TPTP Problem File: SLH0190^1.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_00793_027510__16236350_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1485 ( 519 unt; 210 typ; 0 def)
% Number of atoms : 3723 (1013 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 10654 ( 245 ~; 27 |; 308 &;8438 @)
% ( 0 <=>;1636 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 1102 (1102 >; 0 *; 0 +; 0 <<)
% Number of symbols : 198 ( 195 usr; 23 con; 0-4 aty)
% Number of variables : 3214 ( 98 ^;2916 !; 200 ?;3214 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:10.820
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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% Explicit typings (195)
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo6227097192321305471_nat_o: set_set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
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_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $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_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5298321079317455902at_nat: ( $o > set_nat_nat ) > ( $o > set_nat_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le6539261115178940645et_nat: ( $o > set_set_nat ) > ( $o > set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le8326115459943588763et_nat: ( $o > set_set_set_nat ) > ( $o > set_set_set_nat ) > $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_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_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_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > 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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_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__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_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__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8569768528772619084at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1454916318497077779at_nat: ( set_set_nat > nat ) > set_set_set_nat > 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_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_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_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1140352010380016476at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or6631954706645296601et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_or6241808948974817167et_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_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_Us____,type,
us: set_nat ).
thf(sy_v_Vs____,type,
vs: set_nat ).
thf(sy_v_Ws____,type,
ws: 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_f____,type,
f: nat > nat ).
thf(sy_v_fstt____,type,
fstt: set_nat > nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_pair____,type,
pair: nat > set_nat ).
thf(sy_v_r____,type,
r: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_si____,type,
si2: nat > nat ).
thf(sy_v_sndd____,type,
sndd: set_nat > nat ).
thf(sy_v_ti____,type,
ti: nat > nat ).
thf(sy_v_u____,type,
u: nat > nat ).
thf(sy_v_w____,type,
w: nat > nat ).
% Relevant facts (1271)
thf(fact_0__092_060open_062finite_AUs_092_060close_062,axiom,
finite_finite_nat @ us ).
% \<open>finite Us\<close>
thf(fact_1_p0,axiom,
p != zero_zero_nat ).
% p0
thf(fact_2_rq,axiom,
ord_less_eq_nat @ p @ r ).
% rq
thf(fact_3_s__def,axiom,
( s2
= ( finite_card_nat @ vs ) ) ).
% s_def
thf(fact_4_assms_I3_J,axiom,
( y
= ( clique4095374090462327202g_step @ p @ x ) ) ).
% assms(3)
thf(fact_5_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_6_S_I3_J,axiom,
( ( finite_card_set_nat @ s )
= p ) ).
% S(3)
thf(fact_7_Us__def,axiom,
( us
= ( image_nat_nat @ u @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ).
% Us_def
thf(fact_8_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_9_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_10_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_11_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_12_fin__Vs,axiom,
finite_finite_nat @ vs ).
% fin_Vs
thf(fact_13_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_14_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_15_card__Vs,axiom,
ord_less_eq_nat @ ( finite_card_nat @ vs ) @ l ).
% card_Vs
thf(fact_16_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_17_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_18_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_19_lessThan__iff,axiom,
! [I: set_set_nat,K: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or6631954706645296601et_nat @ K ) )
= ( ord_less_set_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_20_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_21_lessThan__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_22_lessThan__iff,axiom,
! [I: set_set_set_nat,K: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I @ ( set_or6241808948974817167et_nat @ K ) )
= ( ord_le152980574450754630et_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_23_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_24_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_25_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_26_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_36_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_37_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_38_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_39_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_40_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_nat,R: set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B4: nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_41_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_nat,R: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B3: set_nat] :
( ( member_set_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_42_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_set_nat,R: set_set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ A2 )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B4: nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_43_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_nat,R: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat > nat,A22: nat > nat,B4: nat] :
( ( member_nat_nat @ A1 @ A2 )
=> ( ( member_nat_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_44_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_nat,A2: set_nat,R: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B3: set_set_nat] :
( ( member_set_set_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: set_set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_set_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_45_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
=> ? [B3: set_nat] :
( ( member_set_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B4: set_nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_46_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_nat,A2: set_nat,R: nat > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B3: nat > nat] :
( ( member_nat_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat > nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_47_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_nat,A2: set_set_nat,R: set_nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
=> ? [B3: set_set_nat] :
( ( member_set_set_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B4: set_set_nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_set_set_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_48_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_set_set_nat,R: set_set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ A2 )
=> ? [B3: set_nat] :
( ( member_set_nat @ B3 @ B2 )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B4: set_nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_49_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_50_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_51_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_52_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_53_numbers__def,axiom,
clique3652268606331196573umbers = set_ord_lessThan_nat ).
% numbers_def
thf(fact_54_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_55_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_56_i__props_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_nat @ s2 @ ( si2 @ I ) ) ) ).
% i_props(4)
thf(fact_57_i__props_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_nat @ ( si2 @ I ) @ l ) ) ).
% i_props(5)
thf(fact_58_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_59_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_60_card_Oinfinite,axiom,
! [A2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_61_card_Oinfinite,axiom,
! [A2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_card_set_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_62_card_Oinfinite,axiom,
! [A2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_card_nat_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_63_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_64_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_65_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_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_67_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X4: set_set_nat] : ( member_set_set_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X4: nat > nat] : ( member_nat_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_sf__precond,axiom,
! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( clique8462013130872731469t_v_gs @ x ) )
=> ( ( finite_finite_nat @ X3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X3 ) @ l ) ) ) ).
% sf_precond
thf(fact_72_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_73_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_74_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_75_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_76_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_77_finite__imageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_78_finite__imageI,axiom,
! [F: set_nat,H: nat > set_nat] :
( ( finite_finite_nat @ F )
=> ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_79_finite__imageI,axiom,
! [F: set_set_nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ F )
=> ( finite_finite_nat @ ( image_set_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_80_finite__imageI,axiom,
! [F: set_nat,H: nat > set_set_nat] :
( ( finite_finite_nat @ F )
=> ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_81_finite__imageI,axiom,
! [F: set_nat,H: nat > nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ H @ F ) ) ) ).
% finite_imageI
thf(fact_82_finite__imageI,axiom,
! [F: set_set_set_nat,H: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ F )
=> ( finite_finite_nat @ ( image_1454916318497077779at_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_83_finite__imageI,axiom,
! [F: set_set_nat,H: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ F )
=> ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_84_finite__imageI,axiom,
! [F: set_nat_nat,H: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_85_finite__imageI,axiom,
! [F: set_set_set_nat,H: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ F )
=> ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_86_finite__imageI,axiom,
! [F: set_set_nat,H: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ F )
=> ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_87_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_88_v__gs__mono,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_mono
thf(fact_89__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_90_finX,axiom,
finite6739761609112101331et_nat @ x ).
% finX
thf(fact_91_finS,axiom,
finite1152437895449049373et_nat @ s ).
% finS
thf(fact_92_fin1,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% fin1
thf(fact_93_S_I1_J,axiom,
ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) ).
% S(1)
thf(fact_94_L,axiom,
ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) ).
% L
thf(fact_95_X,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% X
thf(fact_96_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_97_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_98_finite__psubset__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ! [B5: set_set_nat] :
( ( ord_less_set_set_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_99_finite__psubset__induct,axiom,
! [A2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ! [A4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A4 )
=> ( ! [B5: set_nat_nat] :
( ( ord_less_set_nat_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_100_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 )
=> ( ! [B5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_101_all__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_102_all__subset__image,axiom,
! [F2: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_set_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_103_all__subset__image,axiom,
! [F2: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_set_nat_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ A2 )
=> ( P @ ( image_set_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_104_all__subset__image,axiom,
! [F2: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B6 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_2194112158459175443et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_105_all__subset__image,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ A2 )
=> ( P @ ( image_7916887816326733075et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_106_all__subset__image,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B6 @ A2 )
=> ( P @ ( image_1454916318497077779at_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_107_all__subset__image,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_108_all__subset__image,axiom,
! [F2: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_109_all__subset__image,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B6 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ A2 )
=> ( P @ ( image_6725021117256019401et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_110_all__subset__image,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B6 @ A2 )
=> ( P @ ( image_5842784325960735177et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_111_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_112_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_113_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_114_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_115_infinite__super,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S @ T )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ T ) ) ) ).
% infinite_super
thf(fact_116_infinite__super,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T ) ) ) ).
% infinite_super
thf(fact_117_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_118_infinite__super,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ T )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ T ) ) ) ).
