TPTP Problem File: SLH0434^1.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_01772_067300__16418486_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1526 ( 751 unt; 240 typ; 0 def)
% Number of atoms : 3327 (1228 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 10737 ( 283 ~; 45 |; 323 &;8906 @)
% ( 0 <=>;1180 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 1102 (1102 >; 0 *; 0 +; 0 <<)
% Number of symbols : 223 ( 220 usr; 33 con; 0-5 aty)
% Number of variables : 3471 ( 415 ^;2896 !; 160 ?;3471 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:51:05.450
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Mo6069479339911551325at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Mo5013373542560054436et_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
monoto5483634261523599098et_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
monoto5348676645462821012mula_a: $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__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
monoto8276428299528460797at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Nat__Onat_J_J,type,
set_Mo2931508685415655150la_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto7244996872745832772et_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_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
set_Mo2626137824023173004mula_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Nat__Onat_J,type,
monoto4181647612830706830la_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_Itf__a_J,type,
monotone_mformula_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (220)
thf(sy_c_Assumptions__and__Approximations_OL0,type,
assumptions_and_L0: nat ).
thf(sy_c_Assumptions__and__Approximations_OL0_H,type,
assumptions_and_L02: nat ).
thf(sy_c_Assumptions__and__Approximations_OM0,type,
assumptions_and_M0: nat ).
thf(sy_c_Assumptions__and__Approximations_OM0_H,type,
assumptions_and_M02: nat ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OClique,type,
clique6749503327923060270Clique: set_nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
clique5786534781347292306Graphs: set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique134924887794942129at_nat: set_nat_nat > set_nat_nat > set_set_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_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
clique8906516429304539640et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique1181040904276305582et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001tf__a,type,
clique9072761800073521420prod_a: set_a > set_a > set_set_a ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC,type,
clique3210737319928189260st_ACC: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC__cf,type,
clique951075384711337423ACC_cf: nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OC,type,
clique5033774636164728462irst_C: nat > ( nat > nat ) > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
clique363107459185959606CLIQUE: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_ONEG,type,
clique3210737375870294875st_NEG: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060F_062,type,
clique2971579238625216137irst_F: nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060G_062l,type,
clique7840962075309931874st_G_l: nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_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_Oodot,type,
clique5469973757772500719t_odot: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodotl,type,
clique7966186356931407165_odotl: nat > nat > set_set_set_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_Clique__Large__Monotone__Circuits_Oforth__assumptions_001tf__a,type,
clique8563529963003110213ions_a: nat > nat > nat > set_a > ( a > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__cf__mf_001tf__a,type,
clique8961599393750669800f_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__mf_001tf__a,type,
clique4708818501384062891C_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR_001tf__a,type,
clique3873310923663319714_APR_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR__rel_001tf__a,type,
clique5870032674357670943_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET_001tf__a,type,
clique6509092761774629891_SET_a: ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET__rel_001tf__a,type,
clique834332680210058238_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001tf__a,type,
clique5987991184601036204th_A_a: set_a > set_Mo2626137824023173004mula_a ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060theta_062_092_060_094sub_062g_001tf__a,type,
clique3148831351753978868ta_g_a: set_a > ( a > set_nat ) > set_set_nat > a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__neg_001tf__a,type,
clique6623365555141101007_neg_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__neg__rel_001tf__a,type,
clique6353239774569474354_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__pos_001tf__a,type,
clique8538548958085942603_pos_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__pos__rel_001tf__a,type,
clique4465983624924118198_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Odeviate__neg_001tf__a,type,
clique2019076642914533763_neg_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Odeviate__pos_001tf__a,type,
clique3934260045859375359_pos_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oeval__g_001tf__a,type,
clique5859573001277246426al_g_a: set_a > ( a > set_nat ) > ( a > $o ) > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oeval__gs_001tf__a,type,
clique835570645587132141l_gs_a: set_a > ( a > set_nat ) > ( a > $o ) > set_set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU,type,
clique2699557479641037314nd_PLU: nat > nat > nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cap,type,
