TPTP Problem File: SLH0372^1.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 : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_00639_020268__16211580_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1466 ( 573 unt; 189 typ; 0 def)
% Number of atoms : 3805 (1004 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 9988 ( 361 ~; 66 |; 330 &;7595 @)
% ( 0 <=>;1636 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 805 ( 805 >; 0 *; 0 +; 0 <<)
% Number of symbols : 177 ( 174 usr; 21 con; 0-4 aty)
% Number of variables : 3301 ( 392 ^;2747 !; 162 ?;3301 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:48:53.476
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (174)
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thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_OClique,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC__cf,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OC,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_ONEG,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060F_062,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060G_062l,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060P_062L_092_060G_062l,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodot,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oplucking__step,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
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clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
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thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
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_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
bot_bo193956671110832956et_nat: set_set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_less_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le466346588697744319_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le3616423863276227763_nat_o: ( set_set_nat > $o ) > ( set_set_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $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_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le5374289575490365114at_nat: set_se7880254595028141658at_nat > set_se7880254595028141658at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3495481059733392331at_nat: set_se3873067930692246379at_nat > set_se3873067930692246379at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le4731320016863163777at_nat: set_se8003284279568041249at_nat > set_se8003284279568041249at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J_J,type,
ord_le2853704879392749623at_nat: set_se7521423693449168855at_nat > set_se7521423693449168855at_nat > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
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__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
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__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1770121190487188718at_nat: ( nat > nat ) > ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or9117062992132219044at_nat: set_nat_nat > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or5410080298493297259et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_or659464924768625697et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Sunflower_Osunflower_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
sunflower_nat_nat: set_set_nat_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Nat__Onat,type,
sunflower_nat: set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Nat__Onat_J,type,
sunflower_set_nat: set_set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sunflo2680516271513359689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
sunflo111067583121249275at_nat: set_se7880254595028141658at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sunflo1841451327523575948at_nat: set_se3873067930692246379at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
sunflo6650083805840251970at_nat: set_se8003284279568041249at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
sunflo3853689026006497528at_nat: set_se7521423693449168855at_nat > $o ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_fChoice_001t__Set__Oset_It__Nat__Onat_J,type,
fChoice_set_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_fChoice_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
fChoice_set_set_nat: ( set_set_nat > $o ) > set_set_nat ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member968451730063008059at_nat: set_Su8808554476274791844at_nat > set_se7880254595028141658at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1869216328726507724at_nat: set_Sum_sum_nat_nat > set_se3873067930692246379at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member5374901640408327554at_nat: set_Su8059080322890262379at_nat > set_se8003284279568041249at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
member5638249034155602744at_nat: set_Su1440016900418933025at_nat > set_se7521423693449168855at_nat > $o ).
thf(sy_v_G____,type,
g: nat > set_set_nat ).
thf(sy_v_Gs____,type,
gs: set_set_nat ).
thf(sy_v_S____,type,
s: set_set_nat ).
thf(sy_v_Si____,type,
si: nat > set_nat ).
thf(sy_v_U____,type,
u: set_set_set_nat ).
thf(sy_v_Vs____,type,
vs: set_nat ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1269)
thf(fact_0__C1_C_I2_J,axiom,
member_nat @ j @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ).
% "1"(2)
thf(fact_1__C1_C_I1_J,axiom,
member_nat @ i @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ).
% "1"(1)
thf(fact_2__C1_C_I3_J,axiom,
( ( g @ i )
= ( g @ j ) ) ).
% "1"(3)
thf(fact_3_p0,axiom,
p != zero_zero_nat ).
% p0
thf(fact_4_Si,axiom,
bij_betw_nat_set_nat @ si @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ).
% Si
thf(fact_5_G_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( clique5033774636164728513irst_v @ ( g @ I ) )
= ( si @ I ) ) ) ).
% G(2)
thf(fact_6_first__assumptions_O_092_060P_062L_092_060G_062l_Ocong,axiom,
clique2294137941332549862_L_G_l = clique2294137941332549862_L_G_l ).
% first_assumptions.\<P>L\<G>l.cong
thf(fact_7_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_8_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_9_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_10_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_11_G__def,axiom,
( g
= ( ^ [I2: nat] :
( fChoice_set_set_nat
@ ^ [Gb: set_set_nat] :
( ( member_set_set_nat @ Gb @ x )
& ( ( clique5033774636164728513irst_v @ Gb )
= ( si @ I2 ) ) ) ) ) ) ).
% G_def
thf(fact_12_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_13_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_14_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_15_G_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ x ) ) ).
% G(1)
thf(fact_16__092_060open_062_092_060exists_062h_O_Abij__betw_Ah_A_1230_O_O_060p_125_AS_092_060close_062,axiom,
? [H: nat > set_nat] : ( bij_betw_nat_set_nat @ H @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ) ).
% \<open>\<exists>h. bij_betw h {0..<p} S\<close>
thf(fact_17__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Si_O_Abij__betw_ASi_A_1230_O_O_060p_125_AS_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Si: nat > set_nat] :
~ ( bij_betw_nat_set_nat @ Si @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ) ).
% \<open>\<And>thesis. (\<And>Si. bij_betw Si {0..<p} S \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_18_assms_I3_J,axiom,
( y
= ( clique4095374090462327202g_step @ p @ x ) ) ).
% assms(3)
thf(fact_19_G_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ s ) ) ).
% G(4)
thf(fact_20_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_21_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_22_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_23_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_24_some__equality,axiom,
! [P: set_set_nat > $o,A: set_set_nat] :
( ( P @ A )
=> ( ! [X: set_set_nat] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_set_set_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_25_some__eq__trivial,axiom,
! [X2: set_set_nat] :
( ( fChoice_set_set_nat
@ ^ [Y: set_set_nat] : ( Y = X2 ) )
= X2 ) ).
% some_eq_trivial
thf(fact_26_some__sym__eq__trivial,axiom,
! [X2: set_set_nat] :
( ( fChoice_set_set_nat
@ ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z )
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_27_U__def,axiom,
( u
= ( collect_set_set_nat
@ ^ [E: set_set_nat] :
( ( member_set_set_nat @ E @ x )
& ( member_set_nat @ ( clique5033774636164728513irst_v @ E ) @ s ) ) ) ) ).
% U_def
thf(fact_28_G_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ u ) ) ).
% G(3)
thf(fact_29_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_30_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_31_S_I2_J,axiom,
sunflower_nat @ s ).
% S(2)
thf(fact_32_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_33_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_34_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_35_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_36_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_37_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_38_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_39_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_40_mem__Collect__eq,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A @ ( collect_set_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: set_set_set_nat,P: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ A @ ( collec7201453139178570183et_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_51_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_52_Collect__cong,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X: set_set_set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec7201453139178570183et_nat @ P )
= ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_53_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_54_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_55_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less_nat @ M3 @ N )
| ( ord_less_nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_56_some__eq__imp,axiom,
! [P: set_set_nat > $o,A: set_set_nat,B: set_set_nat] :
( ( ( fChoice_set_set_nat @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_57_tfl__some,axiom,
! [P2: set_set_nat > $o,X4: set_set_nat] :
( ( P2 @ X4 )
=> ( P2 @ ( fChoice_set_set_nat @ P2 ) ) ) ).
% tfl_some
thf(fact_58_Eps__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( fChoice_set_set_nat @ P )
= ( fChoice_set_set_nat @ Q ) ) ) ).
% Eps_cong
thf(fact_59_someI,axiom,
! [P: set_set_nat > $o,X2: set_set_nat] :
( ( P @ X2 )
=> ( P @ ( fChoice_set_set_nat @ P ) ) ) ).
% someI
thf(fact_60_some1__equality,axiom,
! [P: set_set_nat > $o,A: set_set_nat] :
( ? [X4: set_set_nat] :
( ( P @ X4 )
& ! [Y4: set_set_nat] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_set_set_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_61_some__eq__ex,axiom,
! [P: set_set_nat > $o] :
( ( P @ ( fChoice_set_set_nat @ P ) )
= ( ? [X5: set_set_nat] : ( P @ X5 ) ) ) ).
% some_eq_ex
thf(fact_62_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: nat] :
( ( ( member_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_63_someI2__bex,axiom,
! [A2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: set_nat] :
( ( ( member_set_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_64_someI2__bex,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_65_someI2__bex,axiom,
! [A2: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: set_set_nat] :
( ( ( member_set_set_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_66_someI2__ex,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ? [X_1: set_set_nat] : ( P @ X_1 )
=> ( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_set_set_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_67_someI__ex,axiom,
! [P: set_set_nat > $o] :
( ? [X_1: set_set_nat] : ( P @ X_1 )
=> ( P @ ( fChoice_set_set_nat @ P ) ) ) ).
% someI_ex
thf(fact_68_someI2,axiom,
! [P: set_set_nat > $o,A: set_set_nat,Q: set_set_nat > $o] :
( ( P @ A )
=> ( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_set_set_nat @ P ) ) ) ) ).
% someI2
thf(fact_69_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_70_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_71_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_72_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_73_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_74_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_75_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_76_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_77_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_78_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_79_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_80_UX,axiom,
ord_le9131159989063066194et_nat @ u @ x ).
% UX
thf(fact_81_vplus__dsU,axiom,
( ( clique8462013130872731469t_v_gs @ u )
= s ) ).
% vplus_dsU
thf(fact_82_S_I3_J,axiom,
( ( finite_card_set_nat @ s )
= p ) ).
% S(3)
thf(fact_83__092_060open_062Y_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
y != bot_bo7198184520161983622et_nat ).
% \<open>Y \<noteq> {}\<close>
thf(fact_84_Snempty,axiom,
s != bot_bot_set_set_nat ).
% Snempty
thf(fact_85_Unempty,axiom,
u != bot_bo7198184520161983622et_nat ).
% Unempty
thf(fact_86_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_87_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_88_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_89_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_90__092_060open_062card_A_Iv__gs_AU_J_A_061_Acard_AS_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) )
= ( finite_card_set_nat @ s ) ) ).