% infinite_super
thf(fact_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_all__finite__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_128_all__finite__subset__image,axiom,
! [F2: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_set_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_129_all__finite__subset__image,axiom,
! [F2: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_set_nat_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A2 ) )
=> ( P @ ( image_set_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_130_all__finite__subset__image,axiom,
! [F2: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B6: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_2194112158459175443et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_131_all__finite__subset__image,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A2 ) )
=> ( P @ ( image_7916887816326733075et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_132_all__finite__subset__image,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ A2 ) )
=> ( P @ ( image_1454916318497077779at_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_133_all__finite__subset__image,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_134_all__finite__subset__image,axiom,
! [F2: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat_nat2 @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_135_all__finite__subset__image,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B6: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A2 ) )
=> ( P @ ( image_6725021117256019401et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_136_all__finite__subset__image,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ A2 ) )
=> ( P @ ( image_5842784325960735177et_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_137_ex__finite__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_138_ex__finite__subset__image,axiom,
! [F2: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_set_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_139_ex__finite__subset__image,axiom,
! [F2: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_set_nat_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A2 )
& ( P @ ( image_set_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_140_ex__finite__subset__image,axiom,
! [F2: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ? [B6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_2194112158459175443et_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_141_ex__finite__subset__image,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A2 )
& ( P @ ( image_7916887816326733075et_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_142_ex__finite__subset__image,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ A2 )
& ( P @ ( image_1454916318497077779at_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_143_ex__finite__subset__image,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_144_ex__finite__subset__image,axiom,
! [F2: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat_nat2 @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_145_ex__finite__subset__image,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ? [B6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A2 )
& ( P @ ( image_6725021117256019401et_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_146_ex__finite__subset__image,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ A2 )
& ( P @ ( image_5842784325960735177et_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_147_finite__subset__image,axiom,
! [B2: set_nat,F2: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( finite_finite_nat @ C2 )
& ( B2
= ( image_nat_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_148_finite__subset__image,axiom,
! [B2: set_set_nat,F2: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( finite_finite_nat @ C2 )
& ( B2
= ( image_nat_set_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_149_finite__subset__image,axiom,
! [B2: set_nat,F2: set_nat > nat,A2: set_set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F2 @ A2 ) )
=> ? [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
& ( finite1152437895449049373et_nat @ C2 )
& ( B2
= ( image_set_nat_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_150_finite__subset__image,axiom,
! [B2: set_set_set_nat,F2: nat > set_set_nat,A2: set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( finite_finite_nat @ C2 )
& ( B2
= ( image_2194112158459175443et_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_151_finite__subset__image,axiom,
! [B2: set_set_nat,F2: set_nat > set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ? [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
& ( finite1152437895449049373et_nat @ C2 )
& ( B2
= ( image_7916887816326733075et_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_152_finite__subset__image,axiom,
! [B2: set_nat,F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
=> ? [C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
& ( finite6739761609112101331et_nat @ C2 )
& ( B2
= ( image_1454916318497077779at_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_153_finite__subset__image,axiom,
! [B2: set_nat,F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ? [C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
& ( finite2115694454571419734at_nat @ C2 )
& ( B2
= ( image_nat_nat_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_154_finite__subset__image,axiom,
! [B2: set_nat_nat,F2: nat > nat > nat,A2: set_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( finite_finite_nat @ C2 )
& ( B2
= ( image_nat_nat_nat2 @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_155_finite__subset__image,axiom,
! [B2: set_set_set_nat,F2: set_nat > set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
=> ? [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
& ( finite1152437895449049373et_nat @ C2 )
& ( B2
= ( image_6725021117256019401et_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_156_finite__subset__image,axiom,
! [B2: set_set_nat,F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ? [C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
& ( finite6739761609112101331et_nat @ C2 )
& ( B2
= ( image_5842784325960735177et_nat @ F2 @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_157_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_158_finite__surj,axiom,
! [A2: set_nat,B2: set_set_nat,F2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_159_finite__surj,axiom,
! [A2: set_set_nat,B2: set_nat,F2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_160_finite__surj,axiom,
! [A2: set_nat,B2: set_set_set_nat,F2: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_161_finite__surj,axiom,
! [A2: set_set_nat,B2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_162_finite__surj,axiom,
! [A2: set_set_set_nat,B2: set_nat,F2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_163_finite__surj,axiom,
! [A2: set_nat_nat,B2: set_nat,F2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_164_finite__surj,axiom,
! [A2: set_nat,B2: set_nat_nat,F2: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_165_finite__surj,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,F2: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
=> ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_166_finite__surj,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,F2: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_set_nat,C3: nat] :
( ! [G: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G @ F )
=> ( ( finite6739761609112101331et_nat @ G )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G ) @ C3 ) ) )
=> ( ( finite6739761609112101331et_nat @ F )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_180_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_nat,C3: nat] :
( ! [G: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ F )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C3 ) ) )
=> ( ( finite1152437895449049373et_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_181_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C3: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_182_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat_nat,C3: nat] :
( ! [G: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ G @ F )
=> ( ( finite2115694454571419734at_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ G ) @ C3 ) ) )
=> ( ( finite2115694454571419734at_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_183_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ S ) )
=> ~ ! [T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T2 @ S )
=> ( ( ( finite1149291290879098388et_nat @ T2 )
= N )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_184_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S ) )
=> ~ ! [T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T2 @ S )
=> ( ( ( finite_card_set_nat @ T2 )
= N )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_185_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_186_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ S ) )
=> ~ ! [T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ T2 @ S )
=> ( ( ( finite_card_nat_nat @ T2 )
= N )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_187_exists__subset__between,axiom,
! [A2: set_set_set_nat,N: nat,C3: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ C3 ) )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C3 )
=> ( ( finite6739761609112101331et_nat @ C3 )
=> ? [B7: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B7 )
& ( ord_le9131159989063066194et_nat @ B7 @ C3 )
& ( ( finite1149291290879098388et_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_188_exists__subset__between,axiom,
! [A2: set_set_nat,N: nat,C3: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C3 ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( finite1152437895449049373et_nat @ C3 )
=> ? [B7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B7 )
& ( ord_le6893508408891458716et_nat @ B7 @ C3 )
& ( ( finite_card_set_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_189_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C3 )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_190_exists__subset__between,axiom,
! [A2: set_nat_nat,N: nat,C3: set_nat_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ C3 ) )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C3 )
=> ( ( finite2115694454571419734at_nat @ C3 )
=> ? [B7: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B7 )
& ( ord_le9059583361652607317at_nat @ B7 @ C3 )
& ( ( finite_card_nat_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_191_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_192_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_193_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_194_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_195_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_196_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_197_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_198_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_199_surj__card__le,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_200_surj__card__le,axiom,
! [A2: set_nat,B2: set_set_nat,F2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_201_surj__card__le,axiom,
! [A2: set_set_nat,B2: set_nat,F2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_202_surj__card__le,axiom,
! [A2: set_nat,B2: set_set_set_nat,F2: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_203_surj__card__le,axiom,
! [A2: set_set_nat,B2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_204_surj__card__le,axiom,
! [A2: set_set_set_nat,B2: set_nat,F2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_205_surj__card__le,axiom,
! [A2: set_nat_nat,B2: set_nat,F2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_206_surj__card__le,axiom,
! [A2: set_nat,B2: set_nat_nat,F2: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_207_surj__card__le,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,F2: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_208_surj__card__le,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,F2: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_209_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_210_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_211_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_212_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_213_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_214_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_215_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_216_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_217_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_218_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_219_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_220_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_221_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_222_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_223_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_224_subset__eq__atLeast0__lessThan__finite,axiom,
! [N2: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_225_subset__eq__atLeast0__lessThan__card,axiom,
! [N2: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N2 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_226_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_227_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_228_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_229_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_230_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_231_finite__has__minimal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( ord_less_eq_nat_nat @ X2 @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_232_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_233_finite__has__minimal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A2 )
& ( ord_le9131159989063066194et_nat @ X2 @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_234_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( ord_le6893508408891458716et_nat @ X2 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_235_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_236_finite__has__minimal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
& ( ord_le9059583361652607317at_nat @ X2 @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_237_finite__has__maximal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( ord_less_eq_nat_nat @ A @ X2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_238_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_239_finite__has__maximal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A2 )
& ( ord_le9131159989063066194et_nat @ A @ X2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_240_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_241_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_242_finite__has__maximal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
& ( ord_le9059583361652607317at_nat @ A @ X2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_243_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_244_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_245_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_246_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_247_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_248_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_249_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_250_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_251_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_252_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_253_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_254_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_255_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_256_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_257_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
| ( M4 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_258_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_259_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
& ( M4 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_260_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_261_card__image__le,axiom,
! [A2: set_nat,F2: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_262_card__image__le,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_263_card__image__le,axiom,
! [A2: set_nat,F2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_264_card__image__le,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_265_card__image__le,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_266_card__image__le,axiom,
! [A2: set_set_nat,F2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F2 @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_267_card__image__le,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_268_card__image__le,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > set_set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_269_card__image__le,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ F2 @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_270_card__image__le,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_7432509271690132940et_nat @ F2 @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_271_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_272_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_273_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_274_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_275__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_276_finite__v__gs__Gl,axiom,
! [X5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ).
% finite_v_gs_Gl
thf(fact_277__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_278_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_279_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_280_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_281_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_282_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_283_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_284_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_285_Si,axiom,
bij_betw_nat_set_nat @ si @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ).
% Si
thf(fact_286_inj,axiom,
inj_on_nat_nat @ u @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ).
% inj
thf(fact_287_second__assumptions_OLp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).
% second_assumptions.Lp
thf(fact_288_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_289_S_I2_J,axiom,
sunflower_nat @ s ).
% S(2)
thf(fact_290_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_291_image__eqI,axiom,
! [B: nat,F2: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_292_image__eqI,axiom,
! [B: nat,F2: set_nat > nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_set_nat_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_293_image__eqI,axiom,
! [B: set_nat,F2: nat > set_nat,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_294_image__eqI,axiom,
! [B: nat,F2: set_set_nat > nat,X: set_set_nat,A2: set_set_set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_1454916318497077779at_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_295_image__eqI,axiom,
! [B: set_nat,F2: set_nat > set_nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_296_image__eqI,axiom,
! [B: set_set_nat,F2: nat > set_set_nat,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_297_image__eqI,axiom,
! [B: nat > nat,F2: nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_298_image__eqI,axiom,
! [B: nat,F2: ( nat > nat ) > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_299_image__eqI,axiom,
! [B: set_nat,F2: set_set_nat > set_nat,X: set_set_nat,A2: set_set_set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_300_image__eqI,axiom,
! [B: set_set_nat,F2: set_nat > set_set_nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_301_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_302_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_303_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_304_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_305_subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ X2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_306_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_307_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_308_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ X2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_309_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_310_Vsm,axiom,
ord_less_eq_set_nat @ vs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ).
% Vsm
thf(fact_311__092_060open_062Us_A_092_060subseteq_062_A_091m_093_092_060close_062,axiom,
ord_less_eq_set_nat @ us @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ).
% \<open>Us \<subseteq> [m]\<close>
thf(fact_312__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_313__092_060open_062_092_060exists_062h_O_Abij__betw_Ah_A_1230_O_O_060p_125_AS_092_060close_062,axiom,
? [H3: nat > set_nat] : ( bij_betw_nat_set_nat @ H3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ) ).
% \<open>\<exists>h. bij_betw h {0..<p} S\<close>
thf(fact_314_first__assumptions_Okm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_315_first__assumptions_Omp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_316_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_317_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_318_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_319_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_320_first__assumptions_Ok,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_321_first__assumptions_Okp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ K ) ) ).
% first_assumptions.kp
thf(fact_322_first__assumptions_Opl,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ P2 ) ) ).
% first_assumptions.pl
thf(fact_323_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_324_second__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ).
% second_assumptions.axioms(1)
thf(fact_325_bij__betw__finite,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_326_bij__betw__finite,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F2 @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_327_bij__betw__finite,axiom,
! [F2: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F2 @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_328_bij__betw__finite,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F2 @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_329_bij__betw__finite,axiom,
! [F2: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_330_bij__betw__finite,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( bij_be6199415091885040644at_nat @ F2 @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_331_bij__betw__finite,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F2 @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_332_bij__betw__finite,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F2 @ A2 @ B2 )
=> ( ( finite2115694454571419734at_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_333_bij__betw__finite,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F2 @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_334_bij__betw__finite,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F2 @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_335_bij__betw__same__card,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_336_bij__betw__same__card,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F2 @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_337_bij__betw__same__card,axiom,
! [F2: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F2 @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_338_bij__betw__same__card,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F2 @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_339_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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 ) )
=> ? [H3: nat > nat] : ( bij_betw_nat_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_351_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 ) )
=> ? [H3: nat > set_nat] : ( bij_betw_nat_set_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_352_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 ) )
=> ? [H3: set_nat > nat] : ( bij_betw_set_nat_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_353_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 ) )
=> ? [H3: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_354_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 ) )
=> ? [H3: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_355_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 ) )
=> ? [H3: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_356_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 ) )
=> ? [H3: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_357_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 ) )
=> ? [H3: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_358_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 ) )
=> ? [H3: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_359_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 ) )
=> ? [H3: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ H3 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_360_finite__imageD,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_361_finite__imageD,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( finite_finite_nat @ ( image_set_nat_nat @ F2 @ A2 ) )
=> ( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_362_finite__imageD,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_363_finite__imageD,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( finite_finite_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
=> ( ( inj_on7365807742884704127at_nat @ F2 @ A2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_364_finite__imageD,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( finite_finite_nat @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ( ( inj_on_nat_nat_nat @ F2 @ A2 )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_365_finite__imageD,axiom,
! [F2: nat > set_set_nat,A2: set_nat] :
( ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_366_finite__imageD,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_367_finite__imageD,axiom,
! [F2: nat > nat > nat,A2: set_nat] :
( ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ( ( inj_on_nat_nat_nat2 @ F2 @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_368_finite__imageD,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
=> ( ( inj_on2776966659131765557et_nat @ F2 @ A2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_369_finite__imageD,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_370_finite__image__iff,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_371_finite__image__iff,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( finite_finite_nat @ ( image_set_nat_nat @ F2 @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_372_finite__image__iff,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_373_finite__image__iff,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F2 @ A2 )
=> ( ( finite_finite_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_374_finite__image__iff,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F2 @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat_nat @ F2 @ A2 ) )
= ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_375_finite__image__iff,axiom,
! [F2: nat > set_set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_376_finite__image__iff,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_377_finite__image__iff,axiom,
! [F2: nat > nat > nat,A2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F2 @ A2 )
=> ( ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_378_finite__image__iff,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F2 @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_379_finite__image__iff,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_380_card__image,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image
thf(fact_381_card__image,axiom,
! [F2: nat > set_set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_382_card__image,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_383_card__image,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( finite_card_nat @ ( image_set_nat_nat @ F2 @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ).
