_Omap__ident__strong,axiom,
! [T: relati7126052417554554232at_nat,F1: nat > nat,F2: nat > nat] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati6321887899146193334at_nat @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati2567137625159299127at_nat @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati9000828793121449555at_nat @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_138_fmla_Omap__ident__strong,axiom,
! [T: relati6521690034043994866list_a,F1: nat > nat,F2: list_a > list_a] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati3899910203474895390list_a @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3314577606465187421list_a @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati4590563632004032561list_a @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_139_fmla_Omap__ident__strong,axiom,
! [T: relati8428538799459208780_a_nat,F1: list_a > list_a,F2: nat > nat] :
( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati6972889302305856_a_nat @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati8645012329147373695_a_nat @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati5126268176769250097at_nat @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_140_fmla_Omap__ident__strong,axiom,
! [T: relati5436652119368751390list_a,F1: list_a > list_a,F2: list_a > list_a] :
( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati4910493452150799124list_a @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3206271568683928597list_a @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati9032153196947279887list_a @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_141_fmla_Omap__ident__strong,axiom,
! [T: relational_fmla_a_b,F1: a > a,F2: b > b] :
( ! [Z1: a] :
( ( member_a @ Z1 @ ( relati3071123380395136021la_a_b @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: b] :
( ( member_b @ Z2 @ ( relati8924981150291758614la_a_b @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati4520850492397955663_a_b_b @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_142_fmla_Omap__ident__strong,axiom,
! [T: relati7098373631445621294et_nat,F1: nat > nat,F2: set_nat > set_nat] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati7069291191333231980et_nat @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( relati1785867741103292397et_nat @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati7694559721178626495et_nat @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_143_fmla_Omap__ident__strong,axiom,
! [T: relati2966265381950730542at_nat,F1: set_nat > set_nat,F2: nat > nat] :
( ! [Z1: set_nat] :
( ( member_set_nat @ Z1 @ ( relati3503656286677862764at_nat @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati7443604873302698989at_nat @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati5520000673013508543at_nat @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_144_fmla_Omap__ident__strong,axiom,
! [T: relati3346113412121145042list_a,F1: nat > nat,F2: set_list_a > set_list_a] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati5613704384950755838list_a @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: set_list_a] :
( ( member_set_list_a @ Z2 @ ( relati6609543971511857725list_a @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati8279081407899341873list_a @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_145_fmla_Omap__ident__strong,axiom,
! [T: relati1431216687008021250et_nat,F1: list_a > list_a,F2: set_nat > set_nat] :
( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati7378392412954097654et_nat @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( relati8374231999515199541et_nat @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati1915489029210003101et_nat @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_146_fmla_Omap__ident__strong,axiom,
! [T: relati3319051887937143740list_a,F1: set_nat > set_nat,F2: list_a > list_a] :
( ! [Z1: set_nat] :
( ( member_set_nat @ Z1 @ ( relati6409115001455769320list_a @ T ) )
=> ( ( F1 @ Z1 )
= Z1 ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati7404954588016871207list_a @ T ) )
=> ( ( F2 @ Z2 )
= Z2 ) )
=> ( ( relati4680159878445029789list_a @ F1 @ F2 @ T )
= T ) ) ) ).
% fmla.map_ident_strong
thf(fact_147_fmla_Oinj__map__strong,axiom,
! [X4: relational_fmla_a_b,Xa: relational_fmla_a_b,F1: a > a,F1a: a > a,F2: b > b,F2a: b > b] :
( ! [Z1: a,Z1a: a] :
( ( member_a @ Z1 @ ( relati3071123380395136021la_a_b @ X4 ) )
=> ( ( member_a @ Z1a @ ( relati3071123380395136021la_a_b @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: b,Z2a: b] :
( ( member_b @ Z2 @ ( relati8924981150291758614la_a_b @ X4 ) )
=> ( ( member_b @ Z2a @ ( relati8924981150291758614la_a_b @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( relati4520850492397955663_a_b_b @ F1 @ F2 @ X4 )
= ( relati4520850492397955663_a_b_b @ F1a @ F2a @ Xa ) )
=> ( X4 = Xa ) ) ) ) ).
% fmla.inj_map_strong
thf(fact_148_fmla_Oinj__map__strong,axiom,
! [X4: relati5436652119368751390list_a,Xa: relati5436652119368751390list_a,F1: list_a > list_a,F1a: list_a > list_a,F2: list_a > list_a,F2a: list_a > list_a] :
( ! [Z1: list_a,Z1a: list_a] :
( ( member_list_a @ Z1 @ ( relati4910493452150799124list_a @ X4 ) )
=> ( ( member_list_a @ Z1a @ ( relati4910493452150799124list_a @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: list_a,Z2a: list_a] :
( ( member_list_a @ Z2 @ ( relati3206271568683928597list_a @ X4 ) )
=> ( ( member_list_a @ Z2a @ ( relati3206271568683928597list_a @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( relati9032153196947279887list_a @ F1 @ F2 @ X4 )
= ( relati9032153196947279887list_a @ F1a @ F2a @ Xa ) )
=> ( X4 = Xa ) ) ) ) ).
% fmla.inj_map_strong
thf(fact_149_fmla_Oinj__map__strong,axiom,
! [X4: relati8428538799459208780_a_nat,Xa: relati8428538799459208780_a_nat,F1: list_a > list_a,F1a: list_a > list_a,F2: nat > nat,F2a: nat > nat] :
( ! [Z1: list_a,Z1a: list_a] :
( ( member_list_a @ Z1 @ ( relati6972889302305856_a_nat @ X4 ) )
=> ( ( member_list_a @ Z1a @ ( relati6972889302305856_a_nat @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: nat,Z2a: nat] :
( ( member_nat @ Z2 @ ( relati8645012329147373695_a_nat @ X4 ) )
=> ( ( member_nat @ Z2a @ ( relati8645012329147373695_a_nat @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( relati5126268176769250097at_nat @ F1 @ F2 @ X4 )
= ( relati5126268176769250097at_nat @ F1a @ F2a @ Xa ) )
=> ( X4 = Xa ) ) ) ) ).
% fmla.inj_map_strong
thf(fact_150_fmla_Oinj__map__strong,axiom,
! [X4: relati6521690034043994866list_a,Xa: relati6521690034043994866list_a,F1: nat > nat,F1a: nat > nat,F2: list_a > list_a,F2a: list_a > list_a] :
( ! [Z1: nat,Z1a: nat] :
( ( member_nat @ Z1 @ ( relati3899910203474895390list_a @ X4 ) )
=> ( ( member_nat @ Z1a @ ( relati3899910203474895390list_a @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: list_a,Z2a: list_a] :
( ( member_list_a @ Z2 @ ( relati3314577606465187421list_a @ X4 ) )
=> ( ( member_list_a @ Z2a @ ( relati3314577606465187421list_a @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( relati4590563632004032561list_a @ F1 @ F2 @ X4 )
= ( relati4590563632004032561list_a @ F1a @ F2a @ Xa ) )
=> ( X4 = Xa ) ) ) ) ).
% fmla.inj_map_strong
thf(fact_151_fmla_Oinj__map__strong,axiom,
! [X4: relati7126052417554554232at_nat,Xa: relati7126052417554554232at_nat,F1: nat > nat,F1a: nat > nat,F2: nat > nat,F2a: nat > nat] :
( ! [Z1: nat,Z1a: nat] :
( ( member_nat @ Z1 @ ( relati6321887899146193334at_nat @ X4 ) )
=> ( ( member_nat @ Z1a @ ( relati6321887899146193334at_nat @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: nat,Z2a: nat] :
( ( member_nat @ Z2 @ ( relati2567137625159299127at_nat @ X4 ) )
=> ( ( member_nat @ Z2a @ ( relati2567137625159299127at_nat @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( relati9000828793121449555at_nat @ F1 @ F2 @ X4 )
= ( relati9000828793121449555at_nat @ F1a @ F2a @ Xa ) )
=> ( X4 = Xa ) ) ) ) ).
% fmla.inj_map_strong
thf(fact_152_fmla_Omap__cong0,axiom,
! [X4: relational_fmla_a_b,F1: a > a,G1: a > a,F2: b > b,G2: b > b] :
( ! [Z1: a] :
( ( member_a @ Z1 @ ( relati3071123380395136021la_a_b @ X4 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: b] :
( ( member_b @ Z2 @ ( relati8924981150291758614la_a_b @ X4 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati4520850492397955663_a_b_b @ F1 @ F2 @ X4 )
= ( relati4520850492397955663_a_b_b @ G1 @ G2 @ X4 ) ) ) ) ).
% fmla.map_cong0
thf(fact_153_fmla_Omap__cong0,axiom,
! [X4: relati5436652119368751390list_a,F1: list_a > list_a,G1: list_a > list_a,F2: list_a > list_a,G2: list_a > list_a] :
( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati4910493452150799124list_a @ X4 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3206271568683928597list_a @ X4 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati9032153196947279887list_a @ F1 @ F2 @ X4 )
= ( relati9032153196947279887list_a @ G1 @ G2 @ X4 ) ) ) ) ).
% fmla.map_cong0
thf(fact_154_fmla_Omap__cong0,axiom,
! [X4: relati8428538799459208780_a_nat,F1: list_a > list_a,G1: list_a > list_a,F2: nat > nat,G2: nat > nat] :
( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati6972889302305856_a_nat @ X4 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati8645012329147373695_a_nat @ X4 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati5126268176769250097at_nat @ F1 @ F2 @ X4 )
= ( relati5126268176769250097at_nat @ G1 @ G2 @ X4 ) ) ) ) ).
% fmla.map_cong0
thf(fact_155_fmla_Omap__cong0,axiom,
! [X4: relati6521690034043994866list_a,F1: nat > nat,G1: nat > nat,F2: list_a > list_a,G2: list_a > list_a] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati3899910203474895390list_a @ X4 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3314577606465187421list_a @ X4 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati4590563632004032561list_a @ F1 @ F2 @ X4 )
= ( relati4590563632004032561list_a @ G1 @ G2 @ X4 ) ) ) ) ).
% fmla.map_cong0
thf(fact_156_fmla_Omap__cong0,axiom,
! [X4: relati7126052417554554232at_nat,F1: nat > nat,G1: nat > nat,F2: nat > nat,G2: nat > nat] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati6321887899146193334at_nat @ X4 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati2567137625159299127at_nat @ X4 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati9000828793121449555at_nat @ F1 @ F2 @ X4 )
= ( relati9000828793121449555at_nat @ G1 @ G2 @ X4 ) ) ) ) ).
% fmla.map_cong0
thf(fact_157_fmla_Omap__cong,axiom,
! [X4: relational_fmla_a_b,Ya: relational_fmla_a_b,F1: a > a,G1: a > a,F2: b > b,G2: b > b] :
( ( X4 = Ya )
=> ( ! [Z1: a] :
( ( member_a @ Z1 @ ( relati3071123380395136021la_a_b @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: b] :
( ( member_b @ Z2 @ ( relati8924981150291758614la_a_b @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati4520850492397955663_a_b_b @ F1 @ F2 @ X4 )
= ( relati4520850492397955663_a_b_b @ G1 @ G2 @ Ya ) ) ) ) ) ).
% fmla.map_cong
thf(fact_158_fmla_Omap__cong,axiom,
! [X4: relati5436652119368751390list_a,Ya: relati5436652119368751390list_a,F1: list_a > list_a,G1: list_a > list_a,F2: list_a > list_a,G2: list_a > list_a] :
( ( X4 = Ya )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati4910493452150799124list_a @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3206271568683928597list_a @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati9032153196947279887list_a @ F1 @ F2 @ X4 )
= ( relati9032153196947279887list_a @ G1 @ G2 @ Ya ) ) ) ) ) ).
% fmla.map_cong
thf(fact_159_fmla_Omap__cong,axiom,
! [X4: relati8428538799459208780_a_nat,Ya: relati8428538799459208780_a_nat,F1: list_a > list_a,G1: list_a > list_a,F2: nat > nat,G2: nat > nat] :
( ( X4 = Ya )
=> ( ! [Z1: list_a] :
( ( member_list_a @ Z1 @ ( relati6972889302305856_a_nat @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati8645012329147373695_a_nat @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati5126268176769250097at_nat @ F1 @ F2 @ X4 )
= ( relati5126268176769250097at_nat @ G1 @ G2 @ Ya ) ) ) ) ) ).
% fmla.map_cong
thf(fact_160_fmla_Omap__cong,axiom,
! [X4: relati6521690034043994866list_a,Ya: relati6521690034043994866list_a,F1: nat > nat,G1: nat > nat,F2: list_a > list_a,G2: list_a > list_a] :
( ( X4 = Ya )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati3899910203474895390list_a @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( relati3314577606465187421list_a @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati4590563632004032561list_a @ F1 @ F2 @ X4 )
= ( relati4590563632004032561list_a @ G1 @ G2 @ Ya ) ) ) ) ) ).
% fmla.map_cong
thf(fact_161_fmla_Omap__cong,axiom,
! [X4: relati7126052417554554232at_nat,Ya: relati7126052417554554232at_nat,F1: nat > nat,G1: nat > nat,F2: nat > nat,G2: nat > nat] :
( ( X4 = Ya )
=> ( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( relati6321887899146193334at_nat @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( relati2567137625159299127at_nat @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( relati9000828793121449555at_nat @ F1 @ F2 @ X4 )
= ( relati9000828793121449555at_nat @ G1 @ G2 @ Ya ) ) ) ) ) ).
% fmla.map_cong
thf(fact_162_insert__absorb2,axiom,
! [X4: nat,A: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ X4 @ A ) )
= ( insert_nat @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_163_insert__absorb2,axiom,
! [X4: list_a,A: set_list_a] :
( ( insert_list_a @ X4 @ ( insert_list_a @ X4 @ A ) )
= ( insert_list_a @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_164_insert__absorb2,axiom,
! [X4: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X4 @ ( insert_set_nat @ X4 @ A ) )
= ( insert_set_nat @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_165_insert__absorb2,axiom,
! [X4: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a @ X4 @ ( insert_set_list_a @ X4 @ A ) )
= ( insert_set_list_a @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_166_insert__iff,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_167_insert__iff,axiom,
! [A2: set_list_a,B: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ A2 @ ( insert_set_list_a @ B @ A ) )
= ( ( A2 = B )
| ( member_set_list_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_168_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_169_insert__iff,axiom,
! [A2: list_a,B: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
= ( ( A2 = B )
| ( member_list_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_170_insertCI,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_171_insertCI,axiom,
! [A2: set_list_a,B2: set_set_list_a,B: set_list_a] :
( ( ~ ( member_set_list_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_list_a @ A2 @ ( insert_set_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_172_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_173_insertCI,axiom,
! [A2: list_a,B2: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_174_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_175_singletonI,axiom,
! [A2: set_list_a] : ( member_set_list_a @ A2 @ ( insert_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) ) ).