clique1591571987438064265eg_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cup,type,
clique1591571987439376245eg_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cap,type,
clique3314026705535538693os_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cup,type,
clique3314026705536850673os_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcap,type,
clique2586627118206219037_sqcap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcup,type,
clique2586627118207531017_sqcup: nat > nat > nat > set_set_set_nat > 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_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Monotone____Formula__Omformula_Itf__a_J,type,
finite694752133686741613mula_a: set_Mo2626137824023173004mula_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite3586981331298542604at_nat: set_set_nat_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_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
finite5926941155766903689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_betw_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be2321430536510320189et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a,type,
bij_betw_nat_nat_a: ( ( nat > nat ) > a ) > set_nat_nat > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Monotone____Formula__Omformula_Itf__a_J_001t__Monotone____Formula__Omformula_Itf__a_J,type,
bij_be6841836389319334180mula_a: ( monotone_mformula_a > monotone_mformula_a ) > set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Monotone____Formula__Omformula_Itf__a_J_001t__Nat__Onat,type,
bij_be4432079924810155166_a_nat: ( monotone_mformula_a > nat ) > set_Mo2626137824023173004mula_a > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Monotone____Formula__Omformula_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be1996446448813045844et_nat: ( monotone_mformula_a > set_nat ) > set_Mo2626137824023173004mula_a > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Monotone____Formula__Omformula_Itf__a_J_001tf__a,type,
bij_be7751601365550188976la_a_a: ( monotone_mformula_a > a ) > set_Mo2626137824023173004mula_a > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_betw_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_be3458689793592806333at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Monotone____Formula__Omformula_Itf__a_J,type,
bij_be3858092191235522516mula_a: ( set_nat > monotone_mformula_a ) > set_set_nat > set_Mo2626137824023173004mula_a > $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_001tf__a,type,
bij_betw_set_nat_a: ( set_nat > a ) > set_set_nat > set_a > $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_001tf__a,type,
bij_be3032674665972365258_nat_a: ( set_set_nat > a ) > set_set_set_nat > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_betw_a_nat_nat: ( a > nat > nat ) > set_a > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Monotone____Formula__Omformula_Itf__a_J,type,
bij_be1655973440275287390mula_a: ( a > monotone_mformula_a ) > set_a > set_Mo2626137824023173004mula_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Nat__Onat,type,
bij_betw_a_nat: ( a > nat ) > set_a > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
bij_betw_a_set_nat: ( a > set_nat ) > set_a > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be2639851105560558660et_nat: ( a > set_set_nat ) > set_a > set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_a_set_nat: ( a > set_nat ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_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_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_HOL_Oundefined_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
undefi6751788150640612746et_nat: set_set_set_nat ).
thf(sy_c_If_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_nat_nat: $o > ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
if_set_set_nat: $o > set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
inf_inf_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Monotone____Formula__Omformula_Itf__a_J_M_Eo_J,type,
inf_in6543652950005596391la_a_o: ( monotone_mformula_a > $o ) > ( monotone_mformula_a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
inf_in2551356467856225537_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
sup_sup_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Monotone____Formula__Omformula_Itf__a_J_M_Eo_J,type,
sup_su6423506137380736461la_a_o: ( monotone_mformula_a > $o ) > ( monotone_mformula_a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
sup_su5917979686466268903_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_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_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Monotone__Formula_Ocs_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monotone_cs_nat_nat: monoto8276428299528460797at_nat > nat ).
thf(sy_c_Monotone__Formula_Ocs_001t__Nat__Onat,type,
monotone_cs_nat: monoto4181647612830706830la_nat > nat ).
thf(sy_c_Monotone__Formula_Ocs_001t__Set__Oset_It__Nat__Onat_J,type,
monotone_cs_set_nat: monoto7244996872745832772et_nat > nat ).
thf(sy_c_Monotone__Formula_Ocs_001tf__a,type,
monotone_cs_a: monotone_mformula_a > nat ).
thf(sy_c_Monotone__Formula_Oeval_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto2874428580947315297at_nat: ( ( nat > nat ) > $o ) > monoto8276428299528460797at_nat > $o ).
thf(sy_c_Monotone__Formula_Oeval_001t__Monotone____Formula__Omformula_Itf__a_J,type,
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thf(sy_c_Monotone__Formula_Oeval_001t__Nat__Onat,type,
monotone_eval_nat: ( nat > $o ) > monoto4181647612830706830la_nat > $o ).
thf(sy_c_Monotone__Formula_Oeval_001t__Set__Oset_It__Nat__Onat_J,type,
monoto7255863275561024424et_nat: ( set_nat > $o ) > monoto7244996872745832772et_nat > $o ).