% card_image
thf(fact_384_card__image,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image
thf(fact_385_card__image,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( finite_card_set_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_386_card__image,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ).
% card_image
thf(fact_387_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P2: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_388_second__assumptions_OLm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K ) @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).
% second_assumptions.Lm
thf(fact_389_endo__inj__surj,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F2 @ A2 ) @ A2 )
=> ( ( inj_on2040386338155636715et_nat @ F2 @ A2 )
=> ( ( image_7884819252390400639et_nat @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_390_endo__inj__surj,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ A2 )
=> ( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( image_7916887816326733075et_nat @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_391_endo__inj__surj,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ A2 )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( image_nat_nat @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_392_endo__inj__surj,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ A2 )
=> ( ( inj_on2461717442902640625at_nat @ F2 @ A2 )
=> ( ( image_3205354838064109189at_nat @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_393_inj__on__finite,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_394_inj__on__finite,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_395_inj__on__finite,axiom,
! [F2: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_396_inj__on__finite,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_397_inj__on__finite,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_398_inj__on__finite,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( inj_on7365807742884704127at_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_399_inj__on__finite,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( inj_on_nat_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_400_inj__on__finite,axiom,
! [F2: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( inj_on_nat_nat_nat2 @ F2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_401_inj__on__finite,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_402_inj__on__finite,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_403_finite__surj__inj,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ ( image_7884819252390400639et_nat @ F2 @ A2 ) )
=> ( inj_on2040386338155636715et_nat @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_404_finite__surj__inj,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_405_finite__surj__inj,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_406_finite__surj__inj,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ ( image_3205354838064109189at_nat @ F2 @ A2 ) )
=> ( inj_on2461717442902640625at_nat @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_407_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F2: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on8105003582846801791et_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_408_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_409_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on_nat_set_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_410_eq__card__imp__inj__on,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) )
=> ( inj_on7365807742884704127at_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_411_eq__card__imp__inj__on,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) )
=> ( inj_on1894729867836481333et_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_412_eq__card__imp__inj__on,axiom,
! [A2: set_set_nat,F2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_set_nat_nat @ F2 @ A2 ) )
= ( finite_card_set_nat @ A2 ) )
=> ( inj_on_set_nat_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_413_eq__card__imp__inj__on,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
= ( finite_card_set_nat @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_414_eq__card__imp__inj__on,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > set_set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ( finite1149291290879098388et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) )
=> ( inj_on4164537515518332398et_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_415_eq__card__imp__inj__on,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_nat_nat_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) )
=> ( inj_on_nat_nat_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_416_eq__card__imp__inj__on,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_7432509271690132940et_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) )
=> ( inj_on3232216700808548664et_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_417_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F2: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
= ( ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_418_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
= ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_419_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_set_nat @ F2 @ A2 )
= ( ( finite_card_set_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_420_inj__on__iff__eq__card,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on7365807742884704127at_nat @ F2 @ A2 )
= ( ( finite_card_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_421_inj__on__iff__eq__card,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
= ( ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_422_inj__on__iff__eq__card,axiom,
! [A2: set_set_nat,F2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on_set_nat_nat @ F2 @ A2 )
= ( ( finite_card_nat @ ( image_set_nat_nat @ F2 @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_423_inj__on__iff__eq__card,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
= ( ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_424_inj__on__iff__eq__card,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > set_set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( inj_on4164537515518332398et_nat @ F2 @ A2 )
= ( ( finite1149291290879098388et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_425_inj__on__iff__eq__card,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( inj_on_nat_nat_nat @ F2 @ A2 )
= ( ( finite_card_nat @ ( image_nat_nat_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_426_inj__on__iff__eq__card,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( inj_on3232216700808548664et_nat @ F2 @ A2 )
= ( ( finite_card_set_nat @ ( image_7432509271690132940et_nat @ F2 @ A2 ) )
= ( finite_card_nat_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_427_pigeonhole,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) )
=> ~ ( inj_on4164537515518332398et_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_428_pigeonhole,axiom,
! [F2: nat > set_set_nat,A2: set_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on8105003582846801791et_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_429_pigeonhole,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_430_pigeonhole,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F2 @ A2 ) ) @ ( finite_card_set_nat @ A2 ) )
=> ~ ( inj_on_set_nat_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_431_pigeonhole,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ~ ( inj_on1894729867836481333et_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_432_pigeonhole,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_set_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_433_pigeonhole,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) ) @ ( finite_card_set_nat @ A2 ) )
=> ~ ( inj_on4604407203859583615et_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_434_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ ( image_nat_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_435_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F2: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ ( image_set_nat_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_436_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F2 @ X ) @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_437_imageI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,F2: set_set_nat > nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ ( image_1454916318497077779at_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_438_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F2: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F2 @ X ) @ ( image_7916887816326733075et_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_439_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > set_set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_set_nat @ ( F2 @ X ) @ ( image_2194112158459175443et_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_440_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F2 @ X ) @ ( image_nat_nat_nat2 @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_441_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F2: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ ( image_nat_nat_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_442_imageI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,F2: set_set_nat > set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F2 @ X ) @ ( image_5842784325960735177et_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_443_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F2: set_nat > set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_set_nat @ ( F2 @ X ) @ ( image_6725021117256019401et_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_444_image__iff,axiom,
! [Z: set_set_nat,F2: nat > set_set_nat,A2: set_nat] :
( ( member_set_set_nat @ Z @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_445_image__iff,axiom,
! [Z: set_set_nat,F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat] :
( ( member_set_set_nat @ Z @ ( image_9186907679027735170et_nat @ F2 @ A2 ) )
= ( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_446_image__iff,axiom,
! [Z: set_nat,F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( member_set_nat @ Z @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_447_image__iff,axiom,
! [Z: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F2 @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_448_bex__imageD,axiom,
! [F2: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ X3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_449_bex__imageD,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
& ( P @ X3 ) )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_450_bex__imageD,axiom,
! [F2: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
& ( P @ X3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_451_bex__imageD,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( image_9186907679027735170et_nat @ F2 @ A2 ) )
& ( P @ X3 ) )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_452_image__cong,axiom,
! [M2: set_set_set_nat,N2: set_set_set_nat,F2: set_set_nat > set_nat,G2: set_set_nat > set_nat] :
( ( M2 = N2 )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ N2 )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_5842784325960735177et_nat @ F2 @ M2 )
= ( image_5842784325960735177et_nat @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_453_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F2: nat > nat,G2: nat > nat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_nat_nat @ F2 @ M2 )
= ( image_nat_nat @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_454_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F2: nat > set_set_nat,G2: nat > set_set_nat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_2194112158459175443et_nat @ F2 @ M2 )
= ( image_2194112158459175443et_nat @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_455_image__cong,axiom,
! [M2: set_nat_nat,N2: set_nat_nat,F2: ( nat > nat ) > set_set_nat,G2: ( nat > nat ) > set_set_nat] :
( ( M2 = N2 )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ N2 )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_9186907679027735170et_nat @ F2 @ M2 )
= ( image_9186907679027735170et_nat @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_456_ball__imageD,axiom,
! [F2: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F2 @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_457_ball__imageD,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( P @ ( F2 @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_458_ball__imageD,axiom,
! [F2: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F2 @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_459_ball__imageD,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( image_9186907679027735170et_nat @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( P @ ( F2 @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_460_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F2: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_461_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: nat,F2: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_462_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_nat,F2: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_463_rev__image__eqI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,B: nat,F2: set_set_nat > nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_1454916318497077779at_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_464_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: set_nat,F2: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_465_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_set_nat,F2: nat > set_set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_466_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat,F2: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_467_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat,F2: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_468_rev__image__eqI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,B: set_nat,F2: set_set_nat > set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_469_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: set_set_nat,F2: set_nat > set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_470_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 ) )
= ( ! [X4: set_set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_471_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_472_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_473_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X4: nat > nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_474_set__eq__subset,axiom,
( ( ^ [Y5: set_set_set_nat,Z2: set_set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_475_set__eq__subset,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_476_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_477_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_478_subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C3 )
=> ( ord_le9131159989063066194et_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_479_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_480_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_481_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_482_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X2: set_set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_483_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_484_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_485_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_486_subset__refl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_487_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_488_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_489_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_490_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] :
! [T4: set_set_nat] :
( ( member_set_set_nat @ T4 @ A5 )
=> ( member_set_set_nat @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_491_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
! [T4: set_nat] :
( ( member_set_nat @ T4 @ A5 )
=> ( member_set_nat @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_492_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A5 )
=> ( member_nat @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_493_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
! [T4: nat > nat] :
( ( member_nat_nat @ T4 @ A5 )
=> ( member_nat_nat @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_494_equalityD2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_495_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_496_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_497_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_498_equalityD1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_499_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_500_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_501_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_502_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] :
! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A5 )
=> ( member_set_set_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_503_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A5 )
=> ( member_set_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_504_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_505_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A5 )
=> ( member_nat_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_in__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_set_set_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_515_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_516_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_517_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_518_psubset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C3 )
=> ( ord_le152980574450754630et_nat @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_519_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_520_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_521_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_522_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_523_ex__bij__betw__nat__finite,axiom,
! [M2: set_nat] :
( ( finite_finite_nat @ M2 )
=> ? [H3: nat > nat] : ( bij_betw_nat_nat @ H3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ M2 ) ) @ M2 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_524_ex__bij__betw__nat__finite,axiom,
! [M2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ M2 )
=> ? [H3: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ M2 ) ) @ M2 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_525_ex__bij__betw__nat__finite,axiom,
! [M2: set_set_nat] :
( ( finite1152437895449049373et_nat @ M2 )
=> ? [H3: nat > set_nat] : ( bij_betw_nat_set_nat @ H3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_set_nat @ M2 ) ) @ M2 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_526_ex__bij__betw__nat__finite,axiom,
! [M2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ M2 )
=> ? [H3: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ H3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat_nat @ M2 ) ) @ M2 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_527_card__bij__eq,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_528_card__bij__eq,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_set_nat,G2: set_nat > nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_set_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_529_card__bij__eq,axiom,
! [F2: set_nat > nat,A2: set_set_nat,B2: set_nat,G2: nat > set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_set_nat @ G2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_530_card__bij__eq,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat,G2: set_set_nat > nat] :
( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on7365807742884704127at_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_531_card__bij__eq,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,G2: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on4604407203859583615et_nat @ G2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_532_card__bij__eq,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat,G2: nat > set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on8105003582846801791et_nat @ G2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_533_card__bij__eq,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,G2: nat > nat > nat] :
( ( inj_on_nat_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat_nat2 @ G2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ G2 @ B2 ) @ A2 )
=> ( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_534_card__bij__eq,axiom,
! [F2: nat > nat > nat,A2: set_nat,B2: set_nat_nat,G2: ( nat > nat ) > nat] :
( ( inj_on_nat_nat_nat2 @ F2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_535_card__bij__eq,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat,G2: set_set_nat > set_nat] :
( ( inj_on2776966659131765557et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on1894729867836481333et_nat @ G2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_536_card__bij__eq,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat,G2: set_nat > set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on2776966659131765557et_nat @ G2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ G2 @ B2 ) @ A2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_537_surjective__iff__injective__gen,axiom,
! [S: set_nat,T: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat @ T ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on_nat_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_538_surjective__iff__injective__gen,axiom,
! [S: set_nat,T: set_set_nat,F2: nat > set_nat] :
( ( finite_finite_nat @ S )
=> ( ( finite1152437895449049373et_nat @ T )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_set_nat @ T ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ T )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on_nat_set_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_539_surjective__iff__injective__gen,axiom,
! [S: set_set_nat,T: set_nat,F2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ( ( finite_card_set_nat @ S )
= ( finite_card_nat @ T ) )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on_set_nat_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_540_surjective__iff__injective__gen,axiom,
! [S: set_nat,T: set_set_set_nat,F2: nat > set_set_nat] :
( ( finite_finite_nat @ S )
=> ( ( finite6739761609112101331et_nat @ T )
=> ( ( ( finite_card_nat @ S )
= ( finite1149291290879098388et_nat @ T ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ T )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on8105003582846801791et_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_541_surjective__iff__injective__gen,axiom,
! [S: set_set_nat,T: set_set_nat,F2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( finite1152437895449049373et_nat @ T )
=> ( ( ( finite_card_set_nat @ S )
= ( finite_card_set_nat @ T ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ T )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on4604407203859583615et_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_542_surjective__iff__injective__gen,axiom,
! [S: set_set_set_nat,T: set_nat,F2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ( ( finite1149291290879098388et_nat @ S )
= ( finite_card_nat @ T ) )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ? [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on7365807742884704127at_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_543_surjective__iff__injective__gen,axiom,
! [S: set_nat_nat,T: set_nat,F2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ( ( finite_card_nat_nat @ S )
= ( finite_card_nat @ T ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ? [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on_nat_nat_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_544_surjective__iff__injective__gen,axiom,
! [S: set_nat,T: set_nat_nat,F2: nat > nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite2115694454571419734at_nat @ T )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat_nat @ T ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ S ) @ T )
=> ( ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ T )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on_nat_nat_nat2 @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_545_surjective__iff__injective__gen,axiom,
! [S: set_set_nat,T: set_set_set_nat,F2: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( finite6739761609112101331et_nat @ T )
=> ( ( ( finite_card_set_nat @ S )
= ( finite1149291290879098388et_nat @ T ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ T )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on2776966659131765557et_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_546_surjective__iff__injective__gen,axiom,
! [S: set_set_set_nat,T: set_set_nat,F2: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( finite1152437895449049373et_nat @ T )
=> ( ( ( finite1149291290879098388et_nat @ S )
= ( finite_card_set_nat @ T ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ S ) @ T )
=> ( ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ T )
=> ? [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ S )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on1894729867836481333et_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_547_first__assumptions_OL_Ocong,axiom,
assump1710595444109740301irst_L = assump1710595444109740301irst_L ).
% first_assumptions.L.cong
thf(fact_548_card__le__inj,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F4: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A2 ) @ B2 )
& ( inj_on_nat_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_549_card__le__inj,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ? [F4: nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F4 @ A2 ) @ B2 )
& ( inj_on_nat_set_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_550_card__le__inj,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F4: set_nat > nat] :
( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F4 @ A2 ) @ B2 )
& ( inj_on_set_nat_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_551_card__le__inj,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ? [F4: nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F4 @ A2 ) @ B2 )
& ( inj_on8105003582846801791et_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_552_card__le__inj,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ? [F4: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F4 @ A2 ) @ B2 )
& ( inj_on4604407203859583615et_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_553_card__le__inj,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F4: set_set_nat > nat] :
( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F4 @ A2 ) @ B2 )
& ( inj_on7365807742884704127at_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_554_card__le__inj,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F4: ( nat > nat ) > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F4 @ A2 ) @ B2 )
& ( inj_on_nat_nat_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_555_card__le__inj,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ? [F4: nat > nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F4 @ A2 ) @ B2 )
& ( inj_on_nat_nat_nat2 @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_556_card__le__inj,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ? [F4: set_nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F4 @ A2 ) @ B2 )
& ( inj_on2776966659131765557et_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_557_card__le__inj,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ? [F4: set_set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F4 @ A2 ) @ B2 )
& ( inj_on1894729867836481333et_nat @ F4 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_558_card__inj__on__le,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat,B2: set_set_set_nat] :
( ( inj_on4164537515518332398et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_559_card__inj__on__le,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_560_card__inj__on__le,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_561_card__inj__on__le,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_562_card__inj__on__le,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_563_card__inj__on__le,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_564_card__inj__on__le,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_565_card__inj__on__le,axiom,
! [F2: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_566_card__inj__on__le,axiom,
! [F2: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( inj_on_nat_nat_nat2 @ F2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_567_card__inj__on__le,axiom,
! [F2: set_nat > nat > nat,A2: set_set_nat,B2: set_nat_nat] :
( ( inj_on4369475957891034808at_nat @ F2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_8569768528772619084at_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_568_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_569_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F3: nat > set_nat] :
( ( inj_on_nat_set_nat @ F3 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_570_inj__on__iff__card__le,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: set_nat > nat] :
( ( inj_on_set_nat_nat @ F3 @ A2 )
& ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_571_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F3: nat > set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F3 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_572_inj__on__iff__card__le,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F3: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F3 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_573_inj__on__iff__card__le,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: set_set_nat > nat] :
( ( inj_on7365807742884704127at_nat @ F3 @ A2 )
& ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_574_inj__on__iff__card__le,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: ( nat > nat ) > nat] :
( ( inj_on_nat_nat_nat @ F3 @ A2 )
& ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_575_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ? [F3: nat > nat > nat] :
( ( inj_on_nat_nat_nat2 @ F3 @ A2 )
& ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_576_inj__on__iff__card__le,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F3: set_nat > set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F3 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_577_inj__on__iff__card__le,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F3: set_set_nat > set_nat] :
( ( inj_on1894729867836481333et_nat @ F3 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_578_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_579_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F2: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A2 ) @ ( image_set_nat_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_580_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ ( image_nat_set_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_581_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F2: set_set_nat > nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) @ ( image_1454916318497077779at_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_582_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F2: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ ( image_7916887816326733075et_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_583_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ ( image_2194112158459175443et_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_584_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ ( image_nat_nat_nat2 @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_585_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F2: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ ( image_nat_nat_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_586_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F2: set_set_nat > set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ ( image_5842784325960735177et_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_587_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F2: set_nat > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) @ ( image_6725021117256019401et_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_588_image__subsetI,axiom,
! [A2: set_nat,F2: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_589_image__subsetI,axiom,
! [A2: set_nat,F2: nat > set_nat,B2: set_set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_590_image__subsetI,axiom,
! [A2: set_set_nat,F2: set_nat > nat,B2: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_591_image__subsetI,axiom,
! [A2: set_nat,F2: nat > set_set_nat,B2: set_set_set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_592_image__subsetI,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_593_image__subsetI,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > nat,B2: set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_594_image__subsetI,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat,B2: set_nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_595_image__subsetI,axiom,
! [A2: set_nat,F2: nat > nat > nat,B2: set_nat_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_596_image__subsetI,axiom,
! [A2: set_set_nat,F2: set_nat > set_set_nat,B2: set_set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_597_image__subsetI,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_nat,B2: set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_598_subset__imageE,axiom,
! [B2: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_599_subset__imageE,axiom,
! [B2: set_set_nat,F2: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_600_subset__imageE,axiom,
! [B2: set_nat,F2: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( B2
!= ( image_set_nat_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_601_subset__imageE,axiom,
! [B2: set_set_set_nat,F2: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_2194112158459175443et_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_602_subset__imageE,axiom,
! [B2: set_set_nat,F2: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_603_subset__imageE,axiom,
! [B2: set_nat,F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( B2
!= ( image_1454916318497077779at_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_604_subset__imageE,axiom,
! [B2: set_nat,F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_nat_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_605_subset__imageE,axiom,
! [B2: set_nat_nat,F2: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_nat_nat2 @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_606_subset__imageE,axiom,
! [B2: set_set_set_nat,F2: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( B2
!= ( image_6725021117256019401et_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_607_subset__imageE,axiom,
! [B2: set_set_nat,F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( B2
!= ( image_5842784325960735177et_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_608_image__subset__iff,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_set_set_nat @ ( F2 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_609_image__subset__iff,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) @ B2 )
= ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_set_set_nat @ ( F2 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_610_image__subset__iff,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ B2 )
= ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ( member_set_nat @ ( F2 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_611_image__subset__iff,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F2 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_612_subset__image__iff,axiom,
! [B2: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_613_subset__image__iff,axiom,
! [B2: set_set_nat,F2: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_614_subset__image__iff,axiom,
! [B2: set_nat,F2: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F2 @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_615_subset__image__iff,axiom,
! [B2: set_set_set_nat,F2: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_2194112158459175443et_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_616_subset__image__iff,axiom,
! [B2: set_set_nat,F2: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F2 @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_617_subset__image__iff,axiom,
! [B2: set_nat,F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F2 @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_1454916318497077779at_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_618_subset__image__iff,axiom,
! [B2: set_nat,F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_619_subset__image__iff,axiom,
! [B2: set_nat_nat,F2: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat2 @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_620_subset__image__iff,axiom,
! [B2: set_set_set_nat,F2: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F2 @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_6725021117256019401et_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_621_subset__image__iff,axiom,
! [B2: set_set_nat,F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_5842784325960735177et_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_622_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_623_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
( ( ord_less_set_set_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_624_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_625_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_less_set_nat_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_626_subset__psubset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C3 )
=> ( ord_le152980574450754630et_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_627_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C3 )
=> ( ord_less_set_set_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_628_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_629_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C3 )
=> ( ord_less_set_nat_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_630_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ B6 )
& ~ ( ord_le9131159989063066194et_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_631_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B6 )
& ~ ( ord_le6893508408891458716et_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_632_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_633_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B6 )
& ~ ( ord_le9059583361652607317at_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_634_psubset__subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C3 )
=> ( ord_le152980574450754630et_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_635_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C3 )
=> ( ord_less_set_set_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_636_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_637_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C3 )
=> ( ord_less_set_nat_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_638_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_639_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_640_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_641_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_642_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_643_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_644_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_645_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_646_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_647_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_648_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_649_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_650_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G3: set_set_nat] :
? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ X6 )
& ( ord_le6893508408891458716et_nat @ X4 @ G3 ) ) ) ) ).