% singletonI
thf(fact_176_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_177_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_178_singletonI,axiom,
! [A2: list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_179_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_180_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B3: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B3 ) )
& ~ ( member_set_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_181_mk__disjoint__insert,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ A2 @ A )
=> ? [B3: set_set_list_a] :
( ( A
= ( insert_set_list_a @ A2 @ B3 ) )
& ~ ( member_set_list_a @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_182_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B3: set_nat] :
( ( A
= ( insert_nat @ A2 @ B3 ) )
& ~ ( member_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_183_mk__disjoint__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ? [B3: set_list_a] :
( ( A
= ( insert_list_a @ A2 @ B3 ) )
& ~ ( member_list_a @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_184_insert__commute,axiom,
! [X4: nat,Y3: nat,A: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ Y3 @ A ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_185_insert__commute,axiom,
! [X4: list_a,Y3: list_a,A: set_list_a] :
( ( insert_list_a @ X4 @ ( insert_list_a @ Y3 @ A ) )
= ( insert_list_a @ Y3 @ ( insert_list_a @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_186_insert__commute,axiom,
! [X4: set_nat,Y3: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X4 @ ( insert_set_nat @ Y3 @ A ) )
= ( insert_set_nat @ Y3 @ ( insert_set_nat @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_187_insert__commute,axiom,
! [X4: set_list_a,Y3: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a @ X4 @ ( insert_set_list_a @ Y3 @ A ) )
= ( insert_set_list_a @ Y3 @ ( insert_set_list_a @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_188_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_set_nat] :
( ( A
= ( insert_set_nat @ B @ C2 ) )
& ~ ( member_set_nat @ B @ C2 )
& ( B2
= ( insert_set_nat @ A2 @ C2 ) )
& ~ ( member_set_nat @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_189_insert__eq__iff,axiom,
! [A2: set_list_a,A: set_set_list_a,B: set_list_a,B2: set_set_list_a] :
( ~ ( member_set_list_a @ A2 @ A )
=> ( ~ ( member_set_list_a @ B @ B2 )
=> ( ( ( insert_set_list_a @ A2 @ A )
= ( insert_set_list_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_set_list_a] :
( ( A
= ( insert_set_list_a @ B @ C2 ) )
& ~ ( member_set_list_a @ B @ C2 )
& ( B2
= ( insert_set_list_a @ A2 @ C2 ) )
& ~ ( member_set_list_a @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_190_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_nat] :
( ( A
= ( insert_nat @ B @ C2 ) )
& ~ ( member_nat @ B @ C2 )
& ( B2
= ( insert_nat @ A2 @ C2 ) )
& ~ ( member_nat @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_191_insert__eq__iff,axiom,
! [A2: list_a,A: set_list_a,B: list_a,B2: set_list_a] :
( ~ ( member_list_a @ A2 @ A )
=> ( ~ ( member_list_a @ B @ B2 )
=> ( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_list_a] :
( ( A
= ( insert_list_a @ B @ C2 ) )
& ~ ( member_list_a @ B @ C2 )
& ( B2
= ( insert_list_a @ A2 @ C2 ) )
& ~ ( member_list_a @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_192_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_193_insert__absorb,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ A2 @ A )
=> ( ( insert_set_list_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_194_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_195_insert__absorb,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( insert_list_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_196_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( member_set_list_a @ A2 @ ( collect_set_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_199_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
! [A: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X: set_list_a] : ( member_set_list_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_204_Collect__cong,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_205_Collect__cong,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_206_insert__ident,axiom,
! [X4: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ~ ( member_set_nat @ X4 @ B2 )
=> ( ( ( insert_set_nat @ X4 @ A )
= ( insert_set_nat @ X4 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_207_insert__ident,axiom,
! [X4: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ~ ( member_set_list_a @ X4 @ A )
=> ( ~ ( member_set_list_a @ X4 @ B2 )
=> ( ( ( insert_set_list_a @ X4 @ A )
= ( insert_set_list_a @ X4 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_208_insert__ident,axiom,
! [X4: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ~ ( member_nat @ X4 @ B2 )
=> ( ( ( insert_nat @ X4 @ A )
= ( insert_nat @ X4 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_209_insert__ident,axiom,
! [X4: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X4 @ A )
=> ( ~ ( member_list_a @ X4 @ B2 )
=> ( ( ( insert_list_a @ X4 @ A )
= ( insert_list_a @ X4 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_210_Set_Oset__insert,axiom,
! [X4: set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ! [B3: set_set_nat] :
( ( A
= ( insert_set_nat @ X4 @ B3 ) )
=> ( member_set_nat @ X4 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_211_Set_Oset__insert,axiom,
! [X4: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ X4 @ A )
=> ~ ! [B3: set_set_list_a] :
( ( A
= ( insert_set_list_a @ X4 @ B3 ) )
=> ( member_set_list_a @ X4 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_212_Set_Oset__insert,axiom,
! [X4: nat,A: set_nat] :
( ( member_nat @ X4 @ A )
=> ~ ! [B3: set_nat] :
( ( A
= ( insert_nat @ X4 @ B3 ) )
=> ( member_nat @ X4 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_213_Set_Oset__insert,axiom,
! [X4: list_a,A: set_list_a] :
( ( member_list_a @ X4 @ A )
=> ~ ! [B3: set_list_a] :
( ( A
= ( insert_list_a @ X4 @ B3 ) )
=> ( member_list_a @ X4 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_214_insertI2,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_215_insertI2,axiom,
! [A2: set_list_a,B2: set_set_list_a,B: set_list_a] :
( ( member_set_list_a @ A2 @ B2 )
=> ( member_set_list_a @ A2 @ ( insert_set_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_216_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_217_insertI2,axiom,
! [A2: list_a,B2: set_list_a,B: list_a] :
( ( member_list_a @ A2 @ B2 )
=> ( member_list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_218_insertI1,axiom,
! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_219_insertI1,axiom,
! [A2: set_list_a,B2: set_set_list_a] : ( member_set_list_a @ A2 @ ( insert_set_list_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_220_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_221_insertI1,axiom,
! [A2: list_a,B2: set_list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_222_insertE,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_223_insertE,axiom,
! [A2: set_list_a,B: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ A2 @ ( insert_set_list_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_list_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_224_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_225_insertE,axiom,
! [A2: list_a,B: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_list_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_226_fmla_Osimps_I65_J,axiom,
! [F1: list_a > list_a,F2: list_a > list_a,X2: $o] :
( ( relati9032153196947279887list_a @ F1 @ F2 @ ( relati5295267155521767763list_a @ X2 ) )
= ( relati5295267155521767763list_a @ X2 ) ) ).
% fmla.simps(65)
thf(fact_227_fmla_Osimps_I65_J,axiom,
! [F1: list_a > list_a,F2: nat > nat,X2: $o] :
( ( relati5126268176769250097at_nat @ F1 @ F2 @ ( relati2772089302019829953_a_nat @ X2 ) )
= ( relati2772089302019829953_a_nat @ X2 ) ) ).
% fmla.simps(65)
thf(fact_228_fmla_Osimps_I65_J,axiom,
! [F1: nat > nat,F2: list_a > list_a,X2: $o] :
( ( relati4590563632004032561list_a @ F1 @ F2 @ ( relati6665026616192419487list_a @ X2 ) )
= ( relati6665026616192419487list_a @ X2 ) ) ).
% fmla.simps(65)
thf(fact_229_fmla_Osimps_I65_J,axiom,
! [F1: nat > nat,F2: nat > nat,X2: $o] :
( ( relati9000828793121449555at_nat @ F1 @ F2 @ ( relati4833799250026832501at_nat @ X2 ) )
= ( relati4833799250026832501at_nat @ X2 ) ) ).
% fmla.simps(65)
thf(fact_230_fmla_Osimps_I65_J,axiom,
! [F1: a > a,F2: b > b,X2: $o] :
( ( relati4520850492397955663_a_b_b @ F1 @ F2 @ ( relational_Bool_a_b @ X2 ) )
= ( relational_Bool_a_b @ X2 ) ) ).
% fmla.simps(65)
thf(fact_231_singleton__inject,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( insert_set_nat @ A2 @ bot_bot_set_set_nat )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_232_singleton__inject,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( insert_set_list_a @ A2 @ bot_bo3186585308812441520list_a )
= ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_233_singleton__inject,axiom,
! [A2: b,B: b] :
( ( ( insert_b @ A2 @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_234_singleton__inject,axiom,
! [A2: a,B: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_235_singleton__inject,axiom,
! [A2: list_a,B: list_a] :
( ( ( insert_list_a @ A2 @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_236_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_237_insert__not__empty,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ A2 @ A )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_238_insert__not__empty,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a @ A2 @ A )
!= bot_bo3186585308812441520list_a ) ).
% insert_not_empty
thf(fact_239_insert__not__empty,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ A )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_240_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_241_insert__not__empty,axiom,
! [A2: list_a,A: set_list_a] :
( ( insert_list_a @ A2 @ A )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_242_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_243_doubleton__eq__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ( insert_set_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C @ ( insert_set_nat @ D @ bot_bot_set_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_244_doubleton__eq__iff,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a,D: set_list_a] :
( ( ( insert_set_list_a @ A2 @ ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) )
= ( insert_set_list_a @ C @ ( insert_set_list_a @ D @ bot_bo3186585308812441520list_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_245_doubleton__eq__iff,axiom,
! [A2: b,B: b,C: b,D: b] :
( ( ( insert_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_246_doubleton__eq__iff,axiom,
! [A2: a,B: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_247_doubleton__eq__iff,axiom,
! [A2: list_a,B: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_248_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_249_singleton__iff,axiom,
! [B: set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_250_singleton__iff,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( member_set_list_a @ B @ ( insert_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_251_singleton__iff,axiom,
! [B: b,A2: b] :
( ( member_b @ B @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_252_singleton__iff,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_253_singleton__iff,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_254_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_255_singletonD,axiom,
! [B: set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_256_singletonD,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( member_set_list_a @ B @ ( insert_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_257_singletonD,axiom,
! [B: b,A2: b] :
( ( member_b @ B @ ( insert_b @ A2 @ bot_bot_set_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_258_singletonD,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_259_singletonD,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_260_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_261_is__singleton__the__elem,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
( A3
= ( insert_set_nat @ ( the_elem_set_nat @ A3 ) @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_262_is__singleton__the__elem,axiom,
( is_sin8525870043004244056list_a
= ( ^ [A3: set_set_list_a] :
( A3
= ( insert_set_list_a @ ( the_elem_set_list_a @ A3 ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_263_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A3: set_b] :
( A3
= ( insert_b @ ( the_elem_b @ A3 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_264_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_265_is__singleton__the__elem,axiom,
( is_singleton_list_a
= ( ^ [A3: set_list_a] :
( A3
= ( insert_list_a @ ( the_elem_list_a @ A3 ) @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_266_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_267_the__elem__eq,axiom,
! [X4: set_nat] :
( ( the_elem_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_268_the__elem__eq,axiom,
! [X4: set_list_a] :
( ( the_elem_set_list_a @ ( insert_set_list_a @ X4 @ bot_bo3186585308812441520list_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_269_the__elem__eq,axiom,
! [X4: b] :
( ( the_elem_b @ ( insert_b @ X4 @ bot_bot_set_b ) )
= X4 ) ).
% the_elem_eq
thf(fact_270_the__elem__eq,axiom,
! [X4: a] :
( ( the_elem_a @ ( insert_a @ X4 @ bot_bot_set_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_271_the__elem__eq,axiom,
! [X4: list_a] :
( ( the_elem_list_a @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_272_the__elem__eq,axiom,
! [X4: nat] :
( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_273_fmla_Opred__set,axiom,
( relati3660035184769383399la_a_b
= ( ^ [P12: a > $o,P22: b > $o,X: relational_fmla_a_b] :
( ! [Y4: a] :
( ( member_a @ Y4 @ ( relati3071123380395136021la_a_b @ X ) )
=> ( P12 @ Y4 ) )
& ! [Y4: b] :
( ( member_b @ Y4 @ ( relati8924981150291758614la_a_b @ X ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% fmla.pred_set
thf(fact_274_fmla_Opred__set,axiom,
( relati4741328337081057510list_a
= ( ^ [P12: list_a > $o,P22: list_a > $o,X: relati5436652119368751390list_a] :
( ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( relati4910493452150799124list_a @ X ) )
=> ( P12 @ Y4 ) )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( relati3206271568683928597list_a @ X ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% fmla.pred_set
thf(fact_275_fmla_Opred__set,axiom,
( relati6982367164109063406_a_nat
= ( ^ [P12: list_a > $o,P22: nat > $o,X: relati8428538799459208780_a_nat] :
( ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( relati6972889302305856_a_nat @ X ) )
=> ( P12 @ Y4 ) )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( relati8645012329147373695_a_nat @ X ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% fmla.pred_set
thf(fact_276_fmla_Opred__set,axiom,
( relati1651932441426877132list_a
= ( ^ [P12: nat > $o,P22: list_a > $o,X: relati6521690034043994866list_a] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ ( relati3899910203474895390list_a @ X ) )
=> ( P12 @ Y4 ) )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( relati3314577606465187421list_a @ X ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% fmla.pred_set
thf(fact_277_fmla_Opred__set,axiom,
( relati6725209092851823240at_nat
= ( ^ [P12: nat > $o,P22: nat > $o,X: relati7126052417554554232at_nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ ( relati6321887899146193334at_nat @ X ) )
=> ( P12 @ Y4 ) )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( relati2567137625159299127at_nat @ X ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% fmla.pred_set
thf(fact_278_fmla_Osimps_I127_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a] :
( ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) )
= ( insert_b @ X11 @ bot_bot_set_b ) ) ).
% fmla.simps(127)
thf(fact_279_fmla_Osimps_I127_J,axiom,
! [X11: list_a,X12: list_R4687676760361139541list_a] :
( ( relati3206271568683928597list_a @ ( relati3964196027911924042list_a @ X11 @ X12 ) )
= ( insert_list_a @ X11 @ bot_bot_set_list_a ) ) ).
% fmla.simps(127)
thf(fact_280_fmla_Osimps_I127_J,axiom,
! [X11: nat,X12: list_R4687676760361139541list_a] :
( ( relati8645012329147373695_a_nat @ ( relati2221024441978789736list_a @ X11 @ X12 ) )
= ( insert_nat @ X11 @ bot_bot_set_nat ) ) ).
% fmla.simps(127)
thf(fact_281_fmla_Osimps_I127_J,axiom,
! [X11: list_a,X12: list_R114826772386431851rm_nat] :
( ( relati3314577606465187421list_a @ ( relati7551459164660976010_a_nat @ X11 @ X12 ) )
= ( insert_list_a @ X11 @ bot_bot_set_list_a ) ) ).
% fmla.simps(127)
thf(fact_282_fmla_Osimps_I127_J,axiom,
! [X11: nat,X12: list_R114826772386431851rm_nat] :
( ( relati2567137625159299127at_nat @ ( relati5908763325135257836at_nat @ X11 @ X12 ) )
= ( insert_nat @ X11 @ bot_bot_set_nat ) ) ).
% fmla.simps(127)
thf(fact_283_totalp__on__singleton,axiom,
! [X4: set_nat,R: set_nat > set_nat > $o] : ( totalp_on_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) @ R ) ).
% totalp_on_singleton
thf(fact_284_totalp__on__singleton,axiom,
! [X4: set_list_a,R: set_list_a > set_list_a > $o] : ( totalp_on_set_list_a @ ( insert_set_list_a @ X4 @ bot_bo3186585308812441520list_a ) @ R ) ).
% totalp_on_singleton
thf(fact_285_totalp__on__singleton,axiom,
! [X4: b,R: b > b > $o] : ( totalp_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ R ) ).
% totalp_on_singleton
thf(fact_286_totalp__on__singleton,axiom,
! [X4: a,R: a > a > $o] : ( totalp_on_a @ ( insert_a @ X4 @ bot_bot_set_a ) @ R ) ).
% totalp_on_singleton
thf(fact_287_totalp__on__singleton,axiom,
! [X4: list_a,R: list_a > list_a > $o] : ( totalp_on_list_a @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) @ R ) ).
% totalp_on_singleton
thf(fact_288_totalp__on__singleton,axiom,
! [X4: nat,R: nat > nat > $o] : ( totalp_on_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) @ R ) ).
% totalp_on_singleton
thf(fact_289_pairwise__singleton,axiom,
! [P: set_nat > set_nat > $o,A: set_nat] : ( pairwise_set_nat @ P @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% pairwise_singleton
thf(fact_290_pairwise__singleton,axiom,
! [P: set_list_a > set_list_a > $o,A: set_list_a] : ( pairwise_set_list_a @ P @ ( insert_set_list_a @ A @ bot_bo3186585308812441520list_a ) ) ).