thf(sy_c_Monotone__Formula_Oeval_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto5763065145529399390et_nat: ( set_set_nat > $o ) > monoto5483634261523599098et_nat > $o ).
thf(sy_c_Monotone__Formula_Oeval_001tf__a,type,
monotone_eval_a: ( a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_Monotone__Formula_Omformula_OConj_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto3849437543655978646at_nat: monoto8276428299528460797at_nat > monoto8276428299528460797at_nat > monoto8276428299528460797at_nat ).
thf(sy_c_Monotone__Formula_Omformula_OConj_001t__Nat__Onat,type,
monotone_Conj_nat: monoto4181647612830706830la_nat > monoto4181647612830706830la_nat > monoto4181647612830706830la_nat ).
thf(sy_c_Monotone__Formula_Omformula_OConj_001t__Set__Oset_It__Nat__Onat_J,type,
monoto3675431328128845661et_nat: monoto7244996872745832772et_nat > monoto7244996872745832772et_nat > monoto7244996872745832772et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OConj_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto78699555928797203et_nat: monoto5483634261523599098et_nat > monoto5483634261523599098et_nat > monoto5483634261523599098et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OConj_001tf__a,type,
monotone_Conj_a: monotone_mformula_a > monotone_mformula_a > monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_ODisj_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto6252637820927860106at_nat: monoto8276428299528460797at_nat > monoto8276428299528460797at_nat > monoto8276428299528460797at_nat ).
thf(sy_c_Monotone__Formula_Omformula_ODisj_001t__Nat__Onat,type,
monotone_Disj_nat: monoto4181647612830706830la_nat > monoto4181647612830706830la_nat > monoto4181647612830706830la_nat ).
thf(sy_c_Monotone__Formula_Omformula_ODisj_001t__Set__Oset_It__Nat__Onat_J,type,
monoto2996447309290675281et_nat: monoto7244996872745832772et_nat > monoto7244996872745832772et_nat > monoto7244996872745832772et_nat ).
thf(sy_c_Monotone__Formula_Omformula_ODisj_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto5378078033476056327et_nat: monoto5483634261523599098et_nat > monoto5483634261523599098et_nat > monoto5483634261523599098et_nat ).
thf(sy_c_Monotone__Formula_Omformula_ODisj_001tf__a,type,
monotone_Disj_a: monotone_mformula_a > monotone_mformula_a > monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_OFALSE_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto3735746735589746005at_nat: monoto8276428299528460797at_nat ).
thf(sy_c_Monotone__Formula_Omformula_OFALSE_001t__Nat__Onat,type,
monotone_FALSE_nat: monoto4181647612830706830la_nat ).
thf(sy_c_Monotone__Formula_Omformula_OFALSE_001t__Set__Oset_It__Nat__Onat_J,type,
monoto2388303931541111964et_nat: monoto7244996872745832772et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OFALSE_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto6214072352461320530et_nat: monoto5483634261523599098et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OFALSE_001tf__a,type,
monotone_FALSE_a: monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_OTRUE_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto581402444256252508at_nat: monoto8276428299528460797at_nat ).
thf(sy_c_Monotone__Formula_Omformula_OTRUE_001t__Nat__Onat,type,
monotone_TRUE_nat: monoto4181647612830706830la_nat ).
thf(sy_c_Monotone__Formula_Omformula_OTRUE_001t__Set__Oset_It__Nat__Onat_J,type,
monoto7549873196617247779et_nat: monoto7244996872745832772et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OTRUE_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto5104785069271071961et_nat: monoto5483634261523599098et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OTRUE_001tf__a,type,
monotone_TRUE_a: monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_OVar_001tf__a,type,
monotone_Var_a: a > monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Otf__mformula_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto6982705221701894448at_nat: set_Mo6069479339911551325at_nat ).
thf(sy_c_Monotone__Formula_Otf__mformula_001t__Nat__Onat,type,
monoto1762321936899617089la_nat: set_Mo2931508685415655150la_nat ).
thf(sy_c_Monotone__Formula_Otf__mformula_001t__Set__Oset_It__Nat__Onat_J,type,
monoto1591516381014137079et_nat: set_Mo5013373542560054436et_nat ).
thf(sy_c_Monotone__Formula_Otf__mformula_001tf__a,type,
monoto4877036962378694605mula_a: set_Mo2626137824023173004mula_a ).
