TPTP Problem File: SLH0007^1.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_00787_027282__16236244_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1418 ( 522 unt; 146 typ; 0 def)
% Number of atoms : 3891 (1261 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10767 ( 365 ~; 55 |; 249 &;8382 @)
% ( 0 <=>;1716 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 868 ( 868 >; 0 *; 0 +; 0 <<)
% Number of symbols : 138 ( 135 usr; 21 con; 0-3 aty)
% Number of variables : 3650 ( 234 ^;3306 !; 110 ?;3650 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:10.860
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_nat_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
set_nat_set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
set_set_set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_nat_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
set_set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (135)
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_OL,type,
assump1710595444109740301irst_L: nat > nat > nat ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oaccepts,type,
clique3686358387679108662ccepts: set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oplucking__step,type,
clique4095374090462327202g_step: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov__gs,type,
clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
comple6520635616725053857_nat_o: set_nat_set_nat_o > ( nat > set_nat ) > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
comple6797894177231197998et_nat: set_nat_set_nat > nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
comple7674251194079574480_nat_o: set_set_nat_o > set_nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
comple7924776856948413338_nat_o: set_set_set_nat_o > set_set_nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
comple5153742063261271012et_nat: set_set_nat_set_nat > set_nat_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
comple8067742441731897515et_nat: set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
bij_be5082831075535440701at_nat: ( ( nat > set_nat ) > nat ) > set_nat_set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
bij_be8549092308015455677et_nat: ( nat > nat > set_nat ) > set_nat > set_nat_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bij_betw_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be6938610931847138308et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
bij_betw_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be3438014552859920132et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be5767359585022399418et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
bij_be6199415091885040644at_nat: ( set_set_nat > nat ) > set_set_set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be4885122793727115194et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be1917187662166652016et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on8105003582846801791et_nat: ( nat > set_set_nat ) > set_nat > $o ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
piE_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_8060664002660188164et_nat: set_nat_set_nat > set_nat_set_nat > set_nat_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo8210142506433397254_nat_o: ( nat > set_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__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo4007787791999405887et_nat: set_nat_set_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_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__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_nat_set_nat: ( nat > set_nat ) > ( nat > set_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__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le7745323766158300927et_nat: set_nat_set_nat > set_nat_set_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_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6195038898401538645et_nat: ( nat > set_nat ) > ( nat > set_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__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le1585852046946910987et_nat: set_nat_set_nat > set_nat_set_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_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_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
collect_nat_set_nat: ( ( nat > set_nat ) > $o ) > set_nat_set_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_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
image_6262123972677179520et_nat: ( ( ( nat > set_nat ) > $o ) > set_nat_set_nat ) > set_nat_set_nat_o > set_set_nat_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_970537773860477644at_nat: ( ( nat > set_nat ) > nat ) > set_nat_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_8304670887732450946et_nat: ( ( nat > set_nat ) > set_nat ) > set_nat_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7290825263825464120et_nat: ( ( nat > set_nat ) > set_set_nat ) > set_nat_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_4687162037615663680et_nat: ( ( set_nat > $o ) > set_set_nat ) > set_set_nat_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_3164711303094801856et_nat: ( ( set_set_nat > $o ) > set_set_set_nat ) > set_set_set_nat_o > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
image_4436799006340492620et_nat: ( nat > nat > set_nat ) > set_nat > set_nat_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__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__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_5738044413236618185et_nat: ( nat > set_set_set_nat ) > set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
image_1110766093228767069et_nat: ( set_nat_set_nat > set_nat_set_nat ) > set_set_nat_set_nat > set_set_nat_set_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__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_4583741654806091647et_nat: ( set_nat > set_set_set_nat ) > set_set_nat > set_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_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_6473237745780476395et_nat: ( set_set_set_nat > set_set_set_nat ) > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
insert_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > set_nat_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__Set__Oset_It__Nat__Onat_J_J,type,
set_or6252518528881150372et_nat: ( nat > set_nat ) > ( nat > set_nat ) > set_nat_set_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_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or5410080298493297259et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_or659464924768625697et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Sunflower_Osunflower_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
sunflo5599553548652064642et_nat: set_set_nat_set_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_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_nat_set_nat: ( nat > set_nat ) > set_nat_set_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__Set__Oset_It__Nat__Onat_J_J_J,type,
member6710465769566284994et_nat: set_nat_set_nat > set_set_nat_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_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_v_G____,type,
g: nat > set_set_nat ).
thf(sy_v_Gs____,type,
gs: set_set_nat ).
thf(sy_v_S____,type,
s: set_set_nat ).
thf(sy_v_Si____,type,
si: nat > set_nat ).
thf(sy_v_U____,type,
u: set_set_set_nat ).
thf(sy_v_Us____,type,
us: set_nat ).
thf(sy_v_Vs____,type,
vs: set_nat ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_f____,type,
f: nat > nat ).
thf(sy_v_fstt____,type,
fstt: set_nat > 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,
u2: nat > nat ).
thf(sy_v_w____,type,
w: nat > nat ).
% Relevant facts (1268)
thf(fact_0__092_060open_062_092_060And_062j_Ai_O_A_092_060lbrakk_062i_A_060_Ap_059_Aj_A_060_Ap_059_Au_Ai_A_061_Au_Aj_059_Ai_A_092_060noteq_062_Aj_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( ord_less_nat @ J @ p )
=> ( ( ( u2 @ I )
= ( u2 @ J ) )
=> ( I = J ) ) ) ) ).
% \<open>\<And>j i. \<lbrakk>i < p; j < p; u i = u j; i \<noteq> j\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_1_p0,axiom,
p != zero_zero_nat ).
% p0
thf(fact_2_Us__def,axiom,
( us
= ( image_nat_nat @ u2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ).
% Us_def
thf(fact_3_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_4_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_5__092_060open_062u_A_092_060equiv_062_A_092_060lambda_062i_O_Afstt_A_Ipair_Ai_J_092_060close_062,axiom,
( u2
= ( ^ [I2: nat] : ( fstt @ ( pair @ I2 ) ) ) ) ).
% \<open>u \<equiv> \<lambda>i. fstt (pair i)\<close>
thf(fact_6_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_7_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_8_rq,axiom,
ord_less_eq_nat @ p @ r ).
% rq
thf(fact_9_Si,axiom,
bij_betw_nat_set_nat @ si @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ).
% Si
thf(fact_10_u__def,axiom,
! [I: nat] :
( ( u2 @ I )
= ( fstt @ ( pair @ I ) ) ) ).
% u_def
thf(fact_11__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_12__092_060open_062_092_060exists_062h_O_Abij__betw_Ah_A_1230_O_O_060p_125_AS_092_060close_062,axiom,
? [H: nat > set_nat] : ( bij_betw_nat_set_nat @ H @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ) ).
% \<open>\<exists>h. bij_betw h {0..<p} S\<close>
thf(fact_13_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_14_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_15_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_16_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_17_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_18_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_19_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_20_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_21_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_22_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_23_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_24_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_25_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_26_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_27_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_28_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_29_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_30_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_31_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_32_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_33_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_34_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_35_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_36_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_37_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_38_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_39_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_40_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_41_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: nat > set_nat,P: ( nat > set_nat ) > $o] :
( ( member_nat_set_nat @ A @ ( collect_nat_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X: set_set_nat] : ( member_set_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_nat_set_nat] :
( ( collect_nat_set_nat
@ ^ [X: nat > set_nat] : ( member_nat_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_48_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_49_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_50_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_51_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_52_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_53_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_54_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_55_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_56_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_57_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_58_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_59_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_60_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_61_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_62_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_63_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_64_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_65_atLeastLessThan__iff,axiom,
! [I: nat > set_nat,L: nat > set_nat,U: nat > set_nat] :
( ( member_nat_set_nat @ I @ ( set_or6252518528881150372et_nat @ L @ U ) )
= ( ( ord_le6195038898401538645et_nat @ L @ I )
& ( ord_less_nat_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_66_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_67_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_68_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_69_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_70_ivl__subset,axiom,
! [I: nat,J: nat,M2: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M2 @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_71_S_I2_J,axiom,
sunflower_nat @ s ).
% S(2)
thf(fact_72_S_I3_J,axiom,
( ( finite_card_set_nat @ s )
= p ) ).
% S(3)
thf(fact_73_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_74_image__eqI,axiom,
! [B: nat,F: set_nat > nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_75_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_76_image__eqI,axiom,
! [B: nat,F: set_set_nat > nat,X2: set_set_nat,A2: set_set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_77_image__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_78_image__eqI,axiom,
! [B: set_set_nat,F: nat > set_set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_79_image__eqI,axiom,
! [B: set_nat,F: set_set_nat > set_nat,X2: set_set_nat,A2: set_set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_80_image__eqI,axiom,
! [B: set_set_nat,F: set_nat > set_set_nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_81_image__eqI,axiom,
! [B: nat > set_nat,F: nat > nat > set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_set_nat @ B @ ( image_4436799006340492620et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_82_image__eqI,axiom,
! [B: nat,F: ( nat > set_nat ) > nat,X2: nat > set_nat,A2: set_nat_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_set_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_83_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_84_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_85_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_86_order__refl,axiom,
! [X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_87_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_88_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_89_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_90_dual__order_Orefl,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_91_i__props_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_nat @ s2 @ ( si2 @ I ) ) ) ).
% i_props(4)
thf(fact_92_Snempty,axiom,
s != bot_bot_set_set_nat ).
% Snempty
thf(fact_93_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( image_nat_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_94_bij__betw__imp__surj__on,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( image_5842784325960735177et_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_95_bij__betw__imp__surj__on,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( image_2194112158459175443et_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_96_bij__betw__imp__surj__on,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( image_nat_set_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_97_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_98_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_99_subsetI,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( member_nat_set_nat @ X3 @ B2 ) )
=> ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_100_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ X3 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_101_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_102_subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_set_set_nat @ X3 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_103_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_104_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_105_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_106_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_107_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_108_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_109_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_110_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_111_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X: set_set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_112_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_113_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_114_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X: set_set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_115_all__not__in__conv,axiom,
! [A2: set_nat_set_nat] :
( ( ! [X: nat > set_nat] :
~ ( member_nat_set_nat @ X @ A2 ) )
= ( A2 = bot_bo4007787791999405887et_nat ) ) ).
% all_not_in_conv
thf(fact_116_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_117_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_118_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X: set_set_nat] :
~ ( member_set_set_nat @ X @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_119_empty__iff,axiom,
! [C: nat > set_nat] :
~ ( member_nat_set_nat @ C @ bot_bo4007787791999405887et_nat ) ).
% empty_iff
thf(fact_120_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_121_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_122_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_123_image__is__empty,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_124_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_125_image__is__empty,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ( image_5842784325960735177et_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_126_image__is__empty,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( ( image_set_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_127_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_128_image__is__empty,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ( image_1454916318497077779at_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_129_image__is__empty,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( ( image_6725021117256019401et_nat @ F @ A2 )
= bot_bo7198184520161983622et_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_130_image__is__empty,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( ( image_2194112158459175443et_nat @ F @ A2 )
= bot_bo7198184520161983622et_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_131_image__is__empty,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( ( image_7884819252390400639et_nat @ F @ A2 )
= bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_132_empty__is__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_133_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_134_empty__is__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_135_empty__is__image,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_136_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_137_empty__is__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( bot_bot_set_nat
= ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_138_empty__is__image,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_139_empty__is__image,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_140_empty__is__image,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_141_image__empty,axiom,
! [F: set_nat > set_nat] :
( ( image_7916887816326733075et_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_142_image__empty,axiom,
! [F: set_nat > nat] :
( ( image_set_nat_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_143_image__empty,axiom,
! [F: set_nat > set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ bot_bot_set_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_144_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_145_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_146_image__empty,axiom,
! [F: nat > set_set_nat] :
( ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_147_image__empty,axiom,
! [F: set_set_nat > set_nat] :
( ( image_5842784325960735177et_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_148_image__empty,axiom,
! [F: set_set_nat > nat] :
( ( image_1454916318497077779at_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_149_image__empty,axiom,
! [F: set_set_nat > set_set_nat] :
( ( image_7884819252390400639et_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_150_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_151_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_152_subset__empty,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_153_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_154_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_155_empty__subsetI,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% empty_subsetI
thf(fact_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_in__mono,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,X2: nat > set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ X2 @ A2 )
=> ( member_nat_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_167_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_168_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_169_in__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_170_subsetD,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C: nat > set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ C @ A2 )
=> ( member_nat_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_subset__eq,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ A3 )
=> ( member_nat_set_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_181_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( member_set_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_182_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_183_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A3 )
=> ( member_set_set_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_184_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_185_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_186_equalityD1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_187_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_188_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_189_equalityD2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_190_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_191_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_192_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_193_subset__iff,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
! [T2: nat > set_nat] :
( ( member_nat_set_nat @ T2 @ A3 )
=> ( member_nat_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_194_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A3 )
=> ( member_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_195_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_196_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T2: set_set_nat] :
( ( member_set_set_nat @ T2 @ A3 )
=> ( member_set_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_197_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_198_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_199_subset__refl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_200_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_201_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_202_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_203_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_204_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_205_subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_206_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_207_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_208_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_209_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_210_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_211_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 ) )
= ( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_212_not__psubset__empty,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_213_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_214_not__psubset__empty,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% not_psubset_empty
thf(fact_215_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_216_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_217_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_218_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_219_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_220_psubset__subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_221_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_222_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_223_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ~ ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_224_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_225_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_226_subset__psubset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_227_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_228_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_229_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_230_ex__in__conv,axiom,
! [A2: set_nat_set_nat] :
( ( ? [X: nat > set_nat] : ( member_nat_set_nat @ X @ A2 ) )
= ( A2 != bot_bo4007787791999405887et_nat ) ) ).