% pairwise_singleton
thf(fact_291_pairwise__singleton,axiom,
! [P: b > b > $o,A: b] : ( pairwise_b @ P @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% pairwise_singleton
thf(fact_292_pairwise__singleton,axiom,
! [P: a > a > $o,A: a] : ( pairwise_a @ P @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_293_pairwise__singleton,axiom,
! [P: list_a > list_a > $o,A: list_a] : ( pairwise_list_a @ P @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% pairwise_singleton
thf(fact_294_pairwise__singleton,axiom,
! [P: nat > nat > $o,A: nat] : ( pairwise_nat @ P @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_295_singleton__insert__inj__eq,axiom,
! [B: set_nat,A2: set_nat,A: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_296_singleton__insert__inj__eq,axiom,
! [B: set_list_a,A2: set_list_a,A: set_set_list_a] :
( ( ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a )
= ( insert_set_list_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le8877086941679407844list_a @ A @ ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_297_singleton__insert__inj__eq,axiom,
! [B: b,A2: b,A: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_298_singleton__insert__inj__eq,axiom,
! [B: a,A2: a,A: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_299_singleton__insert__inj__eq,axiom,
! [B: list_a,A2: list_a,A: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_300_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_301_singleton__insert__inj__eq_H,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A2 = B )
& ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_302_singleton__insert__inj__eq_H,axiom,
! [A2: set_list_a,A: set_set_list_a,B: set_list_a] :
( ( ( insert_set_list_a @ A2 @ A )
= ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) )
= ( ( A2 = B )
& ( ord_le8877086941679407844list_a @ A @ ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_303_singleton__insert__inj__eq_H,axiom,
! [A2: b,A: set_b,B: b] :
( ( ( insert_b @ A2 @ A )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A2 = B )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_304_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_305_singleton__insert__inj__eq_H,axiom,
! [A2: list_a,A: set_list_a,B: list_a] :
( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A2 = B )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_306_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_307_fmla_Opred__inject_I2_J,axiom,
! [P1: list_a > $o,P2: list_a > $o,A2: $o] :
( ( relati4741328337081057510list_a @ P1 @ P2 @ ( relati5295267155521767763list_a @ A2 ) )
= ( top_top_o_o @ A2 ) ) ).
% fmla.pred_inject(2)
thf(fact_308_fmla_Opred__inject_I2_J,axiom,
! [P1: list_a > $o,P2: nat > $o,A2: $o] :
( ( relati6982367164109063406_a_nat @ P1 @ P2 @ ( relati2772089302019829953_a_nat @ A2 ) )
= ( top_top_o_o @ A2 ) ) ).
% fmla.pred_inject(2)
thf(fact_309_fmla_Opred__inject_I2_J,axiom,
! [P1: nat > $o,P2: list_a > $o,A2: $o] :
( ( relati1651932441426877132list_a @ P1 @ P2 @ ( relati6665026616192419487list_a @ A2 ) )
= ( top_top_o_o @ A2 ) ) ).
% fmla.pred_inject(2)
thf(fact_310_fmla_Opred__inject_I2_J,axiom,
! [P1: nat > $o,P2: nat > $o,A2: $o] :
( ( relati6725209092851823240at_nat @ P1 @ P2 @ ( relati4833799250026832501at_nat @ A2 ) )
= ( top_top_o_o @ A2 ) ) ).
% fmla.pred_inject(2)
thf(fact_311_fmla_Opred__inject_I2_J,axiom,
! [P1: a > $o,P2: b > $o,A2: $o] :
( ( relati3660035184769383399la_a_b @ P1 @ P2 @ ( relational_Bool_a_b @ A2 ) )
= ( top_top_o_o @ A2 ) ) ).
% fmla.pred_inject(2)
thf(fact_312_insert__Diff__single,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_313_insert__Diff__single,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a @ A2 @ ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) ) )
= ( insert_set_list_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_314_insert__Diff__single,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( insert_b @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_315_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_316_insert__Diff__single,axiom,
! [A2: list_a,A: set_list_a] :
( ( insert_list_a @ A2 @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_317_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_318_disjoint__insert_I2_J,axiom,
! [A: set_set_nat,B: set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) )
= ( ~ ( member_set_nat @ B @ A )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_319_disjoint__insert_I2_J,axiom,
! [A: set_set_list_a,B: set_list_a,B2: set_set_list_a] :
( ( bot_bo3186585308812441520list_a
= ( inf_in4657809108759609906list_a @ A @ ( insert_set_list_a @ B @ B2 ) ) )
= ( ~ ( member_set_list_a @ B @ A )
& ( bot_bo3186585308812441520list_a
= ( inf_in4657809108759609906list_a @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_320_disjoint__insert_I2_J,axiom,
! [A: set_b,B: b,B2: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A @ ( insert_b @ B @ B2 ) ) )
= ( ~ ( member_b @ B @ A )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_321_disjoint__insert_I2_J,axiom,
! [A: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_322_disjoint__insert_I2_J,axiom,
! [A: set_list_a,B: list_a,B2: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ ( insert_list_a @ B @ B2 ) ) )
= ( ~ ( member_list_a @ B @ A )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_323_disjoint__insert_I2_J,axiom,
! [A: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_324_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_325_order__refl,axiom,
! [X4: set_list_a] : ( ord_le8861187494160871172list_a @ X4 @ X4 ) ).
% order_refl
thf(fact_326_order__refl,axiom,
! [X4: $o > nat] : ( ord_less_eq_o_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_327_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_328_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_329_dual__order_Orefl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_330_dual__order_Orefl,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_331_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_332_subsetI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_333_subsetI,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ( member_set_list_a @ X3 @ B2 ) )
=> ( ord_le8877086941679407844list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_334_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_335_subsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( member_list_a @ X3 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_336_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_337_subset__antisym,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_338_IntI,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_339_IntI,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ A )
=> ( ( member_set_list_a @ C @ B2 )
=> ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_340_IntI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_341_IntI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_342_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_343_Int__iff,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A @ B2 ) )
= ( ( member_set_list_a @ C @ A )
& ( member_set_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_344_Int__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_345_Int__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ( member_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_346_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_347_DiffI,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ A )
=> ( ~ ( member_set_list_a @ C @ B2 )
=> ( member_set_list_a @ C @ ( minus_4782336368215558443list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_348_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_349_DiffI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_350_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_351_Diff__iff,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( minus_4782336368215558443list_a @ A @ B2 ) )
= ( ( member_set_list_a @ C @ A )
& ~ ( member_set_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_352_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_353_Diff__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_354_Diff__idemp,axiom,
! [A: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_355_Diff__idemp,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_356_fmla_Oinject_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,Y11: b,Y12: list_R6823256787227418703term_a] :
( ( ( relational_Pred_b_a @ X11 @ X12 )
= ( relational_Pred_b_a @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fmla.inject(1)
thf(fact_357_top1I,axiom,
! [X4: list_a] : ( top_top_list_a_o @ X4 ) ).
% top1I
thf(fact_358_top1I,axiom,
! [X4: $o] : ( top_top_o_o @ X4 ) ).
% top1I
thf(fact_359_top1I,axiom,
! [X4: nat] : ( top_top_nat_o @ X4 ) ).
% top1I
thf(fact_360_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_361_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_362_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_363_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_364_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_365_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_366_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_367_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_368_ball__empty,axiom,
! [P: b > $o,X5: b] :
( ( member_b @ X5 @ bot_bot_set_b )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_369_ball__empty,axiom,
! [P: a > $o,X5: a] :
( ( member_a @ X5 @ bot_bot_set_a )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_370_ball__empty,axiom,
! [P: list_a > $o,X5: list_a] :
( ( member_list_a @ X5 @ bot_bot_set_list_a )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_371_ball__empty,axiom,
! [P: nat > $o,X5: nat] :
( ( member_nat @ X5 @ bot_bot_set_nat )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_372_insert__subset,axiom,
! [X4: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ A ) @ B2 )
= ( ( member_set_nat @ X4 @ B2 )
& ( ord_le6893508408891458716et_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_373_insert__subset,axiom,
! [X4: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ ( insert_set_list_a @ X4 @ A ) @ B2 )
= ( ( member_set_list_a @ X4 @ B2 )
& ( ord_le8877086941679407844list_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_374_insert__subset,axiom,
! [X4: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A ) @ B2 )
= ( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_375_insert__subset,axiom,
! [X4: list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X4 @ A ) @ B2 )
= ( ( member_list_a @ X4 @ B2 )
& ( ord_le8861187494160871172list_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_376_Diff__empty,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ A @ bot_bot_set_b )
= A ) ).
% Diff_empty
thf(fact_377_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_378_Diff__empty,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Diff_empty
thf(fact_379_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_380_empty__Diff,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_381_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_382_empty__Diff,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_383_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_384_Diff__cancel,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ A @ A )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_385_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_386_Diff__cancel,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ A )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_387_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_388_Int__insert__left__if0,axiom,
! [A2: set_nat,C3: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_set_nat @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_389_Int__insert__left__if0,axiom,
! [A2: set_list_a,C3: set_set_list_a,B2: set_set_list_a] :
( ~ ( member_set_list_a @ A2 @ C3 )
=> ( ( inf_in4657809108759609906list_a @ ( insert_set_list_a @ A2 @ B2 ) @ C3 )
= ( inf_in4657809108759609906list_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_390_Int__insert__left__if0,axiom,
! [A2: nat,C3: set_nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_nat @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_391_Int__insert__left__if0,axiom,
! [A2: list_a,C3: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ A2 @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_392_Int__insert__left__if1,axiom,
! [A2: set_nat,C3: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A2 @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C3 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_393_Int__insert__left__if1,axiom,
! [A2: set_list_a,C3: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ A2 @ C3 )
=> ( ( inf_in4657809108759609906list_a @ ( insert_set_list_a @ A2 @ B2 ) @ C3 )
= ( insert_set_list_a @ A2 @ ( inf_in4657809108759609906list_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_394_Int__insert__left__if1,axiom,
! [A2: nat,C3: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C3 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_395_Int__insert__left__if1,axiom,
! [A2: list_a,C3: set_list_a,B2: set_list_a] :
( ( member_list_a @ A2 @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B2 ) @ C3 )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_396_insert__inter__insert,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ ( insert_set_nat @ A2 @ B2 ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_397_insert__inter__insert,axiom,
! [A2: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( insert_set_list_a @ A2 @ A ) @ ( insert_set_list_a @ A2 @ B2 ) )
= ( insert_set_list_a @ A2 @ ( inf_in4657809108759609906list_a @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_398_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_399_insert__inter__insert,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ A ) @ ( insert_list_a @ A2 @ B2 ) )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_400_Int__insert__right__if0,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_401_Int__insert__right__if0,axiom,
! [A2: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ~ ( member_set_list_a @ A2 @ A )
=> ( ( inf_in4657809108759609906list_a @ A @ ( insert_set_list_a @ A2 @ B2 ) )
= ( inf_in4657809108759609906list_a @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_402_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_403_Int__insert__right__if0,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_404_Int__insert__right__if1,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B2 ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_405_Int__insert__right__if1,axiom,
! [A2: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ A2 @ A )
=> ( ( inf_in4657809108759609906list_a @ A @ ( insert_set_list_a @ A2 @ B2 ) )
= ( insert_set_list_a @ A2 @ ( inf_in4657809108759609906list_a @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_406_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_407_Int__insert__right__if1,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B2 ) )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_408_Int__subset__iff,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( ord_less_eq_set_nat @ C3 @ A )
& ( ord_less_eq_set_nat @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_409_Int__subset__iff,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( ord_le8861187494160871172list_a @ C3 @ A )
& ( ord_le8861187494160871172list_a @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_410_Diff__insert0,axiom,
! [X4: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X4 @ B2 ) )
= ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_411_Diff__insert0,axiom,
! [X4: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ~ ( member_set_list_a @ X4 @ A )
=> ( ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a @ X4 @ B2 ) )
= ( minus_4782336368215558443list_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_412_Diff__insert0,axiom,
! [X4: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_413_Diff__insert0,axiom,
! [X4: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X4 @ A )
=> ( ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X4 @ B2 ) )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_414_insert__Diff1,axiom,
! [X4: set_nat,B2: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A ) @ B2 )
= ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_415_insert__Diff1,axiom,
! [X4: set_list_a,B2: set_set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ X4 @ B2 )
=> ( ( minus_4782336368215558443list_a @ ( insert_set_list_a @ X4 @ A ) @ B2 )
= ( minus_4782336368215558443list_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_416_insert__Diff1,axiom,
! [X4: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_417_insert__Diff1,axiom,
! [X4: list_a,B2: set_list_a,A: set_list_a] :
( ( member_list_a @ X4 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X4 @ A ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_418_rrb__simps_I2_J,axiom,
! [P3: b,Ts: list_R6823256787227418703term_a] : ( relational_rrb_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ).
% rrb_simps(2)
thf(fact_419_top__apply,axiom,
( top_top_list_a_o
= ( ^ [X: list_a] : top_top_o ) ) ).
% top_apply
thf(fact_420_top__apply,axiom,
( top_top_o_o
= ( ^ [X: $o] : top_top_o ) ) ).
% top_apply
thf(fact_421_top__apply,axiom,
( top_top_nat_o
= ( ^ [X: nat] : top_top_o ) ) ).
% top_apply
thf(fact_422_cpropagated__simps_I2_J,axiom,
! [P3: b,Ts: list_R6823256787227418703term_a] : ( relati1591879772219623554ed_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ).
% cpropagated_simps(2)
thf(fact_423_insert__disjoint_I1_J,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B2 )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B2 )
& ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_424_insert__disjoint_I1_J,axiom,
! [A2: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ ( insert_set_list_a @ A2 @ A ) @ B2 )
= bot_bo3186585308812441520list_a )
= ( ~ ( member_set_list_a @ A2 @ B2 )
& ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_425_insert__disjoint_I1_J,axiom,
! [A2: b,A: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B2 )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B2 )
& ( ( inf_inf_set_b @ A @ B2 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_426_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B2 )
& ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_427_insert__disjoint_I1_J,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ A ) @ B2 )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A2 @ B2 )
& ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_428_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B2 )
& ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_429_insert__disjoint_I2_J,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_set_nat @ A2 @ B2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_430_insert__disjoint_I2_J,axiom,
! [A2: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( bot_bo3186585308812441520list_a
= ( inf_in4657809108759609906list_a @ ( insert_set_list_a @ A2 @ A ) @ B2 ) )
= ( ~ ( member_set_list_a @ A2 @ B2 )
& ( bot_bo3186585308812441520list_a
= ( inf_in4657809108759609906list_a @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_431_insert__disjoint_I2_J,axiom,
! [A2: b,A: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B2 ) )
= ( ~ ( member_b @ A2 @ B2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_432_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B2 ) )
= ( ~ ( member_a @ A2 @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_433_insert__disjoint_I2_J,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ A ) @ B2 ) )
= ( ~ ( member_list_a @ A2 @ B2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_434_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_nat @ A2 @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_435_disjoint__insert_I1_J,axiom,
! [B2: set_set_nat,A2: set_nat,A: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A2 @ A ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B2 )
& ( ( inf_inf_set_set_nat @ B2 @ A )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_436_disjoint__insert_I1_J,axiom,
! [B2: set_set_list_a,A2: set_list_a,A: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ B2 @ ( insert_set_list_a @ A2 @ A ) )
= bot_bo3186585308812441520list_a )
= ( ~ ( member_set_list_a @ A2 @ B2 )
& ( ( inf_in4657809108759609906list_a @ B2 @ A )
= bot_bo3186585308812441520list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_437_disjoint__insert_I1_J,axiom,
! [B2: set_b,A2: b,A: set_b] :
( ( ( inf_inf_set_b @ B2 @ ( insert_b @ A2 @ A ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B2 )
& ( ( inf_inf_set_b @ B2 @ A )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_438_disjoint__insert_I1_J,axiom,
! [B2: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B2 )
& ( ( inf_inf_set_a @ B2 @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_439_disjoint__insert_I1_J,axiom,
! [B2: set_list_a,A2: list_a,A: set_list_a] :
( ( ( inf_inf_set_list_a @ B2 @ ( insert_list_a @ A2 @ A ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A2 @ B2 )
& ( ( inf_inf_set_list_a @ B2 @ A )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_440_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_441_Diff__eq__empty__iff,axiom,
! [A: set_b,B2: set_b] :
( ( ( minus_minus_set_b @ A @ B2 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_442_Diff__eq__empty__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_443_Diff__eq__empty__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( minus_646659088055828811list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_444_Diff__eq__empty__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_445_Diff__disjoint,axiom,
! [A: set_b,B2: set_b] :
( ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B2 @ A ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_446_Diff__disjoint,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B2 @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_447_Diff__disjoint,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ A ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_448_Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_449_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_450_le__cases3,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_451_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_452_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z3: set_list_a] : ( Y5 = Z3 ) )
= ( ^ [X: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y4 )
& ( ord_le8861187494160871172list_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_453_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: $o > nat,Z3: $o > nat] : ( Y5 = Z3 ) )
= ( ^ [X: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y4 )
& ( ord_less_eq_o_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_454_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_455_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_456_ord__eq__le__trans,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( A2 = B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_457_ord__eq__le__trans,axiom,
! [A2: $o > nat,B: $o > nat,C: $o > nat] :
( ( A2 = B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_458_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_459_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_460_ord__le__eq__trans,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_461_ord__le__eq__trans,axiom,
! [A2: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_462_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_463_order__antisym,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_464_order__antisym,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_465_order__antisym,axiom,
! [X4: $o > nat,Y3: $o > nat] :
( ( ord_less_eq_o_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_o_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_466_order__antisym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_467_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_468_order_Otrans,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_469_order_Otrans,axiom,
! [A2: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_470_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_471_order__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z )
=> ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_472_order__trans,axiom,
! [X4: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ Y3 @ Z )
=> ( ord_le8861187494160871172list_a @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_473_order__trans,axiom,
! [X4: $o > nat,Y3: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_o_nat @ Y3 @ Z )
=> ( ord_less_eq_o_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_474_order__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_475_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_476_top__greatest,axiom,
! [A2: set_set_list_a] : ( ord_le8877086941679407844list_a @ A2 @ top_to7106483174946246804list_a ) ).