% \<open>card (v_gs U) = card S\<close>
thf(fact_91_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_92_atLeastLessThan__iff,axiom,
! [I: nat > nat,L: nat > nat,U: nat > nat] :
( ( member_nat_nat @ I @ ( set_or1770121190487188718at_nat @ L @ U ) )
= ( ( ord_less_eq_nat_nat @ L @ I )
& ( ord_less_nat_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_93_atLeastLessThan__iff,axiom,
! [I: set_set_set_nat,L: set_set_set_nat,U: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I @ ( set_or659464924768625697et_nat @ L @ U ) )
= ( ( ord_le9131159989063066194et_nat @ L @ I )
& ( ord_le152980574450754630et_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_94_atLeastLessThan__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or5410080298493297259et_nat @ L @ U ) )
= ( ( ord_le6893508408891458716et_nat @ L @ I )
& ( ord_less_set_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_95_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_96_atLeastLessThan__iff,axiom,
! [I: set_nat_nat,L: set_nat_nat,U: set_nat_nat] :
( ( member_set_nat_nat @ I @ ( set_or9117062992132219044at_nat @ L @ U ) )
= ( ( ord_le9059583361652607317at_nat @ L @ I )
& ( ord_less_set_nat_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_97_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_98_ivl__subset,axiom,
! [I: nat,J: nat,M3: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M3 @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M3 @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_99_atLeastLessThan__empty,axiom,
! [B: nat > nat,A: nat > nat] :
( ( ord_less_eq_nat_nat @ B @ A )
=> ( ( set_or1770121190487188718at_nat @ A @ B )
= bot_bot_set_nat_nat ) ) ).
% atLeastLessThan_empty
thf(fact_100_atLeastLessThan__empty,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( set_or659464924768625697et_nat @ A @ B )
= bot_bo193956671110832956et_nat ) ) ).
% atLeastLessThan_empty
thf(fact_101_atLeastLessThan__empty,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( set_or5410080298493297259et_nat @ A @ B )
= bot_bo7198184520161983622et_nat ) ) ).
% atLeastLessThan_empty
thf(fact_102_atLeastLessThan__empty,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( set_or3540276404033026485et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_103_atLeastLessThan__empty,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( set_or9117062992132219044at_nat @ A @ B )
= bot_bo7376149671870096959at_nat ) ) ).
% atLeastLessThan_empty
thf(fact_104_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_105_atLeastLessThan__empty__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( set_or5410080298493297259et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ~ ( ord_less_set_set_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_106_atLeastLessThan__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or3540276404033026485et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_set_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_107_atLeastLessThan__empty__iff,axiom,
! [A: nat > nat,B: nat > nat] :
( ( ( set_or1770121190487188718at_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ~ ( ord_less_nat_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_108_atLeastLessThan__empty__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( set_or659464924768625697et_nat @ A @ B )
= bot_bo193956671110832956et_nat )
= ( ~ ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_109_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_110_atLeastLessThan__empty__iff2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( set_or5410080298493297259et_nat @ A @ B ) )
= ( ~ ( ord_less_set_set_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_111_atLeastLessThan__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or3540276404033026485et_nat @ A @ B ) )
= ( ~ ( ord_less_set_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_112_atLeastLessThan__empty__iff2,axiom,
! [A: nat > nat,B: nat > nat] :
( ( bot_bot_set_nat_nat
= ( set_or1770121190487188718at_nat @ A @ B ) )
= ( ~ ( ord_less_nat_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_113_atLeastLessThan__empty__iff2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( bot_bo193956671110832956et_nat
= ( set_or659464924768625697et_nat @ A @ B ) )
= ( ~ ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_114_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B ) )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_115_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_116_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_117_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_118_some__in__eq,axiom,
! [A2: set_set_set_nat] :
( ( member_set_set_nat
@ ( fChoice_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% some_in_eq
thf(fact_119_some__in__eq,axiom,
! [A2: set_set_nat] :
( ( member_set_nat
@ ( fChoice_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_set_nat ) ) ).
% some_in_eq
thf(fact_120_some__in__eq,axiom,
! [A2: set_nat] :
( ( member_nat
@ ( fChoice_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_nat ) ) ).
% some_in_eq
thf(fact_121_some__in__eq,axiom,
! [A2: set_nat_nat] :
( ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% some_in_eq
thf(fact_122_S__def,axiom,
( s
= ( fChoice_set_set_nat
@ ^ [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ S2 )
& ( ( finite_card_set_nat @ S2 )
= p ) ) ) ) ).
% S_def
thf(fact_123__092_060open_062S_A_092_060subseteq_062_Av__gs_AX_A_092_060and_062_Asunflower_AS_A_092_060and_062_Acard_AS_A_061_Ap_092_060close_062,axiom,
( ( ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ s )
& ( ( finite_card_set_nat @ s )
= p ) ) ).
% \<open>S \<subseteq> v_gs X \<and> sunflower S \<and> card S = p\<close>
thf(fact_124_card_Oempty,axiom,
( ( finite1149291290879098388et_nat @ bot_bo7198184520161983622et_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_125_card_Oempty,axiom,
( ( finite_card_set_nat @ bot_bot_set_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_126_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_127_card_Oempty,axiom,
( ( finite_card_nat_nat @ bot_bot_set_nat_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_128_sunflower,axiom,
? [S3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ S3 )
& ( ( finite_card_set_nat @ S3 )
= p ) ) ).
% sunflower
thf(fact_129_subset__empty,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_130_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_131_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_132_subset__empty,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_133_empty__subsetI,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% empty_subsetI
thf(fact_134_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_135_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_136_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_137_S_I1_J,axiom,
ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) ).
% S(1)
thf(fact_138_L,axiom,
ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) ).
% L
thf(fact_139__092_060open_062_092_060And_062A_O_AA_A_092_060subseteq_062_AX_A_092_060Longrightarrow_062_Afinite_AA_092_060close_062,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ x )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% \<open>\<And>A. A \<subseteq> X \<Longrightarrow> finite A\<close>
thf(fact_140_fin1,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% fin1
thf(fact_141_v__mono,axiom,
! [G: set_set_nat,H4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H4 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H4 ) ) ) ).
% v_mono
thf(fact_142_finX,axiom,
finite6739761609112101331et_nat @ x ).
% finX
thf(fact_143_finS,axiom,
finite1152437895449049373et_nat @ s ).
% finS
thf(fact_144_finU,axiom,
finite6739761609112101331et_nat @ u ).
% finU
thf(fact_145_v__gs__mono,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X6 ) @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ).
% v_gs_mono
thf(fact_146_subset__antisym,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_147_subset__antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_148_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_149_subset__antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_150_subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_set_nat @ X @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_151_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_152_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_153_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_154_psubsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_155_psubsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_156_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_157_psubsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_158_empty__Collect__eq,axiom,
! [P: set_set_set_nat > $o] :
( ( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat @ P ) )
= ( ! [X3: set_set_set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_159_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X3: set_set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_160_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_161_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_162_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_163_Collect__empty__eq,axiom,
! [P: set_set_set_nat > $o] :
( ( ( collec7201453139178570183et_nat @ P )
= bot_bo193956671110832956et_nat )
= ( ! [X3: set_set_set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_164_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_165_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_166_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_167_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_168_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X3: set_set_nat] :
~ ( member_set_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_169_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_170_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_171_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_172_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_173_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_174_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_175_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_176_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_177_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_178_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_179_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_180_finite__Collect__conjI,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
| ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ Q ) ) )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_181_finite__Collect__conjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
| ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_182_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_183_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_184_finite__Collect__conjI,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
| ( finite2115694454571419734at_nat @ ( collect_nat_nat @ Q ) ) )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_185_finite__Collect__disjI,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
& ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_186_finite__Collect__disjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
& ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_187_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_188_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_189_finite__Collect__disjI,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
& ( finite2115694454571419734at_nat @ ( collect_nat_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_190_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_191_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_192_card_Oinfinite,axiom,
! [A2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_193_card_Oinfinite,axiom,
! [A2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_card_set_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_194_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_195_card_Oinfinite,axiom,
! [A2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_card_nat_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_196_finite__Collect__subsets,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [B3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_197_finite__Collect__subsets,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_198_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_199_finite__Collect__subsets,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( finite3586981331298542604at_nat
@ ( collect_set_nat_nat
@ ^ [B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_200_card__0__eq,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ) ).
% card_0_eq
thf(fact_201_card__0__eq,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite_card_set_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_nat ) ) ) ).
% card_0_eq
thf(fact_202_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_203_card__0__eq,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ) ).
% card_0_eq
thf(fact_204_not__finite__existsD,axiom,
! [P: set_set_set_nat > $o] :
( ~ ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
=> ? [X_12: set_set_set_nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_205_not__finite__existsD,axiom,
! [P: set_set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ? [X_12: set_set_nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_206_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_12: set_nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_207_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_12: nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_208_not__finite__existsD,axiom,
! [P: ( nat > nat ) > $o] :
( ~ ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
=> ? [X_12: nat > nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_209_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_210_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_211_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_set_nat,R: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_212_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_nat,B2: set_nat,R: set_set_nat > nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_213_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_set_nat,R: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_214_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_set_set_nat,R: nat > set_set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_215_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat_nat,R: nat > ( nat > nat ) > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_216_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat,B2: set_nat,R: ( nat > nat ) > nat > $o] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_217_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_set_nat,B2: set_nat,R: set_set_set_nat > nat > $o] :
( ~ ( finite5926941155766903689et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [A3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_218_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,R: set_set_nat > set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ A2 )
& ( R @ A3 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_219_finite__psubset__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ! [B4: set_set_nat] :
( ( ord_less_set_set_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_220_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_221_finite__psubset__induct,axiom,
! [A2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ! [A4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A4 )
=> ( ! [B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_222_finite__psubset__induct,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [A4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A4 )
=> ( ! [B4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_223_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_224_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_nat,R2: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_225_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_nat,R2: set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B6: nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_226_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_nat,A2: set_nat,R2: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B5: set_set_nat] :
( ( member_set_set_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: set_set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_227_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_set_nat,R2: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B6: set_nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_228_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_set_nat,R2: set_set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B6: nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_229_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_nat,R2: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: nat > nat] :
( ( member_nat_nat @ A5 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: nat > nat,A22: nat > nat,B6: nat] :
( ( member_nat_nat @ A1 @ A2 )
=> ( ( member_nat_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_230_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_nat,A2: set_nat,R2: nat > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B5: nat > nat] :
( ( member_nat_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat > nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_231_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_nat,A2: set_set_nat,R2: set_nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [B5: set_set_nat] :
( ( member_set_set_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B6: set_set_nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_set_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_232_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_set_set_nat,R2: set_set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
=> ? [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
& ( R2 @ A5 @ B5 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B6: set_nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_233_finite__has__minimal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( ord_less_eq_nat_nat @ X @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_234_finite__has__minimal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ( ord_le9131159989063066194et_nat @ X @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_235_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ X @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_236_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_237_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_238_finite__has__minimal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ( ord_le9059583361652607317at_nat @ X @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_239_finite__has__maximal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( ord_less_eq_nat_nat @ A @ X )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_240_finite__has__maximal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ( ord_le9131159989063066194et_nat @ A @ X )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_241_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_242_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ A @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_243_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_244_finite__has__maximal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ( ord_le9059583361652607317at_nat @ A @ X )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_245_rev__finite__subset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_246_rev__finite__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_247_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_248_rev__finite__subset,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_249_infinite__super,axiom,
! [S4: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S4 @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S4 )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_250_infinite__super,axiom,
! [S4: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S4 @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S4 )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_251_infinite__super,axiom,
! [S4: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ T2 )
=> ( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_252_infinite__super,axiom,
! [S4: set_nat_nat,T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S4 @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S4 )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_253_finite__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_254_finite__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_255_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_256_finite__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_257_infinite__imp__nonempty,axiom,
! [S4: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S4 )
=> ( S4 != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_258_infinite__imp__nonempty,axiom,
! [S4: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S4 )
=> ( S4 != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_259_infinite__imp__nonempty,axiom,
! [S4: set_nat] :
( ~ ( finite_finite_nat @ S4 )
=> ( S4 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_260_infinite__imp__nonempty,axiom,
! [S4: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S4 )
=> ( S4 != bot_bot_set_nat_nat ) ) ).