% ex_in_conv
thf(fact_231_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_232_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_233_ex__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ? [X: set_set_nat] : ( member_set_set_nat @ X @ A2 ) )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_234_equals0I,axiom,
! [A2: set_nat_set_nat] :
( ! [Y2: nat > set_nat] :
~ ( member_nat_set_nat @ Y2 @ A2 )
=> ( A2 = bot_bo4007787791999405887et_nat ) ) ).
% equals0I
thf(fact_235_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_236_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_237_equals0I,axiom,
! [A2: set_set_set_nat] :
( ! [Y2: set_set_nat] :
~ ( member_set_set_nat @ Y2 @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_238_equals0D,axiom,
! [A2: set_nat_set_nat,A: nat > set_nat] :
( ( A2 = bot_bo4007787791999405887et_nat )
=> ~ ( member_nat_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_239_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_240_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_241_equals0D,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( A2 = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_242_emptyE,axiom,
! [A: nat > set_nat] :
~ ( member_nat_set_nat @ A @ bot_bo4007787791999405887et_nat ) ).
% emptyE
thf(fact_243_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_244_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_245_emptyE,axiom,
! [A: set_set_nat] :
~ ( member_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_246_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_247_bot_Oextremum__uniqueI,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_248_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_249_bot_Oextremum__uniqueI,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_250_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_251_bot_Oextremum__unique,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_252_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_253_bot_Oextremum__unique,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_254_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_255_bot_Oextremum,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% bot.extremum
thf(fact_256_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_257_bot_Oextremum,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% bot.extremum
thf(fact_258_bot_Onot__eq__extremum,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
= ( ord_less_set_set_nat @ bot_bot_set_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_259_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_260_bot_Onot__eq__extremum,axiom,
! [A: set_set_set_nat] :
( ( A != bot_bo7198184520161983622et_nat )
= ( ord_le152980574450754630et_nat @ bot_bo7198184520161983622et_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_261_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_262_bot_Oextremum__strict,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% bot.extremum_strict
thf(fact_263_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_264_bot_Oextremum__strict,axiom,
! [A: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% bot.extremum_strict
thf(fact_265_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_266_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_267_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_268_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_5842784325960735177et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_269_subset__image__iff,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_270_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_271_subset__image__iff,axiom,
! [B2: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_1454916318497077779at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_272_subset__image__iff,axiom,
! [B2: set_set_set_nat,F: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_6725021117256019401et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_273_subset__image__iff,axiom,
! [B2: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_2194112158459175443et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_274_subset__image__iff,axiom,
! [B2: set_set_set_nat,F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_7884819252390400639et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_275_image__subset__iff,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_276_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_277_image__subset__iff,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_set_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_278_subset__imageE,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_279_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_280_subset__imageE,axiom,
! [B2: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( B2
!= ( image_5842784325960735177et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_281_subset__imageE,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_set_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_282_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_283_subset__imageE,axiom,
! [B2: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( B2
!= ( image_1454916318497077779at_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_284_subset__imageE,axiom,
! [B2: set_set_set_nat,F: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_6725021117256019401et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_285_subset__imageE,axiom,
! [B2: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_2194112158459175443et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_286_subset__imageE,axiom,
! [B2: set_set_set_nat,F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( B2
!= ( image_7884819252390400639et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_287_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_288_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_289_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_290_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_291_image__subsetI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat,B2: set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_292_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_set_nat,B2: set_set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_293_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > set_nat,B2: set_nat_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1585852046946910987et_nat @ ( image_4436799006340492620et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_294_image__subsetI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat,B2: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_295_image__subsetI,axiom,
! [A2: set_nat_set_nat,F: ( nat > set_nat ) > nat,B2: set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_970537773860477644at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_296_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,B2: set_set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_297_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_298_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_299_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_300_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_301_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_302_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_303_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_304_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_305_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_306_bij__betw__empty2,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty2
thf(fact_307_bij__betw__empty2,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_308_bij__betw__empty2,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty2
thf(fact_309_bij__betw__empty2,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty2
thf(fact_310_bij__betw__empty2,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_311_bij__betw__empty2,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty2
thf(fact_312_bij__betw__empty2,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty2
thf(fact_313_bij__betw__empty2,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_314_bij__betw__empty2,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( bij_be1917187662166652016et_nat @ F @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty2
thf(fact_315_bij__betw__empty1,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ bot_bot_set_set_nat @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_316_bij__betw__empty1,axiom,
! [F: set_nat > nat,A2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ bot_bot_set_set_nat @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_317_bij__betw__empty1,axiom,
! [F: set_nat > set_set_nat,A2: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ bot_bot_set_set_nat @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty1
thf(fact_318_bij__betw__empty1,axiom,
! [F: nat > set_nat,A2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ bot_bot_set_nat @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_319_bij__betw__empty1,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ bot_bot_set_nat @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_320_bij__betw__empty1,axiom,
! [F: nat > set_set_nat,A2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ bot_bot_set_nat @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty1
thf(fact_321_bij__betw__empty1,axiom,
! [F: set_set_nat > set_nat,A2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ bot_bo7198184520161983622et_nat @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_322_bij__betw__empty1,axiom,
! [F: set_set_nat > nat,A2: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ bot_bo7198184520161983622et_nat @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_323_bij__betw__empty1,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( bij_be1917187662166652016et_nat @ F @ bot_bo7198184520161983622et_nat @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty1
thf(fact_324_bij__betw__byWitness,axiom,
! [A2: set_set_nat,F2: set_nat > set_nat,F: set_nat > set_nat,A4: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F2 @ A4 ) @ A2 )
=> ( bij_be3438014552859920132et_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_325_bij__betw__byWitness,axiom,
! [A2: set_nat,F2: set_nat > nat,F: nat > set_nat,A4: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F2 @ A4 ) @ A2 )
=> ( bij_betw_nat_set_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_326_bij__betw__byWitness,axiom,
! [A2: set_set_set_nat,F2: set_nat > set_set_nat,F: set_set_nat > set_nat,A4: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ A4 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F2 @ A4 ) @ A2 )
=> ( bij_be4885122793727115194et_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_327_bij__betw__byWitness,axiom,
! [A2: set_set_nat,F2: nat > set_nat,F: set_nat > nat,A4: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ A2 )
=> ( bij_betw_set_nat_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_328_bij__betw__byWitness,axiom,
! [A2: set_nat,F2: nat > nat,F: nat > nat,A4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_329_bij__betw__byWitness,axiom,
! [A2: set_set_set_nat,F2: nat > set_set_nat,F: set_set_nat > nat,A4: set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ A4 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F2 @ A4 ) @ A2 )
=> ( bij_be6199415091885040644at_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_330_bij__betw__byWitness,axiom,
! [A2: set_set_nat,F2: set_set_nat > set_nat,F: set_nat > set_set_nat,A4: set_set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F2 @ A4 ) @ A2 )
=> ( bij_be5767359585022399418et_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_331_bij__betw__byWitness,axiom,
! [A2: set_nat,F2: set_set_nat > nat,F: nat > set_set_nat,A4: set_set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F2 @ A4 ) @ A2 )
=> ( bij_be6938610931847138308et_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_332_bij__betw__byWitness,axiom,
! [A2: set_set_set_nat,F2: set_set_nat > set_set_nat,F: set_set_nat > set_set_nat,A4: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ( F2 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A4 )
=> ( ( F @ ( F2 @ X3 ) )
= X3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ A4 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F2 @ A4 ) @ A2 )
=> ( bij_be1917187662166652016et_nat @ F @ A2 @ A4 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_333_bij__betw__subset,axiom,
! [F: nat > nat,A2: set_nat,A4: set_nat,B2: set_nat,B4: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ A4 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat @ F @ B2 )
= B4 )
=> ( bij_betw_nat_nat @ F @ B2 @ B4 ) ) ) ) ).
% bij_betw_subset
thf(fact_334_bij__betw__subset,axiom,
! [F: nat > set_set_nat,A2: set_nat,A4: set_set_set_nat,B2: set_nat,B4: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ A4 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_2194112158459175443et_nat @ F @ B2 )
= B4 )
=> ( bij_be6938610931847138308et_nat @ F @ B2 @ B4 ) ) ) ) ).
% bij_betw_subset
thf(fact_335_bij__betw__subset,axiom,
! [F: nat > set_nat,A2: set_nat,A4: set_set_nat,B2: set_nat,B4: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ A4 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_set_nat @ F @ B2 )
= B4 )
=> ( bij_betw_nat_set_nat @ F @ B2 @ B4 ) ) ) ) ).
% bij_betw_subset
thf(fact_336_bij__betw__subset,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,A4: set_set_nat,B2: set_set_set_nat,B4: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ A4 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ( image_5842784325960735177et_nat @ F @ B2 )
= B4 )
=> ( bij_be4885122793727115194et_nat @ F @ B2 @ B4 ) ) ) ) ).