% top_greatest
thf(fact_477_top__greatest,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ top_top_set_set_nat ) ).
% top_greatest
thf(fact_478_top__greatest,axiom,
! [A2: list_a > $o] : ( ord_less_eq_list_a_o @ A2 @ top_top_list_a_o ) ).
% top_greatest
thf(fact_479_top__greatest,axiom,
! [A2: $o > $o] : ( ord_less_eq_o_o @ A2 @ top_top_o_o ) ).
% top_greatest
thf(fact_480_top__greatest,axiom,
! [A2: nat > $o] : ( ord_less_eq_nat_o @ A2 @ top_top_nat_o ) ).
% top_greatest
thf(fact_481_top__greatest,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ top_top_set_list_a ) ).
% top_greatest
thf(fact_482_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_483_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_484_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_list_a,Z3: set_list_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ B5 @ A5 )
& ( ord_le8861187494160871172list_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_485_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: $o > nat,Z3: $o > nat] : ( Y5 = Z3 ) )
= ( ^ [A5: $o > nat,B5: $o > nat] :
( ( ord_less_eq_o_nat @ B5 @ A5 )
& ( ord_less_eq_o_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_486_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_487_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_488_dual__order_Oantisym,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_489_dual__order_Oantisym,axiom,
! [B: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A2 )
=> ( ( ord_less_eq_o_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_490_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_491_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_492_dual__order_Otrans,axiom,
! [B: set_list_a,A2: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_le8861187494160871172list_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_493_dual__order_Otrans,axiom,
! [B: $o > nat,A2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A2 )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_494_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_495_top_Oextremum__unique,axiom,
! [A2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ top_to7106483174946246804list_a @ A2 )
= ( A2 = top_to7106483174946246804list_a ) ) ).
% top.extremum_unique
thf(fact_496_top_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ top_top_set_set_nat @ A2 )
= ( A2 = top_top_set_set_nat ) ) ).
% top.extremum_unique
thf(fact_497_top_Oextremum__unique,axiom,
! [A2: list_a > $o] :
( ( ord_less_eq_list_a_o @ top_top_list_a_o @ A2 )
= ( A2 = top_top_list_a_o ) ) ).
% top.extremum_unique
thf(fact_498_top_Oextremum__unique,axiom,
! [A2: $o > $o] :
( ( ord_less_eq_o_o @ top_top_o_o @ A2 )
= ( A2 = top_top_o_o ) ) ).
% top.extremum_unique
thf(fact_499_top_Oextremum__unique,axiom,
! [A2: nat > $o] :
( ( ord_less_eq_nat_o @ top_top_nat_o @ A2 )
= ( A2 = top_top_nat_o ) ) ).
% top.extremum_unique
thf(fact_500_top_Oextremum__unique,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ top_top_set_list_a @ A2 )
= ( A2 = top_top_set_list_a ) ) ).
% top.extremum_unique
thf(fact_501_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_502_top_Oextremum__uniqueI,axiom,
! [A2: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ top_to7106483174946246804list_a @ A2 )
=> ( A2 = top_to7106483174946246804list_a ) ) ).
% top.extremum_uniqueI
thf(fact_503_top_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ top_top_set_set_nat @ A2 )
=> ( A2 = top_top_set_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_504_top_Oextremum__uniqueI,axiom,
! [A2: list_a > $o] :
( ( ord_less_eq_list_a_o @ top_top_list_a_o @ A2 )
=> ( A2 = top_top_list_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_505_top_Oextremum__uniqueI,axiom,
! [A2: $o > $o] :
( ( ord_less_eq_o_o @ top_top_o_o @ A2 )
=> ( A2 = top_top_o_o ) ) ).
% top.extremum_uniqueI
thf(fact_506_top_Oextremum__uniqueI,axiom,
! [A2: nat > $o] :
( ( ord_less_eq_nat_o @ top_top_nat_o @ A2 )
=> ( A2 = top_top_nat_o ) ) ).
% top.extremum_uniqueI
thf(fact_507_top_Oextremum__uniqueI,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ top_top_set_list_a @ A2 )
=> ( A2 = top_top_set_list_a ) ) ).
% top.extremum_uniqueI
thf(fact_508_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_509_IntE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ~ ( member_set_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_510_IntE,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A @ B2 ) )
=> ~ ( ( member_set_list_a @ C @ A )
=> ~ ( member_set_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_511_IntE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_512_IntE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ~ ( member_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_513_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_514_DiffE,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( minus_4782336368215558443list_a @ A @ B2 ) )
=> ~ ( ( member_set_list_a @ C @ A )
=> ( member_set_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_515_DiffE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_516_DiffE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_517_IntD1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_518_IntD1,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A @ B2 ) )
=> ( member_set_list_a @ C @ A ) ) ).
% IntD1
thf(fact_519_IntD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_520_IntD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% IntD1
thf(fact_521_IntD2,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_522_IntD2,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( inf_in4657809108759609906list_a @ A @ B2 ) )
=> ( member_set_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_523_IntD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_524_IntD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( member_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_525_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_526_DiffD1,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( minus_4782336368215558443list_a @ A @ B2 ) )
=> ( member_set_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_527_DiffD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_528_DiffD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_529_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ~ ( member_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_530_DiffD2,axiom,
! [C: set_list_a,A: set_set_list_a,B2: set_set_list_a] :
( ( member_set_list_a @ C @ ( minus_4782336368215558443list_a @ A @ B2 ) )
=> ~ ( member_set_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_531_DiffD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_532_DiffD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_533_in__mono,axiom,
! [A: set_set_nat,B2: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_534_in__mono,axiom,
! [A: set_set_list_a,B2: set_set_list_a,X4: set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B2 )
=> ( ( member_set_list_a @ X4 @ A )
=> ( member_set_list_a @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_535_in__mono,axiom,
! [A: set_nat,B2: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_536_in__mono,axiom,
! [A: set_list_a,B2: set_list_a,X4: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ X4 @ A )
=> ( member_list_a @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_537_subsetD,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_538_subsetD,axiom,
! [A: set_set_list_a,B2: set_set_list_a,C: set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B2 )
=> ( ( member_set_list_a @ C @ A )
=> ( member_set_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_539_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_540_subsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_541_Ball__def,axiom,
( ball_set_nat
= ( ^ [A3: set_set_nat,P4: set_nat > $o] :
! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( P4 @ X ) ) ) ) ).
% Ball_def
thf(fact_542_Ball__def,axiom,
( ball_set_list_a
= ( ^ [A3: set_set_list_a,P4: set_list_a > $o] :
! [X: set_list_a] :
( ( member_set_list_a @ X @ A3 )
=> ( P4 @ X ) ) ) ) ).
% Ball_def
thf(fact_543_Ball__def,axiom,
( ball_nat
= ( ^ [A3: set_nat,P4: nat > $o] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( P4 @ X ) ) ) ) ).
% Ball_def
thf(fact_544_Ball__def,axiom,
( ball_list_a
= ( ^ [A3: set_list_a,P4: list_a > $o] :
! [X: list_a] :
( ( member_list_a @ X @ A3 )
=> ( P4 @ X ) ) ) ) ).
% Ball_def
thf(fact_545_Int__Diff,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C3 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_546_Int__Diff,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_547_Int__mono,axiom,
! [A: set_nat,C3: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_548_Int__mono,axiom,
! [A: set_list_a,C3: set_list_a,B2: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_549_Diff__Int2,axiom,
! [A: set_nat,C3: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C3 ) @ ( inf_inf_set_nat @ B2 @ C3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_550_Diff__Int2,axiom,
! [A: set_list_a,C3: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C3 ) @ ( inf_inf_set_list_a @ B2 @ C3 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_551_Diff__mono,axiom,
! [A: set_nat,C3: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_552_Diff__mono,axiom,
! [A: set_list_a,C3: set_list_a,D2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_553_Int__assoc,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C3 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_554_Int__assoc,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_555_equalityE,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_556_equalityE,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_557_pairwiseD,axiom,
! [R: set_nat > set_nat > $o,S: set_set_nat,X4: set_nat,Y3: set_nat] :
( ( pairwise_set_nat @ R @ S )
=> ( ( member_set_nat @ X4 @ S )
=> ( ( member_set_nat @ Y3 @ S )
=> ( ( X4 != Y3 )
=> ( R @ X4 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_558_pairwiseD,axiom,
! [R: set_list_a > set_list_a > $o,S: set_set_list_a,X4: set_list_a,Y3: set_list_a] :
( ( pairwise_set_list_a @ R @ S )
=> ( ( member_set_list_a @ X4 @ S )
=> ( ( member_set_list_a @ Y3 @ S )
=> ( ( X4 != Y3 )
=> ( R @ X4 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_559_pairwiseD,axiom,
! [R: nat > nat > $o,S: set_nat,X4: nat,Y3: nat] :
( ( pairwise_nat @ R @ S )
=> ( ( member_nat @ X4 @ S )
=> ( ( member_nat @ Y3 @ S )
=> ( ( X4 != Y3 )
=> ( R @ X4 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_560_pairwiseD,axiom,
! [R: list_a > list_a > $o,S: set_list_a,X4: list_a,Y3: list_a] :
( ( pairwise_list_a @ R @ S )
=> ( ( member_list_a @ X4 @ S )
=> ( ( member_list_a @ Y3 @ S )
=> ( ( X4 != Y3 )
=> ( R @ X4 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_561_pairwiseI,axiom,
! [S: set_set_nat,R: set_nat > set_nat > $o] :
( ! [X3: set_nat,Y: set_nat] :
( ( member_set_nat @ X3 @ S )
=> ( ( member_set_nat @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) )
=> ( pairwise_set_nat @ R @ S ) ) ).
% pairwiseI
thf(fact_562_pairwiseI,axiom,
! [S: set_set_list_a,R: set_list_a > set_list_a > $o] :
( ! [X3: set_list_a,Y: set_list_a] :
( ( member_set_list_a @ X3 @ S )
=> ( ( member_set_list_a @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) )
=> ( pairwise_set_list_a @ R @ S ) ) ).
% pairwiseI
thf(fact_563_pairwiseI,axiom,
! [S: set_nat,R: nat > nat > $o] :
( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ S )
=> ( ( member_nat @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) )
=> ( pairwise_nat @ R @ S ) ) ).
% pairwiseI
thf(fact_564_pairwiseI,axiom,
! [S: set_list_a,R: list_a > list_a > $o] :
( ! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ S )
=> ( ( member_list_a @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) )
=> ( pairwise_list_a @ R @ S ) ) ).
% pairwiseI
thf(fact_565_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B6: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( member_set_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_566_subset__eq,axiom,
( ord_le8877086941679407844list_a
= ( ^ [A3: set_set_list_a,B6: set_set_list_a] :
! [X: set_list_a] :
( ( member_set_list_a @ X @ A3 )
=> ( member_set_list_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_567_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B6: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_568_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B6: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A3 )
=> ( member_list_a @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_569_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_570_Int__absorb,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_571_Int__lower1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_572_Int__lower1,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_573_Int__lower2,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_574_Int__lower2,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_575_equalityD1,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_576_equalityD1,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_577_Set_OequalityD2,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% Set.equalityD2
thf(fact_578_Set_OequalityD2,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A ) ) ).
% Set.equalityD2
thf(fact_579_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_580_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B6: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A3 )
=> ( member_list_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_581_Collect__mono,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_582_Collect__mono,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_583_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_584_Collect__mono__iff,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_585_Collect__mono__iff,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q2 ) )
= ( ! [X: list_a] :
( ( P @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_586_Int__Collect__mono,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_587_Int__Collect__mono,axiom,
! [A: set_list_a,B2: set_list_a,P: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B2 @ ( collect_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_588_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_589_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_590_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_591_order__eq__refl,axiom,
! [X4: nat,Y3: nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_592_linorder__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_linear
thf(fact_593_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_594_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_595_linorder__le__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_le_cases
thf(fact_596_order__antisym__conv,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_597_totalp__on__le,axiom,
! [A: set_nat] : ( totalp_on_nat @ A @ ord_less_eq_nat ) ).
% totalp_on_le
thf(fact_598_totalp__onD,axiom,
! [A: set_nat,R: nat > nat > $o,X4: nat,Y3: nat] :
( ( totalp_on_nat @ A @ R )
=> ( ( member_nat @ X4 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( R @ X4 @ Y3 )
| ( R @ Y3 @ X4 ) ) ) ) ) ) ).
% totalp_onD
thf(fact_599_totalp__onD,axiom,
! [A: set_list_a,R: list_a > list_a > $o,X4: list_a,Y3: list_a] :
( ( totalp_on_list_a @ A @ R )
=> ( ( member_list_a @ X4 @ A )
=> ( ( member_list_a @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( R @ X4 @ Y3 )
| ( R @ Y3 @ X4 ) ) ) ) ) ) ).
% totalp_onD
thf(fact_600_totalp__onI,axiom,
! [A: set_nat,R: nat > nat > $o] :
( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( X3 != Y )
=> ( ( R @ X3 @ Y )
| ( R @ Y @ X3 ) ) ) ) )
=> ( totalp_on_nat @ A @ R ) ) ).
% totalp_onI
thf(fact_601_totalp__onI,axiom,
! [A: set_list_a,R: list_a > list_a > $o] :
( ! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ( member_list_a @ Y @ A )
=> ( ( X3 != Y )
=> ( ( R @ X3 @ Y )
| ( R @ Y @ X3 ) ) ) ) )
=> ( totalp_on_list_a @ A @ R ) ) ).