% infinite_imp_nonempty
thf(fact_261_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_262_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_263_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_264_finite_OemptyI,axiom,
finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).
% finite.emptyI
thf(fact_265_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_266_bij__betw__finite,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_267_bij__betw__finite,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_268_bij__betw__finite,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_269_bij__betw__finite,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_270_bij__betw__finite,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_271_bij__betw__finite,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_272_bij__betw__finite,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_273_bij__betw__finite,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( finite2115694454571419734at_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_274_bij__betw__finite,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_275_bij__betw__finite,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_276_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_set_nat,C2: nat] :
( ! [G2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G2 @ F2 )
=> ( ( finite6739761609112101331et_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G2 ) @ C2 ) ) )
=> ( ( finite6739761609112101331et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_277_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_nat,C2: nat] :
( ! [G2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G2 @ F2 )
=> ( ( finite1152437895449049373et_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G2 ) @ C2 ) ) )
=> ( ( finite1152437895449049373et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_278_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_279_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat_nat,C2: nat] :
( ! [G2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ G2 @ F2 )
=> ( ( finite2115694454571419734at_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ G2 ) @ C2 ) ) )
=> ( ( finite2115694454571419734at_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_280_obtain__subset__with__card__n,axiom,
! [N: nat,S4: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ S4 ) )
=> ~ ! [T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T3 @ S4 )
=> ( ( ( finite1149291290879098388et_nat @ T3 )
= N )
=> ~ ( finite6739761609112101331et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_281_obtain__subset__with__card__n,axiom,
! [N: nat,S4: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S4 ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S4 )
=> ( ( ( finite_card_set_nat @ T3 )
= N )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_282_obtain__subset__with__card__n,axiom,
! [N: nat,S4: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S4 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S4 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_283_obtain__subset__with__card__n,axiom,
! [N: nat,S4: set_nat_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ S4 ) )
=> ~ ! [T3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ T3 @ S4 )
=> ( ( ( finite_card_nat_nat @ T3 )
= N )
=> ~ ( finite2115694454571419734at_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_284_exists__subset__between,axiom,
! [A2: set_set_set_nat,N: nat,C2: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ C2 ) )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( finite6739761609112101331et_nat @ C2 )
=> ? [B7: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B7 )
& ( ord_le9131159989063066194et_nat @ B7 @ C2 )
& ( ( finite1149291290879098388et_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_285_exists__subset__between,axiom,
! [A2: set_set_nat,N: nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( finite1152437895449049373et_nat @ C2 )
=> ? [B7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B7 )
& ( ord_le6893508408891458716et_nat @ B7 @ C2 )
& ( ( finite_card_set_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_286_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C2 )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_287_exists__subset__between,axiom,
! [A2: set_nat_nat,N: nat,C2: set_nat_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ C2 ) )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( finite2115694454571419734at_nat @ C2 )
=> ? [B7: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B7 )
& ( ord_le9059583361652607317at_nat @ B7 @ C2 )
& ( ( finite_card_nat_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_288_card__seteq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_289_card__seteq,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_290_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_291_card__seteq,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_292_card__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_293_card__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_294_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_295_card__mono,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_296_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A6 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_297_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_298_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_299_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A6 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_300_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_301_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_302_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_303_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_304_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_305_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_306_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_307_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
| ( M = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_308_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_309_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
& ( M != N4 ) ) ) ) ).
% nat_less_le
thf(fact_310_finite__has__minimal,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_311_finite__has__minimal,axiom,
! [A2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_312_finite__has__minimal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_313_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_314_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_315_finite__has__minimal,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_316_finite__has__maximal,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_317_finite__has__maximal,axiom,
! [A2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_318_finite__has__maximal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_319_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_320_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_321_finite__has__maximal,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_322_infinite__arbitrarily__large,axiom,
! [A2: set_set_set_nat,N: nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ? [B7: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B7 )
& ( ( finite1149291290879098388et_nat @ B7 )
= N )
& ( ord_le9131159989063066194et_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_323_infinite__arbitrarily__large,axiom,
! [A2: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ? [B7: set_set_nat] :
( ( finite1152437895449049373et_nat @ B7 )
& ( ( finite_card_set_nat @ B7 )
= N )
& ( ord_le6893508408891458716et_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_324_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_325_infinite__arbitrarily__large,axiom,
! [A2: set_nat_nat,N: nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ? [B7: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B7 )
& ( ( finite_card_nat_nat @ B7 )
= N )
& ( ord_le9059583361652607317at_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_326_card__subset__eq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_327_card__subset__eq,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_328_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_329_card__subset__eq,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_330_psubset__card__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_331_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_332_psubset__card__mono,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_333_psubset__card__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_334_card__eq__0__iff,axiom,
! [A2: set_set_set_nat] :
( ( ( finite1149291290879098388et_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
| ~ ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_335_card__eq__0__iff,axiom,
! [A2: set_set_nat] :
( ( ( finite_card_set_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_nat )
| ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_336_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_337_card__eq__0__iff,axiom,
! [A2: set_nat_nat] :
( ( ( finite_card_nat_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat_nat )
| ~ ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_338_card__psubset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_339_card__psubset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_340_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_341_card__psubset,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_342_card__ge__0__finite,axiom,
! [A2: set_set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ A2 ) )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_343_card__ge__0__finite,axiom,
! [A2: set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
=> ( finite1152437895449049373et_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_344_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_345_card__ge__0__finite,axiom,
! [A2: set_nat_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat_nat @ A2 ) )
=> ( finite2115694454571419734at_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_346_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ~ ( P @ I4 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_347_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: nat > nat] : ( bij_betw_nat_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_348_bij__betw__iff__card,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: set_nat > nat] : ( bij_betw_set_nat_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_349_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F3: nat > set_nat] : ( bij_betw_nat_set_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_350_bij__betw__iff__card,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ F3 @ A2 @ B2 ) )
= ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_351_bij__betw__iff__card,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F3: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_352_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F3: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_353_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ? [F3: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ F3 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_354_bij__betw__iff__card,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_355_bij__betw__iff__card,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F3: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ F3 @ A2 @ B2 ) )
= ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_356_bij__betw__iff__card,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F3: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ F3 @ A2 @ B2 ) )
= ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_357_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_358_finite__same__card__bij,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: set_nat > nat] : ( bij_betw_set_nat_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_359_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ? [H: nat > set_nat] : ( bij_betw_nat_set_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_360_finite__same__card__bij,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_361_finite__same__card__bij,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ? [H: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_362_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ? [H: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_363_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) )
=> ? [H: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_364_finite__same__card__bij,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_365_finite__same__card__bij,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ? [H: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_366_finite__same__card__bij,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ? [H: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_367_atLeastLessThan0,axiom,
! [M3: nat] :
( ( set_or4665077453230672383an_nat @ M3 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_368_card__gt__0__iff,axiom,
! [A2: set_set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ A2 ) )
= ( ( A2 != bot_bo7198184520161983622et_nat )
& ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_369_card__gt__0__iff,axiom,
! [A2: set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
= ( ( A2 != bot_bot_set_set_nat )
& ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_370_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_371_card__gt__0__iff,axiom,
! [A2: set_nat_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat_nat )
& ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_372_Collect__mono__iff,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) )
= ( ! [X3: set_set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_373_Collect__mono__iff,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_374_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_375_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_376_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_377_set__eq__subset,axiom,
( ( ^ [Y2: set_set_set_nat,Z: set_set_set_nat] : ( Y2 = Z ) )
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A6 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_378_set__eq__subset,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_379_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A6: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_380_set__eq__subset,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_381_subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_382_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_383_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_384_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_385_Collect__mono,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X: set_set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_386_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_387_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_388_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_389_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_390_subset__refl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_391_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_392_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_393_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_394_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
! [T4: set_set_nat] :
( ( member_set_set_nat @ T4 @ A6 )
=> ( member_set_set_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_395_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
! [T4: set_nat] :
( ( member_set_nat @ T4 @ A6 )
=> ( member_set_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_396_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A6 )
=> ( member_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_397_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
! [T4: nat > nat] :
( ( member_nat_nat @ T4 @ A6 )
=> ( member_nat_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_398_equalityD2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_399_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_400_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_401_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_402_equalityD1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_403_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_404_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_405_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_406_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A6 )
=> ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_407_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_408_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_409_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A6 )
=> ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_410_equalityE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_411_equalityE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_412_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_413_equalityE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_414_subsetD,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_415_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_416_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_417_subsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_418_in__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_419_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_420_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_421_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_422_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A6 @ B3 )
| ( A6 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_423_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A6 @ B3 )
| ( A6 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_424_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A6 @ B3 )
| ( A6 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_425_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A6 @ B3 )
| ( A6 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_426_subset__psubset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_427_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_428_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_429_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C2 )
=> ( ord_less_set_nat_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_430_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A6 @ B3 )
& ~ ( ord_le9131159989063066194et_nat @ B3 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_431_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_432_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_433_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B3 )
& ~ ( ord_le9059583361652607317at_nat @ B3 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_434_psubset__subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_435_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_436_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_437_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_less_set_nat_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_438_psubset__imp__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_439_psubset__imp__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_440_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_441_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_442_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A6 @ B3 )
& ( A6 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_443_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ B3 )
& ( A6 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_444_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B3 )
& ( A6 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_445_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B3 )
& ( A6 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_446_psubsetE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_447_psubsetE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_448_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_449_psubsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_450_ex__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ? [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_451_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_452_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_453_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_454_equals0I,axiom,
! [A2: set_set_set_nat] :
( ! [Y4: set_set_nat] :
~ ( member_set_set_nat @ Y4 @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_455_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_456_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_457_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_458_equals0D,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( A2 = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_459_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_460_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_461_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_462_emptyE,axiom,
! [A: set_set_nat] :
~ ( member_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_463_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_464_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_465_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_466_not__psubset__empty,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_467_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_468_not__psubset__empty,axiom,
! [A2: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_469_not__psubset__empty,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% not_psubset_empty
thf(fact_470_ex__bij__betw__nat__finite,axiom,
! [M4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ M4 )
=> ? [H: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_471_ex__bij__betw__nat__finite,axiom,
! [M4: set_set_nat] :
( ( finite1152437895449049373et_nat @ M4 )
=> ? [H: nat > set_nat] : ( bij_betw_nat_set_nat @ H @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_set_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_472_ex__bij__betw__nat__finite,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_473_ex__bij__betw__nat__finite,axiom,
! [M4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ M4 )
=> ? [H: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ H @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_474_Collect__subset,axiom,
! [A2: set_set_set_set_nat,P: set_set_set_nat > $o] :
( ord_le572741076514265352et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_475_Collect__subset,axiom,
! [A2: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_476_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_477_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_478_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_479_empty__def,axiom,
( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : $false ) ) ).