% bij_betw_subset
thf(fact_337_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_338_order__antisym__conv,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_339_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_340_order__antisym__conv,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_341_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_342_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_343_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_344_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_345_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_346_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_347_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_348_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_349_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_350_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_351_ord__le__eq__subst,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_352_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_353_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_354_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_355_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_356_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_357_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_358_ord__eq__le__subst,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9131159989063066194et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_359_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_360_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_361_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_362_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_363_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_364_order__eq__refl,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( X2 = Y )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_365_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_366_order__eq__refl,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( X2 = Y )
=> ( ord_le9131159989063066194et_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_367_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_368_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_369_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_370_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_371_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_372_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_373_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_374_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_375_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_376_order__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_377_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_378_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_379_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_380_order__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_381_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_382_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_383_order__subst1,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_384_order__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_385_order__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_386_order__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9131159989063066194et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_387_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_388_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_389_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_390_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_set_set_nat,B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ B5 )
& ( ord_le9131159989063066194et_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_391_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_392_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_393_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_394_antisym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_395_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_396_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_397_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_398_dual__order_Otrans,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( ord_le9131159989063066194et_nat @ C @ B )
=> ( ord_le9131159989063066194et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_399_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_400_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_401_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_402_dual__order_Oantisym,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_403_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_404_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A5 )
& ( ord_le6893508408891458716et_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_405_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_406_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_set_set_nat,B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ A5 )
& ( ord_le9131159989063066194et_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_407_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_408_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_409_order__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_410_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_411_order__trans,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( ord_le9131159989063066194et_nat @ Y @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_412_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_413_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_414_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_415_order_Otrans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_416_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_417_order__antisym,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_418_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_419_order__antisym,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( ord_le9131159989063066194et_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_420_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_421_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_422_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_423_ord__le__eq__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_424_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_425_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_426_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_427_ord__eq__le__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( A = B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_428_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_429_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [X: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y5 )
& ( ord_le6893508408891458716et_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_430_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_431_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [X: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y5 )
& ( ord_le9131159989063066194et_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_432_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_433_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_434_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_435_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_436_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_437_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_438_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_439_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_440_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_441_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_442_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_443_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_444_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_445_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_446_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_447_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_448_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_449_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_450_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_451_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_452_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_453_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_454_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_455_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_456_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_457_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_458_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_459_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_460_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_461_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_462_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_463_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_464_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_465_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_466_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_467_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_468_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_469_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_470_rev__image__eqI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B: nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_471_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_472_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: set_set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_473_rev__image__eqI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B: set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_474_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_475_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat > set_nat,F: nat > nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_set_nat @ B @ ( image_4436799006340492620et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_476_rev__image__eqI,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,B: nat,F: ( nat > set_nat ) > nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_477_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_478_ball__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_479_ball__imageD,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_480_image__cong,axiom,
! [M4: set_set_set_nat,N4: set_set_set_nat,F: set_set_nat > set_nat,G: set_set_nat > set_nat] :
( ( M4 = N4 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5842784325960735177et_nat @ F @ M4 )
= ( image_5842784325960735177et_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_481_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M4 )
= ( image_nat_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_482_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( M4 = N4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_2194112158459175443et_nat @ F @ M4 )
= ( image_2194112158459175443et_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_483_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_484_bex__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_485_bex__imageD,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_486_image__iff,axiom,
! [Z2: set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( member_set_set_nat @ Z2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_487_image__iff,axiom,
! [Z2: set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( member_set_nat @ Z2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_488_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_489_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_490_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_491_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_492_imageI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_493_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_494_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F @ X2 ) @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_495_imageI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_496_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F @ X2 ) @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_497_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_set_nat @ ( F @ X2 ) @ ( image_4436799006340492620et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_498_imageI,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,F: ( nat > set_nat ) > nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_499_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M4: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_500_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F3: nat > nat,A3: set_nat,B3: set_nat] :
? [G2: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( member_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_501_bij__betw__iff__bijections,axiom,
( bij_betw_set_nat_nat
= ( ^ [F3: set_nat > nat,A3: set_set_nat,B3: set_nat] :
? [G2: nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( ( member_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( member_set_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_502_bij__betw__iff__bijections,axiom,
( bij_betw_nat_set_nat
= ( ^ [F3: nat > set_nat,A3: set_nat,B3: set_set_nat] :
? [G2: set_nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_set_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: set_nat] :
( ( member_set_nat @ X @ B3 )
=> ( ( member_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_503_bij__betw__iff__bijections,axiom,
( bij_be6938610931847138308et_nat
= ( ^ [F3: nat > set_set_nat,A3: set_nat,B3: set_set_set_nat] :
? [G2: set_set_nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_set_set_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ B3 )
=> ( ( member_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_504_bij__betw__iff__bijections,axiom,
( bij_be3438014552859920132et_nat
= ( ^ [F3: set_nat > set_nat,A3: set_set_nat,B3: set_set_nat] :
? [G2: set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( ( member_set_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: set_nat] :
( ( member_set_nat @ X @ B3 )
=> ( ( member_set_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_505_bij__betw__iff__bijections,axiom,
( bij_be6199415091885040644at_nat
= ( ^ [F3: set_set_nat > nat,A3: set_set_set_nat,B3: set_nat] :
? [G2: nat > set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A3 )
=> ( ( member_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( member_set_set_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_506_bij__betw__iff__bijections,axiom,
( bij_be5767359585022399418et_nat
= ( ^ [F3: set_nat > set_set_nat,A3: set_set_nat,B3: set_set_set_nat] :
? [G2: set_set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( ( member_set_set_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ B3 )
=> ( ( member_set_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_507_bij__betw__iff__bijections,axiom,
( bij_be4885122793727115194et_nat
= ( ^ [F3: set_set_nat > set_nat,A3: set_set_set_nat,B3: set_set_nat] :
? [G2: set_nat > set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A3 )
=> ( ( member_set_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: set_nat] :
( ( member_set_nat @ X @ B3 )
=> ( ( member_set_set_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_508_bij__betw__iff__bijections,axiom,
( bij_be5082831075535440701at_nat
= ( ^ [F3: ( nat > set_nat ) > nat,A3: set_nat_set_nat,B3: set_nat] :
? [G2: nat > nat > set_nat] :
( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ A3 )
=> ( ( member_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( member_nat_set_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_509_bij__betw__iff__bijections,axiom,
( bij_be8549092308015455677et_nat
= ( ^ [F3: nat > nat > set_nat,A3: set_nat,B3: set_nat_set_nat] :
? [G2: ( nat > set_nat ) > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_nat_set_nat @ ( F3 @ X ) @ B3 )
& ( ( G2 @ ( F3 @ X ) )
= X ) ) )
& ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ B3 )
=> ( ( member_nat @ ( G2 @ X ) @ A3 )
& ( ( F3 @ ( G2 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_510_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_511_bij__betw__apply,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat,A: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_512_bij__betw__apply,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,A: nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_513_bij__betw__apply,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat,A: set_set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_514_bij__betw__apply,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,A: set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_515_bij__betw__apply,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat,A: nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_516_bij__betw__apply,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat,A: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_517_bij__betw__apply,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat,A: set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ B2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_set_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_518_bij__betw__apply,axiom,
! [F: nat > nat > set_nat,A2: set_nat,B2: set_nat_set_nat,A: nat] :
( ( bij_be8549092308015455677et_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_519_bij__betw__apply,axiom,
! [F: ( nat > set_nat ) > nat,A2: set_nat_set_nat,B2: set_nat,A: nat > set_nat] :
( ( bij_be5082831075535440701at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_set_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_520_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > set_nat,G: nat > set_nat,A4: set_set_nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_betw_nat_set_nat @ F @ A2 @ A4 )
= ( bij_betw_nat_set_nat @ G @ A2 @ A4 ) ) ) ).
% bij_betw_cong
thf(fact_521_bij__betw__ball,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,Phi: set_nat > $o] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( Phi @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( Phi @ ( F @ X ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_522_bij__betw__inv,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ? [G3: nat > set_nat] : ( bij_betw_nat_set_nat @ G3 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_523_bij__betw__inv,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ? [G3: set_nat > nat] : ( bij_betw_set_nat_nat @ G3 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_524_bij__betwE,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_set_nat @ ( F @ X5 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_525_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_526_order__le__imp__less__or__eq,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_less_set_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_527_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_528_order__le__imp__less__or__eq,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( ord_le152980574450754630et_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_529_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_530_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_531_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_532_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_533_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le152980574450754630et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_534_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_535_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_536_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_537_order__less__le__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_538_order__less__le__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_539_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_540_order__less__le__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_541_order__less__le__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_542_order__less__le__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_543_order__less__le__subst1,axiom,
! [A: nat,F: set_set_set_nat > nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_544_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_545_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_546_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_547_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_548_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_549_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_550_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_551_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_552_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_553_order__le__less__subst2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_554_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_555_order__le__less__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_556_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_557_order__le__less__subst1,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat,B: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le152980574450754630et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le152980574450754630et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_558_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_559_order__less__le__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_less_set_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_560_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_561_order__less__le__trans,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y )
=> ( ( ord_le9131159989063066194et_nat @ Y @ Z2 )
=> ( ord_le152980574450754630et_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_562_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_563_order__le__less__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_less_set_set_nat @ Y @ Z2 )
=> ( ord_less_set_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_564_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_565_order__le__less__trans,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( ord_le152980574450754630et_nat @ Y @ Z2 )
=> ( ord_le152980574450754630et_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_566_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_567_order__neq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A != B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_568_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_569_order__neq__le__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A != B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_570_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_571_order__le__neq__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_572_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_573_order__le__neq__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_574_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_575_order__less__imp__le,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_576_order__less__imp__le,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_577_order__less__imp__le,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y )
=> ( ord_le9131159989063066194et_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_578_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_579_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_580_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_581_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_582_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_583_order__less__le,axiom,
( ord_le152980574450754630et_nat
= ( ^ [X: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_584_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_585_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X: set_set_nat,Y5: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_586_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_587_order__le__less,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_588_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_589_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_590_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_591_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_592_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_593_order_Ostrict__implies__order,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_594_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_595_order_Ostrict__implies__order,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_596_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_597_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B5: set_set_nat,A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A5 )
& ~ ( ord_le6893508408891458716et_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_598_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ~ ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_599_dual__order_Ostrict__iff__not,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B5: set_set_set_nat,A5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ A5 )
& ~ ( ord_le9131159989063066194et_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_600_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_601_dual__order_Ostrict__trans2,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_602_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_603_dual__order_Ostrict__trans2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B @ A )
=> ( ( ord_le9131159989063066194et_nat @ C @ B )
=> ( ord_le152980574450754630et_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_604_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_605_dual__order_Ostrict__trans1,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_606_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_607_dual__order_Ostrict__trans1,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( ord_le152980574450754630et_nat @ C @ B )
=> ( ord_le152980574450754630et_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_608_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_609_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B5: set_set_nat,A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_610_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_611_dual__order_Ostrict__iff__order,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B5: set_set_set_nat,A5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_612_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_613_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A5: set_set_nat] :
( ( ord_less_set_set_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_614_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_set_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_615_dual__order_Oorder__iff__strict,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B5: set_set_set_nat,A5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_616_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_617_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ~ ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_618_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_619_order_Ostrict__iff__not,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A5: set_set_set_nat,B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ B5 )
& ~ ( ord_le9131159989063066194et_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_620_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_621_order_Ostrict__trans2,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_622_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_623_order_Ostrict__trans2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_624_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_625_order_Ostrict__trans1,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_626_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_627_order_Ostrict__trans1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_628_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_629_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_630_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_631_order_Ostrict__iff__order,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A5: set_set_set_nat,B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_632_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_633_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_634_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_635_order_Oorder__iff__strict,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A5: set_set_set_nat,B5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_636_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_637_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_638_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y5 )
& ~ ( ord_le6893508408891458716et_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_639_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_640_less__le__not__le,axiom,
( ord_le152980574450754630et_nat
= ( ^ [X: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y5 )
& ~ ( ord_le9131159989063066194et_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_641_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_642_antisym__conv2,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_643_antisym__conv2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_644_antisym__conv2,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( ~ ( ord_le152980574450754630et_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_645_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_646_antisym__conv1,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_647_antisym__conv1,axiom,
! [X2: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_648_antisym__conv1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ~ ( ord_le152980574450754630et_nat @ X2 @ Y )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_649_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_650_nless__le,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A @ B ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_651_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_652_nless__le,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ~ ( ord_le152980574450754630et_nat @ A @ B ) )
= ( ~ ( ord_le9131159989063066194et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_653_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_654_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_655_leD,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ~ ( ord_less_set_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_656_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_657_leD,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X2 )
=> ~ ( ord_le152980574450754630et_nat @ X2 @ Y ) ) ).
% leD
thf(fact_658_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_659_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_660_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_661_card_Oempty,axiom,
( ( finite_card_set_nat @ bot_bot_set_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_662_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_663_card_Oempty,axiom,
( ( finite1149291290879098388et_nat @ bot_bo7198184520161983622et_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_664_i__props_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_nat @ ( si2 @ I ) @ l ) ) ).
% i_props(5)
thf(fact_665_ex__card,axiom,
! [N: nat,A2: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ A2 ) )
=> ? [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ A2 )
& ( ( finite_card_set_nat @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_666_ex__card,axiom,
! [N: nat,A2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A2 ) )
=> ? [S2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ A2 )
& ( ( finite_card_nat @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_667_ex__card,axiom,
! [N: nat,A2: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ A2 ) )
=> ? [S2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ A2 )
& ( ( finite1149291290879098388et_nat @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_668_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_669_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G4: set_set_nat] :
? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X6 )
& ( ord_le6893508408891458716et_nat @ X @ G4 ) ) ) ) ).
% accepts_def
thf(fact_670_empty__sunflower,axiom,
sunflower_nat @ bot_bot_set_set_nat ).
% empty_sunflower
thf(fact_671_empty__sunflower,axiom,
sunflower_set_nat @ bot_bo7198184520161983622et_nat ).
% empty_sunflower
thf(fact_672_bij__betw__same__card,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_673_bij__betw__same__card,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_674_bij__betw__same__card,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_675_bij__betw__same__card,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_676_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_677_bij__betw__same__card,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_678_bij__betw__same__card,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_679_bij__betw__same__card,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_680_bij__betw__same__card,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bij_be1917187662166652016et_nat @ F @ A2 @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_681_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_682_subset__emptyI,axiom,
! [A2: set_nat_set_nat] :
( ! [X3: nat > set_nat] :
~ ( member_nat_set_nat @ X3 @ A2 )
=> ( ord_le1585852046946910987et_nat @ A2 @ bot_bo4007787791999405887et_nat ) ) ).
% subset_emptyI
thf(fact_683_subset__emptyI,axiom,
! [A2: set_set_nat] :
( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_684_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_685_subset__emptyI,axiom,
! [A2: set_set_set_nat] :
( ! [X3: set_set_nat] :
~ ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ).