% totalp_onI
thf(fact_602_Int__Diff__disjoint,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ A @ B2 ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_603_Int__Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ A @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_604_Diff__triv,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_605_Diff__triv,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_606_subset__Diff__insert,axiom,
! [A: set_nat,B2: set_nat,X4: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X4 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C3 ) )
& ~ ( member_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_607_subset__Diff__insert,axiom,
! [A: set_list_a,B2: set_list_a,X4: list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( minus_646659088055828811list_a @ B2 @ ( insert_list_a @ X4 @ C3 ) ) )
= ( ( ord_le8861187494160871172list_a @ A @ ( minus_646659088055828811list_a @ B2 @ C3 ) )
& ~ ( member_list_a @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_608_Diff__single__insert,axiom,
! [A: set_list_a,X4: list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) @ B2 )
=> ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_609_Diff__single__insert,axiom,
! [A: set_nat,X4: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_610_subset__insert__iff,axiom,
! [A: set_list_a,X4: list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X4 @ B2 ) )
= ( ( ( member_list_a @ X4 @ A )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) @ B2 ) )
& ( ~ ( member_list_a @ X4 @ A )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_611_subset__insert__iff,axiom,
! [A: set_nat,X4: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B2 ) )
= ( ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_612_fmla_Odistinct_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X2: $o] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Bool_a_b @ X2 ) ) ).
% fmla.distinct(1)
thf(fact_613_pairwise__alt,axiom,
( pairwise_list_a
= ( ^ [R2: list_a > list_a > $o,S2: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ S2 )
=> ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( minus_646659088055828811list_a @ S2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) )
=> ( R2 @ X @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_614_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R2: nat > nat > $o,S2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ S2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
=> ( R2 @ X @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_615_Int__emptyI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ( member_list_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_616_Int__emptyI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_617_disjoint__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ~ ( member_list_a @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_618_disjoint__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_619_Int__empty__left,axiom,
! [B2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B2 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_620_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_621_Int__empty__right,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_622_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_623_disjoint__iff__not__equal,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ! [Y4: list_a] :
( ( member_list_a @ Y4 @ B2 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_624_disjoint__iff__not__equal,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B2 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_625_Int__insert__left,axiom,
! [A2: nat,C3: set_nat,B2: set_nat] :
( ( ( member_nat @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C3 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) )
& ( ~ ( member_nat @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_626_Int__insert__left,axiom,
! [A2: list_a,C3: set_list_a,B2: set_list_a] :
( ( ( member_list_a @ A2 @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B2 ) @ C3 )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ B2 @ C3 ) ) ) )
& ( ~ ( member_list_a @ A2 @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_627_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_628_Int__insert__right,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ( ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B2 ) )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) )
& ( ~ ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_629_insert__Diff__if,axiom,
! [X4: nat,B2: set_nat,A: set_nat] :
( ( ( member_nat @ X4 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) )
& ( ~ ( member_nat @ X4 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B2 )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_630_insert__Diff__if,axiom,
! [X4: list_a,B2: set_list_a,A: set_list_a] :
( ( ( member_list_a @ X4 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X4 @ A ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) )
& ( ~ ( member_list_a @ X4 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X4 @ A ) @ B2 )
= ( insert_list_a @ X4 @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_631_bot_Oextremum,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% bot.extremum
thf(fact_632_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_633_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_634_bot_Oextremum__unique,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_635_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_636_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_637_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
=> ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_638_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_639_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_640_subset__insert,axiom,
! [X4: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_641_subset__insert,axiom,
! [X4: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X4 @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X4 @ B2 ) )
= ( ord_le8861187494160871172list_a @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_642_pairwise__empty,axiom,
! [P: list_a > list_a > $o] : ( pairwise_list_a @ P @ bot_bot_set_list_a ) ).
% pairwise_empty
thf(fact_643_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_644_pairwise__insert,axiom,
! [R3: nat > nat > $o,X4: nat,S3: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat @ X4 @ S3 ) )
= ( ! [Y4: nat] :
( ( ( member_nat @ Y4 @ S3 )
& ( Y4 != X4 ) )
=> ( ( R3 @ X4 @ Y4 )
& ( R3 @ Y4 @ X4 ) ) )
& ( pairwise_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_645_pairwise__insert,axiom,
! [R3: list_a > list_a > $o,X4: list_a,S3: set_list_a] :
( ( pairwise_list_a @ R3 @ ( insert_list_a @ X4 @ S3 ) )
= ( ! [Y4: list_a] :
( ( ( member_list_a @ Y4 @ S3 )
& ( Y4 != X4 ) )
=> ( ( R3 @ X4 @ Y4 )
& ( R3 @ Y4 @ X4 ) ) )
& ( pairwise_list_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_646_totalp__on__empty,axiom,
! [R: list_a > list_a > $o] : ( totalp_on_list_a @ bot_bot_set_list_a @ R ) ).
% totalp_on_empty
thf(fact_647_totalp__on__empty,axiom,
! [R: nat > nat > $o] : ( totalp_on_nat @ bot_bot_set_nat @ R ) ).
% totalp_on_empty
thf(fact_648_Diff__insert__absorb,axiom,
! [X4: list_a,A: set_list_a] :
( ~ ( member_list_a @ X4 @ A )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X4 @ A ) @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_649_Diff__insert__absorb,axiom,
! [X4: nat,A: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_650_Diff__insert2,axiom,
! [A: set_list_a,A2: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_651_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_652_insert__Diff,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( insert_list_a @ A2 @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_653_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_654_Diff__insert,axiom,
! [A: set_list_a,A2: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_655_Diff__insert,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_656_subset__singleton__iff,axiom,
! [X6: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ X6 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( ( X6 = bot_bot_set_list_a )
| ( X6
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_657_subset__singleton__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_658_subset__singletonD,axiom,
! [A: set_list_a,X4: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
=> ( ( A = bot_bot_set_list_a )
| ( A
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_659_subset__singletonD,axiom,
! [A: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_660_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_661_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B ) )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_662_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_663_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_664_top__eq__inf__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X4 @ Y3 ) )
= ( ( X4 = top_top_set_nat )
& ( Y3 = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_665_inf__eq__top__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ X4 @ Y3 )
= top_top_set_nat )
= ( ( X4 = top_top_set_nat )
& ( Y3 = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_666_inf__top__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ top_top_set_nat )
= X4 ) ).
% inf_top_right
thf(fact_667_inf__top__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X4 )
= X4 ) ).
% inf_top_left
thf(fact_668_inf__bot__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X4 )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_669_inf__bot__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_670_inf__bot__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_671_inf__bot__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_672_Int__UNIV,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_673_le__inf__iff,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y3 @ Z ) )
= ( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_674_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_675_Diff__UNIV,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ top_top_set_list_a )
= bot_bot_set_list_a ) ).
% Diff_UNIV
thf(fact_676_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_677_empty__not__UNIV,axiom,
bot_bot_set_list_a != top_top_set_list_a ).
% empty_not_UNIV
thf(fact_678_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_679_insert__UNIV,axiom,
! [X4: nat] :
( ( insert_nat @ X4 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_680_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_681_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_682_Int__UNIV__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_683_totalpI,axiom,
! [R: nat > nat > $o] :
( ! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ( R @ X3 @ Y )
| ( R @ Y @ X3 ) ) )
=> ( totalp_on_nat @ top_top_set_nat @ R ) ) ).
% totalpI
thf(fact_684_totalpD,axiom,
! [R: nat > nat > $o,X4: nat,Y3: nat] :
( ( totalp_on_nat @ top_top_set_nat @ R )
=> ( ( X4 != Y3 )
=> ( ( R @ X4 @ Y3 )
| ( R @ Y3 @ X4 ) ) ) ) ).
% totalpD
thf(fact_685_top__empty__eq,axiom,
( top_top_list_a_o
= ( ^ [X: list_a] : ( member_list_a @ X @ top_top_set_list_a ) ) ) ).
% top_empty_eq
thf(fact_686_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_687_top__set__def,axiom,
( top_top_set_list_a
= ( collect_list_a @ top_top_list_a_o ) ) ).
% top_set_def
thf(fact_688_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_689_inf__sup__ord_I2_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_690_inf__sup__ord_I1_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_691_inf__le1,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_692_inf__le2,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_693_le__infE,axiom,
! [X4: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X4 @ A2 )
=> ~ ( ord_less_eq_nat @ X4 @ B ) ) ) ).
% le_infE
thf(fact_694_le__infI,axiom,
! [X4: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ B )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_695_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_696_le__infI1,axiom,
! [A2: nat,X4: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_697_le__infI2,axiom,
! [B: nat,X4: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_698_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_699_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_700_inf__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y3: nat] :
( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_701_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( inf_inf_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_702_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_703_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_704_inf__absorb1,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( inf_inf_nat @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_705_inf__absorb2,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( inf_inf_nat @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_706_inf_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_707_inf_OboundedI,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_708_inf__greatest,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_709_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_710_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_711_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_712_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_713_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_714_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_715_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_716_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_717_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_718_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X4 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_719_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_720_insert__remove__id,axiom,
! [X4: list_a,X6: set_list_a] :
( ( member_list_a @ X4 @ X6 )
=> ( X6
= ( insert_list_a @ X4 @ ( minus_646659088055828811list_a @ X6 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) ).
% insert_remove_id
thf(fact_721_insert__remove__id,axiom,
! [X4: nat,X6: set_nat] :
( ( member_nat @ X4 @ X6 )
=> ( X6
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ X6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_722_diff__shunt__var,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( ( minus_646659088055828811list_a @ X4 @ Y3 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_723_diff__shunt__var,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_724_Ball__Collect,axiom,
( ball_nat
= ( ^ [A3: set_nat,P4: nat > $o] : ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P4 ) ) ) ) ).
% Ball_Collect
thf(fact_725_Ball__Collect,axiom,
( ball_list_a
= ( ^ [A3: set_list_a,P4: list_a > $o] : ( ord_le8861187494160871172list_a @ A3 @ ( collect_list_a @ P4 ) ) ) ) ).
% Ball_Collect
thf(fact_726_insert__subsetI,axiom,
! [X4: nat,A: set_nat,X6: set_nat] :
( ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_727_insert__subsetI,axiom,
! [X4: list_a,A: set_list_a,X6: set_list_a] :
( ( member_list_a @ X4 @ A )
=> ( ( ord_le8861187494160871172list_a @ X6 @ A )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ X4 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_728_boolean__algebra_Oconj__one__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ top_top_set_nat )
= X4 ) ).
% boolean_algebra.conj_one_right
thf(fact_729_UNIV__I,axiom,
! [X4: list_a] : ( member_list_a @ X4 @ top_top_set_list_a ) ).
% UNIV_I
thf(fact_730_UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_731_UNIV__eq__I,axiom,
! [A: set_list_a] :
( ! [X3: list_a] : ( member_list_a @ X3 @ A )
=> ( top_top_set_list_a = A ) ) ).
% UNIV_eq_I
thf(fact_732_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_733_UNIV__witness,axiom,
? [X3: list_a] : ( member_list_a @ X3 @ top_top_set_list_a ) ).
% UNIV_witness
thf(fact_734_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_735_subset__emptyI,axiom,
! [A: set_list_a] :
( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A )
=> ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a ) ) ).
% subset_emptyI
thf(fact_736_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_737_iso__tuple__UNIV__I,axiom,
! [X4: list_a] : ( member_list_a @ X4 @ top_top_set_list_a ) ).
% iso_tuple_UNIV_I
thf(fact_738_iso__tuple__UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_739_remove__def,axiom,
( remove_list_a
= ( ^ [X: list_a,A3: set_list_a] : ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% remove_def
thf(fact_740_remove__def,axiom,
( remove_nat
= ( ^ [X: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_741_member__remove,axiom,
! [X4: nat,Y3: nat,A: set_nat] :
( ( member_nat @ X4 @ ( remove_nat @ Y3 @ A ) )
= ( ( member_nat @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_742_member__remove,axiom,
! [X4: list_a,Y3: list_a,A: set_list_a] :
( ( member_list_a @ X4 @ ( remove_list_a @ Y3 @ A ) )
= ( ( member_list_a @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_743_diff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_744_psubset__insert__iff,axiom,
! [A: set_list_a,X4: list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A @ ( insert_list_a @ X4 @ B2 ) )
= ( ( ( member_list_a @ X4 @ B2 )
=> ( ord_less_set_list_a @ A @ B2 ) )
& ( ~ ( member_list_a @ X4 @ B2 )
=> ( ( ( member_list_a @ X4 @ A )
=> ( ord_less_set_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) @ B2 ) )
& ( ~ ( member_list_a @ X4 @ A )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_745_psubset__insert__iff,axiom,
! [A: set_nat,X4: nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X4 @ B2 ) )
= ( ( ( member_nat @ X4 @ B2 )
=> ( ord_less_set_nat @ A @ B2 ) )
& ( ~ ( member_nat @ X4 @ B2 )
=> ( ( ( member_nat @ X4 @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_746_subset__Compl__singleton,axiom,
! [A: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( uminus7925729386456332763list_a @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( ~ ( member_list_a @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_747_subset__Compl__singleton,axiom,
! [A: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_748_Greatest__equality,axiom,
! [P: nat > $o,X4: nat] :
( ( P @ X4 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( order_Greatest_nat @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_749_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_750_ComplI,axiom,
! [C: list_a,A: set_list_a] :
( ~ ( member_list_a @ C @ A )
=> ( member_list_a @ C @ ( uminus7925729386456332763list_a @ A ) ) ) ).
% ComplI
thf(fact_751_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_752_Compl__iff,axiom,
! [C: list_a,A: set_list_a] :
( ( member_list_a @ C @ ( uminus7925729386456332763list_a @ A ) )
= ( ~ ( member_list_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_753_boolean__algebra_Ocompl__one,axiom,
( ( uminus7925729386456332763list_a @ top_top_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.compl_one
thf(fact_754_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_755_boolean__algebra_Ocompl__zero,axiom,
( ( uminus7925729386456332763list_a @ bot_bot_set_list_a )
= top_top_set_list_a ) ).
% boolean_algebra.compl_zero
thf(fact_756_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_757_inf__compl__bot__left1,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X4 ) @ ( inf_inf_set_list_a @ X4 @ Y3 ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_left1
thf(fact_758_inf__compl__bot__left1,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( inf_inf_set_nat @ X4 @ Y3 ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_759_inf__compl__bot__left2,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X4 ) @ Y3 ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_left2
thf(fact_760_inf__compl__bot__left2,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_761_inf__compl__bot__right,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y3 @ ( uminus7925729386456332763list_a @ X4 ) ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_right
thf(fact_762_inf__compl__bot__right,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X4 ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_763_boolean__algebra_Oconj__cancel__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X4 ) @ X4 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_764_boolean__algebra_Oconj__cancel__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ X4 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_765_boolean__algebra_Oconj__cancel__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( uminus7925729386456332763list_a @ X4 ) )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_766_boolean__algebra_Oconj__cancel__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ X4 ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_767_Compl__disjoint2,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ A ) @ A )
= bot_bot_set_list_a ) ).
% Compl_disjoint2
thf(fact_768_Compl__disjoint2,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_769_Compl__disjoint,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( uminus7925729386456332763list_a @ A ) )
= bot_bot_set_list_a ) ).
% Compl_disjoint
thf(fact_770_Compl__disjoint,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_771_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_772_less__imp__neq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_773_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_774_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_775_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_776_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_777_antisym__conv3,axiom,
! [Y3: nat,X4: nat] :
( ~ ( ord_less_nat @ Y3 @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_778_linorder__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ( X4 != Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_cases
thf(fact_779_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_780_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_781_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_782_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_783_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_784_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X4 )
| ( X4 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_785_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_786_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_787_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_788_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_789_ComplD,axiom,
! [C: list_a,A: set_list_a] :
( ( member_list_a @ C @ ( uminus7925729386456332763list_a @ A ) )
=> ~ ( member_list_a @ C @ A ) ) ).
% ComplD
thf(fact_790_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_791_psubsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_792_linorder__neqE,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_793_order__less__asym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_794_linorder__neq__iff,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
= ( ( ord_less_nat @ X4 @ Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_795_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_796_order__less__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_797_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_798_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_799_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_800_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_801_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_802_order__less__not__sym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_803_order__less__imp__triv,axiom,
! [X4: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_804_linorder__less__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_less_linear
thf(fact_805_order__less__imp__not__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_806_order__less__imp__not__eq2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_807_order__less__imp__not__less,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_808_leD,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y3 ) ) ).