% empty_def
thf(fact_480_empty__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat
@ ^ [X3: set_set_nat] : $false ) ) ).
% empty_def
thf(fact_481_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X3: set_nat] : $false ) ) ).
% empty_def
thf(fact_482_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_483_empty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X3: nat > nat] : $false ) ) ).
% empty_def
thf(fact_484_bij__betw__same__card,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_485_bij__betw__same__card,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_486_bij__betw__same__card,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_487_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_488_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X5: set_set_set_nat,G3: set_set_nat] :
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X5 )
& ( ord_le6893508408891458716et_nat @ X3 @ G3 ) ) ) ) ).
% accepts_def
thf(fact_489_sunflower__card__subset__lift,axiom,
! [K2: nat,C: nat,R2: nat,F2: set_set_set_set_nat] :
( ! [G2: set_se7521423693449168855at_nat] :
( ! [X4: set_Su1440016900418933025at_nat] :
( ( member5638249034155602744at_nat @ X4 @ G2 )
=> ( ( finite8770298478261192322at_nat @ X4 )
& ( ( finite8251389301641259331at_nat @ X4 )
= K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite7696428214769936121at_nat @ G2 ) )
=> ? [S5: set_se7521423693449168855at_nat] :
( ( ord_le2853704879392749623at_nat @ S5 @ G2 )
& ( sunflo3853689026006497528at_nat @ S5 )
& ( ( finite7696428214769936121at_nat @ S5 )
= R2 ) ) ) )
=> ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ F2 )
=> ( ( finite6739761609112101331et_nat @ X )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ X ) @ K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite8805468973633305546et_nat @ F2 ) )
=> ? [S3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ S3 @ F2 )
& ( sunflo2680516271513359689et_nat @ S3 )
& ( ( finite8805468973633305546et_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_490_sunflower__card__subset__lift,axiom,
! [K2: nat,C: nat,R2: nat,F2: set_set_nat_nat] :
( ! [G2: set_se7880254595028141658at_nat] :
( ! [X4: set_Su8808554476274791844at_nat] :
( ( member968451730063008059at_nat @ X4 @ G2 )
=> ( ( finite5967121830935861893at_nat @ X4 )
& ( ( finite2091696060772798406at_nat @ X4 )
= K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite5641098376000219004at_nat @ G2 ) )
=> ? [S5: set_se7880254595028141658at_nat] :
( ( ord_le5374289575490365114at_nat @ S5 @ G2 )
& ( sunflo111067583121249275at_nat @ S5 )
& ( ( finite5641098376000219004at_nat @ S5 )
= R2 ) ) ) )
=> ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ F2 )
=> ( ( finite2115694454571419734at_nat @ X )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ X ) @ K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite5893285860794289869at_nat @ F2 ) )
=> ? [S3: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ S3 @ F2 )
& ( sunflower_nat_nat @ S3 )
& ( ( finite5893285860794289869at_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_491_sunflower__card__subset__lift,axiom,
! [K2: nat,C: nat,R2: nat,F2: set_set_set_nat] :
( ! [G2: set_se8003284279568041249at_nat] :
( ! [X4: set_Su8059080322890262379at_nat] :
( ( member5374901640408327554at_nat @ X4 @ G2 )
=> ( ( finite2491568536608231884at_nat @ X4 )
& ( ( finite8413070326521870477at_nat @ X4 )
= K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite7758422657562484035at_nat @ G2 ) )
=> ? [S5: set_se8003284279568041249at_nat] :
( ( ord_le4731320016863163777at_nat @ S5 @ G2 )
& ( sunflo6650083805840251970at_nat @ S5 )
& ( ( finite7758422657562484035at_nat @ S5 )
= R2 ) ) ) )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ F2 )
=> ( ( finite1152437895449049373et_nat @ X )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ X ) @ K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite1149291290879098388et_nat @ F2 ) )
=> ? [S3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S3 @ F2 )
& ( sunflower_set_nat @ S3 )
& ( ( finite1149291290879098388et_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_492_sunflower__card__subset__lift,axiom,
! [K2: nat,C: nat,R2: nat,F2: set_set_nat] :
( ! [G2: set_se3873067930692246379at_nat] :
( ! [X4: set_Sum_sum_nat_nat] :
( ( member1869216328726507724at_nat @ X4 @ G2 )
=> ( ( finite6187706683773761046at_nat @ X4 )
& ( ( finite8494011213269508311at_nat @ X4 )
= K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite2024029949821234317at_nat @ G2 ) )
=> ? [S5: set_se3873067930692246379at_nat] :
( ( ord_le3495481059733392331at_nat @ S5 @ G2 )
& ( sunflo1841451327523575948at_nat @ S5 )
& ( ( finite2024029949821234317at_nat @ S5 )
= R2 ) ) ) )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ F2 )
=> ( ( finite_finite_nat @ X )
& ( ord_less_eq_nat @ ( finite_card_nat @ X ) @ K2 ) ) )
=> ( ( ord_less_nat @ C @ ( finite_card_set_nat @ F2 ) )
=> ? [S3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ F2 )
& ( sunflower_nat @ S3 )
& ( ( finite_card_set_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_493_card__Vs,axiom,
ord_less_eq_nat @ ( finite_card_nat @ vs ) @ l ).
% card_Vs
thf(fact_494_X,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% X
thf(fact_495_sf__precond,axiom,
! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( clique8462013130872731469t_v_gs @ x ) )
=> ( ( finite_finite_nat @ X4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X4 ) @ l ) ) ) ).
% sf_precond
thf(fact_496__092_060P_062L_092_060G_062l__def,axiom,
( ( clique2294137941332549862_L_G_l @ l @ p @ k )
= ( collec7201453139178570183et_nat
@ ^ [X5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ ( assump1710595444109740301irst_L @ l @ p ) ) ) ) ) ).
% \<P>L\<G>l_def
thf(fact_497_filter__preserves__multiset,axiom,
! [M4: set_set_set_nat > nat,P: set_set_set_nat > $o] :
( ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_498_filter__preserves__multiset,axiom,
! [M4: set_set_nat > nat,P: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_499_filter__preserves__multiset,axiom,
! [M4: set_nat > nat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_500_filter__preserves__multiset,axiom,
! [M4: nat > nat,P: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_501_filter__preserves__multiset,axiom,
! [M4: ( nat > nat ) > nat,P: ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_502_ex__min__if__finite,axiom,
! [S4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ S4 )
=> ( ( S4 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ S4 )
& ~ ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ S4 )
& ( ord_less_set_set_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_503_ex__min__if__finite,axiom,
! [S4: set_set_nat] :
( ( finite1152437895449049373et_nat @ S4 )
=> ( ( S4 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ S4 )
& ~ ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ S4 )
& ( ord_less_set_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_504_ex__min__if__finite,axiom,
! [S4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S4 )
=> ( ( S4 != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ S4 )
& ~ ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S4 )
& ( ord_less_nat_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_505_ex__min__if__finite,axiom,
! [S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ( S4 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ S4 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S4 )
& ( ord_less_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_506_ex__min__if__finite,axiom,
! [S4: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ S4 )
=> ( ( S4 != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ S4 )
& ~ ? [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ S4 )
& ( ord_le152980574450754630et_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_507_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_508_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_509_fin__Vs,axiom,
finite_finite_nat @ vs ).
% fin_Vs
thf(fact_510_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_511_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_512_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_513_order__refl,axiom,
! [X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_514_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_515_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_516_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_517_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_518_dual__order_Orefl,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_519_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_520_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_521_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_522_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_523_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_524_U,axiom,
ord_le9131159989063066194et_nat @ u @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% U
thf(fact_525__092_060open_062Y_A_092_060subseteq_062_A_092_060G_062l_092_060close_062,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% \<open>Y \<subseteq> \<G>l\<close>
thf(fact_526_finite__v__gs__Gl,axiom,
! [X6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X6 ) ) ) ).
% finite_v_gs_Gl
thf(fact_527_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_528_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_529_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_530_acceptsI,axiom,
! [D2: set_set_nat,G: set_set_nat,X6: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G )
=> ( ( member_set_set_nat @ D2 @ X6 )
=> ( clique3686358387679108662ccepts @ X6 @ G ) ) ) ).
% acceptsI
thf(fact_531_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_532_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_533_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_534_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_535_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_536_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_537_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M4: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_538_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_539_psubset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_540_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_541_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_542_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A6 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_543_less__set__def,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( ord_le466346588697744319_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A6 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_544_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_545_psubsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_546_psubsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_547_psubsetD,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_548_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_549_bot__set__def,axiom,
( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat @ bot_bo5536612546450143305_nat_o ) ) ).