% accepts_def
thf(fact_651_sunflower__card__subset__lift,axiom,
! [K: nat,C: nat,R: nat,F: set_set_set_set_nat] :
( ! [G: set_se7521423693449168855at_nat] :
( ! [X3: set_Su1440016900418933025at_nat] :
( ( member5638249034155602744at_nat @ X3 @ G )
=> ( ( finite8770298478261192322at_nat @ X3 )
& ( ( finite8251389301641259331at_nat @ X3 )
= K ) ) )
=> ( ( ord_less_nat @ C @ ( finite7696428214769936121at_nat @ G ) )
=> ? [S4: set_se7521423693449168855at_nat] :
( ( ord_le2853704879392749623at_nat @ S4 @ G )
& ( sunflo3853689026006497528at_nat @ S4 )
& ( ( finite7696428214769936121at_nat @ S4 )
= R ) ) ) )
=> ( ! [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ F )
=> ( ( finite6739761609112101331et_nat @ X2 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ X2 ) @ K ) ) )
=> ( ( ord_less_nat @ C @ ( finite8805468973633305546et_nat @ F ) )
=> ? [S3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ S3 @ F )
& ( sunflo2680516271513359689et_nat @ S3 )
& ( ( finite8805468973633305546et_nat @ S3 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_652_sunflower__card__subset__lift,axiom,
! [K: nat,C: nat,R: nat,F: set_set_nat_nat] :
( ! [G: set_se7880254595028141658at_nat] :
( ! [X3: set_Su8808554476274791844at_nat] :
( ( member968451730063008059at_nat @ X3 @ G )
=> ( ( finite5967121830935861893at_nat @ X3 )
& ( ( finite2091696060772798406at_nat @ X3 )
= K ) ) )
=> ( ( ord_less_nat @ C @ ( finite5641098376000219004at_nat @ G ) )
=> ? [S4: set_se7880254595028141658at_nat] :
( ( ord_le5374289575490365114at_nat @ S4 @ G )
& ( sunflo111067583121249275at_nat @ S4 )
& ( ( finite5641098376000219004at_nat @ S4 )
= R ) ) ) )
=> ( ! [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ F )
=> ( ( finite2115694454571419734at_nat @ X2 )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ X2 ) @ K ) ) )
=> ( ( ord_less_nat @ C @ ( finite5893285860794289869at_nat @ F ) )
=> ? [S3: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ S3 @ F )
& ( sunflower_nat_nat @ S3 )
& ( ( finite5893285860794289869at_nat @ S3 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_653_sunflower__card__subset__lift,axiom,
! [K: nat,C: nat,R: nat,F: set_set_set_nat] :
( ! [G: set_se8003284279568041249at_nat] :
( ! [X3: set_Su8059080322890262379at_nat] :
( ( member5374901640408327554at_nat @ X3 @ G )
=> ( ( finite2491568536608231884at_nat @ X3 )
& ( ( finite8413070326521870477at_nat @ X3 )
= K ) ) )
=> ( ( ord_less_nat @ C @ ( finite7758422657562484035at_nat @ G ) )
=> ? [S4: set_se8003284279568041249at_nat] :
( ( ord_le4731320016863163777at_nat @ S4 @ G )
& ( sunflo6650083805840251970at_nat @ S4 )
& ( ( finite7758422657562484035at_nat @ S4 )
= R ) ) ) )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ F )
=> ( ( finite1152437895449049373et_nat @ X2 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ X2 ) @ K ) ) )
=> ( ( ord_less_nat @ C @ ( finite1149291290879098388et_nat @ F ) )
=> ? [S3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S3 @ F )
& ( sunflower_set_nat @ S3 )
& ( ( finite1149291290879098388et_nat @ S3 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_654_sunflower__card__subset__lift,axiom,
! [K: nat,C: nat,R: nat,F: set_set_nat] :
( ! [G: set_se3873067930692246379at_nat] :
( ! [X3: set_Sum_sum_nat_nat] :
( ( member1869216328726507724at_nat @ X3 @ G )
=> ( ( finite6187706683773761046at_nat @ X3 )
& ( ( finite8494011213269508311at_nat @ X3 )
= K ) ) )
=> ( ( ord_less_nat @ C @ ( finite2024029949821234317at_nat @ G ) )
=> ? [S4: set_se3873067930692246379at_nat] :
( ( ord_le3495481059733392331at_nat @ S4 @ G )
& ( sunflo1841451327523575948at_nat @ S4 )
& ( ( finite2024029949821234317at_nat @ S4 )
= R ) ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ F )
=> ( ( finite_finite_nat @ X2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X2 ) @ K ) ) )
=> ( ( ord_less_nat @ C @ ( finite_card_set_nat @ F ) )
=> ? [S3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ F )
& ( sunflower_nat @ S3 )
& ( ( finite_card_set_nat @ S3 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_655_finite__v__gs,axiom,
! [X5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ).
% finite_v_gs
thf(fact_656_XD,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% XD
thf(fact_657_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_658_finite__members___092_060G_062,axiom,
! [G4: set_set_nat] :
( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G4 ) ) ).
% finite_members_\<G>
thf(fact_659_Schroeder__Bernstein,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: nat > nat] : ( bij_betw_nat_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_660_Schroeder__Bernstein,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_set_nat,G2: set_nat > nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_set_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: nat > set_nat] : ( bij_betw_nat_set_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_661_Schroeder__Bernstein,axiom,
! [F2: set_nat > nat,A2: set_set_nat,B2: set_nat,G2: nat > set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_set_nat @ G2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: set_nat > nat] : ( bij_betw_set_nat_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_662_Schroeder__Bernstein,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat,G2: set_set_nat > nat] :
( ( inj_on8105003582846801791et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on7365807742884704127at_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_663_Schroeder__Bernstein,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,G2: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on4604407203859583615et_nat @ G2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_664_Schroeder__Bernstein,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat,G2: nat > set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on8105003582846801791et_nat @ G2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_665_Schroeder__Bernstein,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,G2: nat > nat > nat] :
( ( inj_on_nat_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat_nat2 @ G2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ G2 @ B2 ) @ A2 )
=> ? [H3: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_666_Schroeder__Bernstein,axiom,
! [F2: nat > nat > nat,A2: set_nat,B2: set_nat_nat,G2: ( nat > nat ) > nat] :
( ( inj_on_nat_nat_nat2 @ F2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_667_Schroeder__Bernstein,axiom,
! [F2: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat,G2: set_set_nat > set_nat] :
( ( inj_on2776966659131765557et_nat @ F2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on1894729867836481333et_nat @ G2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_668_Schroeder__Bernstein,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat,G2: set_nat > set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on2776966659131765557et_nat @ G2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ G2 @ B2 ) @ A2 )
=> ? [H3: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_669_ACC__cf___092_060F_062,axiom,
! [X5: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_670_GsGl,axiom,
member_set_set_nat @ gs @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% GsGl
thf(fact_671_ACC__cf__mono,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_mono
thf(fact_672_GsG,axiom,
member_set_set_nat @ gs @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% GsG
thf(fact_673_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_674_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_675_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_676_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_677_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_678_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_679_acceptsI,axiom,
! [D2: set_set_nat,G4: set_set_nat,X5: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G4 )
=> ( ( member_set_set_nat @ D2 @ X5 )
=> ( clique3686358387679108662ccepts @ X5 @ G4 ) ) ) ).
% acceptsI
thf(fact_680_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_681_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_682_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_683_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_684_finite__ACC,axiom,
! [X5: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) ) ).
% finite_ACC
thf(fact_685_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_686_i__props_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( ti @ I )
= ( minus_minus_nat @ ( si2 @ I ) @ s2 ) ) ) ).
% i_props(3)
thf(fact_687_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_688_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_689_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_690_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_691_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_692_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_693_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_694_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_695_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_696_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_697_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_698_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_699_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_700_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_701_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_702_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_703_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_704_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_705_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_706_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_707_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X6: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_708_le__rel__bool__arg__iff,axiom,
( ord_le8326115459943588763et_nat
= ( ^ [X6: $o > set_set_set_nat,Y7: $o > set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le9131159989063066194et_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_709_le__rel__bool__arg__iff,axiom,
( ord_le6539261115178940645et_nat
= ( ^ [X6: $o > set_set_nat,Y7: $o > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le6893508408891458716et_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_710_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X6: $o > set_nat,Y7: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_711_le__rel__bool__arg__iff,axiom,
( ord_le5298321079317455902at_nat
= ( ^ [X6: $o > set_nat_nat,Y7: $o > set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le9059583361652607317at_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_712_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_713_first__assumptions_Okml,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_714_ex__bij__betw__finite__nat,axiom,
! [M2: set_nat] :
( ( finite_finite_nat @ M2 )
=> ? [H3: nat > nat] : ( bij_betw_nat_nat @ H3 @ M2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ M2 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_715_ex__bij__betw__finite__nat,axiom,
! [M2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ M2 )
=> ? [H3: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H3 @ M2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ M2 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_716_ex__bij__betw__finite__nat,axiom,
! [M2: set_set_nat] :
( ( finite1152437895449049373et_nat @ M2 )
=> ? [H3: set_nat > nat] : ( bij_betw_set_nat_nat @ H3 @ M2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_set_nat @ M2 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_717_ex__bij__betw__finite__nat,axiom,
! [M2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ M2 )
=> ? [H3: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ H3 @ M2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat_nat @ M2 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_718_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,G4: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3686358387679108662ccepts @ X5 @ G4 )
= ( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ X5 )
& ( ord_le6893508408891458716et_nat @ X4 @ G4 ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_719_first__assumptions_OacceptsI,axiom,
! [L: nat,P2: nat,K: nat,D2: set_set_nat,G4: set_set_nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ G4 )
=> ( ( member_set_set_nat @ D2 @ X5 )
=> ( clique3686358387679108662ccepts @ X5 @ G4 ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_720_sunflower__def,axiom,
( sunflo2680516271513359689et_nat
= ( ^ [S5: set_set_set_set_nat] :
! [X4: set_set_nat] :
( ? [A5: set_set_set_nat,B6: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A5 @ S5 )
& ( member2946998982187404937et_nat @ B6 @ S5 )
& ( A5 != B6 )
& ( member_set_set_nat @ X4 @ A5 )
& ( member_set_set_nat @ X4 @ B6 ) )
=> ! [A5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A5 @ S5 )
=> ( member_set_set_nat @ X4 @ A5 ) ) ) ) ) ).