% leD
thf(fact_809_leI,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% leI
thf(fact_810_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_811_antisym__conv1,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_812_antisym__conv2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_813_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_814_not__le__imp__less,axiom,
! [Y3: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ord_less_nat @ X4 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_815_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_816_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_817_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_818_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_819_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_820_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_821_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_822_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_823_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_824_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_825_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_826_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_827_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_828_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_829_linorder__not__le,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_not_le
thf(fact_830_linorder__not__less,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_not_less
thf(fact_831_order__less__imp__le,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_832_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_833_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_834_order__le__less__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_835_order__less__le__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_836_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_837_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_838_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_839_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_840_linorder__le__less__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_841_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_842_bot_Oextremum__strict,axiom,
! [A2: set_list_a] :
~ ( ord_less_set_list_a @ A2 @ bot_bot_set_list_a ) ).
% bot.extremum_strict
thf(fact_843_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_844_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_845_bot_Onot__eq__extremum,axiom,
! [A2: set_list_a] :
( ( A2 != bot_bot_set_list_a )
= ( ord_less_set_list_a @ bot_bot_set_list_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_846_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_847_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_848_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_849_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_850_not__psubset__empty,axiom,
! [A: set_list_a] :
~ ( ord_less_set_list_a @ A @ bot_bot_set_list_a ) ).
% not_psubset_empty
thf(fact_851_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_852_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_853_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_854_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( A5
= ( inf_inf_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_855_inf_Ostrict__boundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_856_inf_Oabsorb4,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_857_inf_Oabsorb3,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_858_less__infI2,axiom,
! [B: nat,X4: nat,A2: nat] :
( ( ord_less_nat @ B @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X4 ) ) ).
% less_infI2
thf(fact_859_less__infI1,axiom,
! [A2: nat,X4: nat,B: nat] :
( ( ord_less_nat @ A2 @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X4 ) ) ).
% less_infI1
thf(fact_860_psubset__imp__ex__mem,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_861_psubset__imp__ex__mem,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A @ B2 )
=> ? [B4: list_a] : ( member_list_a @ B4 @ ( minus_646659088055828811list_a @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_862_totalp__on__less,axiom,
! [A: set_nat] : ( totalp_on_nat @ A @ ord_less_nat ) ).
% totalp_on_less
thf(fact_863_Compl__empty__eq,axiom,
( ( uminus7925729386456332763list_a @ bot_bot_set_list_a )
= top_top_set_list_a ) ).
% Compl_empty_eq
thf(fact_864_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_865_Compl__UNIV__eq,axiom,
( ( uminus7925729386456332763list_a @ top_top_set_list_a )
= bot_bot_set_list_a ) ).
% Compl_UNIV_eq
thf(fact_866_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_867_inf__cancel__left1,axiom,
! [X4: set_list_a,A2: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ A2 ) @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X4 ) @ B ) )
= bot_bot_set_list_a ) ).
% inf_cancel_left1
thf(fact_868_inf__cancel__left1,axiom,
! [X4: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X4 @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_869_inf__cancel__left2,axiom,
! [X4: set_list_a,A2: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X4 ) @ A2 ) @ ( inf_inf_set_list_a @ X4 @ B ) )
= bot_bot_set_list_a ) ).
% inf_cancel_left2
thf(fact_870_inf__cancel__left2,axiom,
! [X4: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ A2 ) @ ( inf_inf_set_nat @ X4 @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_871_subset__Compl__self__eq,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( uminus7925729386456332763list_a @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% subset_Compl_self_eq
thf(fact_872_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_873_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_874_inf__shunt,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( ( inf_inf_set_list_a @ X4 @ Y3 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X4 @ ( uminus7925729386456332763list_a @ Y3 ) ) ) ).
% inf_shunt
thf(fact_875_inf__shunt,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% inf_shunt
thf(fact_876_disjoint__eq__subset__Compl,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ ( uminus7925729386456332763list_a @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_877_disjoint__eq__subset__Compl,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_878_Compl__insert,axiom,
! [X4: list_a,A: set_list_a] :
( ( uminus7925729386456332763list_a @ ( insert_list_a @ X4 @ A ) )
= ( minus_646659088055828811list_a @ ( uminus7925729386456332763list_a @ A ) @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ).
% Compl_insert
thf(fact_879_Compl__insert,axiom,
! [X4: nat,A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ A ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_880_GreatestI2__order,axiom,
! [P: nat > $o,X4: nat,Q2: nat > $o] :
( ( P @ X4 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_881_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_882_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_883_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_884_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_885_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_886_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_887_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_888_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_889_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_890_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_891_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( ( inf_inf_set_list_a @ X4 @ Y3 )
= bot_bot_set_list_a )
=> ( ( ( sup_sup_set_list_a @ X4 @ Y3 )
= top_top_set_list_a )
=> ( ( uminus7925729386456332763list_a @ X4 )
= Y3 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_892_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X4 @ Y3 )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X4 )
= Y3 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_893_inf_Osemilattice__order__axioms,axiom,
semila1248733672344298208er_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% inf.semilattice_order_axioms
thf(fact_894_Un__empty,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_895_Un__empty,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_896_le__sup__iff,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X4 @ Z )
& ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_897_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_898_sup__bot_Oright__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_899_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_900_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ A2 @ B ) )
= ( ( A2 = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_901_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_902_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_903_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_904_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_905_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_906_sup__eq__bot__iff,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( ( sup_sup_set_list_a @ X4 @ Y3 )
= bot_bot_set_list_a )
= ( ( X4 = bot_bot_set_list_a )
& ( Y3 = bot_bot_set_list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_907_sup__eq__bot__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ( X4 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_908_bot__eq__sup__iff,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ X4 @ Y3 ) )
= ( ( X4 = bot_bot_set_list_a )
& ( Y3 = bot_bot_set_list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_909_bot__eq__sup__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X4 @ Y3 ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_910_sup__bot__right,axiom,
! [X4: set_list_a] :
( ( sup_sup_set_list_a @ X4 @ bot_bot_set_list_a )
= X4 ) ).
% sup_bot_right
thf(fact_911_sup__bot__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_912_sup__bot__left,axiom,
! [X4: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_913_sup__bot__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_914_sup__top__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X4 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_915_sup__top__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_916_boolean__algebra_Odisj__one__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_917_boolean__algebra_Odisj__one__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X4 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_918_sup__compl__top__left1,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( sup_sup_set_nat @ X4 @ Y3 ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_919_sup__compl__top__left2,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_920_boolean__algebra_Odisj__cancel__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ X4 )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_921_boolean__algebra_Odisj__cancel__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ X4 ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_922_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X4: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_923_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_924_le__supE,axiom,
! [A2: nat,B: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X4 )
=> ~ ( ord_less_eq_nat @ B @ X4 ) ) ) ).
% le_supE
thf(fact_925_le__supI,axiom,
! [A2: nat,X4: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ( ord_less_eq_nat @ B @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X4 ) ) ) ).
% le_supI
thf(fact_926_sup__ge1,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% sup_ge1
thf(fact_927_sup__ge2,axiom,
! [Y3: nat,X4: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% sup_ge2
thf(fact_928_le__supI1,axiom,
! [X4: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_929_le__supI2,axiom,
! [X4: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X4 @ B )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_930_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_931_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_932_sup__least,axiom,
! [Y3: nat,X4: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_933_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_934_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_935_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_936_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y3: nat] :
( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_937_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_938_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_939_sup__absorb1,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= X4 ) ) ).
% sup_absorb1
thf(fact_940_sup__absorb2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_941_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_942_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_943_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_944_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_945_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_946_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_947_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_948_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_949_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_950_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_951_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_952_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_953_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_954_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_955_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_956_less__supI2,axiom,
! [X4: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X4 @ B )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_957_less__supI1,axiom,
! [X4: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X4 @ A2 )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_958_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_list_a] :
( ( sup_sup_set_list_a @ X4 @ bot_bot_set_list_a )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_959_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_960_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_961_Un__UNIV__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B2 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_962_Un__empty__right,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Un_empty_right
thf(fact_963_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_964_Un__empty__left,axiom,
! [B2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_965_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_966_distrib__sup__le,axiom,
! [X4: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ ( sup_sup_nat @ X4 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_967_distrib__inf__le,axiom,
! [X4: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ ( inf_inf_nat @ X4 @ Z ) ) @ ( inf_inf_nat @ X4 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_968_sup__cancel__left1,axiom,
! [X4: set_nat,A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ A2 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_969_sup__cancel__left2,axiom,
! [X4: set_nat,A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ A2 ) @ ( sup_sup_set_nat @ X4 @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_970_singleton__Un__iff,axiom,
! [X4: list_a,A: set_list_a,B2: set_list_a] :
( ( ( insert_list_a @ X4 @ bot_bot_set_list_a )
= ( sup_sup_set_list_a @ A @ B2 ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B2
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
& ( B2 = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
& ( B2
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_971_singleton__Un__iff,axiom,
! [X4: nat,A: set_nat,B2: set_nat] :
( ( ( insert_nat @ X4 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( ( A = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_972_Un__singleton__iff,axiom,
! [A: set_list_a,B2: set_list_a,X4: list_a] :
( ( ( sup_sup_set_list_a @ A @ B2 )
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B2
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
& ( B2 = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
& ( B2
= ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_973_Un__singleton__iff,axiom,
! [A: set_nat,B2: set_nat,X4: nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_974_insert__is__Un,axiom,
( insert_list_a
= ( ^ [A5: list_a] : ( sup_sup_set_list_a @ ( insert_list_a @ A5 @ bot_bot_set_list_a ) ) ) ) ).
% insert_is_Un
thf(fact_975_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_976_Compl__partition2,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_977_Compl__partition,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_978_sup__shunt,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y3 )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) ) ).
% sup_shunt
thf(fact_979_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_list_a,X4: set_list_a,Y3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ X4 )
= bot_bot_set_list_a )
=> ( ( ( sup_sup_set_list_a @ A2 @ X4 )
= top_top_set_list_a )
=> ( ( ( inf_inf_set_list_a @ A2 @ Y3 )
= bot_bot_set_list_a )
=> ( ( ( sup_sup_set_list_a @ A2 @ Y3 )
= top_top_set_list_a )
=> ( X4 = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_980_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X4: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X4 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X4 )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y3 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y3 )
= top_top_set_nat )
=> ( X4 = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_981_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea4281390645585673690list_a @ inf_inf_set_list_a @ sup_sup_set_list_a @ uminus7925729386456332763list_a @ bot_bot_set_list_a @ top_top_set_list_a ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_982_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_983_UnCI,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_984_UnCI,axiom,
! [C: list_a,B2: set_list_a,A: set_list_a] :
( ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ A ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_985_Un__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_986_Un__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
| ( member_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_987_UnE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_988_UnE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) )
=> ( ~ ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_989_UnI1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_990_UnI1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_991_UnI2,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_992_UnI2,axiom,
! [C: list_a,B2: set_list_a,A: set_list_a] :
( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_993_sup__bot_Osemilattice__neutr__axioms,axiom,
semila4471139486997153862list_a @ sup_sup_set_list_a @ bot_bot_set_list_a ).
% sup_bot.semilattice_neutr_axioms
thf(fact_994_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1241773964035338532et_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_995_inf__top_Osemilattice__neutr__axioms,axiom,
semila1241773964035338532et_nat @ inf_inf_set_nat @ top_top_set_nat ).
% inf_top.semilattice_neutr_axioms
thf(fact_996_ball__reg,axiom,
! [R: set_nat,P: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ R )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ R )
=> ( Q2 @ X5 ) ) ) ) ).
% ball_reg
thf(fact_997_ball__reg,axiom,
! [R: set_list_a,P: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ R )
=> ( P @ X3 ) )
=> ! [X5: list_a] :
( ( member_list_a @ X5 @ R )
=> ( Q2 @ X5 ) ) ) ) ).
% ball_reg
thf(fact_998_order_Oordering__axioms,axiom,
ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.ordering_axioms
thf(fact_999_order_Opartial__preordering__axioms,axiom,
partia6822818058636336922ng_nat @ ord_less_eq_nat ).
% order.partial_preordering_axioms
thf(fact_1000_sup__bot_Ocomm__monoid__axioms,axiom,
comm_m7296721492685993625list_a @ sup_sup_set_list_a @ bot_bot_set_list_a ).
% sup_bot.comm_monoid_axioms
thf(fact_1001_sup__bot_Ocomm__monoid__axioms,axiom,
comm_monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.comm_monoid_axioms
thf(fact_1002_inf__top_Ocomm__monoid__axioms,axiom,
comm_monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).
% inf_top.comm_monoid_axioms
thf(fact_1003_bdd__above_Opreordering__bdd__axioms,axiom,
condit7935552474144124665dd_nat @ ord_less_eq_nat @ ord_less_nat ).
% bdd_above.preordering_bdd_axioms
thf(fact_1004_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X8: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_1005_classes__disjoint,axiom,
! [X6: set_list_a,Qeq: set_Pr4048851178543822343list_a,Y8: set_list_a] :
( ( member_set_list_a @ X6 @ ( classes_list_a @ Qeq ) )
=> ( ( member_set_list_a @ Y8 @ ( classes_list_a @ Qeq ) )
=> ( ( X6 = Y8 )
| ( ( inf_inf_set_list_a @ X6 @ Y8 )
= bot_bot_set_list_a ) ) ) ) ).
% classes_disjoint
thf(fact_1006_classes__disjoint,axiom,
! [X6: set_nat,Qeq: set_Pr1261947904930325089at_nat,Y8: set_nat] :
( ( member_set_nat @ X6 @ ( classes_nat @ Qeq ) )
=> ( ( member_set_nat @ Y8 @ ( classes_nat @ Qeq ) )
=> ( ( X6 = Y8 )
| ( ( inf_inf_set_nat @ X6 @ Y8 )
= bot_bot_set_nat ) ) ) ) ).
% classes_disjoint
thf(fact_1007_classes__nonempty,axiom,
! [Qeq: set_Pr4048851178543822343list_a] :
~ ( member_set_list_a @ bot_bot_set_list_a @ ( classes_list_a @ Qeq ) ) ).
% classes_nonempty
thf(fact_1008_classes__nonempty,axiom,
! [Qeq: set_Pr1261947904930325089at_nat] :
~ ( member_set_nat @ bot_bot_set_nat @ ( classes_nat @ Qeq ) ) ).
% classes_nonempty
thf(fact_1009_preordering__bdd_Oempty,axiom,
! [Less_eq: list_a > list_a > $o,Less: list_a > list_a > $o] :
( ( condit7729410879213921563list_a @ Less_eq @ Less )
=> ( condit5051389248180226297list_a @ Less_eq @ bot_bot_set_list_a ) ) ).
% preordering_bdd.empty
thf(fact_1010_preordering__bdd_Oempty,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o] :
( ( condit7935552474144124665dd_nat @ Less_eq @ Less )
=> ( condit4013746787832047771dd_nat @ Less_eq @ bot_bot_set_nat ) ) ).
% preordering_bdd.empty
thf(fact_1011_Iio__Int__singleton,axiom,
! [X4: nat,K: nat] :
( ( ( ord_less_nat @ X4 @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X4 @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ) ).
% Iio_Int_singleton
thf(fact_1012_inf__top_Omonoid__axioms,axiom,
monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).
% inf_top.monoid_axioms
thf(fact_1013_sup__bot_Omonoid__axioms,axiom,
monoid_set_list_a @ sup_sup_set_list_a @ bot_bot_set_list_a ).
% sup_bot.monoid_axioms
thf(fact_1014_sup__bot_Omonoid__axioms,axiom,
monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.monoid_axioms
thf(fact_1015_lessThan__subset__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_1016_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1017_Iio__eq__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_1018_Inf__atMostLessThan,axiom,
! [X4: set_list_a] :
( ( ord_less_set_list_a @ top_top_set_list_a @ X4 )
=> ( ( comple903356909981783403list_a @ ( set_or4827425070552584583list_a @ X4 ) )
= bot_bot_set_list_a ) ) ).