% bot_set_def
thf(fact_550_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_551_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_552_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_553_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_554_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_555_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_556_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_557_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_558_subset__eq__atLeast0__lessThan__finite,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_559_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_560_le__cases3,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_561_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_set_nat,Z: set_set_set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
& ( ord_le9131159989063066194et_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_562_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
& ( ord_le6893508408891458716et_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_563_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
& ( ord_less_eq_set_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_564_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_565_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
& ( ord_le9059583361652607317at_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_566_ord__eq__le__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A = B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_567_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_568_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_569_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_570_ord__eq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_571_ord__le__eq__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_572_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_573_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_574_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_575_ord__le__eq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_576_order__antisym,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
=> ( ( ord_le9131159989063066194et_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_577_order__antisym,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_578_order__antisym,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_579_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_580_order__antisym,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_581_order_Otrans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_582_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_583_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_584_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_585_order_Otrans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_586_order__trans,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
=> ( ( ord_le9131159989063066194et_nat @ Y3 @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_587_order__trans,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_588_order__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_589_order__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_590_order__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_591_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_592_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_set_nat,Z: set_set_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_set_set_nat,B8: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B8 @ A3 )
& ( ord_le9131159989063066194et_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_593_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_set_nat,B8: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B8 @ A3 )
& ( ord_le6893508408891458716et_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_594_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat,B8: set_nat] :
( ( ord_less_eq_set_nat @ B8 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_595_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B8: nat] :
( ( ord_less_eq_nat @ B8 @ A3 )
& ( ord_less_eq_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_596_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat_nat,B8: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B8 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_597_dual__order_Oantisym,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_598_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_599_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_600_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_601_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_602_dual__order_Otrans,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( ord_le9131159989063066194et_nat @ C @ B )
=> ( ord_le9131159989063066194et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_603_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_604_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_605_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_606_dual__order_Otrans,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_607_antisym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_608_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_609_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_610_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_611_antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_612_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_set_nat,Z: set_set_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_set_set_nat,B8: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B8 )
& ( ord_le9131159989063066194et_nat @ B8 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_613_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_set_nat,B8: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B8 )
& ( ord_le6893508408891458716et_nat @ B8 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_614_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat,B8: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_615_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B8: nat] :
( ( ord_less_eq_nat @ A3 @ B8 )
& ( ord_less_eq_nat @ B8 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_616_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat_nat,B8: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B8 )
& ( ord_le9059583361652607317at_nat @ B8 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_617_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_618_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_619_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_620_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_621_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_622_order__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_623_order__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_624_order__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_625_order__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_626_order__subst1,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_627_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_628_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_629_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_630_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_631_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_632_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_633_order__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_634_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_635_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_636_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_637_order__eq__refl,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( X2 = Y3 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_638_order__eq__refl,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( X2 = Y3 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_639_order__eq__refl,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_640_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_641_order__eq__refl,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( X2 = Y3 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_642_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_643_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_644_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_645_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_646_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_647_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_648_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_649_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_650_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_651_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_652_ord__eq__le__subst,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_653_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_654_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_655_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_656_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_657_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_658_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_659_ord__le__eq__subst,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_660_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_661_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_662_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_663_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_664_order__antisym__conv,axiom,
! [Y3: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y3 @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_665_order__antisym__conv,axiom,
! [Y3: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_666_order__antisym__conv,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_667_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_668_order__antisym__conv,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_669_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_670_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_671_less__imp__neq,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_672_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_673_order_Oasym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ~ ( ord_le152980574450754630et_nat @ B @ A ) ) ).
% order.asym
thf(fact_674_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_675_ord__eq__less__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A = B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_676_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_677_ord__less__eq__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_678_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_679_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_680_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_681_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_682_dual__order_Oasym,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ~ ( ord_le152980574450754630et_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_683_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_684_dual__order_Oirrefl,axiom,
! [A: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_685_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N4 )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_686_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_687_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_688_order_Ostrict__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_689_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_690_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_691_dual__order_Ostrict__trans,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ( ( ord_le152980574450754630et_nat @ C @ B )
=> ( ord_le152980574450754630et_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_692_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_693_order_Ostrict__implies__not__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_694_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_695_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_696_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_697_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_698_order__less__asym,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ~ ( ord_le152980574450754630et_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_699_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_700_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_701_order__less__asym_H,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ~ ( ord_le152980574450754630et_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_702_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_703_order__less__trans,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( ( ord_le152980574450754630et_nat @ Y3 @ Z2 )
=> ( ord_le152980574450754630et_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_704_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_705_ord__eq__less__subst,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_706_ord__eq__less__subst,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_707_ord__eq__less__subst,axiom,
! [A: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_708_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_709_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_710_ord__less__eq__subst,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_711_ord__less__eq__subst,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_712_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_713_order__less__irrefl,axiom,
! [X2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_714_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_715_order__less__subst1,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_716_order__less__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_717_order__less__subst1,axiom,
! [A: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_718_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_719_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_720_order__less__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_721_order__less__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_722_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_723_order__less__not__sym,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ~ ( ord_le152980574450754630et_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_724_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_725_order__less__imp__triv,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat,P: $o] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( ( ord_le152980574450754630et_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_726_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_727_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_728_order__less__imp__not__eq,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_729_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_730_order__less__imp__not__eq2,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_731_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_732_order__less__imp__not__less,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ~ ( ord_le152980574450754630et_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_733_ex__bij__betw__finite__nat,axiom,
! [M4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ M4 )
=> ? [H: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_734_ex__bij__betw__finite__nat,axiom,
! [M4: set_set_nat] :
( ( finite1152437895449049373et_nat @ M4 )
=> ? [H: set_nat > nat] : ( bij_betw_set_nat_nat @ H @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_set_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_735_ex__bij__betw__finite__nat,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_736_ex__bij__betw__finite__nat,axiom,
! [M4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ M4 )
=> ? [H: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ H @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_737_sunflower__def,axiom,
( sunflo2680516271513359689et_nat
= ( ^ [S2: set_set_set_set_nat] :
! [X3: set_set_nat] :
( ? [A6: set_set_set_nat,B3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A6 @ S2 )
& ( member2946998982187404937et_nat @ B3 @ S2 )
& ( A6 != B3 )
& ( member_set_set_nat @ X3 @ A6 )
& ( member_set_set_nat @ X3 @ B3 ) )
=> ! [A6: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A6 @ S2 )
=> ( member_set_set_nat @ X3 @ A6 ) ) ) ) ) ).
% sunflower_def
thf(fact_738_sunflower__def,axiom,
( sunflower_nat_nat
= ( ^ [S2: set_set_nat_nat] :
! [X3: nat > nat] :
( ? [A6: set_nat_nat,B3: set_nat_nat] :
( ( member_set_nat_nat @ A6 @ S2 )
& ( member_set_nat_nat @ B3 @ S2 )
& ( A6 != B3 )
& ( member_nat_nat @ X3 @ A6 )
& ( member_nat_nat @ X3 @ B3 ) )
=> ! [A6: set_nat_nat] :
( ( member_set_nat_nat @ A6 @ S2 )
=> ( member_nat_nat @ X3 @ A6 ) ) ) ) ) ).
% sunflower_def
thf(fact_739_sunflower__def,axiom,
( sunflower_set_nat
= ( ^ [S2: set_set_set_nat] :
! [X3: set_nat] :
( ? [A6: set_set_nat,B3: set_set_nat] :
( ( member_set_set_nat @ A6 @ S2 )
& ( member_set_set_nat @ B3 @ S2 )
& ( A6 != B3 )
& ( member_set_nat @ X3 @ A6 )
& ( member_set_nat @ X3 @ B3 ) )
=> ! [A6: set_set_nat] :
( ( member_set_set_nat @ A6 @ S2 )
=> ( member_set_nat @ X3 @ A6 ) ) ) ) ) ).
% sunflower_def
thf(fact_740_sunflower__def,axiom,
( sunflower_nat
= ( ^ [S2: set_set_nat] :
! [X3: nat] :
( ? [A6: set_nat,B3: set_nat] :
( ( member_set_nat @ A6 @ S2 )
& ( member_set_nat @ B3 @ S2 )
& ( A6 != B3 )
& ( member_nat @ X3 @ A6 )
& ( member_nat @ X3 @ B3 ) )
=> ! [A6: set_nat] :
( ( member_set_nat @ A6 @ S2 )
=> ( member_nat @ X3 @ A6 ) ) ) ) ) ).
% sunflower_def
thf(fact_741_leD,axiom,
! [Y3: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y3 @ X2 )
=> ~ ( ord_le152980574450754630et_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_742_leD,axiom,
! [Y3: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_set_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_743_leD,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_744_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_745_leD,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_nat_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_746_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_747_nless__le,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ~ ( ord_le152980574450754630et_nat @ A @ B ) )
= ( ~ ( ord_le9131159989063066194et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_748_nless__le,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A @ B ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_749_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_750_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_751_nless__le,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A @ B ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_752_antisym__conv1,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ~ ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_753_antisym__conv1,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X2 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_754_antisym__conv1,axiom,
! [X2: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_755_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_756_antisym__conv1,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_757_antisym__conv2,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_le152980574450754630et_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_758_antisym__conv2,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_set_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_759_antisym__conv2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_760_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_761_antisym__conv2,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_nat_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_762_less__le__not__le,axiom,
( ord_le152980574450754630et_nat
= ( ^ [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
& ~ ( ord_le9131159989063066194et_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_763_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
& ~ ( ord_le6893508408891458716et_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_764_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_765_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_766_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
& ~ ( ord_le9059583361652607317at_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_767_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_768_order_Oorder__iff__strict,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B8: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A3 @ B8 )
| ( A3 = B8 ) ) ) ) ).
% order.order_iff_strict
thf(fact_769_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B8: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B8 )
| ( A3 = B8 ) ) ) ) ).
% order.order_iff_strict
thf(fact_770_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B8: set_nat] :
( ( ord_less_set_nat @ A3 @ B8 )
| ( A3 = B8 ) ) ) ) ).
% order.order_iff_strict
thf(fact_771_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B8: nat] :
( ( ord_less_nat @ A3 @ B8 )
| ( A3 = B8 ) ) ) ) ).
% order.order_iff_strict
thf(fact_772_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B8: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B8 )
| ( A3 = B8 ) ) ) ) ).
% order.order_iff_strict
thf(fact_773_order_Ostrict__iff__order,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B8: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B8 )
& ( A3 != B8 ) ) ) ) ).
% order.strict_iff_order
thf(fact_774_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B8: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B8 )
& ( A3 != B8 ) ) ) ) ).
% order.strict_iff_order
thf(fact_775_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B8: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B8 )
& ( A3 != B8 ) ) ) ) ).
% order.strict_iff_order
thf(fact_776_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B8: nat] :
( ( ord_less_eq_nat @ A3 @ B8 )
& ( A3 != B8 ) ) ) ) ).
% order.strict_iff_order
thf(fact_777_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B8: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B8 )
& ( A3 != B8 ) ) ) ) ).