% subset_emptyI
thf(fact_686_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_687_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_688_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_689_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_690_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_691_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_692_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_693_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_694_acceptsI,axiom,
! [D2: set_set_nat,G5: set_set_nat,X7: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G5 )
=> ( ( member_set_set_nat @ D2 @ X7 )
=> ( clique3686358387679108662ccepts @ X7 @ G5 ) ) ) ).
% acceptsI
thf(fact_695_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_696_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_697_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_698_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_699_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_700_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_701_psubsetD,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C: nat > set_nat] :
( ( ord_le7745323766158300927et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ C @ A2 )
=> ( member_nat_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_702_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_703_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_704_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_705_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_706_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_707_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_708_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_709_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_710_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_711_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_712_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_713_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_714_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_715_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_716_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_717_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_718_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_719_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_720_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_721_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_722_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_723_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_724_bij__betwI_H,axiom,
! [X7: set_nat,F: nat > nat,Y6: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( member_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ Y6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_betw_nat_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_725_bij__betwI_H,axiom,
! [X7: set_set_nat,F: set_nat > nat,Y6: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ( member_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ Y6 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_betw_set_nat_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_726_bij__betwI_H,axiom,
! [X7: set_nat,F: nat > set_nat,Y6: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( member_set_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ Y6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_betw_nat_set_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_727_bij__betwI_H,axiom,
! [X7: set_set_set_nat,F: set_set_nat > nat,Y6: set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ! [Y2: set_set_nat] :
( ( member_set_set_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ( member_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ Y6 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be6199415091885040644at_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_728_bij__betwI_H,axiom,
! [X7: set_set_nat,F: set_nat > set_nat,Y6: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ( member_set_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ Y6 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be3438014552859920132et_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_729_bij__betwI_H,axiom,
! [X7: set_nat,F: nat > set_set_nat,Y6: set_set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( member_set_set_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: set_set_nat] :
( ( member_set_set_nat @ Y2 @ Y6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be6938610931847138308et_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_730_bij__betwI_H,axiom,
! [X7: set_set_set_nat,F: set_set_nat > set_nat,Y6: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ! [Y2: set_set_nat] :
( ( member_set_set_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ( member_set_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ Y6 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be4885122793727115194et_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_731_bij__betwI_H,axiom,
! [X7: set_set_nat,F: set_nat > set_set_nat,Y6: set_set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ( member_set_set_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: set_set_nat] :
( ( member_set_set_nat @ Y2 @ Y6 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be5767359585022399418et_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_732_bij__betwI_H,axiom,
! [X7: set_nat,F: nat > nat > set_nat,Y6: set_nat_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( member_nat_set_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: nat > set_nat] :
( ( member_nat_set_nat @ Y2 @ Y6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be8549092308015455677et_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_733_bij__betwI_H,axiom,
! [X7: set_nat_set_nat,F: ( nat > set_nat ) > nat,Y6: set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X7 )
=> ! [Y2: nat > set_nat] :
( ( member_nat_set_nat @ Y2 @ X7 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X7 )
=> ( member_nat @ ( F @ X3 ) @ Y6 ) )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ Y6 )
=> ? [X5: nat > set_nat] :
( ( member_nat_set_nat @ X5 @ X7 )
& ( Y2
= ( F @ X5 ) ) ) )
=> ( bij_be5082831075535440701at_nat @ F @ X7 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_734_sunflower__def,axiom,
( sunflo2680516271513359689et_nat
= ( ^ [S3: set_set_set_set_nat] :
! [X: set_set_nat] :
( ? [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A3 @ S3 )
& ( member2946998982187404937et_nat @ B3 @ S3 )
& ( A3 != B3 )
& ( member_set_set_nat @ X @ A3 )
& ( member_set_set_nat @ X @ B3 ) )
=> ! [A3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A3 @ S3 )
=> ( member_set_set_nat @ X @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_735_sunflower__def,axiom,
( sunflo5599553548652064642et_nat
= ( ^ [S3: set_set_nat_set_nat] :
! [X: nat > set_nat] :
( ? [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ A3 @ S3 )
& ( member6710465769566284994et_nat @ B3 @ S3 )
& ( A3 != B3 )
& ( member_nat_set_nat @ X @ A3 )
& ( member_nat_set_nat @ X @ B3 ) )
=> ! [A3: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ A3 @ S3 )
=> ( member_nat_set_nat @ X @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_736_sunflower__def,axiom,
( sunflower_set_nat
= ( ^ [S3: set_set_set_nat] :
! [X: set_nat] :
( ? [A3: set_set_nat,B3: set_set_nat] :
( ( member_set_set_nat @ A3 @ S3 )
& ( member_set_set_nat @ B3 @ S3 )
& ( A3 != B3 )
& ( member_set_nat @ X @ A3 )
& ( member_set_nat @ X @ B3 ) )
=> ! [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ S3 )
=> ( member_set_nat @ X @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_737_sunflower__def,axiom,
( sunflower_nat
= ( ^ [S3: set_set_nat] :
! [X: nat] :
( ? [A3: set_nat,B3: set_nat] :
( ( member_set_nat @ A3 @ S3 )
& ( member_set_nat @ B3 @ S3 )
& ( A3 != B3 )
& ( member_nat @ X @ A3 )
& ( member_nat @ X @ B3 ) )
=> ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S3 )
=> ( member_nat @ X @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_738_sunflower__subset,axiom,
! [F4: set_set_nat,G5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ F4 @ G5 )
=> ( ( sunflower_nat @ G5 )
=> ( sunflower_nat @ F4 ) ) ) ).
% sunflower_subset
thf(fact_739_sunflower__subset,axiom,
! [F4: set_set_set_nat,G5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ F4 @ G5 )
=> ( ( sunflower_set_nat @ G5 )
=> ( sunflower_set_nat @ F4 ) ) ) ).
% sunflower_subset
thf(fact_740_all__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_741_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_742_all__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( P @ ( image_5842784325960735177et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_743_all__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_744_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_745_all__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_746_all__subset__image,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_6725021117256019401et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_747_all__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_748_all__subset__image,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( P @ ( image_7884819252390400639et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_749_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_750_diff__shunt__var,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X2 @ Y )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_751_diff__shunt__var,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_752_diff__shunt__var,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ X2 @ Y )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_753_s__def,axiom,
( s2
= ( finite_card_nat @ vs ) ) ).
% s_def
thf(fact_754_card__Vs,axiom,
ord_less_eq_nat @ ( finite_card_nat @ vs ) @ l ).
% card_Vs
thf(fact_755_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_756_Diff__cancel,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ A2 )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_757_Diff__cancel,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% Diff_cancel
thf(fact_758_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_759_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_760_empty__Diff,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% empty_Diff
thf(fact_761_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_762_Diff__empty,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_763_Diff__empty,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% Diff_empty
thf(fact_764_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_765_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_766_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_767_ivl__diff,axiom,
! [I: nat,N: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M2 ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_768_Diff__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,D2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_769_Diff__mono,axiom,
! [A2: set_nat,C2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_770_Diff__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,D2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ D2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ ( minus_2447799839930672331et_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_771_Diff__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_772_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_773_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_774_double__diff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ( minus_2163939370556025621et_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_775_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_776_double__diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ( minus_2447799839930672331et_nat @ B2 @ ( minus_2447799839930672331et_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_777_psubset__imp__ex__mem,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( ord_le7745323766158300927et_nat @ A2 @ B2 )
=> ? [B6: nat > set_nat] : ( member_nat_set_nat @ B6 @ ( minus_8060664002660188164et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_778_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_779_psubset__imp__ex__mem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ? [B6: set_nat] : ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_780_psubset__imp__ex__mem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ? [B6: set_set_nat] : ( member_set_set_nat @ B6 @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_781_image__diff__subset,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_782_image__diff__subset,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) @ ( image_7916887816326733075et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_783_image__diff__subset,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) @ ( image_5842784325960735177et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_784_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_785_image__diff__subset,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) @ ( image_set_nat_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_786_image__diff__subset,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B2 ) ) @ ( image_1454916318497077779at_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_787_image__diff__subset,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) @ ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_788_image__diff__subset,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B2 ) ) @ ( image_6725021117256019401et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_789_image__diff__subset,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B2 ) ) @ ( image_7884819252390400639et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_790_Vs__def,axiom,
( vs
= ( comple7806235888213564991et_nat @ s ) ) ).
% Vs_def
thf(fact_791_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_792_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_793_Collect__empty__eq__bot,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( P = bot_bo6227097192321305471_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_794_bot__empty__eq,axiom,
( bot_bo8210142506433397254_nat_o
= ( ^ [X: nat > set_nat] : ( member_nat_set_nat @ X @ bot_bo4007787791999405887et_nat ) ) ) ).
% bot_empty_eq
thf(fact_795_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_796_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_797_bot__empty__eq,axiom,
( bot_bo6227097192321305471_nat_o
= ( ^ [X: set_set_nat] : ( member_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) ) ) ).
% bot_empty_eq
thf(fact_798_L,axiom,
ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) ).
% L
thf(fact_799_uw_I6_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( pair @ I )
= ( insert_nat @ ( u2 @ I ) @ ( insert_nat @ ( w @ I ) @ bot_bot_set_nat ) ) ) ) ).
% uw(6)
thf(fact_800_v__mono,axiom,
! [G5: set_set_nat,H4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G5 @ H4 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G5 ) @ ( clique5033774636164728513irst_v @ H4 ) ) ) ).
% v_mono
thf(fact_801_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_802_v__gs__mono,axiom,
! [X7: set_set_set_nat,Y6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X7 @ Y6 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X7 ) @ ( clique8462013130872731469t_v_gs @ Y6 ) ) ) ).
% v_gs_mono
thf(fact_803_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_804_insert__absorb2,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ X2 @ ( insert_set_set_nat @ X2 @ A2 ) )
= ( insert_set_set_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_805_insert__absorb2,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ X2 @ A2 ) )
= ( insert_set_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_806_insert__iff,axiom,
! [A: set_set_nat,B: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_807_insert__iff,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_808_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_809_insert__iff,axiom,
! [A: nat > set_nat,B: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ A @ ( insert_nat_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_810_insertCI,axiom,
! [A: set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( ~ ( member_set_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_811_insertCI,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_812_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_813_insertCI,axiom,
! [A: nat > set_nat,B2: set_nat_set_nat,B: nat > set_nat] :
( ( ~ ( member_nat_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_set_nat @ A @ ( insert_nat_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_814_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_815_Diff__idemp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_816_Diff__idemp,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ B2 )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_817_Diff__iff,axiom,
! [C: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
= ( ( member_nat_set_nat @ C @ A2 )
& ~ ( member_nat_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_818_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_819_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_820_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_821_DiffI,axiom,
! [C: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C @ A2 )
=> ( ~ ( member_nat_set_nat @ C @ B2 )
=> ( member_nat_set_nat @ C @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_822_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_823_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_824_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_825_S_I1_J,axiom,
ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) ).
% S(1)
thf(fact_826_image__insert,axiom,
! [F: nat > nat,A: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_827_image__insert,axiom,
! [F: nat > set_set_nat,A: nat,B2: set_nat] :
( ( image_2194112158459175443et_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_828_image__insert,axiom,
! [F: nat > set_nat,A: nat,B2: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_829_image__insert,axiom,
! [F: set_set_nat > nat,A: set_set_nat,B2: set_set_set_nat] :
( ( image_1454916318497077779at_nat @ F @ ( insert_set_set_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_830_image__insert,axiom,
! [F: set_set_nat > set_set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( image_7884819252390400639et_nat @ F @ ( insert_set_set_nat @ A @ B2 ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_7884819252390400639et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_831_image__insert,axiom,
! [F: set_set_nat > set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( image_5842784325960735177et_nat @ F @ ( insert_set_set_nat @ A @ B2 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_832_image__insert,axiom,
! [F: set_nat > nat,A: set_nat,B2: set_set_nat] :
( ( image_set_nat_nat @ F @ ( insert_set_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_833_image__insert,axiom,
! [F: set_nat > set_set_nat,A: set_nat,B2: set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_6725021117256019401et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_834_image__insert,axiom,
! [F: set_nat > set_nat,A: set_nat,B2: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_835_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_836_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) )
= ( image_set_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_837_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) )
= ( image_nat_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_838_insert__image,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_839_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_840_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_841_insert__image,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_842_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_843_insert__image,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,F: ( nat > set_nat ) > nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_970537773860477644at_nat @ F @ A2 ) )
= ( image_970537773860477644at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_844_insert__image,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( image_7884819252390400639et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_845_insert__subset,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( ord_le1585852046946910987et_nat @ ( insert_nat_set_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat_set_nat @ X2 @ B2 )
& ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_846_insert__subset,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( ( member_set_nat @ X2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_847_insert__subset,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_848_insert__subset,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B2 )
= ( ( member_set_set_nat @ X2 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_849_singletonI,axiom,
! [A: nat > set_nat] : ( member_nat_set_nat @ A @ ( insert_nat_set_nat @ A @ bot_bo4007787791999405887et_nat ) ) ).