% Inf_atMostLessThan
thf(fact_1019_Inf__atMostLessThan,axiom,
! [X4: set_nat] :
( ( ord_less_set_nat @ top_top_set_nat @ X4 )
=> ( ( comple7806235888213564991et_nat @ ( set_or890127255671739683et_nat @ X4 ) )
= bot_bot_set_nat ) ) ).
% Inf_atMostLessThan
thf(fact_1020_ivl__disj__un__singleton_I2_J,axiom,
! [U: nat] :
( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_ord_atMost_nat @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_1021_Iic__subset__Iio__iff,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A2 ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_1022_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_1023_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1024_cInf__singleton,axiom,
! [X4: nat] :
( ( complete_Inf_Inf_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% cInf_singleton
thf(fact_1025_atMost__subset__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X4 ) @ ( set_ord_atMost_nat @ Y3 ) )
= ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_1026_Inf__atMost,axiom,
! [X4: set_list_a] :
( ( comple903356909981783403list_a @ ( set_or6279072120763780779list_a @ X4 ) )
= bot_bot_set_list_a ) ).
% Inf_atMost
thf(fact_1027_Inf__atMost,axiom,
! [X4: set_nat] :
( ( comple7806235888213564991et_nat @ ( set_or4236626031148496127et_nat @ X4 ) )
= bot_bot_set_nat ) ).
% Inf_atMost
thf(fact_1028_not__UNIV__eq__Iic,axiom,
! [H: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_eq_Iic
thf(fact_1029_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_1030_cInf__greatest,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ Z @ X3 ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1031_cInf__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ A2 @ X3 ) )
=> ( ! [Y: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ Y @ X5 ) )
=> ( ord_less_eq_nat @ Y @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1032_cInf__eq__minimum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ Z @ X3 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1033_cInf__eq,axiom,
! [X6: set_nat,A2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ A2 @ X3 ) )
=> ( ! [Y: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ Y @ X5 ) )
=> ( ord_less_eq_nat @ Y @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_1034_wellorder__Inf__le1,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_1035_Inf__nat__def1,axiom,
! [K2: set_nat] :
( ( K2 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K2 ) @ K2 ) ) ).
% Inf_nat_def1
thf(fact_1036_cInf__lessD,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Z )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_1037_atMost__eq__UNIV__iff,axiom,
! [X4: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X4 )
= top_top_set_set_nat )
= ( X4 = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_1038_not__UNIV__le__Iic,axiom,
! [H2: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_le_Iic
thf(fact_1039_Inf__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inf_empty
thf(fact_1040_Inf__UNIV,axiom,
( ( comple903356909981783403list_a @ top_to7106483174946246804list_a )
= bot_bot_set_list_a ) ).
% Inf_UNIV
thf(fact_1041_Inf__UNIV,axiom,
( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
= bot_bot_set_nat ) ).
% Inf_UNIV
thf(fact_1042_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1043_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1044_Inf__top__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1045_Inf__top__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1046_Inter__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inter_empty
thf(fact_1047_Inter__UNIV,axiom,
( ( comple903356909981783403list_a @ top_to7106483174946246804list_a )
= bot_bot_set_list_a ) ).
% Inter_UNIV
thf(fact_1048_Inter__UNIV,axiom,
( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
= bot_bot_set_nat ) ).
% Inter_UNIV
thf(fact_1049_Inf__sup__eq__top__iff,axiom,
! [B2: set_set_nat,A2: set_nat] :
( ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ B2 ) @ A2 )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( sup_sup_set_nat @ X @ A2 )
= top_top_set_nat ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1050_less__eq__cInf__inter,axiom,
! [A: set_nat,B2: set_nat] :
( ( condit1738341127787009408ow_nat @ A )
=> ( ( condit1738341127787009408ow_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A @ B2 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B2 ) ) @ ( complete_Inf_Inf_nat @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1051_bdd__belowI,axiom,
! [A: set_nat,M2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ X3 ) )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_belowI
thf(fact_1052_bdd__below_OI,axiom,
! [A: set_nat,M3: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M3 @ X3 ) )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_below.I
thf(fact_1053_bdd__below__empty,axiom,
condit1738341127787009408ow_nat @ bot_bot_set_nat ).
% bdd_below_empty
thf(fact_1054_cInf__lower2,axiom,
! [X4: nat,X6: set_nat,Y3: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Y3 ) ) ) ) ).
% cInf_lower2
thf(fact_1055_cInf__lower,axiom,
! [X4: nat,X6: set_nat] :
( ( member_nat @ X4 @ X6 )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ X4 ) ) ) ).
% cInf_lower
thf(fact_1056_bdd__below_OE,axiom,
! [A: set_nat] :
( ( condit1738341127787009408ow_nat @ A )
=> ~ ! [M4: nat] :
~ ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( ord_less_eq_nat @ M4 @ X5 ) ) ) ).
% bdd_below.E
thf(fact_1057_bdd__below_Ounfold,axiom,
( condit1738341127787009408ow_nat
= ( ^ [A3: set_nat] :
? [M5: nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ord_less_eq_nat @ M5 @ X ) ) ) ) ).
% bdd_below.unfold
thf(fact_1058_cInf__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( B2 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_nat @ X5 @ B4 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B2 ) ) ) ) ) ).
% cInf_mono
thf(fact_1059_le__cInf__iff,axiom,
! [S: set_nat,A2: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ S )
=> ( ( ord_less_eq_nat @ A2 @ ( complete_Inf_Inf_nat @ S ) )
= ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ord_less_eq_nat @ A2 @ X ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1060_cInf__less__iff,axiom,
! [X6: set_nat,Y3: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Y3 )
= ( ? [X: nat] :
( ( member_nat @ X @ X6 )
& ( ord_less_nat @ X @ Y3 ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_1061_cInf__superset__mono,axiom,
! [A: set_nat,B2: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ B2 ) @ ( complete_Inf_Inf_nat @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1062_cInf__insert,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X6 ) )
= ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ) ).
% cInf_insert
thf(fact_1063_cInf__insert__If,axiom,
! [X6: set_nat,A2: nat] :
( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( ( X6 = bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X6 ) )
= A2 ) )
& ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X6 ) )
= ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_1064_cInf__union__distrib,axiom,
! [A: set_nat,B2: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B2 )
=> ( ( complete_Inf_Inf_nat @ ( sup_sup_set_nat @ A @ B2 ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B2 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_1065_cINF__union,axiom,
! [A: set_list_a,F: list_a > nat,B2: set_list_a] :
( ( A != bot_bot_set_list_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) )
=> ( ( B2 != bot_bot_set_list_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ B2 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ ( sup_sup_set_list_a @ A @ B2 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_1066_cINF__union,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ B2 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_1067_cINF__insert,axiom,
! [A: set_list_a,F: list_a > nat,A2: list_a] :
( ( A != bot_bot_set_list_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ ( insert_list_a @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_1068_cINF__insert,axiom,
! [A: set_nat,F: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( insert_nat @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_1069_image__eqI,axiom,
! [B: nat,F: nat > nat,X4: nat,A: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1070_image__eqI,axiom,
! [B: list_a,F: nat > list_a,X4: nat,A: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A )
=> ( member_list_a @ B @ ( image_nat_list_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1071_image__eqI,axiom,
! [B: nat,F: list_a > nat,X4: list_a,A: set_list_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_list_a @ X4 @ A )
=> ( member_nat @ B @ ( image_list_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1072_image__eqI,axiom,
! [B: list_a,F: list_a > list_a,X4: list_a,A: set_list_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_list_a @ X4 @ A )
=> ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1073_image__empty,axiom,
! [F: list_a > list_a] :
( ( image_list_a_list_a @ F @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_1074_image__empty,axiom,
! [F: list_a > nat] :
( ( image_list_a_nat @ F @ bot_bot_set_list_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1075_image__empty,axiom,
! [F: nat > list_a] :
( ( image_nat_list_a @ F @ bot_bot_set_nat )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_1076_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1077_empty__is__image,axiom,
! [F: list_a > list_a,A: set_list_a] :
( ( bot_bot_set_list_a
= ( image_list_a_list_a @ F @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_1078_empty__is__image,axiom,
! [F: nat > list_a,A: set_nat] :
( ( bot_bot_set_list_a
= ( image_nat_list_a @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1079_empty__is__image,axiom,
! [F: list_a > nat,A: set_list_a] :
( ( bot_bot_set_nat
= ( image_list_a_nat @ F @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_1080_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1081_image__is__empty,axiom,
! [F: list_a > list_a,A: set_list_a] :
( ( ( image_list_a_list_a @ F @ A )
= bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_1082_image__is__empty,axiom,
! [F: nat > list_a,A: set_nat] :
( ( ( image_nat_list_a @ F @ A )
= bot_bot_set_list_a )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1083_image__is__empty,axiom,
! [F: list_a > nat,A: set_list_a] :
( ( ( image_list_a_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_1084_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1085_bdd__below_OI2,axiom,
! [A: set_nat,M3: nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M3 @ ( F @ X3 ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1086_bdd__below_OI2,axiom,
! [A: set_list_a,M3: nat,F: list_a > nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ord_less_eq_nat @ M3 @ ( F @ X3 ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1087_bdd__belowI2,axiom,
! [A: set_nat,M2: nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1088_bdd__belowI2,axiom,
! [A: set_list_a,M2: nat,F: list_a > nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1089_rangeI,axiom,
! [F: nat > nat,X4: nat] : ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1090_rangeI,axiom,
! [F: nat > list_a,X4: nat] : ( member_list_a @ ( F @ X4 ) @ ( image_nat_list_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1091_range__eqI,axiom,
! [B: nat,F: nat > nat,X4: nat] :
( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1092_range__eqI,axiom,
! [B: list_a,F: nat > list_a,X4: nat] :
( ( B
= ( F @ X4 ) )
=> ( member_list_a @ B @ ( image_nat_list_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1093_surjD,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1094_surjE,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1095_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1096_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X: nat] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_1097_imageI,axiom,
! [X4: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_1098_imageI,axiom,
! [X4: nat,A: set_nat,F: nat > list_a] :
( ( member_nat @ X4 @ A )
=> ( member_list_a @ ( F @ X4 ) @ ( image_nat_list_a @ F @ A ) ) ) ).
% imageI
thf(fact_1099_imageI,axiom,
! [X4: list_a,A: set_list_a,F: list_a > nat] :
( ( member_list_a @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( image_list_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_1100_imageI,axiom,
! [X4: list_a,A: set_list_a,F: list_a > list_a] :
( ( member_list_a @ X4 @ A )
=> ( member_list_a @ ( F @ X4 ) @ ( image_list_a_list_a @ F @ A ) ) ) ).
% imageI
thf(fact_1101_rev__image__eqI,axiom,
! [X4: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1102_rev__image__eqI,axiom,
! [X4: nat,A: set_nat,B: list_a,F: nat > list_a] :
( ( member_nat @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_list_a @ B @ ( image_nat_list_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1103_rev__image__eqI,axiom,
! [X4: list_a,A: set_list_a,B: nat,F: list_a > nat] :
( ( member_list_a @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_list_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1104_rev__image__eqI,axiom,
! [X4: list_a,A: set_list_a,B: list_a,F: list_a > list_a] :
( ( member_list_a @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1105_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_1106_image__subsetI,axiom,
! [A: set_nat,F: nat > list_a,B2: set_list_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_1107_image__subsetI,axiom,
! [A: set_list_a,F: list_a > nat,B2: set_nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_list_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_1108_image__subsetI,axiom,
! [A: set_list_a,F: list_a > list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_1109_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_1110_range__subsetD,axiom,
! [F: nat > list_a,B2: set_list_a,I: nat] :
( ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F @ top_top_set_nat ) @ B2 )
=> ( member_list_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_1111_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S2: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1112_Inf__INT__eq,axiom,
( comple1750779994657059634st_a_o
= ( ^ [S2: set_list_a_o,X: list_a] : ( member_list_a @ X @ ( comple903356909981783403list_a @ ( image_5820879363088598756list_a @ collect_list_a @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1113_cINF__greatest,axiom,
! [A: set_list_a,M2: nat,F: list_a > nat] :
( ( A != bot_bot_set_list_a )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1114_cINF__greatest,axiom,
! [A: set_nat,M2: nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1115_range__eq__singletonD,axiom,
! [F: nat > list_a,A2: list_a,X4: nat] :
( ( ( image_nat_list_a @ F @ top_top_set_nat )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
=> ( ( F @ X4 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1116_range__eq__singletonD,axiom,
! [F: nat > nat,A2: nat,X4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X4 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1117_in__image__insert__iff,axiom,
! [B2: set_set_list_a,X4: list_a,A: set_list_a] :
( ! [C5: set_list_a] :
( ( member_set_list_a @ C5 @ B2 )
=> ~ ( member_list_a @ X4 @ C5 ) )
=> ( ( member_set_list_a @ A @ ( image_5749939591322298757list_a @ ( insert_list_a @ X4 ) @ B2 ) )
= ( ( member_list_a @ X4 @ A )
& ( member_set_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1118_in__image__insert__iff,axiom,
! [B2: set_set_nat,X4: nat,A: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ B2 )
=> ~ ( member_nat @ X4 @ C5 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X4 ) @ B2 ) )
= ( ( member_nat @ X4 @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1119_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1120_cINF__lower,axiom,
! [F: nat > nat,A: set_nat,X4: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ ( F @ X4 ) ) ) ) ).
% cINF_lower
thf(fact_1121_cINF__lower,axiom,
! [F: list_a > nat,A: set_list_a,X4: list_a] :
( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) )
=> ( ( member_list_a @ X4 @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) @ ( F @ X4 ) ) ) ) ).
% cINF_lower
thf(fact_1122_cINF__lower2,axiom,
! [F: nat > nat,A: set_nat,X4: nat,U: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_nat @ ( F @ X4 ) @ U )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1123_cINF__lower2,axiom,
! [F: list_a > nat,A: set_list_a,X4: list_a,U: nat] :
( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) )
=> ( ( member_list_a @ X4 @ A )
=> ( ( ord_less_eq_nat @ ( F @ X4 ) @ U )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1124_le__cINF__iff,axiom,
! [A: set_list_a,F: list_a > nat,U: nat] :
( ( A != bot_bot_set_list_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1125_le__cINF__iff,axiom,
! [A: set_nat,F: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1126_cINF__superset__mono,axiom,
! [A: set_list_a,G: list_a > nat,B2: set_list_a,F: list_a > nat] :
( ( A != bot_bot_set_list_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_list_a_nat @ G @ B2 ) )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ G @ B2 ) ) @ ( complete_Inf_Inf_nat @ ( image_list_a_nat @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1127_cINF__superset__mono,axiom,
! [A: set_nat,G: nat > nat,B2: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ B2 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1128_surj__to__nat,axiom,
( ( image_nat_nat @ infinite_to_nat_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_to_nat
thf(fact_1129_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1130_Pow__empty,axiom,
( ( pow_list_a @ bot_bot_set_list_a )
= ( insert_set_list_a @ bot_bot_set_list_a @ bot_bo3186585308812441520list_a ) ) ).
% Pow_empty
thf(fact_1131_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_1132_Pow__singleton__iff,axiom,
! [X6: set_list_a,Y8: set_list_a] :
( ( ( pow_list_a @ X6 )
= ( insert_set_list_a @ Y8 @ bot_bo3186585308812441520list_a ) )
= ( ( X6 = bot_bot_set_list_a )
& ( Y8 = bot_bot_set_list_a ) ) ) ).
% Pow_singleton_iff
thf(fact_1133_Pow__singleton__iff,axiom,
! [X6: set_nat,Y8: set_nat] :
( ( ( pow_nat @ X6 )
= ( insert_set_nat @ Y8 @ bot_bot_set_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
& ( Y8 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_1134_empty__in__Fpow,axiom,
! [A: set_list_a] : ( member_set_list_a @ bot_bot_set_list_a @ ( finite_Fpow_list_a @ A ) ) ).