% order.strict_iff_order
thf(fact_778_order_Ostrict__trans1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_779_order_Ostrict__trans1,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_780_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_781_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_782_order_Ostrict__trans1,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_783_order_Ostrict__trans2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_784_order_Ostrict__trans2,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_785_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_786_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_787_order_Ostrict__trans2,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_788_order_Ostrict__iff__not,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B8: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B8 )
& ~ ( ord_le9131159989063066194et_nat @ B8 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_789_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B8: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B8 )
& ~ ( ord_le6893508408891458716et_nat @ B8 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_790_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B8: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B8 )
& ~ ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_791_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B8: nat] :
( ( ord_less_eq_nat @ A3 @ B8 )
& ~ ( ord_less_eq_nat @ B8 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_792_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B8: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B8 )
& ~ ( ord_le9059583361652607317at_nat @ B8 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_793_dual__order_Oorder__iff__strict,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B8: set_set_set_nat,A3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B8 @ A3 )
| ( A3 = B8 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_794_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B8: set_set_nat,A3: set_set_nat] :
( ( ord_less_set_set_nat @ B8 @ A3 )
| ( A3 = B8 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_795_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B8: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B8 @ A3 )
| ( A3 = B8 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_796_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A3: nat] :
( ( ord_less_nat @ B8 @ A3 )
| ( A3 = B8 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_797_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B8: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B8 @ A3 )
| ( A3 = B8 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_798_dual__order_Ostrict__iff__order,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B8: set_set_set_nat,A3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B8 @ A3 )
& ( A3 != B8 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_799_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B8: set_set_nat,A3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B8 @ A3 )
& ( A3 != B8 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_800_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B8: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B8 @ A3 )
& ( A3 != B8 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_801_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B8: nat,A3: nat] :
( ( ord_less_eq_nat @ B8 @ A3 )
& ( A3 != B8 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_802_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B8: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B8 @ A3 )
& ( A3 != B8 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_803_dual__order_Ostrict__trans1,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( ord_le152980574450754630et_nat @ C @ B )
=> ( ord_le152980574450754630et_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_804_dual__order_Ostrict__trans1,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_805_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_806_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_807_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_less_set_nat_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_808_dual__order_Ostrict__trans2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ( ( ord_le9131159989063066194et_nat @ C @ B )
=> ( ord_le152980574450754630et_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_809_dual__order_Ostrict__trans2,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_810_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_811_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_812_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_813_dual__order_Ostrict__iff__not,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B8: set_set_set_nat,A3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B8 @ A3 )
& ~ ( ord_le9131159989063066194et_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_814_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B8: set_set_nat,A3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B8 @ A3 )
& ~ ( ord_le6893508408891458716et_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_815_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B8: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B8 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_816_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B8: nat,A3: nat] :
( ( ord_less_eq_nat @ B8 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_817_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B8: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B8 @ A3 )
& ~ ( ord_le9059583361652607317at_nat @ A3 @ B8 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_818_order_Ostrict__implies__order,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_819_order_Ostrict__implies__order,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_820_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_821_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_822_order_Ostrict__implies__order,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_823_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_824_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_825_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_826_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_827_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_828_order__le__less,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_829_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_830_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_831_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_832_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_833_order__less__le,axiom,
( ord_le152980574450754630et_nat
= ( ^ [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_834_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_835_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_836_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_837_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_838_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_839_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_840_order__less__imp__le,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_841_order__less__imp__le,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_842_order__less__imp__le,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_843_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_844_order__less__imp__le,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_845_order__le__neq__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_846_order__le__neq__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_847_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_848_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_849_order__le__neq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_850_order__neq__le__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A != B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_851_order__neq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A != B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_852_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_853_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_854_order__neq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A != B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_855_order__le__less__trans,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
=> ( ( ord_le152980574450754630et_nat @ Y3 @ Z2 )
=> ( ord_le152980574450754630et_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_856_order__le__less__trans,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
=> ( ( ord_less_set_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_857_order__le__less__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_858_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_859_order__le__less__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_860_order__less__le__trans,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
=> ( ( ord_le9131159989063066194et_nat @ Y3 @ Z2 )
=> ( ord_le152980574450754630et_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_861_order__less__le__trans,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ Z2 )
=> ( ord_less_set_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_862_order__less__le__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_863_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_864_order__less__le__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_865_order__le__less__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_866_order__le__less__subst1,axiom,
! [A: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_867_order__le__less__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_868_order__le__less__subst1,axiom,
! [A: set_set_nat,F: set_set_set_nat > set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_869_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_870_order__le__less__subst1,axiom,
! [A: set_nat,F: set_set_set_nat > set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_871_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_872_order__le__less__subst1,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_873_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_874_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: set_set_set_nat > set_nat_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_875_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_876_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_877_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_878_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_879_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_880_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_881_order__le__less__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_882_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_883_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_884_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_885_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_886_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_887_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_888_order__less__le__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_889_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_890_order__less__le__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_891_order__less__le__subst1,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_892_order__less__le__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_893_order__less__le__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_894_order__less__le__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_895_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_896_order__less__le__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_le152980574450754630et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_897_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_898_order__less__le__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > set_set_nat,C: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_899_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_900_order__less__le__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > set_nat,C: set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_901_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_902_order__less__le__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_903_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_904_order__less__le__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_905_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_906_order__le__imp__less__or__eq,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
=> ( ( ord_le152980574450754630et_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_907_order__le__imp__less__or__eq,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
=> ( ( ord_less_set_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_908_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_909_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_910_order__le__imp__less__or__eq,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_911_bot_Oextremum,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% bot.extremum
thf(fact_912_bot_Oextremum,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% bot.extremum
thf(fact_913_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_914_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_915_bot_Oextremum,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% bot.extremum
thf(fact_916_bot_Oextremum__unique,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_917_bot_Oextremum__unique,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_918_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_919_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_920_bot_Oextremum__unique,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_921_bot_Oextremum__uniqueI,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_922_bot_Oextremum__uniqueI,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_923_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_924_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_925_bot_Oextremum__uniqueI,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_926_bot_Oextremum__strict,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% bot.extremum_strict
thf(fact_927_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_928_bot_Oextremum__strict,axiom,
! [A: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% bot.extremum_strict
thf(fact_929_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_930_bot_Oextremum__strict,axiom,
! [A: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% bot.extremum_strict
thf(fact_931_bot_Onot__eq__extremum,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
= ( ord_less_set_set_nat @ bot_bot_set_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_932_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_933_bot_Onot__eq__extremum,axiom,
! [A: set_nat_nat] :
( ( A != bot_bot_set_nat_nat )
= ( ord_less_set_nat_nat @ bot_bot_set_nat_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_934_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_935_bot_Onot__eq__extremum,axiom,
! [A: set_set_set_nat] :
( ( A != bot_bo7198184520161983622et_nat )
= ( ord_le152980574450754630et_nat @ bot_bo7198184520161983622et_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_936_sunflower__subset,axiom,
! [F2: set_set_set_nat,G: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ F2 @ G )
=> ( ( sunflower_set_nat @ G )
=> ( sunflower_set_nat @ F2 ) ) ) ).
% sunflower_subset
thf(fact_937_sunflower__subset,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ F2 @ G )
=> ( ( sunflower_nat @ G )
=> ( sunflower_nat @ F2 ) ) ) ).
% sunflower_subset
thf(fact_938_empty__sunflower,axiom,
sunflower_set_nat @ bot_bo7198184520161983622et_nat ).
% empty_sunflower
thf(fact_939_empty__sunflower,axiom,
sunflower_nat @ bot_bot_set_set_nat ).
% empty_sunflower
thf(fact_940_joinl__join,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X6 @ Y5 ) @ ( clique5469973757772500719t_odot @ X6 @ Y5 ) ) ).
% joinl_join
thf(fact_941_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_942_ACC__cf__def,axiom,
! [X6: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ X6 )
= ( collect_nat_nat
@ ^ [F4: nat > nat] :
( ( member_nat_nat @ F4 @ ( clique2971579238625216137irst_F @ k ) )
& ( clique3686358387679108662ccepts @ X6 @ ( clique5033774636164728462irst_C @ k @ F4 ) ) ) ) ) ).
% ACC_cf_def
thf(fact_943_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_944_GsGl,axiom,
member_set_set_nat @ gs @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% GsGl
thf(fact_945_vGs,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ vs ).
% vGs
thf(fact_946_ACC__cf___092_060F_062,axiom,
! [X6: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X6 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_947_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_948_ACC__cf__mono,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ Y5 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X6 ) @ ( clique951075384711337423ACC_cf @ k @ Y5 ) ) ) ).
% ACC_cf_mono
thf(fact_949_finite__ACC,axiom,
! [X6: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X6 ) ) ).
% finite_ACC
thf(fact_950_ACC__cf__I,axiom,
! [F2: nat > nat,X6: set_set_set_nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X6 @ ( clique5033774636164728462irst_C @ k @ F2 ) )
=> ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ k @ X6 ) ) ) ) ).
% ACC_cf_I
thf(fact_951_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_952_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_953__092_060open_062_092_060And_062G_O_A_092_060lbrakk_062G_A_092_060in_062_AACC_AX_059_AG_A_092_060in_062_APOS_092_060rbrakk_062_A_092_060Longrightarrow_062_AG_A_092_060in_062_AACC_AY_092_060close_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ x ) )
=> ( ( member_set_set_nat @ G @ ( clique3326749438856946062irst_K @ k ) )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ y ) ) ) ) ).
% \<open>\<And>G. \<lbrakk>G \<in> ACC X; G \<in> POS\<rbrakk> \<Longrightarrow> G \<in> ACC Y\<close>
thf(fact_954_odotl__def,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X6 @ Y5 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X6 @ Y5 ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_955_Gs__def,axiom,
( gs
= ( clique6722202388162463298od_nat @ vs @ vs ) ) ).
% Gs_def
thf(fact_956_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_957_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_958_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_959_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_960_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_961_Int__iff,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C @ A2 )
& ( member_set_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_962_Int__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( ( member_set_set_nat @ C @ A2 )
& ( member_set_set_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_963_Int__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( ( member_nat_nat @ C @ A2 )
& ( member_nat_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_964_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_965_IntI,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_966_IntI,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( ( member_set_set_nat @ C @ B2 )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_967_IntI,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_968_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_969_ACC__odot,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X6 @ Y5 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X6 ) @ ( clique3210737319928189260st_ACC @ k @ Y5 ) ) ) ).
% ACC_odot
thf(fact_970_Int__subset__iff,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
& ( ord_le9131159989063066194et_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_971_Int__subset__iff,axiom,
! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_972_Int__subset__iff,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_973_Int__subset__iff,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_974_finite__Int,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F2 )
| ( finite1152437895449049373et_nat @ G ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_975_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_976_finite__Int,axiom,
! [F2: set_set_set_nat,G: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F2 )
| ( finite6739761609112101331et_nat @ G ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_977_finite__Int,axiom,
! [F2: set_nat_nat,G: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F2 )
| ( finite2115694454571419734at_nat @ G ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_978__092_060open_062POS_A_092_060inter_062_AACC_AX_A_092_060subseteq_062_AACC_AY_092_060close_062,axiom,
ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ x ) ) @ ( clique3210737319928189260st_ACC @ k @ y ) ).