% singletonI
thf(fact_850_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_851_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_852_singletonI,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ).
% singletonI
thf(fact_853_insert__Diff1,axiom,
! [X2: nat > set_nat,B2: set_nat_set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ X2 @ B2 )
=> ( ( minus_8060664002660188164et_nat @ ( insert_nat_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_8060664002660188164et_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_854_insert__Diff1,axiom,
! [X2: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_855_insert__Diff1,axiom,
! [X2: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_856_insert__Diff1,axiom,
! [X2: set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ B2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_857_Diff__insert0,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ~ ( member_nat_set_nat @ X2 @ A2 )
=> ( ( minus_8060664002660188164et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ B2 ) )
= ( minus_8060664002660188164et_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_858_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_859_Diff__insert0,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_860_Diff__insert0,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B2 ) )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_861_sunflower,axiom,
? [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ S2 )
& ( ( finite_card_set_nat @ S2 )
= p ) ) ).
% sunflower
thf(fact_862_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_863__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_864_singleton__insert__inj__eq_H,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_865_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_866_singleton__insert__inj__eq_H,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_nat] :
( ( ( insert_set_set_nat @ A @ A2 )
= ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) )
= ( ( A = B )
& ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_867_singleton__insert__inj__eq,axiom,
! [B: set_nat,A: set_nat,A2: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_868_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_869_singleton__insert__inj__eq,axiom,
! [B: set_set_nat,A: set_set_nat,A2: set_set_set_nat] :
( ( ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_870_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_871_insert__Diff__single,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_872_insert__Diff__single,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ A @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
= ( insert_set_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_873_Inf__atLeastLessThan,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( comple7806235888213564991et_nat @ ( set_or3540276404033026485et_nat @ X2 @ Y ) )
= X2 ) ) ).
% Inf_atLeastLessThan
thf(fact_874_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_875__092_060open_062w_A_092_060equiv_062_A_092_060lambda_062i_O_Asndd_A_Ipair_Ai_J_092_060close_062,axiom,
( w
= ( ^ [I2: nat] : ( sndd @ ( pair @ I2 ) ) ) ) ).
% \<open>w \<equiv> \<lambda>i. sndd (pair i)\<close>
thf(fact_876_insert__Diff__if,axiom,
! [X2: nat > set_nat,B2: set_nat_set_nat,A2: set_nat_set_nat] :
( ( ( member_nat_set_nat @ X2 @ B2 )
=> ( ( minus_8060664002660188164et_nat @ ( insert_nat_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_8060664002660188164et_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat_set_nat @ X2 @ B2 )
=> ( ( minus_8060664002660188164et_nat @ ( insert_nat_set_nat @ X2 @ A2 ) @ B2 )
= ( insert_nat_set_nat @ X2 @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_877_insert__Diff__if,axiom,
! [X2: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_878_insert__Diff__if,axiom,
! [X2: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X2 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
& ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( insert_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_879_insert__Diff__if,axiom,
! [X2: set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ( member_set_set_nat @ X2 @ B2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) )
& ( ~ ( member_set_set_nat @ X2 @ B2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B2 )
= ( insert_set_set_nat @ X2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_880_DiffD2,axiom,
! [C: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
=> ~ ( member_nat_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_881_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_882_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_883_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_884_DiffD1,axiom,
! [C: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
=> ( member_nat_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_885_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_886_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_887_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_888_DiffE,axiom,
! [C: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_set_nat @ C @ A2 )
=> ( member_nat_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_889_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_890_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_891_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_892_mk__disjoint__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ? [B7: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ A @ B7 ) )
& ~ ( member_set_set_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_893_mk__disjoint__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ? [B7: set_set_nat] :
( ( A2
= ( insert_set_nat @ A @ B7 ) )
& ~ ( member_set_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_894_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B7: set_nat] :
( ( A2
= ( insert_nat @ A @ B7 ) )
& ~ ( member_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_895_mk__disjoint__insert,axiom,
! [A: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ A @ A2 )
=> ? [B7: set_nat_set_nat] :
( ( A2
= ( insert_nat_set_nat @ A @ B7 ) )
& ~ ( member_nat_set_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_896_insert__commute,axiom,
! [X2: nat,Y: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y @ A2 ) )
= ( insert_nat @ Y @ ( insert_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_897_insert__commute,axiom,
! [X2: set_set_nat,Y: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ X2 @ ( insert_set_set_nat @ Y @ A2 ) )
= ( insert_set_set_nat @ Y @ ( insert_set_set_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_898_insert__commute,axiom,
! [X2: set_nat,Y: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ Y @ A2 ) )
= ( insert_set_nat @ Y @ ( insert_set_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_899_insert__eq__iff,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ A @ A2 )
=> ( ~ ( member_set_set_nat @ B @ B2 )
=> ( ( ( insert_set_set_nat @ A @ A2 )
= ( insert_set_set_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ B @ C4 ) )
& ~ ( member_set_set_nat @ B @ C4 )
& ( B2
= ( insert_set_set_nat @ A @ C4 ) )
& ~ ( member_set_set_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_900_insert__eq__iff,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_set_nat] :
( ( A2
= ( insert_set_nat @ B @ C4 ) )
& ~ ( member_set_nat @ B @ C4 )
& ( B2
= ( insert_set_nat @ A @ C4 ) )
& ~ ( member_set_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_901_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B2
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_902_insert__eq__iff,axiom,
! [A: nat > set_nat,A2: set_nat_set_nat,B: nat > set_nat,B2: set_nat_set_nat] :
( ~ ( member_nat_set_nat @ A @ A2 )
=> ( ~ ( member_nat_set_nat @ B @ B2 )
=> ( ( ( insert_nat_set_nat @ A @ A2 )
= ( insert_nat_set_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat_set_nat] :
( ( A2
= ( insert_nat_set_nat @ B @ C4 ) )
& ~ ( member_nat_set_nat @ B @ C4 )
& ( B2
= ( insert_nat_set_nat @ A @ C4 ) )
& ~ ( member_nat_set_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_903_insert__absorb,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( insert_set_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_904_insert__absorb,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_905_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_906_insert__absorb,axiom,
! [A: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ A @ A2 )
=> ( ( insert_nat_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_907_insert__ident,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ~ ( member_set_set_nat @ X2 @ B2 )
=> ( ( ( insert_set_set_nat @ X2 @ A2 )
= ( insert_set_set_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_908_insert__ident,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( ( insert_set_nat @ X2 @ A2 )
= ( insert_set_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_909_insert__ident,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_910_insert__ident,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ~ ( member_nat_set_nat @ X2 @ A2 )
=> ( ~ ( member_nat_set_nat @ X2 @ B2 )
=> ( ( ( insert_nat_set_nat @ X2 @ A2 )
= ( insert_nat_set_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_911_Set_Oset__insert,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ~ ! [B7: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ X2 @ B7 ) )
=> ( member_set_set_nat @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_912_Set_Oset__insert,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ! [B7: set_set_nat] :
( ( A2
= ( insert_set_nat @ X2 @ B7 ) )
=> ( member_set_nat @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_913_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B7: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B7 ) )
=> ( member_nat @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_914_Set_Oset__insert,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ~ ! [B7: set_nat_set_nat] :
( ( A2
= ( insert_nat_set_nat @ X2 @ B7 ) )
=> ( member_nat_set_nat @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_915_insertI2,axiom,
! [A: set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ B2 )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_916_insertI2,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B2 )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_917_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_918_insertI2,axiom,
! [A: nat > set_nat,B2: set_nat_set_nat,B: nat > set_nat] :
( ( member_nat_set_nat @ A @ B2 )
=> ( member_nat_set_nat @ A @ ( insert_nat_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_919_insertI1,axiom,
! [A: set_set_nat,B2: set_set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_920_insertI1,axiom,
! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_921_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_922_insertI1,axiom,
! [A: nat > set_nat,B2: set_nat_set_nat] : ( member_nat_set_nat @ A @ ( insert_nat_set_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_923_insertE,axiom,
! [A: set_set_nat,B: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_924_insertE,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_925_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_926_insertE,axiom,
! [A: nat > set_nat,B: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ A @ ( insert_nat_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_927_insert__subsetI,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,X7: set_nat_set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( ( ord_le1585852046946910987et_nat @ X7 @ A2 )
=> ( ord_le1585852046946910987et_nat @ ( insert_nat_set_nat @ X2 @ X7 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_928_insert__subsetI,axiom,
! [X2: set_nat,A2: set_set_nat,X7: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X7 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ X7 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_929_insert__subsetI,axiom,
! [X2: nat,A2: set_nat,X7: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X7 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X7 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_930_insert__subsetI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,X7: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X7 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X2 @ X7 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_931_subset__insertI2,axiom,
! [A2: set_set_nat,B2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_932_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_933_subset__insertI2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_934_subset__insertI,axiom,
! [B2: set_set_nat,A: set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( insert_set_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_935_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_936_subset__insertI,axiom,
! [B2: set_set_set_nat,A: set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ ( insert_set_set_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_937_subset__insert,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ~ ( member_nat_set_nat @ X2 @ A2 )
=> ( ( ord_le1585852046946910987et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ B2 ) )
= ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_938_subset__insert,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_939_subset__insert,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_940_subset__insert,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B2 ) )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_941_insert__mono,axiom,
! [C2: set_set_nat,D2: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A @ C2 ) @ ( insert_set_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_942_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_943_insert__mono,axiom,
! [C2: set_set_set_nat,D2: set_set_set_nat,A: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ A @ C2 ) @ ( insert_set_set_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_944_subset__Diff__insert,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,X2: nat > set_nat,C2: set_nat_set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ ( minus_8060664002660188164et_nat @ B2 @ ( insert_nat_set_nat @ X2 @ C2 ) ) )
= ( ( ord_le1585852046946910987et_nat @ A2 @ ( minus_8060664002660188164et_nat @ B2 @ C2 ) )
& ~ ( member_nat_set_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_945_subset__Diff__insert,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X2 @ C2 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
& ~ ( member_set_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_946_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_947_subset__Diff__insert,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X2: set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B2 @ ( insert_set_set_nat @ X2 @ C2 ) ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B2 @ C2 ) )
& ~ ( member_set_set_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_948_singletonD,axiom,
! [B: nat > set_nat,A: nat > set_nat] :
( ( member_nat_set_nat @ B @ ( insert_nat_set_nat @ A @ bot_bo4007787791999405887et_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_949_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_950_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_951_singletonD,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_952_singleton__iff,axiom,
! [B: nat > set_nat,A: nat > set_nat] :
( ( member_nat_set_nat @ B @ ( insert_nat_set_nat @ A @ bot_bo4007787791999405887et_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_953_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_954_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_955_singleton__iff,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_956_doubleton__eq__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ( insert_set_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C @ ( insert_set_nat @ D @ bot_bot_set_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_957_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_958_doubleton__eq__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ( insert_set_set_nat @ A @ ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_set_nat @ C @ ( insert_set_set_nat @ D @ bot_bo7198184520161983622et_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_959_insert__not__empty,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ A2 )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_960_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_961_insert__not__empty,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ A @ A2 )
!= bot_bo7198184520161983622et_nat ) ).
% insert_not_empty
thf(fact_962_singleton__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ bot_bot_set_set_nat )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_963_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_964_singleton__inject,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_965_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_966_Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ).
% Diff_insert
thf(fact_967_Diff__insert,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ).