% empty_in_Fpow
thf(fact_1135_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_1136_Pow__bottom,axiom,
! [B2: set_list_a] : ( member_set_list_a @ bot_bot_set_list_a @ ( pow_list_a @ B2 ) ) ).
% Pow_bottom
thf(fact_1137_Pow__bottom,axiom,
! [B2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_1138_atMost__Int__atLeast,axiom,
! [N2: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( set_ord_atLeast_nat @ N2 ) )
= ( insert_nat @ N2 @ bot_bot_set_nat ) ) ).
% atMost_Int_atLeast
thf(fact_1139_atLeast__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_1140_atLeast__empty__triv,axiom,
( ( set_or7033417953538090159list_a @ bot_bot_set_list_a )
= top_to7106483174946246804list_a ) ).
% atLeast_empty_triv
thf(fact_1141_atLeast__empty__triv,axiom,
( ( set_or1731685050470061051et_nat @ bot_bot_set_nat )
= top_top_set_set_nat ) ).
% atLeast_empty_triv
thf(fact_1142_atLeast__subset__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X4 ) @ ( set_ord_atLeast_nat @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_1143_inj__on__insert,axiom,
! [F: list_a > nat,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_nat @ F @ ( insert_list_a @ A2 @ A ) )
= ( ( inj_on_list_a_nat @ F @ A )
& ~ ( member_nat @ ( F @ A2 ) @ ( image_list_a_nat @ F @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1144_inj__on__insert,axiom,
! [F: list_a > list_a,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_list_a @ F @ ( insert_list_a @ A2 @ A ) )
= ( ( inj_on_list_a_list_a @ F @ A )
& ~ ( member_list_a @ ( F @ A2 ) @ ( image_list_a_list_a @ F @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1145_inj__on__insert,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_nat @ F @ A )
& ~ ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1146_inj__on__insert,axiom,
! [F: nat > list_a,A2: nat,A: set_nat] :
( ( inj_on_nat_list_a @ F @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_list_a @ F @ A )
& ~ ( member_list_a @ ( F @ A2 ) @ ( image_nat_list_a @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1147_not__empty__eq__Ici__eq__empty,axiom,
! [L: nat] :
( bot_bot_set_nat
!= ( set_ord_atLeast_nat @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_1148_atLeast__eq__UNIV__iff,axiom,
! [X4: set_list_a] :
( ( ( set_or7033417953538090159list_a @ X4 )
= top_to7106483174946246804list_a )
= ( X4 = bot_bot_set_list_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1149_atLeast__eq__UNIV__iff,axiom,
! [X4: set_nat] :
( ( ( set_or1731685050470061051et_nat @ X4 )
= top_top_set_set_nat )
= ( X4 = bot_bot_set_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1150_atLeast__eq__UNIV__iff,axiom,
! [X4: nat] :
( ( ( set_ord_atLeast_nat @ X4 )
= top_top_set_nat )
= ( X4 = bot_bot_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1151_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1152_inj__on__image__mem__iff,axiom,
! [F: nat > list_a,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_list_a @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_list_a @ ( F @ A2 ) @ ( image_nat_list_a @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1153_inj__on__image__mem__iff,axiom,
! [F: list_a > nat,B2: set_list_a,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_nat @ F @ B2 )
=> ( ( member_list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_list_a_nat @ F @ A ) )
= ( member_list_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1154_inj__on__image__mem__iff,axiom,
! [F: list_a > list_a,B2: set_list_a,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_list_a @ F @ B2 )
=> ( ( member_list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ ( F @ A2 ) @ ( image_list_a_list_a @ F @ A ) )
= ( member_list_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1155_inj__img__insertE,axiom,
! [F: nat > nat,A: set_nat,X4: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ~ ( member_nat @ X4 @ B2 )
=> ( ( ( insert_nat @ X4 @ B2 )
= ( image_nat_nat @ F @ A ) )
=> ~ ! [X9: nat,A7: set_nat] :
( ~ ( member_nat @ X9 @ A7 )
=> ( ( A
= ( insert_nat @ X9 @ A7 ) )
=> ( ( X4
= ( F @ X9 ) )
=> ( B2
!= ( image_nat_nat @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1156_inj__img__insertE,axiom,
! [F: list_a > nat,A: set_list_a,X4: nat,B2: set_nat] :
( ( inj_on_list_a_nat @ F @ A )
=> ( ~ ( member_nat @ X4 @ B2 )
=> ( ( ( insert_nat @ X4 @ B2 )
= ( image_list_a_nat @ F @ A ) )
=> ~ ! [X9: list_a,A7: set_list_a] :
( ~ ( member_list_a @ X9 @ A7 )
=> ( ( A
= ( insert_list_a @ X9 @ A7 ) )
=> ( ( X4
= ( F @ X9 ) )
=> ( B2
!= ( image_list_a_nat @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1157_inj__img__insertE,axiom,
! [F: nat > list_a,A: set_nat,X4: list_a,B2: set_list_a] :
( ( inj_on_nat_list_a @ F @ A )
=> ( ~ ( member_list_a @ X4 @ B2 )
=> ( ( ( insert_list_a @ X4 @ B2 )
= ( image_nat_list_a @ F @ A ) )
=> ~ ! [X9: nat,A7: set_nat] :
( ~ ( member_nat @ X9 @ A7 )
=> ( ( A
= ( insert_nat @ X9 @ A7 ) )
=> ( ( X4
= ( F @ X9 ) )
=> ( B2
!= ( image_nat_list_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1158_inj__img__insertE,axiom,
! [F: list_a > list_a,A: set_list_a,X4: list_a,B2: set_list_a] :
( ( inj_on_list_a_list_a @ F @ A )
=> ( ~ ( member_list_a @ X4 @ B2 )
=> ( ( ( insert_list_a @ X4 @ B2 )
= ( image_list_a_list_a @ F @ A ) )
=> ~ ! [X9: list_a,A7: set_list_a] :
( ~ ( member_list_a @ X9 @ A7 )
=> ( ( A
= ( insert_list_a @ X9 @ A7 ) )
=> ( ( X4
= ( F @ X9 ) )
=> ( B2
!= ( image_list_a_list_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1159_inj__image__mem__iff,axiom,
! [F: list_a > nat,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_nat @ F @ top_top_set_list_a )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_list_a_nat @ F @ A ) )
= ( member_list_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1160_inj__image__mem__iff,axiom,
! [F: list_a > list_a,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_list_a @ F @ top_top_set_list_a )
=> ( ( member_list_a @ ( F @ A2 ) @ ( image_list_a_list_a @ F @ A ) )
= ( member_list_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1161_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1162_inj__image__mem__iff,axiom,
! [F: nat > list_a,A2: nat,A: set_nat] :
( ( inj_on_nat_list_a @ F @ top_top_set_nat )
=> ( ( member_list_a @ ( F @ A2 ) @ ( image_nat_list_a @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1163_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y4: nat] :
( ( B
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1164_range__ex1__eq,axiom,
! [F: nat > list_a,B: list_a] :
( ( inj_on_nat_list_a @ F @ top_top_set_nat )
=> ( ( member_list_a @ B @ ( image_nat_list_a @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y4: nat] :
( ( B
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1165_ivl__disj__un__singleton_I1_J,axiom,
! [L: nat] :
( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or1210151606488870762an_nat @ L ) )
= ( set_ord_atLeast_nat @ L ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_1166_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X4: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X4 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_1167_the__inv__into__into,axiom,
! [F: list_a > nat,A: set_list_a,X4: nat,B2: set_list_a] :
( ( inj_on_list_a_nat @ F @ A )
=> ( ( member_nat @ X4 @ ( image_list_a_nat @ F @ A ) )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( member_list_a @ ( the_in5654461283803998945_a_nat @ A @ F @ X4 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_1168_the__inv__into__into,axiom,
! [F: nat > list_a,A: set_nat,X4: list_a,B2: set_nat] :
( ( inj_on_nat_list_a @ F @ A )
=> ( ( member_list_a @ X4 @ ( image_nat_list_a @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_in324026561121812671list_a @ A @ F @ X4 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_1169_the__inv__into__into,axiom,
! [F: list_a > list_a,A: set_list_a,X4: list_a,B2: set_list_a] :
( ( inj_on_list_a_list_a @ F @ A )
=> ( ( member_list_a @ X4 @ ( image_list_a_list_a @ F @ A ) )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( member_list_a @ ( the_in2841908037993899827list_a @ A @ F @ X4 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_1170_greaterThan__subset__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X4 ) @ ( set_or1210151606488870762an_nat @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_1171_greaterThan__non__empty,axiom,
! [X4: nat] :
( ( set_or1210151606488870762an_nat @ X4 )
!= bot_bot_set_nat ) ).
% greaterThan_non_empty
thf(fact_1172_Ici__subset__Ioi__iff,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ A2 ) @ ( set_or1210151606488870762an_nat @ B ) )
= ( ord_less_nat @ B @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_1173_finite__induct__select,axiom,
! [S: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ S )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [T3: set_list_a] :
( ( ord_less_set_list_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( minus_646659088055828811list_a @ S @ T3 ) )
& ( P @ ( insert_list_a @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1174_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1175_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1176_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1177_finite__Int,axiom,
! [F3: set_nat,G3: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G3 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_1178_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1179_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_1180_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1181_finite__compl,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_1182_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1183_infinite__countable__subset,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_1184_infinite__UNIV,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV
thf(fact_1185_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1186_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1187_ex__new__if__finite,axiom,
! [A: set_list_a] :
( ~ ( finite_finite_list_a @ top_top_set_list_a )
=> ( ( finite_finite_list_a @ A )
=> ? [A4: list_a] :
~ ( member_list_a @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1188_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1189_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1190_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1191_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1192_finite__bind,axiom,
! [S: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ S )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( finite_finite_nat @ ( F @ X3 ) ) )
=> ( finite_finite_nat @ ( bind_nat_nat @ S @ F ) ) ) ) ).
% finite_bind
thf(fact_1193_Diff__infinite__finite,axiom,
! [T4: set_nat,S: set_nat] :
( ( finite_finite_nat @ T4 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1194_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1195_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_1196_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1197_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_1198_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1199_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A: set_list_a] :
( ! [A8: set_list_a] :
( ~ ( finite_finite_list_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F6: set_list_a] :
( ( finite_finite_list_a @ F6 )
=> ( ~ ( member_list_a @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_list_a @ X3 @ F6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1200_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ~ ( member_nat @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ X3 @ F6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1201_finite__ne__induct,axiom,
! [F3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ! [X3: list_a] : ( P @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) )
=> ( ! [X3: list_a,F6: set_list_a] :
( ( finite_finite_list_a @ F6 )
=> ( ( F6 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_list_a @ X3 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1202_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( F6 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ X3 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1203_finite__induct,axiom,
! [F3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F6: set_list_a] :
( ( finite_finite_list_a @ F6 )
=> ( ~ ( member_list_a @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_list_a @ X3 @ F6 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1204_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ~ ( member_nat @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ X3 @ F6 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1205_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A5: set_list_a] :
( ( A5 = bot_bot_set_list_a )
| ? [A3: set_list_a,B5: list_a] :
( ( A5
= ( insert_list_a @ B5 @ A3 ) )
& ( finite_finite_list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1206_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A5
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1207_finite_Ocases,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( A2 != bot_bot_set_list_a )
=> ~ ! [A8: set_list_a] :
( ? [A4: list_a] :
( A2
= ( insert_list_a @ A4 @ A8 ) )
=> ~ ( finite_finite_list_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1208_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A4: nat] :
( A2
= ( insert_nat @ A4 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1209_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1210_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1211_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1212_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1213_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_1214_infinite__super,axiom,
! [S: set_nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S @ T4 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_1215_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_1216_cInf__le__finite,axiom,
! [X6: set_nat,X4: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ X4 ) ) ) ).
% cInf_le_finite
thf(fact_1217_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1218_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A )
& ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1219_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1220_infinite__surj,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ B2 ) )
=> ~ ( finite_finite_nat @ B2 ) ) ) ).
% infinite_surj
thf(fact_1221_finite__surj,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1222_finite__less__Inf__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ A2 @ ( complete_Inf_Inf_nat @ X6 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_nat @ A2 @ X ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_1223_finite__subset__induct_H,axiom,
! [F3: set_list_a,A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A4: list_a,F6: set_list_a] :
( ( finite_finite_list_a @ F6 )
=> ( ( member_list_a @ A4 @ A )
=> ( ( ord_le8861187494160871172list_a @ F6 @ A )
=> ( ~ ( member_list_a @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_list_a @ A4 @ F6 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1224_finite__subset__induct_H,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F6 @ A )
=> ( ~ ( member_nat @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ A4 @ F6 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1225_finite__subset__induct,axiom,
! [F3: set_list_a,A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A4: list_a,F6: set_list_a] :
( ( finite_finite_list_a @ F6 )
=> ( ( member_list_a @ A4 @ A )
=> ( ~ ( member_list_a @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_list_a @ A4 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1226_finite__subset__induct,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ A4 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1227_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1228_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1229_infinite__remove,axiom,
! [S: set_list_a,A2: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_1230_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1231_infinite__coinduct,axiom,
! [X6: set_list_a > $o,A: set_list_a] :
( ( X6 @ A )
=> ( ! [A8: set_list_a] :
( ( X6 @ A8 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ A8 )
& ( ( X6 @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_1232_infinite__coinduct,axiom,
! [X6: set_nat > $o,A: set_nat] :
( ( X6 @ A )
=> ( ! [A8: set_nat] :
( ( X6 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X6 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1233_finite__empty__induct,axiom,
! [A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ( P @ A )
=> ( ! [A4: list_a,A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ( member_list_a @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ A4 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1234_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1235_endo__inj__surj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_1236_inj__on__finite,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_1237_finite__surj__inj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_1238_finite__remove__induct,axiom,
! [B2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ( A8 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A8 @ B2 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A8 )
=> ( P @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1239_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1240_remove__induct,axiom,
! [P: set_list_a > $o,B2: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ( A8 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A8 @ B2 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A8 )
=> ( P @ ( minus_646659088055828811list_a @ A8 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1241_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1242_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1243_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ B4 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1244_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1245_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa2: nat] :
( ( member_nat @ Xa2 @ S )
& ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1246_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X6 )
& ( ord_less_nat @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_1247_finite__ranking__induct,axiom,
! [S: set_list_a,P: set_list_a > $o,F: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X3: list_a,S4: set_list_a] :
( ( finite_finite_list_a @ S4 )
=> ( ! [Y6: list_a] :
( ( member_list_a @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert_list_a @ X3 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1248_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1249_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ X5 @ B4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1250_cofinite__bot,axiom,
( ( cofinite_nat = bot_bot_filter_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% cofinite_bot
thf(fact_1251_arg__min__if__finite_I2_J,axiom,
! [S: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( S != bot_bot_set_list_a )
=> ~ ? [X5: list_a] :
( ( member_list_a @ X5 @ S )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic5043722365632780795_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1252_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1253_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M3: nat] :
( ( P @ X4 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1254_arg__min__least,axiom,
! [S: set_list_a,Y3: list_a,F: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( S != bot_bot_set_list_a )
=> ( ( member_list_a @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5043722365632780795_a_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1255_arg__min__least,axiom,
! [S: set_nat,Y3: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1256_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1257_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1258_Inf__fin_Osingleton,axiom,
! [X4: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% Inf_fin.singleton
thf(fact_1259_Inf__fin_Oinsert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1260_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1261_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_1262_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1263_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1264_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1265_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1266_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1267_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1268_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_1269_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1270_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_1271_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_1272_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_1273_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1274_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1275_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1276_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
% Conjectures (1)
thf(conj_0,conjecture,
( ( relational_eval_a_b @ ( relational_Bool_a_b @ $false ) @ i )
= bot_bot_set_list_a ) ).
%------------------------------------------------------------------------------