% \<open>POS \<inter> ACC X \<subseteq> ACC Y\<close>
thf(fact_979_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_980_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_981_Collect__conj__eq,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in2396666505901392698et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_982_Collect__conj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_983_Collect__conj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in5711780100303410308et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_984_Collect__conj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_985_Int__Collect,axiom,
! [X2: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_986_Int__Collect,axiom,
! [X2: set_set_set_nat,A2: set_set_set_set_nat,P: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ X2 @ ( inf_in2396666505901392698et_nat @ A2 @ ( collec7201453139178570183et_nat @ P ) ) )
= ( ( member2946998982187404937et_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_987_Int__Collect,axiom,
! [X2: set_nat,A2: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_988_Int__Collect,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ X2 @ ( inf_in5711780100303410308et_nat @ A2 @ ( collect_set_set_nat @ P ) ) )
= ( ( member_set_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_989_Int__Collect,axiom,
! [X2: nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ ( collect_nat_nat @ P ) ) )
= ( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_990_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_991_Int__def,axiom,
( inf_in2396666505901392698et_nat
= ( ^ [A6: set_set_set_set_nat,B3: set_set_set_set_nat] :
( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A6 )
& ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_992_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
& ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_993_Int__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A6 )
& ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_994_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A6 )
& ( member_nat_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_995_Int__left__commute,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
= ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_996_Int__left__commute,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_997_Int__left__absorb,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_998_Int__left__absorb,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_999_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B3 @ A6 ) ) ) ).
% Int_commute
thf(fact_1000_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] : ( inf_inf_set_nat_nat @ B3 @ A6 ) ) ) ).
% Int_commute
thf(fact_1001_Int__absorb,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1002_Int__absorb,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1003_Int__assoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1004_Int__assoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1005_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1006_IntD2,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1007_IntD2,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ( member_set_set_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1008_IntD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ( member_nat_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1009_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1010_IntD1,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1011_IntD1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ( member_set_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1012_IntD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1013_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1014_IntE,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ~ ( member_set_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1015_IntE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_set_nat @ C @ A2 )
=> ~ ( member_set_set_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1016_IntE,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1017_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_1018_Int__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1019_Int__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1020_Int__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1021_Int__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1022_Int__lower1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1023_Int__lower1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1024_Int__lower1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1025_Int__lower1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1026_Int__lower2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1027_Int__lower2,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1028_Int__lower2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1029_Int__lower2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1030_Int__absorb1,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1031_Int__absorb1,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1032_Int__absorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1033_Int__absorb1,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1034_Int__absorb2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1035_Int__absorb2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1036_Int__absorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1037_Int__absorb2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1038_Int__greatest,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1039_Int__greatest,axiom,
! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1040_Int__greatest,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1041_Int__greatest,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1042_Int__Collect__mono,axiom,
! [A2: set_set_set_set_nat,B2: set_set_set_set_nat,P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ A2 @ B2 )
=> ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le572741076514265352et_nat @ ( inf_in2396666505901392698et_nat @ A2 @ ( collec7201453139178570183et_nat @ P ) ) @ ( inf_in2396666505901392698et_nat @ B2 @ ( collec7201453139178570183et_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1043_Int__Collect__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ ( collect_set_set_nat @ P ) ) @ ( inf_in5711780100303410308et_nat @ B2 @ ( collect_set_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1044_Int__Collect__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1045_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1046_Int__Collect__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B2 @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1047_Int__emptyI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ~ ( member_set_set_nat @ X @ B2 ) )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat ) ) ).
% Int_emptyI
thf(fact_1048_Int__emptyI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ~ ( member_set_nat @ X @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_1049_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ X @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1050_Int__emptyI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ~ ( member_nat_nat @ X @ B2 ) )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_1051_disjoint__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ~ ( member_set_set_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1052_disjoint__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ~ ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1053_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1054_disjoint__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ~ ( member_nat_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1055_Int__empty__left,axiom,
! [B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ B2 )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_left
thf(fact_1056_Int__empty__left,axiom,
! [B2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B2 )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_1057_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_1058_Int__empty__left,axiom,
! [B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ B2 )
= bot_bot_set_nat_nat ) ).
% Int_empty_left
thf(fact_1059_Int__empty__right,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_right
thf(fact_1060_Int__empty__right,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_1061_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1062_Int__empty__right,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% Int_empty_right
thf(fact_1063_disjoint__iff__not__equal,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ! [Y: set_set_nat] :
( ( member_set_set_nat @ Y @ B2 )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1064_disjoint__iff__not__equal,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ B2 )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1065_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1066_disjoint__iff__not__equal,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ B2 )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1067_ivl__disj__int__two_I3_J,axiom,
! [L: set_set_nat,M3: set_set_nat,U: set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( set_or5410080298493297259et_nat @ L @ M3 ) @ ( set_or5410080298493297259et_nat @ M3 @ U ) )
= bot_bo7198184520161983622et_nat ) ).
% ivl_disj_int_two(3)
thf(fact_1068_ivl__disj__int__two_I3_J,axiom,
! [L: set_nat,M3: set_nat,U: set_nat] :
( ( inf_inf_set_set_nat @ ( set_or3540276404033026485et_nat @ L @ M3 ) @ ( set_or3540276404033026485et_nat @ M3 @ U ) )
= bot_bot_set_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_1069_ivl__disj__int__two_I3_J,axiom,
! [L: nat > nat,M3: nat > nat,U: nat > nat] :
( ( inf_inf_set_nat_nat @ ( set_or1770121190487188718at_nat @ L @ M3 ) @ ( set_or1770121190487188718at_nat @ M3 @ U ) )
= bot_bot_set_nat_nat ) ).
% ivl_disj_int_two(3)
thf(fact_1070_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_1071_second__assumptions_Ov__sameprod__subset,axiom,
! [L: nat,P5: nat,K2: nat,Vs: set_nat] :
( ( assump2881078719466019805ptions @ L @ P5 @ K2 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ) ).
% second_assumptions.v_sameprod_subset
thf(fact_1072_sameprod__mono,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ Y5 )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X6 @ X6 ) @ ( clique1181040904276305582et_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_1073_sameprod__mono,axiom,
! [X6: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X6 @ Y5 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X6 @ X6 ) @ ( clique8906516429304539640et_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_1074_sameprod__mono,axiom,
! [X6: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X6 @ Y5 )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X6 @ X6 ) @ ( clique134924887794942129at_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_1075_sameprod__mono,axiom,
! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X6 @ X6 ) @ ( clique6722202388162463298od_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_1076_sameprod__finite,axiom,
! [X6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X6 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X6 @ X6 ) ) ) ).
% sameprod_finite
thf(fact_1077_sameprod__finite,axiom,
! [X6: set_set_nat] :
( ( finite1152437895449049373et_nat @ X6 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X6 @ X6 ) ) ) ).
% sameprod_finite
thf(fact_1078_sameprod__finite,axiom,
! [X6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X6 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X6 @ X6 ) ) ) ).
% sameprod_finite
thf(fact_1079_sameprod__finite,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X6 @ X6 ) ) ) ).
% sameprod_finite
thf(fact_1080_inf__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_1081_inf__bot__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_1082_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1083_inf__bot__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_1084_inf__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_1085_inf__bot__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_1086_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1087_inf__bot__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_1088_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1089_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1090_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1091_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1092_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1093_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1094_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1095_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1096_le__inf__iff,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y3 @ Z2 ) )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Y3 )
& ( ord_le9131159989063066194et_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1097_le__inf__iff,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y3 @ Z2 ) )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
& ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1098_le__inf__iff,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1099_le__inf__iff,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1100_le__inf__iff,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y3 @ Z2 ) )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
& ( ord_le9059583361652607317at_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1101_ACC__cf__odot,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X6 @ Y5 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X6 ) @ ( clique951075384711337423ACC_cf @ k @ Y5 ) ) ) ).
% ACC_cf_odot
thf(fact_1102_inf_Obounded__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) )
= ( ( ord_le9131159989063066194et_nat @ A @ B )
& ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1103_inf_Obounded__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A @ B )
& ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1104_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1105_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1106_inf_Obounded__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A @ B )
& ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1107_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1108_inf__set__def,axiom,
( inf_in2396666505901392698et_nat
= ( ^ [A6: set_set_set_set_nat,B3: set_set_set_set_nat] :
( collec7201453139178570183et_nat
@ ( inf_in8098123048512461259_nat_o
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A6 )
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1109_inf__set__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A6: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ( inf_inf_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1110_inf__set__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A6: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ( inf_in2551356467856225537_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A6 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1111_inf__set__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A6: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ( inf_inf_nat_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A6 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1112_inf__sup__ord_I2_J,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1113_inf__sup__ord_I2_J,axiom,
! [X2: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1114_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1115_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1116_inf__sup__ord_I2_J,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1117_inf__sup__ord_I1_J,axiom,
! [X2: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1118_inf__sup__ord_I1_J,axiom,
! [X2: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1119_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1120_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1121_inf__sup__ord_I1_J,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1122_inf__le1,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_1123_inf__le1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_1124_inf__le1,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_1125_second__assumptions_OLm,axiom,
! [L: nat,P5: nat,K2: nat] :
( ( assump2881078719466019805ptions @ L @ P5 @ K2 )
=> ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K2 ) @ ( assump1710595444109740301irst_L @ L @ P5 ) ) ) ).
% second_assumptions.Lm
thf(fact_1126_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_1127_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1128_Vsm,axiom,
ord_less_eq_set_nat @ vs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ).
% Vsm
thf(fact_1129_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1130_ACC__union,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X6 @ Y5 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X6 ) @ ( clique3210737319928189260st_ACC @ k @ Y5 ) ) ) ).
% ACC_union
thf(fact_1131_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1132_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1133_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1134_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_1135_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_1136_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1137_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1138_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_1139_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_1140_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_1141_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1142_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1143_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1144_le__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1145_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1146_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1147_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_1148_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1149_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1150_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1151_diff__less__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1152_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1153_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_1154_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1155_less__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1156_numbers2__mono,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X2 ) @ ( clique3652268606331196573umbers @ X2 ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y3 ) @ ( clique3652268606331196573umbers @ Y3 ) ) ) ) ).
% numbers2_mono
thf(fact_1157_first__assumptions_OL_Ocong,axiom,
assump1710595444109740301irst_L = assump1710595444109740301irst_L ).