% Diff_insert
thf(fact_968_insert__Diff,axiom,
! [A: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ A @ A2 )
=> ( ( insert_nat_set_nat @ A @ ( minus_8060664002660188164et_nat @ A2 @ ( insert_nat_set_nat @ A @ bot_bo4007787791999405887et_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_969_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_970_insert__Diff,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_971_insert__Diff,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( insert_set_set_nat @ A @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_972_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_973_Diff__insert2,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_974_Diff__insert2,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_975_Diff__insert__absorb,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat] :
( ~ ( member_nat_set_nat @ X2 @ A2 )
=> ( ( minus_8060664002660188164et_nat @ ( insert_nat_set_nat @ X2 @ A2 ) @ ( insert_nat_set_nat @ X2 @ bot_bo4007787791999405887et_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_976_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_977_Diff__insert__absorb,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_978_Diff__insert__absorb,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_979_in__image__insert__iff,axiom,
! [B2: set_set_nat_set_nat,X2: nat > set_nat,A2: set_nat_set_nat] :
( ! [C3: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ C3 @ B2 )
=> ~ ( member_nat_set_nat @ X2 @ C3 ) )
=> ( ( member6710465769566284994et_nat @ A2 @ ( image_1110766093228767069et_nat @ ( insert_nat_set_nat @ X2 ) @ B2 ) )
= ( ( member_nat_set_nat @ X2 @ A2 )
& ( member6710465769566284994et_nat @ ( minus_8060664002660188164et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ bot_bo4007787791999405887et_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_980_in__image__insert__iff,axiom,
! [B2: set_set_nat,X2: nat,A2: set_nat] :
( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ B2 )
=> ~ ( member_nat @ X2 @ C3 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B2 ) )
= ( ( member_nat @ X2 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_981_in__image__insert__iff,axiom,
! [B2: set_set_set_nat,X2: set_nat,A2: set_set_nat] :
( ! [C3: set_set_nat] :
( ( member_set_set_nat @ C3 @ B2 )
=> ~ ( member_set_nat @ X2 @ C3 ) )
=> ( ( member_set_set_nat @ A2 @ ( image_7884819252390400639et_nat @ ( insert_set_nat @ X2 ) @ B2 ) )
= ( ( member_set_nat @ X2 @ A2 )
& ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_982_in__image__insert__iff,axiom,
! [B2: set_set_set_set_nat,X2: set_set_nat,A2: set_set_set_nat] :
( ! [C3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ C3 @ B2 )
=> ~ ( member_set_set_nat @ X2 @ C3 ) )
=> ( ( member2946998982187404937et_nat @ A2 @ ( image_6473237745780476395et_nat @ ( insert_set_set_nat @ X2 ) @ B2 ) )
= ( ( member_set_set_nat @ X2 @ A2 )
& ( member2946998982187404937et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_983_subset__singleton__iff,axiom,
! [X7: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ X7 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( ( X7 = bot_bot_set_set_nat )
| ( X7
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_984_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_985_subset__singleton__iff,axiom,
! [X7: set_set_set_nat,A: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X7 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
= ( ( X7 = bot_bo7198184520161983622et_nat )
| ( X7
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_986_subset__singletonD,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
=> ( ( A2 = bot_bot_set_set_nat )
| ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_987_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_988_subset__singletonD,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
=> ( ( A2 = bot_bo7198184520161983622et_nat )
| ( A2
= ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% subset_singletonD
thf(fact_989_Diff__single__insert,axiom,
! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_990_Diff__single__insert,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_991_Diff__single__insert,axiom,
! [A2: set_set_set_nat,X2: set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_992_subset__insert__iff,axiom,
! [A2: set_nat_set_nat,X2: nat > set_nat,B2: set_nat_set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ B2 ) )
= ( ( ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le1585852046946910987et_nat @ ( minus_8060664002660188164et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ bot_bo4007787791999405887et_nat ) ) @ B2 ) )
& ( ~ ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_993_subset__insert__iff,axiom,
! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( ( ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_994_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_995_subset__insert__iff,axiom,
! [A2: set_set_set_nat,X2: set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B2 ) )
= ( ( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B2 ) )
& ( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_996_card__insert__le,axiom,
! [A2: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_997_card__insert__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_998_card__insert__le,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] : ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ ( insert_set_set_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_999_card__Diff1__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1000_card__Diff1__le,axiom,
! [A2: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1001_card__Diff1__le,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] : ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1002_psubset__insert__iff,axiom,
! [A2: set_nat_set_nat,X2: nat > set_nat,B2: set_nat_set_nat] :
( ( ord_le7745323766158300927et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ B2 ) )
= ( ( ( member_nat_set_nat @ X2 @ B2 )
=> ( ord_le7745323766158300927et_nat @ A2 @ B2 ) )
& ( ~ ( member_nat_set_nat @ X2 @ B2 )
=> ( ( ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le7745323766158300927et_nat @ ( minus_8060664002660188164et_nat @ A2 @ ( insert_nat_set_nat @ X2 @ bot_bo4007787791999405887et_nat ) ) @ B2 ) )
& ( ~ ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1003_psubset__insert__iff,axiom,
! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( ( ( member_set_nat @ X2 @ B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) )
& ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1004_psubset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ( ( member_nat @ X2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1005_psubset__insert__iff,axiom,
! [A2: set_set_set_nat,X2: set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B2 ) )
= ( ( ( member_set_set_nat @ X2 @ B2 )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) )
& ( ~ ( member_set_set_nat @ X2 @ B2 )
=> ( ( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le152980574450754630et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B2 ) )
& ( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1006__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
= ( ^ [I2: nat] : ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) @ vs ) ) ) ) ).
% \<open>ti \<equiv> \<lambda>i. card (v (G i) - Vs)\<close>
thf(fact_1007__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
= ( ^ [I2: nat] : ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) ) ) ) ).
% \<open>si \<equiv> \<lambda>i. card (v (G i))\<close>
thf(fact_1008_assms_I3_J,axiom,
( y
= ( clique4095374090462327202g_step @ p @ x ) ) ).
% assms(3)
thf(fact_1009_cInf__atLeastLessThan,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_set_nat @ Y @ X2 )
=> ( ( comple7806235888213564991et_nat @ ( set_or3540276404033026485et_nat @ Y @ X2 ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1010_cInf__atLeastLessThan,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_nat @ Y @ X2 )
=> ( ( complete_Inf_Inf_nat @ ( set_or4665077453230672383an_nat @ Y @ X2 ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1011_cInf__singleton,axiom,
! [X2: set_set_nat] :
( ( comple1065008630642458357et_nat @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
= X2 ) ).
% cInf_singleton
thf(fact_1012_cInf__singleton,axiom,
! [X2: set_nat] :
( ( comple7806235888213564991et_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% cInf_singleton
thf(fact_1013_cInf__singleton,axiom,
! [X2: nat] :
( ( complete_Inf_Inf_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% cInf_singleton
thf(fact_1014__092_060open_062Y_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
y != bot_bo7198184520161983622et_nat ).
% \<open>Y \<noteq> {}\<close>
thf(fact_1015_Inter__iff,axiom,
! [A2: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A2 @ ( comple8067742441731897515et_nat @ C2 ) )
= ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ C2 )
=> ( member_set_set_nat @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_1016_Inter__iff,axiom,
! [A2: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A2 @ ( comple1065008630642458357et_nat @ C2 ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ C2 )
=> ( member_set_nat @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_1017_Inter__iff,axiom,
! [A2: nat > set_nat,C2: set_set_nat_set_nat] :
( ( member_nat_set_nat @ A2 @ ( comple5153742063261271012et_nat @ C2 ) )
= ( ! [X: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ X @ C2 )
=> ( member_nat_set_nat @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_1018_Inter__iff,axiom,
! [A2: nat,C2: set_set_nat] :
( ( member_nat @ A2 @ ( comple7806235888213564991et_nat @ C2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ C2 )
=> ( member_nat @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_1019_InterI,axiom,
! [C2: set_set_set_set_nat,A2: set_set_nat] :
( ! [X8: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X8 @ C2 )
=> ( member_set_set_nat @ A2 @ X8 ) )
=> ( member_set_set_nat @ A2 @ ( comple8067742441731897515et_nat @ C2 ) ) ) ).
% InterI
thf(fact_1020_InterI,axiom,
! [C2: set_set_nat_set_nat,A2: nat > set_nat] :
( ! [X8: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ X8 @ C2 )
=> ( member_nat_set_nat @ A2 @ X8 ) )
=> ( member_nat_set_nat @ A2 @ ( comple5153742063261271012et_nat @ C2 ) ) ) ).
% InterI
thf(fact_1021_InterI,axiom,
! [C2: set_set_set_nat,A2: set_nat] :
( ! [X8: set_set_nat] :
( ( member_set_set_nat @ X8 @ C2 )
=> ( member_set_nat @ A2 @ X8 ) )
=> ( member_set_nat @ A2 @ ( comple1065008630642458357et_nat @ C2 ) ) ) ).
% InterI
thf(fact_1022_InterI,axiom,
! [C2: set_set_nat,A2: nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ C2 )
=> ( member_nat @ A2 @ X8 ) )
=> ( member_nat @ A2 @ ( comple7806235888213564991et_nat @ C2 ) ) ) ).
% InterI
thf(fact_1023_w__def,axiom,
! [I: nat] :
( ( w @ I )
= ( sndd @ ( pair @ I ) ) ) ).
% w_def
thf(fact_1024_G_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ x ) ) ).
% G(1)
thf(fact_1025_si__def,axiom,
! [I: nat] :
( ( si2 @ I )
= ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% si_def
thf(fact_1026_G_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ s ) ) ).
% G(4)
thf(fact_1027_uw_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_nat @ ( w @ I ) @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% uw(2)
thf(fact_1028_G_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( clique5033774636164728513irst_v @ ( g @ I ) )
= ( si @ I ) ) ) ).
% G(2)
thf(fact_1029_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_1030_ti__def,axiom,
! [I: nat] :
( ( ti @ I )
= ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ vs ) ) ) ).
% ti_def
thf(fact_1031_SvG,axiom,
( s
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ ( image_2194112158459175443et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ) ).
% SvG
thf(fact_1032_uw_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_nat @ ( u2 @ I ) @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ vs ) ) ) ).
% uw(1)
thf(fact_1033_Inf__nat__def1,axiom,
! [K3: set_nat] :
( ( K3 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K3 ) @ K3 ) ) ).
% Inf_nat_def1
thf(fact_1034_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1035_Inf__INT__eq,axiom,
( comple7924776856948413338_nat_o
= ( ^ [S3: set_set_set_nat_o,X: set_set_nat] : ( member_set_set_nat @ X @ ( comple8067742441731897515et_nat @ ( image_3164711303094801856et_nat @ collect_set_set_nat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1036_Inf__INT__eq,axiom,
( comple7674251194079574480_nat_o
= ( ^ [S3: set_set_nat_o,X: set_nat] : ( member_set_nat @ X @ ( comple1065008630642458357et_nat @ ( image_4687162037615663680et_nat @ collect_set_nat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1037_Inf__INT__eq,axiom,
( comple6520635616725053857_nat_o
= ( ^ [S3: set_nat_set_nat_o,X: nat > set_nat] : ( member_nat_set_nat @ X @ ( comple5153742063261271012et_nat @ ( image_6262123972677179520et_nat @ collect_nat_set_nat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1038_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S3: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1039_singleton__sunflower,axiom,
! [A2: set_nat] : ( sunflower_nat @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singleton_sunflower
thf(fact_1040_singleton__sunflower,axiom,
! [A2: set_set_nat] : ( sunflower_set_nat @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ).
% singleton_sunflower
thf(fact_1041_doubleton__sunflower,axiom,
! [A2: set_nat,B2: set_nat] : ( sunflower_nat @ ( insert_set_nat @ A2 @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) ) ).
% doubleton_sunflower
thf(fact_1042_doubleton__sunflower,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( sunflower_set_nat @ ( insert_set_set_nat @ A2 @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) ) ) ).
% doubleton_sunflower
thf(fact_1043_Sup_OSUP__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_nat > set_nat,D2: set_set_nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_5842784325960735177et_nat @ C2 @ A2 ) )
= ( Sup @ ( image_5842784325960735177et_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1044_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1045_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > set_set_nat,D2: nat > set_set_nat,Sup: set_set_set_nat > set_set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_2194112158459175443et_nat @ C2 @ A2 ) )
= ( Sup @ ( image_2194112158459175443et_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1046_Inf_OINF__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_nat > set_nat,D2: set_set_nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_5842784325960735177et_nat @ C2 @ A2 ) )
= ( Inf @ ( image_5842784325960735177et_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1047_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1048_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > set_set_nat,D2: nat > set_set_nat,Inf: set_set_set_nat > set_set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_2194112158459175443et_nat @ C2 @ A2 ) )
= ( Inf @ ( image_2194112158459175443et_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1049_wellorder__InfI,axiom,
! [K: nat,A2: set_nat] :
( ( member_nat @ K @ A2 )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A2 ) @ A2 ) ) ).
% wellorder_InfI
thf(fact_1050_InterE,axiom,
! [A2: set_set_nat,C2: set_set_set_set_nat,X7: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ ( comple8067742441731897515et_nat @ C2 ) )
=> ( ( member2946998982187404937et_nat @ X7 @ C2 )
=> ( member_set_set_nat @ A2 @ X7 ) ) ) ).
% InterE
thf(fact_1051_InterE,axiom,
! [A2: set_nat,C2: set_set_set_nat,X7: set_set_nat] :
( ( member_set_nat @ A2 @ ( comple1065008630642458357et_nat @ C2 ) )
=> ( ( member_set_set_nat @ X7 @ C2 )
=> ( member_set_nat @ A2 @ X7 ) ) ) ).