thf(sy_c_Monotone__Formula_Ovars_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monoto4799612099597528497at_nat: monoto8276428299528460797at_nat > set_nat_nat ).
thf(sy_c_Monotone__Formula_Ovars_001t__Monotone____Formula__Omformula_Itf__a_J,type,
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thf(sy_c_Monotone__Formula_Ovars_001t__Nat__Onat,type,
monotone_vars_nat: monoto4181647612830706830la_nat > set_nat ).
thf(sy_c_Monotone__Formula_Ovars_001t__Set__Oset_It__Nat__Onat_J,type,
monoto8378391831928444664et_nat: monoto7244996872745832772et_nat > set_set_nat ).
thf(sy_c_Monotone__Formula_Ovars_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto3765512064276419502et_nat: monoto5483634261523599098et_nat > set_set_set_nat ).
thf(sy_c_Monotone__Formula_Ovars_001tf__a,type,
monotone_vars_a: monotone_mformula_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_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_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
ord_le5054881893329012716mula_a: set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OBex_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bex_nat_nat: set_nat_nat > ( ( nat > nat ) > $o ) > $o ).
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thf(sy_c_Set_OBex_001t__Set__Oset_It__Nat__Onat_J,type,
bex_set_nat: set_set_nat > ( set_nat > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bex_set_set_nat: set_set_set_nat > ( set_set_nat > $o ) > $o ).
thf(sy_c_Set_OBex_001tf__a,type,
bex_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_Itf__a_J,type,
accp_M6162913489380515981mula_a: ( monotone_mformula_a > monotone_mformula_a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member435044527007775910at_nat: monoto8276428299528460797at_nat > set_Mo6069479339911551325at_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Nat__Onat_J,type,
member6605755813721784503la_nat: monoto4181647612830706830la_nat > set_Mo2931508685415655150la_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
member7623223977307079021et_nat: monoto7244996872745832772et_nat > set_Mo5013373542560054436et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_Itf__a_J,type,
member535913909593306477mula_a: monotone_mformula_a > set_Mo2626137824023173004mula_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060V_062,type,
v: set_a ).
thf(sy_v__092_060phi_062,type,
phi: monotone_mformula_a ).
thf(sy_v__092_060pi_062,type,
pi: a > set_nat ).
thf(sy_v__092_060psi_062____,type,
psi: monotone_mformula_a ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1276)
thf(fact_0__C_K_C_I1_J,axiom,
member535913909593306477mula_a @ psi @ monoto4877036962378694605mula_a ).
% "*"(1)
thf(fact_1_phi,axiom,
member535913909593306477mula_a @ phi @ ( clique5987991184601036204th_A_a @ v ) ).
% phi
thf(fact_2__C_K_C_I2_J,axiom,
! [Theta: a > $o] :
( ( monotone_eval_a @ Theta @ phi )
= ( monotone_eval_a @ Theta @ psi ) ) ).
% "*"(2)
thf(fact_3__C_K_C_I4_J,axiom,
ord_less_eq_nat @ ( monotone_cs_a @ psi ) @ ( monotone_cs_a @ phi ) ).
% "*"(4)
thf(fact_4__C_K_C_I3_J,axiom,
ord_less_eq_set_a @ ( monotone_vars_a @ psi ) @ ( monotone_vars_a @ phi ) ).
% "*"(3)
thf(fact_5__092_060theta_062_092_060_094sub_062g__def,axiom,
! [G: set_set_nat,X: a] :
( ( clique3148831351753978868ta_g_a @ v @ pi @ G @ X )
= ( ( member_a @ X @ v )
& ( member_set_nat @ ( pi @ X ) @ G ) ) ) ).
% \<theta>\<^sub>g_def
thf(fact_6_forth__assumptions_O_092_060A_062_Ocong,axiom,
clique5987991184601036204th_A_a = clique5987991184601036204th_A_a ).
% forth_assumptions.\<A>.cong
thf(fact_7__092_060A_062__simps_I1_J,axiom,
member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ v ) ).
% \<A>_simps(1)
thf(fact_8_eval__g__def,axiom,
! [Theta2: a > $o,G: set_set_nat] :
( ( clique5859573001277246426al_g_a @ v @ pi @ Theta2 @ G )
= ( ! [X2: a] :
( ( member_a @ X2 @ v )
=> ( ( member_set_nat @ ( pi @ X2 ) @ G )
=> ( Theta2 @ X2 ) ) ) ) ) ).