% sunflower_def
thf(fact_721_sunflower__def,axiom,
( sunflower_nat_nat
= ( ^ [S5: set_set_nat_nat] :
! [X4: nat > nat] :
( ? [A5: set_nat_nat,B6: set_nat_nat] :
( ( member_set_nat_nat @ A5 @ S5 )
& ( member_set_nat_nat @ B6 @ S5 )
& ( A5 != B6 )
& ( member_nat_nat @ X4 @ A5 )
& ( member_nat_nat @ X4 @ B6 ) )
=> ! [A5: set_nat_nat] :
( ( member_set_nat_nat @ A5 @ S5 )
=> ( member_nat_nat @ X4 @ A5 ) ) ) ) ) ).
% sunflower_def
thf(fact_722_sunflower__def,axiom,
( sunflower_set_nat
= ( ^ [S5: set_set_set_nat] :
! [X4: set_nat] :
( ? [A5: set_set_nat,B6: set_set_nat] :
( ( member_set_set_nat @ A5 @ S5 )
& ( member_set_set_nat @ B6 @ S5 )
& ( A5 != B6 )
& ( member_set_nat @ X4 @ A5 )
& ( member_set_nat @ X4 @ B6 ) )
=> ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ S5 )
=> ( member_set_nat @ X4 @ A5 ) ) ) ) ) ).
% sunflower_def
thf(fact_723_sunflower__def,axiom,
( sunflower_nat
= ( ^ [S5: set_set_nat] :
! [X4: nat] :
( ? [A5: set_nat,B6: set_nat] :
( ( member_set_nat @ A5 @ S5 )
& ( member_set_nat @ B6 @ S5 )
& ( A5 != B6 )
& ( member_nat @ X4 @ A5 )
& ( member_nat @ X4 @ B6 ) )
=> ! [A5: set_nat] :
( ( member_set_nat @ A5 @ S5 )
=> ( member_nat @ X4 @ A5 ) ) ) ) ) ).
% sunflower_def
thf(fact_724_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G4: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ G4 ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_725_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_726_first__assumptions_Ofinite__v__gs,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ) ).
% first_assumptions.finite_v_gs
thf(fact_727_sunflower__subset,axiom,
! [F: set_set_set_nat,G4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ F @ G4 )
=> ( ( sunflower_set_nat @ G4 )
=> ( sunflower_set_nat @ F ) ) ) ).
% sunflower_subset
thf(fact_728_sunflower__subset,axiom,
! [F: set_set_nat,G4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ F @ G4 )
=> ( ( sunflower_nat @ G4 )
=> ( sunflower_nat @ F ) ) ) ).
% sunflower_subset
thf(fact_729_odot___092_060G_062,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_730__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_731_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_732_v___092_060G_062,axiom,
! [G4: set_set_nat] :
( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G4 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_733_finite__vG,axiom,
! [G4: set_set_nat] :
( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G4 ) ) ) ).
% finite_vG
thf(fact_734_Gs__def,axiom,
( gs
= ( clique6722202388162463298od_nat @ vs @ vs ) ) ).
% Gs_def
thf(fact_735_i__props_I7_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% i_props(7)
thf(fact_736_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_737_Snempty,axiom,
s != bot_bot_set_set_nat ).
% Snempty
thf(fact_738__092_060open_062si_A_092_060equiv_062_A_092_060lambda_062i_O_Acard_A_Iv_A_IG_Ai_J_J_092_060close_062,axiom,
( si2
= ( ^ [I4: nat] : ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I4 ) ) ) ) ) ).
% \<open>si \<equiv> \<lambda>i. card (v (G i))\<close>
thf(fact_739_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_740_v__mono,axiom,
! [G4: set_set_nat,H4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G4 @ H4 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G4 ) @ ( clique5033774636164728513irst_v @ H4 ) ) ) ).
% v_mono
thf(fact_741__092_060open_062ti_A_092_060equiv_062_A_092_060lambda_062i_O_Acard_A_Iv_A_IG_Ai_J_A_N_AVs_J_092_060close_062,axiom,
( ti
= ( ^ [I4: nat] : ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I4 ) ) @ vs ) ) ) ) ).
% \<open>ti \<equiv> \<lambda>i. card (v (G i) - Vs)\<close>
thf(fact_742_Diff__empty,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_743_Diff__empty,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% Diff_empty
thf(fact_744_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_745_Diff__empty,axiom,
! [A2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% Diff_empty
thf(fact_746_empty__Diff,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A2 )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_747_empty__Diff,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% empty_Diff
thf(fact_748_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_749_empty__Diff,axiom,
! [A2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ bot_bot_set_nat_nat @ A2 )
= bot_bot_set_nat_nat ) ).
% empty_Diff
thf(fact_750_Diff__cancel,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ A2 )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_751_Diff__cancel,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% Diff_cancel
thf(fact_752_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_753_Diff__cancel,axiom,
! [A2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ A2 )
= bot_bot_set_nat_nat ) ).
% Diff_cancel
thf(fact_754_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X4: set_nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_755_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X4: set_set_nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_756_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X4: nat > nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_757_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_758_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X4: set_nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_759_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X4: set_set_nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_760_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X4: nat > nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_761_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_762_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X4: set_nat] :
~ ( member_set_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_763_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X4: set_set_nat] :
~ ( member_set_set_nat @ X4 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_764_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X4: nat > nat] :
~ ( member_nat_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_765_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_766_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_767_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_768_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_769_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_770_finite__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_771_finite__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_772_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_773_finite__Diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_774_finite__Diff2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_775_finite__Diff2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_776_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_777_finite__Diff2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_778_injG,axiom,
inj_on8105003582846801791et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ).
% injG
thf(fact_779_i__props_I6_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( finite1152437895449049373et_nat @ ( g @ I ) ) ) ).
% i_props(6)
thf(fact_780_vGs,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ vs ).
% vGs
thf(fact_781_G_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ x ) ) ).
% G(1)
thf(fact_782_si__def,axiom,
! [I: nat] :
( ( si2 @ I )
= ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% si_def
thf(fact_783_i__props_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% i_props(2)
thf(fact_784_G_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ s ) ) ).
% G(4)
thf(fact_785_image__empty,axiom,
! [F2: nat > nat] :
( ( image_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_786_image__empty,axiom,
! [F2: set_nat > nat] :
( ( image_set_nat_nat @ F2 @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_787_image__empty,axiom,
! [F2: nat > set_nat] :
( ( image_nat_set_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_788_image__empty,axiom,
! [F2: set_nat > set_nat] :
( ( image_7916887816326733075et_nat @ F2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_789_image__empty,axiom,
! [F2: set_set_nat > nat] :
( ( image_1454916318497077779at_nat @ F2 @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_790_image__empty,axiom,
! [F2: ( nat > nat ) > nat] :
( ( image_nat_nat_nat @ F2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_791_image__empty,axiom,
! [F2: nat > set_set_nat] :
( ( image_2194112158459175443et_nat @ F2 @ bot_bot_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_792_image__empty,axiom,
! [F2: nat > nat > nat] :
( ( image_nat_nat_nat2 @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_793_image__empty,axiom,
! [F2: set_nat > set_set_nat] :
( ( image_6725021117256019401et_nat @ F2 @ bot_bot_set_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_794_image__empty,axiom,
! [F2: set_nat > nat > nat] :
( ( image_8569768528772619084at_nat @ F2 @ bot_bot_set_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_795_empty__is__image,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_796_empty__is__image,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_797_empty__is__image,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_798_empty__is__image,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_7916887816326733075et_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_799_empty__is__image,axiom,
! [F2: nat > set_set_nat,A2: set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_2194112158459175443et_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_800_empty__is__image,axiom,
! [F2: nat > nat > nat,A2: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_801_empty__is__image,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( bot_bot_set_nat
= ( image_1454916318497077779at_nat @ F2 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_802_empty__is__image,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_803_empty__is__image,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_5842784325960735177et_nat @ F2 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_804_empty__is__image,axiom,
! [F2: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( bot_bot_set_set_nat
= ( image_7432509271690132940et_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_805_image__is__empty,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F2 @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_806_image__is__empty,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F2 @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_807_image__is__empty,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( ( image_set_nat_nat @ F2 @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_808_image__is__empty,axiom,
! [F2: set_nat > set_nat,A2: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F2 @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_809_image__is__empty,axiom,
! [F2: nat > set_set_nat,A2: set_nat] :
( ( ( image_2194112158459175443et_nat @ F2 @ A2 )
= bot_bo7198184520161983622et_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_810_image__is__empty,axiom,
! [F2: nat > nat > nat,A2: set_nat] :
( ( ( image_nat_nat_nat2 @ F2 @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_811_image__is__empty,axiom,
! [F2: set_set_nat > nat,A2: set_set_set_nat] :
( ( ( image_1454916318497077779at_nat @ F2 @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_812_image__is__empty,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ( image_nat_nat_nat @ F2 @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_813_image__is__empty,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ( image_5842784325960735177et_nat @ F2 @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_814_image__is__empty,axiom,
! [F2: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( ( image_7432509271690132940et_nat @ F2 @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_815_Diff__eq__empty__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_816_Diff__eq__empty__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_817_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_818_Diff__eq__empty__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_819_empty__subsetI,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% empty_subsetI
thf(fact_820_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_821_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_822_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_823_subset__empty,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_824_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_825_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_826_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_827_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_828_v___092_060G_062__2,axiom,
! [G4: set_set_nat] :
( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G4 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G4 ) @ ( clique5033774636164728513irst_v @ G4 ) ) ) ) ).
% v_\<G>_2
thf(fact_829_G_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( clique5033774636164728513irst_v @ ( g @ I ) )
= ( si @ I ) ) ) ).
% G(2)
thf(fact_830_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_831_SvG,axiom,
( s
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ ( image_2194112158459175443et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ) ).
% SvG
thf(fact_832_i__props_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_set_nat @ vs @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% i_props(1)
thf(fact_833_ti__def,axiom,
! [I: nat] :
( ( ti @ I )
= ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ vs ) ) ) ).
% ti_def
thf(fact_834_uw_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_nat @ ( u @ I ) @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ vs ) ) ) ).
% uw(1)
thf(fact_835_Ws__def,axiom,
( ws
= ( minus_minus_set_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ us ) ) ).
% Ws_def
thf(fact_836_card_Oempty,axiom,
( ( finite_card_set_nat @ bot_bot_set_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_837_card_Oempty,axiom,
( ( finite1149291290879098388et_nat @ bot_bo7198184520161983622et_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_838_card_Oempty,axiom,
( ( finite_card_nat_nat @ bot_bot_set_nat_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_839_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_857_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_858_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_859_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_860_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_861_joinl__join,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y2 ) @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) ) ).
% joinl_join
thf(fact_862__092_060open_062f_A_092_060in_062_AACC__cf_AY_A_N_AACC__cf_AX_092_060close_062,axiom,
member_nat_nat @ f @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ y ) @ ( clique951075384711337423ACC_cf @ k @ x ) ) ).