% InterE
thf(fact_1052_InterE,axiom,
! [A2: nat > set_nat,C2: set_set_nat_set_nat,X7: set_nat_set_nat] :
( ( member_nat_set_nat @ A2 @ ( comple5153742063261271012et_nat @ C2 ) )
=> ( ( member6710465769566284994et_nat @ X7 @ C2 )
=> ( member_nat_set_nat @ A2 @ X7 ) ) ) ).
% InterE
thf(fact_1053_InterE,axiom,
! [A2: nat,C2: set_set_nat,X7: set_nat] :
( ( member_nat @ A2 @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X7 @ C2 )
=> ( member_nat @ A2 @ X7 ) ) ) ).
% InterE
thf(fact_1054_InterD,axiom,
! [A2: set_set_nat,C2: set_set_set_set_nat,X7: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ ( comple8067742441731897515et_nat @ C2 ) )
=> ( ( member2946998982187404937et_nat @ X7 @ C2 )
=> ( member_set_set_nat @ A2 @ X7 ) ) ) ).
% InterD
thf(fact_1055_InterD,axiom,
! [A2: set_nat,C2: set_set_set_nat,X7: set_set_nat] :
( ( member_set_nat @ A2 @ ( comple1065008630642458357et_nat @ C2 ) )
=> ( ( member_set_set_nat @ X7 @ C2 )
=> ( member_set_nat @ A2 @ X7 ) ) ) ).
% InterD
thf(fact_1056_InterD,axiom,
! [A2: nat > set_nat,C2: set_set_nat_set_nat,X7: set_nat_set_nat] :
( ( member_nat_set_nat @ A2 @ ( comple5153742063261271012et_nat @ C2 ) )
=> ( ( member6710465769566284994et_nat @ X7 @ C2 )
=> ( member_nat_set_nat @ A2 @ X7 ) ) ) ).
% InterD
thf(fact_1057_InterD,axiom,
! [A2: nat,C2: set_set_nat,X7: set_nat] :
( ( member_nat @ A2 @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X7 @ C2 )
=> ( member_nat @ A2 @ X7 ) ) ) ).
% InterD
thf(fact_1058_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1059_wellorder__Inf__le1,axiom,
! [K: nat,A2: set_nat] :
( ( member_nat @ K @ A2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A2 ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_1060_cInf__eq,axiom,
! [X7: set_nat,A: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( ord_less_eq_nat @ A @ X3 ) )
=> ( ! [Y2: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X7 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) )
=> ( ord_less_eq_nat @ Y2 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X7 )
= A ) ) ) ).
% cInf_eq
thf(fact_1061_cInf__eq__minimum,axiom,
! [Z2: nat > set_nat,X7: set_nat_set_nat] :
( ( member_nat_set_nat @ Z2 @ X7 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X7 )
=> ( ord_le6195038898401538645et_nat @ Z2 @ X3 ) )
=> ( ( comple6797894177231197998et_nat @ X7 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1062_cInf__eq__minimum,axiom,
! [Z2: set_set_nat,X7: set_set_set_nat] :
( ( member_set_set_nat @ Z2 @ X7 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ( ord_le6893508408891458716et_nat @ Z2 @ X3 ) )
=> ( ( comple1065008630642458357et_nat @ X7 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1063_cInf__eq__minimum,axiom,
! [Z2: set_set_set_nat,X7: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ Z2 @ X7 )
=> ( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ X7 )
=> ( ord_le9131159989063066194et_nat @ Z2 @ X3 ) )
=> ( ( comple8067742441731897515et_nat @ X7 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1064_cInf__eq__minimum,axiom,
! [Z2: set_nat,X7: set_set_nat] :
( ( member_set_nat @ Z2 @ X7 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ( ord_less_eq_set_nat @ Z2 @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ X7 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1065_cInf__eq__minimum,axiom,
! [Z2: nat,X7: set_nat] :
( ( member_nat @ Z2 @ X7 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ( ( complete_Inf_Inf_nat @ X7 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1066_Inf__greatest,axiom,
! [A2: set_nat_set_nat,Z2: nat > set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Z2 @ X3 ) )
=> ( ord_le6195038898401538645et_nat @ Z2 @ ( comple6797894177231197998et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1067_Inf__greatest,axiom,
! [A2: set_set_set_nat,Z2: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Z2 @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ Z2 @ ( comple1065008630642458357et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1068_Inf__greatest,axiom,
! [A2: set_set_set_set_nat,Z2: set_set_set_nat] :
( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ Z2 @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Z2 @ ( comple8067742441731897515et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1069_Inf__greatest,axiom,
! [A2: set_set_nat,Z2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ Z2 @ X3 ) )
=> ( ord_less_eq_set_nat @ Z2 @ ( comple7806235888213564991et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1070_le__Inf__iff,axiom,
! [B: set_set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( comple1065008630642458357et_nat @ A2 ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_1071_le__Inf__iff,axiom,
! [B: set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( comple8067742441731897515et_nat @ A2 ) )
= ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
=> ( ord_le9131159989063066194et_nat @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_1072_le__Inf__iff,axiom,
! [B: set_nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_1073_Inf__lower2,axiom,
! [U: nat > set_nat,A2: set_nat_set_nat,V: nat > set_nat] :
( ( member_nat_set_nat @ U @ A2 )
=> ( ( ord_le6195038898401538645et_nat @ U @ V )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1074_Inf__lower2,axiom,
! [U: set_set_nat,A2: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ U @ V )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1075_Inf__lower2,axiom,
! [U: set_set_set_nat,A2: set_set_set_set_nat,V: set_set_set_nat] :
( ( member2946998982187404937et_nat @ U @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ U @ V )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1076_Inf__lower2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ V )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1077_Inf__lower,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_1078_Inf__lower,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_1079_Inf__lower,axiom,
! [X2: set_set_set_nat,A2: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_1080_Inf__lower,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_1081_Inf__mono,axiom,
! [B2: set_nat_set_nat,A2: set_nat_set_nat] :
( ! [B6: nat > set_nat] :
( ( member_nat_set_nat @ B6 @ B2 )
=> ? [X5: nat > set_nat] :
( ( member_nat_set_nat @ X5 @ A2 )
& ( ord_le6195038898401538645et_nat @ X5 @ B6 ) ) )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ ( comple6797894177231197998et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1082_Inf__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ! [B6: set_set_nat] :
( ( member_set_set_nat @ B6 @ B2 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
& ( ord_le6893508408891458716et_nat @ X5 @ B6 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1083_Inf__mono,axiom,
! [B2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ! [B6: set_set_set_nat] :
( ( member2946998982187404937et_nat @ B6 @ B2 )
=> ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ A2 )
& ( ord_le9131159989063066194et_nat @ X5 @ B6 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ ( comple8067742441731897515et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1084_Inf__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ B2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ X5 @ B6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1085_Inf__eqI,axiom,
! [A2: set_nat_set_nat,X2: nat > set_nat] :
( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ A2 )
=> ( ord_le6195038898401538645et_nat @ X2 @ I3 ) )
=> ( ! [Y2: nat > set_nat] :
( ! [I4: nat > set_nat] :
( ( member_nat_set_nat @ I4 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Y2 @ I4 ) )
=> ( ord_le6195038898401538645et_nat @ Y2 @ X2 ) )
=> ( ( comple6797894177231197998et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1086_Inf__eqI,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ I3 ) )
=> ( ! [Y2: set_set_nat] :
( ! [I4: set_set_nat] :
( ( member_set_set_nat @ I4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ I4 ) )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X2 ) )
=> ( ( comple1065008630642458357et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1087_Inf__eqI,axiom,
! [A2: set_set_set_set_nat,X2: set_set_set_nat] :
( ! [I3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ I3 ) )
=> ( ! [Y2: set_set_set_nat] :
( ! [I4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I4 @ A2 )
=> ( ord_le9131159989063066194et_nat @ Y2 @ I4 ) )
=> ( ord_le9131159989063066194et_nat @ Y2 @ X2 ) )
=> ( ( comple8067742441731897515et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1088_Inf__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ I3 ) )
=> ( ! [Y2: set_nat] :
( ! [I4: set_nat] :
( ( member_set_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ Y2 @ I4 ) )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1089_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > set_set_nat,D2: nat > set_set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ C2 @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1090_INF__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_nat > set_nat,D2: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ C2 @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1091_INF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat > set_nat,D2: set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ C2 @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1092_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > set_nat,D2: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C2 @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1093_INF__cong,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C2: ( nat > set_nat ) > set_nat,D2: ( nat > set_nat ) > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ C2 @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1094_INF__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_nat > nat,D2: set_set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_1454916318497077779at_nat @ C2 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_1454916318497077779at_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1095_INF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat > nat,D2: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_set_nat_nat @ C2 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_set_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1096_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C2 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1097_INF__cong,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C2: ( nat > set_nat ) > nat,D2: ( nat > set_nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_970537773860477644at_nat @ C2 @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_970537773860477644at_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1098_Inter__lower,axiom,
! [B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1099_Inter__lower,axiom,
! [B2: set_set_set_nat,A2: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1100_Inter__lower,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1101_Inter__greatest,axiom,
! [A2: set_set_set_nat,C2: set_set_nat] :
( ! [X8: set_set_nat] :
( ( member_set_set_nat @ X8 @ A2 )
=> ( ord_le6893508408891458716et_nat @ C2 @ X8 ) )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( comple1065008630642458357et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1102_Inter__greatest,axiom,
! [A2: set_set_set_set_nat,C2: set_set_set_nat] :
( ! [X8: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X8 @ A2 )
=> ( ord_le9131159989063066194et_nat @ C2 @ X8 ) )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( comple8067742441731897515et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1103_Inter__greatest,axiom,
! [A2: set_set_nat,C2: set_nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ C2 @ X8 ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1104_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1105_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_set_nat,F: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ( ord_le6893508408891458716et_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1106_INF__eq,axiom,
! [A2: set_set_nat,B2: set_nat,G: nat > set_nat,F: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1107_INF__eq,axiom,
! [A2: set_nat,B2: set_set_nat,G: set_nat > set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1108_INF__eq,axiom,
! [A2: set_set_nat,B2: set_nat,G: nat > set_set_nat,F: set_nat > set_set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ( ord_le6893508408891458716et_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1109_INF__eq,axiom,
! [A2: set_nat,B2: set_set_nat,G: set_nat > set_set_nat,F: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B2 )
& ( ord_le6893508408891458716et_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1110_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_set_set_nat,F: nat > set_set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ( ord_le9131159989063066194et_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ F @ A2 ) )
= ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1111_INF__eq,axiom,
! [A2: set_set_set_nat,B2: set_nat,G: nat > set_nat,F: set_set_nat > set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1112_INF__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat,G: set_nat > set_nat,F: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1113_INF__eq,axiom,
! [A2: set_nat,B2: set_set_set_nat,G: set_set_nat > set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_set_nat] :
( ( member_set_set_nat @ J2 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1114_Inf__superset__mono,axiom,
! [B2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ ( comple8067742441731897515et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1115_Inf__superset__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1116_Inf__superset__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1117_Inf__less__eq,axiom,
! [A2: set_nat_set_nat,U: nat > set_nat] :
( ! [V2: nat > set_nat] :
( ( member_nat_set_nat @ V2 @ A2 )
=> ( ord_le6195038898401538645et_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo4007787791999405887et_nat )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1118_Inf__less__eq,axiom,
! [A2: set_set_set_nat,U: set_set_nat] :
( ! [V2: set_set_nat] :
( ( member_set_set_nat @ V2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1119_Inf__less__eq,axiom,
! [A2: set_set_set_set_nat,U: set_set_set_nat] :
( ! [V2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ V2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1120_Inf__less__eq,axiom,
! [A2: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ V2 @ U ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1121_cInf__greatest,axiom,
! [X7: set_nat_set_nat,Z2: nat > set_nat] :
( ( X7 != bot_bo4007787791999405887et_nat )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X7 )
=> ( ord_le6195038898401538645et_nat @ Z2 @ X3 ) )
=> ( ord_le6195038898401538645et_nat @ Z2 @ ( comple6797894177231197998et_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_1122_cInf__greatest,axiom,
! [X7: set_set_set_nat,Z2: set_set_nat] :
( ( X7 != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ( ord_le6893508408891458716et_nat @ Z2 @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ Z2 @ ( comple1065008630642458357et_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_1123_cInf__greatest,axiom,
! [X7: set_set_set_set_nat,Z2: set_set_set_nat] :
( ( X7 != bot_bo193956671110832956et_nat )
=> ( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ X7 )
=> ( ord_le9131159989063066194et_nat @ Z2 @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Z2 @ ( comple8067742441731897515et_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_1124_cInf__greatest,axiom,
! [X7: set_set_nat,Z2: set_nat] :
( ( X7 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ( ord_less_eq_set_nat @ Z2 @ X3 ) )
=> ( ord_less_eq_set_nat @ Z2 @ ( comple7806235888213564991et_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_1125_cInf__greatest,axiom,
! [X7: set_nat,Z2: nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ( ord_less_eq_nat @ Z2 @ ( complete_Inf_Inf_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_1126_cInf__eq__non__empty,axiom,
! [X7: set_nat_set_nat,A: nat > set_nat] :
( ( X7 != bot_bo4007787791999405887et_nat )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X7 )
=> ( ord_le6195038898401538645et_nat @ A @ X3 ) )
=> ( ! [Y2: nat > set_nat] :
( ! [X5: nat > set_nat] :
( ( member_nat_set_nat @ X5 @ X7 )
=> ( ord_le6195038898401538645et_nat @ Y2 @ X5 ) )
=> ( ord_le6195038898401538645et_nat @ Y2 @ A ) )
=> ( ( comple6797894177231197998et_nat @ X7 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1127_cInf__eq__non__empty,axiom,
! [X7: set_set_set_nat,A: set_set_nat] :
( ( X7 != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X7 )
=> ( ord_le6893508408891458716et_nat @ A @ X3 ) )
=> ( ! [Y2: set_set_nat] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ X7 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X5 ) )
=> ( ord_le6893508408891458716et_nat @ Y2 @ A ) )
=> ( ( comple1065008630642458357et_nat @ X7 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1128_cInf__eq__non__empty,axiom,
! [X7: set_set_set_set_nat,A: set_set_set_nat] :
( ( X7 != bot_bo193956671110832956et_nat )
=> ( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ X7 )
=> ( ord_le9131159989063066194et_nat @ A @ X3 ) )
=> ( ! [Y2: set_set_set_nat] :
( ! [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ X7 )
=> ( ord_le9131159989063066194et_nat @ Y2 @ X5 ) )
=> ( ord_le9131159989063066194et_nat @ Y2 @ A ) )
=> ( ( comple8067742441731897515et_nat @ X7 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1129_cInf__eq__non__empty,axiom,
! [X7: set_set_nat,A: set_nat] :
( ( X7 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
=> ( ord_less_eq_set_nat @ A @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ X7 )
=> ( ord_less_eq_set_nat @ Y2 @ X5 ) )
=> ( ord_less_eq_set_nat @ Y2 @ A ) )
=> ( ( comple7806235888213564991et_nat @ X7 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1130_cInf__eq__non__empty,axiom,
! [X7: set_nat,A: nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ( ord_less_eq_nat @ A @ X3 ) )
=> ( ! [Y2: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X7 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) )
=> ( ord_less_eq_nat @ Y2 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X7 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1131_cInf__lessD,axiom,
! [X7: set_nat,Z2: nat] :
( ( X7 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X7 ) @ Z2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X7 )
& ( ord_less_nat @ X3 @ Z2 ) ) ) ) ).