% first_assumptions.L.cong
thf(fact_1158_second__assumptions_OLp,axiom,
! [L: nat,P5: nat,K2: nat] :
( ( assump2881078719466019805ptions @ L @ P5 @ K2 )
=> ( ord_less_nat @ P5 @ ( assump1710595444109740301irst_L @ L @ P5 ) ) ) ).
% second_assumptions.Lp
thf(fact_1159__092_060G_062l__def,axiom,
( ( clique7840962075309931874st_G_l @ l @ k )
= ( collect_set_set_nat
@ ^ [G3: set_set_nat] :
( ( member_set_set_nat @ G3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ord_less_eq_nat @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G3 ) ) @ l ) ) ) ) ).
% \<G>l_def
thf(fact_1160__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [K4: set_set_nat] :
( ( member_set_set_nat @ K4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K4 ) )
= k )
& ( K4
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K4 ) @ ( clique5033774636164728513irst_v @ K4 ) ) ) ) ) ) ).
% \<K>_def
thf(fact_1161_finite__v__gs,axiom,
! [X6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X6 ) ) ) ).
% finite_v_gs
thf(fact_1162__092_060K_062__altdef,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
& ( ( finite_card_nat @ V )
= k ) ) ) ) ).
% \<K>_altdef
thf(fact_1163_v__union,axiom,
! [G: set_set_nat,H4: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H4 ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H4 ) ) ) ).
% v_union
thf(fact_1164_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_1165_v__gs__union,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X6 @ Y5 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X6 ) @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ).
% v_gs_union
thf(fact_1166_ACC__cf__union,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X6 @ Y5 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X6 ) @ ( clique951075384711337423ACC_cf @ k @ Y5 ) ) ) ).
% ACC_cf_union
thf(fact_1167_vplus__dsXU,axiom,
( ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) )
= ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) ) ).
% vplus_dsXU
thf(fact_1168_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_1169_finite__members___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ).
% finite_members_\<G>
thf(fact_1170_GsG,axiom,
member_set_set_nat @ gs @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% GsG
thf(fact_1171__092_060open_062card_A_Iv__gs_AX_A_N_Av__gs_AU_J_A_061_Acard_A_Iv__gs_AX_J_A_N_Acard_A_Iv__gs_AU_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) ) ) ) ).
% \<open>card (v_gs X - v_gs U) = card (v_gs X) - card (v_gs U)\<close>
thf(fact_1172_finite__vG,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ).
% finite_vG
thf(fact_1173_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_1174__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_1175_odot___092_060G_062,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X6 @ Y5 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_1176_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_1177__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
= ( collect_set_set_nat
@ ^ [G3: set_set_nat] : ( ord_le6893508408891458716et_nat @ G3 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<G>_def
thf(fact_1178_ACC__def,axiom,
! [X6: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ X6 )
= ( collect_set_set_nat
@ ^ [G3: set_set_nat] :
( ( member_set_set_nat @ G3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( clique3686358387679108662ccepts @ X6 @ G3 ) ) ) ) ).
% ACC_def
thf(fact_1179_XD,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% XD
thf(fact_1180__092_060open_062U_A_092_060subseteq_062_A_092_060G_062_092_060close_062,axiom,
ord_le9131159989063066194et_nat @ u @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<open>U \<subseteq> \<G>\<close>
thf(fact_1181_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_1182_SvG,axiom,
( s
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ ( image_2194112158459175443et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ) ).
% SvG
thf(fact_1183_union___092_060G_062,axiom,
! [G: set_set_nat,H4: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ H4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H4 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% union_\<G>
thf(fact_1184_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_1185_ACC__I,axiom,
! [G: set_set_nat,X6: set_set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X6 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ X6 ) ) ) ) ).
% ACC_I
thf(fact_1186_Graphs__def,axiom,
( clique5786534781347292306Graphs
= ( ^ [V: set_nat] :
( collect_set_set_nat
@ ^ [G3: set_set_nat] : ( ord_le6893508408891458716et_nat @ G3 @ ( clique6722202388162463298od_nat @ V @ V ) ) ) ) ) ).
% Graphs_def
thf(fact_1187_Clique__def,axiom,
( clique6749503327923060270Clique
= ( ^ [V: set_nat,K3: nat] :
( collect_set_set_nat
@ ^ [G3: set_set_nat] :
( ( member_set_set_nat @ G3 @ ( clique5786534781347292306Graphs @ V ) )
& ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ C3 @ C3 ) @ G3 )
& ( ( finite_card_nat @ C3 )
= K3 ) ) ) ) ) ) ).
% Clique_def
thf(fact_1188__092_060open_062card_A_Iv__gs_AY_J_A_061_Acard_A_Iv__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) )
= ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% \<open>card (v_gs Y) = card (v_gs (X - U \<union> {Gs}))\<close>
thf(fact_1189_card__v__gs__join,axiom,
! [X6: set_set_set_nat,Y5: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X6 @ Y5 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X6 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_1190_odot__def,axiom,
( clique5469973757772500719t_odot
= ( ^ [X5: set_set_set_nat,Y7: set_set_set_nat] :
( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D3: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D3 @ E ) )
& ( member_set_set_nat @ D3 @ X5 )
& ( member_set_set_nat @ E @ Y7 ) ) ) ) ) ).
% odot_def
thf(fact_1191_mult__cancel2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M3 @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M3 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1192_mult__cancel1,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M3 )
= ( times_times_nat @ K2 @ N ) )
= ( ( M3 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1193_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1194_mult__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1195_mult__less__cancel2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M3 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1196_nat__0__less__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1197_Y,axiom,
( y
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ).
% Y
thf(fact_1198__092_060open_062v__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_A_061_Av__gs_A_IX_A_N_AU_J_A_092_060union_062_Av__gs_A_123Gs_125_092_060close_062,axiom,
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) @ ( clique8462013130872731469t_v_gs @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% \<open>v_gs (X - U \<union> {Gs}) = v_gs (X - U) \<union> v_gs {Gs}\<close>
thf(fact_1199_mult__le__cancel2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1200_diff__mult__distrib2,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M3 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1201_diff__mult__distrib,axiom,
! [M3: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1202_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1203_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_1204_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_1205_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1206_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1207_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1208_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1209_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1210_plucking__step__def,axiom,
! [X6: set_set_set_nat] :
( ( clique4095374090462327202g_step @ p @ X6 )
= ( sup_su4213647025997063966et_nat
@ ( minus_2447799839930672331et_nat @ X6
@ ( collect_set_set_nat
@ ^ [E: set_set_nat] :
( ( member_set_set_nat @ E @ X6 )
& ( member_set_nat @ ( clique5033774636164728513irst_v @ E )
@ ( fChoice_set_set_nat
@ ^ [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( clique8462013130872731469t_v_gs @ X6 ) )
& ( sunflower_nat @ S2 )
& ( ( finite_card_set_nat @ S2 )
= p ) ) ) ) ) ) )
@ ( insert_set_set_nat
@ ( clique6722202388162463298od_nat
@ ( comple7806235888213564991et_nat
@ ( fChoice_set_set_nat
@ ^ [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( clique8462013130872731469t_v_gs @ X6 ) )
& ( sunflower_nat @ S2 )
& ( ( finite_card_set_nat @ S2 )
= p ) ) ) )
@ ( comple7806235888213564991et_nat
@ ( fChoice_set_set_nat
@ ^ [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( clique8462013130872731469t_v_gs @ X6 ) )
& ( sunflower_nat @ S2 )
& ( ( finite_card_set_nat @ S2 )
= p ) ) ) ) )
@ bot_bo7198184520161983622et_nat ) ) ) ).
% plucking_step_def
thf(fact_1211_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1212_v__def,axiom,
( clique5033774636164728513irst_v
= ( ^ [G3: set_set_nat] :
( collect_nat
@ ^ [X3: nat] :
? [Y: nat] : ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ G3 ) ) ) ) ).
% v_def
thf(fact_1213_Vs__def,axiom,
( vs
= ( comple7806235888213564991et_nat @ s ) ) ).
% Vs_def
thf(fact_1214_C__def,axiom,
! [F: nat > nat] :
( ( clique5033774636164728462irst_C @ k @ F )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat,Y: nat] :
( ( Uu
= ( insert_nat @ X3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( F @ X3 )
!= ( F @ Y ) ) ) ) ) ).
% C_def
thf(fact_1215__092_060open_062v__gs_A_123Gs_125_A_061_A_123v_AGs_125_092_060close_062,axiom,
( ( clique8462013130872731469t_v_gs @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ).
% \<open>v_gs {Gs} = {v Gs}\<close>
thf(fact_1216_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M3 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1217_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y3: nat,X2: nat] :
( ( ( ord_less_nat @ C @ Y3 )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y3 )
=> ( ( ( ord_less_nat @ X2 @ Y3 )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_1218_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M3 )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M3 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1219_nat__mult__eq__cancel1,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M3 )
= ( times_times_nat @ K2 @ N ) )
= ( M3 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1220_nat__mult__less__cancel1,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1221_nat__mult__le__cancel1,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1222__092_060open_062card_A_Iv__gs_A_IX_A_N_AU_J_A_092_060union_062_A_123v_AGs_125_J_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_Acard_A_123v_AGs_125_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ ( finite_card_set_nat @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) ).
% \<open>card (v_gs (X - U) \<union> {v Gs}) \<le> card (v_gs (X - U)) + card {v Gs}\<close>
thf(fact_1223_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1224_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1225_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1226_nat__add__left__cancel__less,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1227_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1228_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1229_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1230_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1231_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1232_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1233_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1234_diff__cancel2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1235_Nat_Odiff__cancel,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1236_add__mult__distrib2,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1237_add__mult__distrib,axiom,
! [M3: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M3 @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1238_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1239_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K2: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1240_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1241_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1242_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1243_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1244_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1245_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1246_add__diff__inverse__nat,axiom,
! [M3: nat,N: nat] :
( ~ ( ord_less_nat @ M3 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M3 @ N ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1247_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1248_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1249_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1250_add__leE,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1251_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_1252_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_1253_add__leD1,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_1254_add__leD2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1255_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1256_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1257_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1258_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1259_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1260_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1261_less__add__eq__less,axiom,
! [K2: nat,L: nat,M3: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1262_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1263_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1264_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1265_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1266_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1267_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1268_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
% Helper facts (7)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X5: nat] : ( P @ X5 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: set_nat > $o] :
( ( P @ ( fChoice_set_nat @ P ) )
= ( ? [X5: set_nat] : ( P @ X5 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X5: nat > nat] : ( P @ X5 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [P: set_set_nat > $o] :
( ( P @ ( fChoice_set_set_nat @ P ) )
= ( ? [X5: set_set_nat] : ( P @ X5 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( si @ i )
= ( si @ j ) ) ).
%------------------------------------------------------------------------------