% \<open>f \<in> ACC_cf Y - ACC_cf X\<close>
thf(fact_863_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_864_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_865_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_866_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_867_ex__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ? [X4: set_set_nat] : ( member_set_set_nat @ X4 @ A2 ) )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_868_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X4: nat > nat] : ( member_nat_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_869_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_870_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_871_equals0I,axiom,
! [A2: set_set_set_nat] :
( ! [Y3: set_set_nat] :
~ ( member_set_set_nat @ Y3 @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_872_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y3: nat > nat] :
~ ( member_nat_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_873_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_874_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_875_equals0D,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( A2 = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_876_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_877_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_878_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_879_emptyE,axiom,
! [A: set_set_nat] :
~ ( member_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_880_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_881_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_882_double__diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C3 )
=> ( ( minus_2447799839930672331et_nat @ B2 @ ( minus_2447799839930672331et_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_883_double__diff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C3 )
=> ( ( minus_2163939370556025621et_nat @ B2 @ ( minus_2163939370556025621et_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_884_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_885_double__diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C3 )
=> ( ( minus_8121590178497047118at_nat @ B2 @ ( minus_8121590178497047118at_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_886_Diff__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_887_Diff__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_888_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_889_Diff__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_890_Diff__mono,axiom,
! [A2: set_set_set_nat,C3: set_set_set_nat,D2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C3 )
=> ( ( ord_le9131159989063066194et_nat @ D2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ ( minus_2447799839930672331et_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_891_Diff__mono,axiom,
! [A2: set_set_nat,C3: set_set_nat,D2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_892_Diff__mono,axiom,
! [A2: set_nat,C3: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_893_Diff__mono,axiom,
! [A2: set_nat_nat,C3: set_nat_nat,D2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C3 )
=> ( ( ord_le9059583361652607317at_nat @ D2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ ( minus_8121590178497047118at_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_894_Diff__infinite__finite,axiom,
! [T: set_set_set_nat,S: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ T )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_895_Diff__infinite__finite,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( finite1152437895449049373et_nat @ T )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_896_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_897_Diff__infinite__finite,axiom,
! [T: set_nat_nat,S: set_nat_nat] :
( ( finite2115694454571419734at_nat @ T )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_898_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_899_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_900_finite_OemptyI,axiom,
finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).
% finite.emptyI
thf(fact_901_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_902_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_903_infinite__imp__nonempty,axiom,
! [S: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ( S != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_904_infinite__imp__nonempty,axiom,
! [S: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ( S != bot_bot_set_nat_nat ) ) ).
% infinite_imp_nonempty
thf(fact_905_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_906_psubset__imp__ex__mem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_907_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_908_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ? [B4: nat > nat] : ( member_nat_nat @ B4 @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_909_psubset__imp__ex__mem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ? [B4: set_set_nat] : ( member_set_set_nat @ B4 @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_910_second__assumptions_Ov__sameprod__subset,axiom,
! [L: nat,P2: nat,K: nat,Vs: set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ) ).
% second_assumptions.v_sameprod_subset
thf(fact_911_not__psubset__empty,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_912_not__psubset__empty,axiom,
! [A2: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_913_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_914_not__psubset__empty,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% not_psubset_empty
thf(fact_915_empty__sunflower,axiom,
sunflower_nat @ bot_bot_set_set_nat ).
% empty_sunflower
thf(fact_916_empty__sunflower,axiom,
sunflower_set_nat @ bot_bo7198184520161983622et_nat ).
% empty_sunflower
thf(fact_917_image__diff__subset,axiom,
! [F2: nat > set_set_nat,A2: set_nat,B2: set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F2 @ A2 ) @ ( image_2194112158459175443et_nat @ F2 @ B2 ) ) @ ( image_2194112158459175443et_nat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_918_image__diff__subset,axiom,
! [F2: ( nat > nat ) > set_set_nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_9186907679027735170et_nat @ F2 @ A2 ) @ ( image_9186907679027735170et_nat @ F2 @ B2 ) ) @ ( image_9186907679027735170et_nat @ F2 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_919_image__diff__subset,axiom,
! [F2: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F2 @ A2 ) @ ( image_5842784325960735177et_nat @ F2 @ B2 ) ) @ ( image_5842784325960735177et_nat @ F2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_920_image__diff__subset,axiom,
! [F2: nat > set_nat,A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ ( image_nat_set_nat @ F2 @ B2 ) ) @ ( image_nat_set_nat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_921_image__diff__subset,axiom,
! [F2: ( nat > nat ) > set_nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_7432509271690132940et_nat @ F2 @ A2 ) @ ( image_7432509271690132940et_nat @ F2 @ B2 ) ) @ ( image_7432509271690132940et_nat @ F2 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_922_image__diff__subset,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B2 ) ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_923_image__diff__subset,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ ( image_nat_nat_nat @ F2 @ B2 ) ) @ ( image_nat_nat_nat @ F2 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_924_image__diff__subset,axiom,
! [F2: nat > nat > nat,A2: set_nat,B2: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ ( image_nat_nat_nat2 @ F2 @ B2 ) ) @ ( image_nat_nat_nat2 @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_925_image__diff__subset,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ ( image_3205354838064109189at_nat @ F2 @ B2 ) ) @ ( image_3205354838064109189at_nat @ F2 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_926_finite__has__maximal,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat_nat )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_927_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_928_finite__has__maximal,axiom,
! [A2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ? [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_929_finite__has__maximal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_930_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_931_finite__has__maximal,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_932_finite__has__minimal,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat_nat )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_933_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_934_finite__has__minimal,axiom,
! [A2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ? [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_935_finite__has__minimal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_936_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_937_finite__has__minimal,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_938_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ X5 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X5 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_939_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P2: nat,K: nat,G4: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G4 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G4 ) @ ( clique5033774636164728513irst_v @ G4 ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_940_card__le__sym__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_941_card__le__sym__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_942_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_943_card__le__sym__Diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_944_card__less__sym__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_945_card__less__sym__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_946_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_947_card__less__sym__Diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_948_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_949_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_950_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_951_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_952_first__assumptions_Ov__mono,axiom,
! [L: nat,P2: nat,K: nat,G4: set_set_nat,H4: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G4 @ H4 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G4 ) @ ( clique5033774636164728513irst_v @ H4 ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_953_numbers2__mono,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y ) @ ( clique3652268606331196573umbers @ Y ) ) ) ) ).
% numbers2_mono
thf(fact_954_card__Diff__subset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_955_card__Diff__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_956_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_957_card__Diff__subset,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_958_diff__card__le__card__Diff,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_959_diff__card__le__card__Diff,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_960_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_961_diff__card__le__card__Diff,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_962_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P2: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_963_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_964_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_965_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_966_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_967_sameprod__mono,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) @ ( clique1181040904276305582et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_968_sameprod__mono,axiom,
! [X5: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) @ ( clique8906516429304539640et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_969_sameprod__mono,axiom,
! [X5: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X5 @ Y2 )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X5 @ X5 ) @ ( clique134924887794942129at_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_970_sameprod__mono,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) @ ( clique6722202388162463298od_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_971_sameprod__finite,axiom,
! [X5: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X5 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_972_sameprod__finite,axiom,
! [X5: set_set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_973_sameprod__finite,axiom,
! [X5: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X5 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_974_sameprod__finite,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_975_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_976_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P2: nat,K: nat,G4: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G4 ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_977_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G4: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G4 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_978_first__assumptions_O_092_060K_062___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.\<K>_\<G>
thf(fact_979_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_980_first__assumptions_ONEG___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.NEG_\<G>
thf(fact_981_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_982_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_983_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_984__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,
! [G4: set_set_nat] :
( ( member_set_set_nat @ G4 @ ( clique3210737319928189260st_ACC @ k @ x ) )
=> ( ( member_set_set_nat @ G4 @ ( clique3326749438856946062irst_K @ k ) )
=> ( member_set_set_nat @ G4 @ ( clique3210737319928189260st_ACC @ k @ y ) ) ) ) ).
% \<open>\<And>G. \<lbrakk>G \<in> ACC X; G \<in> POS\<rbrakk> \<Longrightarrow> G \<in> ACC Y\<close>
thf(fact_985_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_986_card__v__gs__join,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_987__092_060open_062Y_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
y != bot_bo7198184520161983622et_nat ).
% \<open>Y \<noteq> {}\<close>
thf(fact_988_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_989_DiffI,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 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_990_DiffI,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 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_991_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_992_DiffI,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 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_993_Diff__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( ( member_set_set_nat @ C @ A2 )
& ~ ( member_set_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_994_Diff__iff,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C @ A2 )
& ~ ( member_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_995_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_996_Diff__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat @ C @ A2 )
& ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_997_Diff__idemp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_998_Diff__idemp,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ B2 )
= ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_999_f,axiom,
member_nat_nat @ f @ ( clique2971579238625216137irst_F @ k ) ).
% f
thf(fact_1000__092_060open_062C_Af_A_092_060in_062_ANEG_092_060close_062,axiom,
member_set_set_nat @ ( clique5033774636164728462irst_C @ k @ f ) @ ( clique3210737375870294875st_NEG @ k ) ).
% \<open>C f \<in> NEG\<close>
thf(fact_1001_GsCf,axiom,
ord_le6893508408891458716et_nat @ gs @ ( clique5033774636164728462irst_C @ k @ f ) ).
% GsCf
thf(fact_1002_Cf,axiom,
member_set_set_nat @ ( clique5033774636164728462irst_C @ k @ f ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% Cf
thf(fact_1003_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1004_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1005_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1006_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1007_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1008_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_1009_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_1010_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_1011_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1012_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1013_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1014_ACC__cf__I,axiom,
! [F: nat > nat,X5: set_set_set_nat] :
( ( member_nat_nat @ F @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ ( clique5033774636164728462irst_C @ k @ F ) )
=> ( member_nat_nat @ F @ ( clique951075384711337423ACC_cf @ k @ X5 ) ) ) ) ).
% ACC_cf_I
thf(fact_1015_ACC__I,axiom,
! [G4: set_set_nat,X5: set_set_set_nat] :
( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ G4 )
=> ( member_set_set_nat @ G4 @ ( clique3210737319928189260st_ACC @ k @ X5 ) ) ) ) ).
% ACC_I
thf(fact_1016_DiffE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1017_DiffE,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1018_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1019_DiffE,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1020_DiffD1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ( member_set_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1021_DiffD1,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1022_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1023_DiffD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1024_DiffD2,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ~ ( member_set_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1025_DiffD2,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( member_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1026_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1027_DiffD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1028_first__assumptions_OACC__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1029_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_1030_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1031_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_1032_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1033_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1034_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A6: nat,B8: nat] : ( times_times_nat @ B8 @ A6 ) ) ) ).