% cInf_lessD
thf(fact_1132_INF__eq__const,axiom,
! [I5: set_nat,F: nat > set_set_nat,X2: set_set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_1133_INF__eq__const,axiom,
! [I5: set_nat_set_nat,F: ( nat > set_nat ) > set_nat,X2: set_nat] :
( ( I5 != bot_bo4007787791999405887et_nat )
=> ( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_1134_INF__eq__const,axiom,
! [I5: set_set_nat,F: set_nat > set_nat,X2: set_nat] :
( ( I5 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_1135_INF__eq__const,axiom,
! [I5: set_nat,F: nat > set_nat,X2: set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_1136_INF__eq__const,axiom,
! [I5: set_set_set_nat,F: set_set_nat > set_nat,X2: set_nat] :
( ( I5 != bot_bo7198184520161983622et_nat )
=> ( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_1137_Inter__anti__mono,axiom,
! [B2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ ( comple8067742441731897515et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1138_Inter__anti__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1139_Inter__anti__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1140_Inter__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ! [X8: set_set_nat] :
( ( member_set_set_nat @ X8 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X8 @ B2 ) )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1141_Inter__subset,axiom,
! [A2: set_set_set_set_nat,B2: set_set_set_nat] :
( ! [X8: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X8 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X8 @ B2 ) )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1142_Inter__subset,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ X8 @ B2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1143_cINF__greatest,axiom,
! [A2: set_nat,M2: nat,F: nat > nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1144_cINF__greatest,axiom,
! [A2: set_nat,M2: set_nat,F: nat > set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_set_nat @ M2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1145_cINF__greatest,axiom,
! [A2: set_set_nat,M2: nat,F: set_nat > nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1146_cINF__greatest,axiom,
! [A2: set_nat,M2: set_set_nat,F: nat > set_set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_le6893508408891458716et_nat @ M2 @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1147_cINF__greatest,axiom,
! [A2: set_set_nat,M2: set_nat,F: set_nat > set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_set_nat @ M2 @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1148_cINF__greatest,axiom,
! [A2: set_set_set_nat,M2: nat,F: set_set_nat > nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1149_cINF__greatest,axiom,
! [A2: set_set_nat,M2: set_set_nat,F: set_nat > set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_le6893508408891458716et_nat @ M2 @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1150_cINF__greatest,axiom,
! [A2: set_nat,M2: set_set_set_nat,F: nat > set_set_set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_le9131159989063066194et_nat @ M2 @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1151_cINF__greatest,axiom,
! [A2: set_set_set_nat,M2: set_nat,F: set_set_nat > set_nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_set_nat @ M2 @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1152_cINF__greatest,axiom,
! [A2: set_nat_set_nat,M2: nat,F: ( nat > set_nat ) > nat] :
( ( A2 != bot_bo4007787791999405887et_nat )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M2 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1153_INF__eq__iff,axiom,
! [I5: set_nat,F: nat > set_nat,C: set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1154_INF__eq__iff,axiom,
! [I5: set_nat,F: nat > set_set_nat,C: set_set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1155_INF__eq__iff,axiom,
! [I5: set_set_nat,F: set_nat > set_nat,C: set_nat] :
( ( I5 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ I5 ) )
= C )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1156_INF__eq__iff,axiom,
! [I5: set_set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( I5 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ F @ I5 ) )
= C )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1157_INF__eq__iff,axiom,
! [I5: set_nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le9131159989063066194et_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ F @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1158_INF__eq__iff,axiom,
! [I5: set_set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( I5 != bot_bo7198184520161983622et_nat )
=> ( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ I5 ) )
= C )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1159_INF__eq__iff,axiom,
! [I5: set_set_set_nat,F: set_set_nat > set_set_nat,C: set_set_nat] :
( ( I5 != bot_bo7198184520161983622et_nat )
=> ( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ I5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple1065008630642458357et_nat @ ( image_7884819252390400639et_nat @ F @ I5 ) )
= C )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1160_INF__eq__iff,axiom,
! [I5: set_set_nat,F: set_nat > set_set_set_nat,C: set_set_set_nat] :
( ( I5 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I5 )
=> ( ord_le9131159989063066194et_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple8067742441731897515et_nat @ ( image_4583741654806091647et_nat @ F @ I5 ) )
= C )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1161_INF__eq__iff,axiom,
! [I5: set_nat_set_nat,F: ( nat > set_nat ) > set_nat,C: set_nat] :
( ( I5 != bot_bo4007787791999405887et_nat )
=> ( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ F @ I5 ) )
= C )
= ( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1162_INF__eq__iff,axiom,
! [I5: set_nat_set_nat,F: ( nat > set_nat ) > set_set_nat,C: set_set_nat] :
( ( I5 != bot_bo4007787791999405887et_nat )
=> ( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ I5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple1065008630642458357et_nat @ ( image_7290825263825464120et_nat @ F @ I5 ) )
= C )
= ( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1163_sf__precond,axiom,
! [X5: set_nat] :
( ( member_set_nat @ X5 @ ( clique8462013130872731469t_v_gs @ x ) )
=> ( ( finite_finite_nat @ X5 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X5 ) @ l ) ) ) ).
% sf_precond
thf(fact_1164_i__props_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% i_props(2)
thf(fact_1165_Pi,axiom,
member_nat_set_nat @ pair @ ( piE_nat_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ g ) ).
% Pi
thf(fact_1166__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_1167_i__props_I6_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( finite1152437895449049373et_nat @ ( g @ I ) ) ) ).
% i_props(6)
thf(fact_1168_fin__Vs,axiom,
finite_finite_nat @ vs ).
% fin_Vs
thf(fact_1169_finS,axiom,
finite1152437895449049373et_nat @ s ).
% finS
thf(fact_1170_fin1,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% fin1
thf(fact_1171_vGs,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ vs ).
% vGs
thf(fact_1172_finite__imageI,axiom,
! [F4: set_set_set_nat,H2: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ F4 )
=> ( finite6739761609112101331et_nat @ ( image_7884819252390400639et_nat @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_1173_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1174_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1175_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1176_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1177_subset__eq__atLeast0__lessThan__finite,axiom,
! [N4: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1178_Gs__def,axiom,
( gs
= ( clique6722202388162463298od_nat @ vs @ vs ) ) ).
% Gs_def
thf(fact_1179_uw_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( f @ ( u2 @ I ) )
= ( f @ ( w @ I ) ) ) ) ).
% uw(4)
thf(fact_1180_injG,axiom,
inj_on8105003582846801791et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ).
% injG
thf(fact_1181_finX,axiom,
finite6739761609112101331et_nat @ x ).
% finX
thf(fact_1182__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_1183_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_1184_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1185_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1186_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1187_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1188_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1189_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1190_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1191_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1192_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1193_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_1194_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_1195_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_1196_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1197_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1198_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1199_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_1200_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_1201_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1202_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1203_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1204_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1205_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1206_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1207_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1208_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1209_choose__mult,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_1210_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1211_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1212_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_1213__092_060open_062card_A_Iv__gs_AX_A_N_Av__gs_AU_J_A_061_Acard_A_Iv__gs_AX_J_A_N_Acard_A_Iv__gs_AU_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) ) ) ) ).
% \<open>card (v_gs X - v_gs U) = card (v_gs X) - card (v_gs U)\<close>
thf(fact_1214_G_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ u ) ) ).
% G(3)
thf(fact_1215_Unempty,axiom,
u != bot_bo7198184520161983622et_nat ).
% Unempty
thf(fact_1216_finU,axiom,
finite6739761609112101331et_nat @ u ).
% finU
thf(fact_1217_UX,axiom,
ord_le9131159989063066194et_nat @ u @ x ).
% UX
thf(fact_1218_vplus__dsU,axiom,
( ( clique8462013130872731469t_v_gs @ u )
= s ) ).
% vplus_dsU
thf(fact_1219_r__def,axiom,
( r
= ( finite1149291290879098388et_nat @ u ) ) ).
% r_def
thf(fact_1220__092_060open_062card_A_Iv__gs_AU_J_A_061_Acard_AS_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) )
= ( finite_card_set_nat @ s ) ) ).
% \<open>card (v_gs U) = card S\<close>
thf(fact_1221_vplus__dsXU,axiom,
( ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) )
= ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) ) ).
% vplus_dsXU
thf(fact_1222__092_060open_062card_A_Iv__gs_AY_J_A_061_Acard_A_Iv__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) )
= ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% \<open>card (v_gs Y) = card (v_gs (X - U \<union> {Gs}))\<close>
thf(fact_1223_Y,axiom,
( y
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ).
% Y
thf(fact_1224__092_060open_062card_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_Acard_A_123v_AGs_125_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ ( finite_card_set_nat @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ one_one_nat ) ).
% \<open>card (v_gs (X - U)) + card {v Gs} \<le> card (v_gs (X - U)) + 1\<close>
thf(fact_1225_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1226_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1227_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1228_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1229_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_1230_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1231_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1232_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1233_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1234_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_1235_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_1236_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_1237__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_1238_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1239_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N2: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1240_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1241_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1242_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1243_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1244_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1245_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_1246_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_1247_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_1248_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_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_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_1251_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1252_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_1253_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1254_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1255_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1256_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1257_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1258_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1259_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1260_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_1261_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_1262_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1263_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1264_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1265_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1266_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1267_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
% 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,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) )
=> ( ( ( u2 @ X3 )
= ( u2 @ Xa ) )
=> ( X3 = Xa ) ) ) ) ).
%------------------------------------------------------------------------------