% mult.commute
thf(fact_1035_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1036_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_1037_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_1038_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_1039_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_1040_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1041_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1042_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1043_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1044_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1045_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1046_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1047_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1048_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1049_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1050_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1051_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1052_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1053_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1054_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1055_first__assumptions_Ov__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_1056_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_1057_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1058_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1059_first__assumptions_ONEG__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1060_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1061_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P2: nat,K: nat,F: nat > nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_nat_nat @ F @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ ( clique5033774636164728462irst_C @ K @ F ) )
=> ( member_nat_nat @ F @ ( clique951075384711337423ACC_cf @ K @ X5 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1062_first__assumptions_OACC__I,axiom,
! [L: nat,P2: nat,K: nat,G4: set_set_nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ G4 )
=> ( member_set_set_nat @ G4 @ ( clique3210737319928189260st_ACC @ K @ X5 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_1063_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y2 ) @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1064_first__assumptions_Ocard__v__gs__join,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat,Z3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ) ) ) ) ).
% first_assumptions.card_v_gs_join
thf(fact_1065_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1066_uw_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( f @ ( u @ I ) )
= ( f @ ( w @ I ) ) ) ) ).
% uw(4)
thf(fact_1067_uw_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_nat @ ( w @ I ) @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% uw(2)
thf(fact_1068_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1069__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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_ACC__odot,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X5 ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ).
% ACC_odot
thf(fact_1079_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_1080_Int__subset__iff,axiom,
! [C3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
& ( ord_le9131159989063066194et_nat @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1081_Int__subset__iff,axiom,
! [C3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
& ( ord_le6893508408891458716et_nat @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1082_Int__subset__iff,axiom,
! [C3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( ord_less_eq_set_nat @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1083_Int__subset__iff,axiom,
! [C3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( ( ord_le9059583361652607317at_nat @ C3 @ A2 )
& ( ord_le9059583361652607317at_nat @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1084_finite__Int,axiom,
! [F: set_nat,G4: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G4 ) ) ) ).
% finite_Int
thf(fact_1085_finite__Int,axiom,
! [F: set_set_set_nat,G4: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F )
| ( finite6739761609112101331et_nat @ G4 ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F @ G4 ) ) ) ).
% finite_Int
thf(fact_1086_finite__Int,axiom,
! [F: set_nat_nat,G4: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F )
| ( finite2115694454571419734at_nat @ G4 ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F @ G4 ) ) ) ).
% finite_Int
thf(fact_1087_finite__Int,axiom,
! [F: set_set_nat,G4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F )
| ( finite1152437895449049373et_nat @ G4 ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F @ G4 ) ) ) ).
% finite_Int
thf(fact_1088_odotl__def,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1089_Diff__disjoint,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
= bot_bot_set_set_nat ) ).
% Diff_disjoint
thf(fact_1090_Diff__disjoint,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) )
= bot_bo7198184520161983622et_nat ) ).
% Diff_disjoint
thf(fact_1091_Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1092_Diff__disjoint,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) )
= bot_bot_set_nat_nat ) ).
% Diff_disjoint
thf(fact_1093_Int__left__commute,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C3 ) )
= ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_1094_Int__left__commute,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C3 ) )
= ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_1095_Int__left__commute,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C3 ) )
= ( inf_inf_set_set_nat @ B2 @ ( inf_inf_set_set_nat @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_1096_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_1097_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_1098_Int__left__absorb,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1099_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A5: set_set_set_nat,B6: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B6 @ A5 ) ) ) ).
% Int_commute
thf(fact_1100_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] : ( inf_inf_set_nat_nat @ B6 @ A5 ) ) ) ).
% Int_commute
thf(fact_1101_Int__commute,axiom,
( inf_inf_set_set_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] : ( inf_inf_set_set_nat @ B6 @ A5 ) ) ) ).
% Int_commute
thf(fact_1102_Int__absorb,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1103_Int__absorb,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1104_Int__absorb,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1105_Int__assoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C3 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_1106_Int__assoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_1107_Int__assoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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 )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_1121_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 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_1122_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_1123_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 )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_1124_Int__greatest,axiom,
! [C3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C3 @ B2 )
=> ( ord_le9131159989063066194et_nat @ C3 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1125_Int__greatest,axiom,
! [C3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C3 @ B2 )
=> ( ord_le6893508408891458716et_nat @ C3 @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1126_Int__greatest,axiom,
! [C3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C3 @ B2 )
=> ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1127_Int__greatest,axiom,
! [C3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ C3 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1128_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_1129_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_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_Int__mono,axiom,
! [A2: set_set_set_nat,C3: set_set_set_nat,B2: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C3 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1145_Int__mono,axiom,
! [A2: set_set_nat,C3: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1146_Int__mono,axiom,
! [A2: set_nat,C3: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1147_Int__mono,axiom,
! [A2: set_nat_nat,C3: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C3 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1148_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 )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ! [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B2 )
=> ( X4 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1149_disjoint__iff__not__equal,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ! [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ B2 )
=> ( X4 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1150_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 )
= ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ! [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ B2 )
=> ( X4 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1151_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ B2 )
=> ( X4 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1152_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_1153_Int__empty__right,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_right
thf(fact_1154_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_1155_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_1156_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_1157_Int__empty__left,axiom,
! [B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ B2 )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_left
thf(fact_1158_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_1159_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_1160_disjoint__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ~ ( member_set_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1161_disjoint__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ~ ( member_set_set_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1162_disjoint__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ~ ( member_nat_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1163_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1164_Int__emptyI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ( member_set_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_1165_Int__emptyI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ~ ( member_set_set_nat @ X2 @ B2 ) )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat ) ) ).
% Int_emptyI
thf(fact_1166_Int__emptyI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ~ ( member_nat_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_1167_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1168_Diff__Int__distrib2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ C3 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C3 ) @ ( inf_in5711780100303410308et_nat @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1169_Diff__Int__distrib2,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ C3 )
= ( minus_2163939370556025621et_nat @ ( inf_inf_set_set_nat @ A2 @ C3 ) @ ( inf_inf_set_set_nat @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1170_Diff__Int__distrib2,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C3 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1171_Diff__Int__distrib2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ C3 )
= ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A2 @ C3 ) @ ( inf_inf_set_nat_nat @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1172_Diff__Int__distrib,axiom,
! [C3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ C3 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ C3 @ A2 ) @ ( inf_in5711780100303410308et_nat @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1173_Diff__Int__distrib,axiom,
! [C3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ C3 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( inf_inf_set_set_nat @ C3 @ A2 ) @ ( inf_inf_set_set_nat @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1174_Diff__Int__distrib,axiom,
! [C3: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ C3 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C3 @ A2 ) @ ( inf_inf_set_nat @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1175_Diff__Int__distrib,axiom,
! [C3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ C3 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ C3 @ A2 ) @ ( inf_inf_set_nat_nat @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1176_Diff__Diff__Int,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1177_Diff__Diff__Int,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1178_Diff__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1179_Diff__Diff__Int,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1180_Diff__Int2,axiom,
! [A2: set_set_set_nat,C3: set_set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C3 ) @ ( inf_in5711780100303410308et_nat @ B2 @ C3 ) )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1181_Diff__Int2,axiom,
! [A2: set_set_nat,C3: set_set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( inf_inf_set_set_nat @ A2 @ C3 ) @ ( inf_inf_set_set_nat @ B2 @ C3 ) )
= ( minus_2163939370556025621et_nat @ ( inf_inf_set_set_nat @ A2 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1182_Diff__Int2,axiom,
! [A2: set_nat,C3: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ ( inf_inf_set_nat @ B2 @ C3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1183_Diff__Int2,axiom,
! [A2: set_nat_nat,C3: set_nat_nat,B2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A2 @ C3 ) @ ( inf_inf_set_nat_nat @ B2 @ C3 ) )
= ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A2 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1184_first__assumptions_OACC__odot,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X5 ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1185_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1186_first__assumptions_Oodotl__def,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1187_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1188_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1189_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1190_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1191_uw1,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( member_set_nat @ ( insert_nat @ ( u @ I ) @ ( insert_nat @ ( w @ I ) @ bot_bot_set_nat ) ) @ ( g @ I ) )
& ( member_nat @ ( u @ I ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
& ( member_nat @ ( w @ I ) @ ( minus_minus_set_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ us ) )
& ( ( f @ ( u @ I ) )
= ( f @ ( w @ I ) ) ) ) ) ).
% uw1
thf(fact_1192_uw_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_nat @ ( f @ ( w @ I ) ) @ ( clique3652268606331196573umbers @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% uw(5)
thf(fact_1193__092_060open_062f_A_092_060in_062_AACC__cf_A_123Gs_125_092_060close_062,axiom,
member_nat_nat @ f @ ( clique951075384711337423ACC_cf @ k @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ).
% \<open>f \<in> ACC_cf {Gs}\<close>
thf(fact_1194_ACC__cf__odot,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_odot
thf(fact_1195_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1196_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1197_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1198_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1199_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1200_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1201_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X5 @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1202_uw_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_nat @ ( pair @ I ) @ ( inf_inf_set_set_nat @ ( g @ I ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% uw(3)
thf(fact_1203_uw_I6_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( pair @ I )
= ( insert_nat @ ( u @ I ) @ ( insert_nat @ ( w @ I ) @ bot_bot_set_nat ) ) ) ) ).
% uw(6)
thf(fact_1204__092_060open_062card_A_Iv__gs_AY_J_A_092_060le_062_Acard_A_Iv__gs_AX_J_A_N_Ap_A_L_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ p ) @ one_one_nat ) ).
% \<open>card (v_gs Y) \<le> card (v_gs X) - p + 1\<close>
thf(fact_1205_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1206__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_1207_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1208_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1209_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1210_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1211_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1212_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1213_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1214_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1215_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1216__092_060open_062w_A_092_060equiv_062_A_092_060lambda_062i_O_Asndd_A_Ipair_Ai_J_092_060close_062,axiom,
( w
= ( ^ [I4: nat] : ( sndd @ ( pair @ I4 ) ) ) ) ).
% \<open>w \<equiv> \<lambda>i. sndd (pair i)\<close>
thf(fact_1217__092_060open_062u_A_092_060equiv_062_A_092_060lambda_062i_O_Afstt_A_Ipair_Ai_J_092_060close_062,axiom,
( u
= ( ^ [I4: nat] : ( fstt @ ( pair @ I4 ) ) ) ) ).
% \<open>u \<equiv> \<lambda>i. fstt (pair i)\<close>
thf(fact_1218_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1219_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1220_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1221_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1222_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1223_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1224_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1225_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1226_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1227_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1228_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1229_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1230_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1231_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1232_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1233_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1234_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1235_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1236_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1237_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1238_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1239_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1240_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1241_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1242_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1243_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1244_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1245_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1246_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F2 @ M3 ) @ ( F2 @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1247_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1248_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1249_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1250_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1251_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1252_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1253_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1254_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1255_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1256_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1257_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1258_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1259_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1260_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1261_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1262_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1263_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1264_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
=> ( A2
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1265_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1266_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1267_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N5: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N5 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1268_first__assumptions_Olm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1269_u__def,axiom,
! [I: nat] :
( ( u @ I )
= ( fstt @ ( pair @ I ) ) ) ).
% u_def
thf(fact_1270_w__def,axiom,
! [I: nat] :
( ( w @ I )
= ( sndd @ ( pair @ I ) ) ) ).
% w_def
% Helper facts (3)
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,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( finite_card_nat @ us )
= p ) ).
%------------------------------------------------------------------------------