% eval_g_def
thf(fact_9__092_060A_062__simps_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ v ) ) ) ) ).
% \<A>_simps(4)
thf(fact_10__092_060A_062__simps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ v ) ) ) ) ).
% \<A>_simps(3)
thf(fact_11__092_060A_062__simps_I2_J,axiom,
! [X: a] :
( ( member535913909593306477mula_a @ ( monotone_Var_a @ X ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( member_a @ X @ v ) ) ).
% \<A>_simps(2)
thf(fact_12_forth__assumptions_O_092_060theta_062_092_060_094sub_062g_Ocong,axiom,
clique3148831351753978868ta_g_a = clique3148831351753978868ta_g_a ).
% forth_assumptions.\<theta>\<^sub>g.cong
thf(fact_13_solution,axiom,
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ X3 @ ( clique363107459185959606CLIQUE @ k ) )
= ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ X3 ) @ phi ) ) ) ).
% solution
thf(fact_14_inj__on___092_060pi_062,axiom,
inj_on_a_set_nat @ pi @ v ).
% inj_on_\<pi>
thf(fact_15_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_16_eval__gs__def,axiom,
! [Theta2: a > $o,X4: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ X4 )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X4 )
& ( clique5859573001277246426al_g_a @ v @ pi @ Theta2 @ X2 ) ) ) ) ).
% eval_gs_def
thf(fact_17__092_060A_062__def,axiom,
( ( clique5987991184601036204th_A_a @ v )
= ( collec4794253742848188331mula_a
@ ^ [Phi2: monotone_mformula_a] : ( ord_less_eq_set_a @ ( monotone_vars_a @ Phi2 ) @ v ) ) ) ).
% \<A>_def
thf(fact_18_eval__set,axiom,
! [Phi: monotone_mformula_a,Theta2: a > $o] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ( monotone_eval_a @ Theta2 @ Phi )
= ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ) ) ).
% eval_set
thf(fact_19_vars,axiom,
( member535913909593306477mula_a @ phi
@ ( collec4794253742848188331mula_a
@ ^ [Phi2: monotone_mformula_a] : ( ord_less_eq_set_a @ ( monotone_vars_a @ Phi2 ) @ v ) ) ) ).
% vars
thf(fact_20__092_060open_062_092_060exists_062_092_060psi_062_092_060in_062tf__mformula_O_A_I_092_060forall_062_092_060theta_062_O_Aeval_A_092_060theta_062_A_092_060phi_062_A_061_Aeval_A_092_060theta_062_A_092_060psi_062_J_A_092_060and_062_Avars_A_092_060psi_062_A_092_060subseteq_062_Avars_A_092_060phi_062_A_092_060and_062_Acs_A_092_060psi_062_A_092_060le_062_Acs_A_092_060phi_062_092_060close_062,axiom,
? [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ monoto4877036962378694605mula_a )
& ! [Theta: a > $o] :
( ( monotone_eval_a @ Theta @ phi )
= ( monotone_eval_a @ Theta @ X5 ) )
& ( ord_less_eq_set_a @ ( monotone_vars_a @ X5 ) @ ( monotone_vars_a @ phi ) )
& ( ord_less_eq_nat @ ( monotone_cs_a @ X5 ) @ ( monotone_cs_a @ phi ) ) ) ).
% \<open>\<exists>\<psi>\<in>tf_mformula. (\<forall>\<theta>. eval \<theta> \<phi> = eval \<theta> \<psi>) \<and> vars \<psi> \<subseteq> vars \<phi> \<and> cs \<psi> \<le> cs \<phi>\<close>
thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060psi_062_O_A_092_060lbrakk_062_092_060psi_062_A_092_060in_062_Atf__mformula_059_A_092_060forall_062_092_060theta_062_O_Aeval_A_092_060theta_062_A_092_060phi_062_A_061_Aeval_A_092_060theta_062_A_092_060psi_062_059_Avars_A_092_060psi_062_A_092_060subseteq_062_Avars_A_092_060phi_062_059_Acs_A_092_060psi_062_A_092_060le_062_Acs_A_092_060phi_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Psi2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Psi2 @ monoto4877036962378694605mula_a )
=> ( ! [Theta: a > $o] :
( ( monotone_eval_a @ Theta @ phi )
= ( monotone_eval_a @ Theta @ Psi2 ) )
=> ( ( ord_less_eq_set_a @ ( monotone_vars_a @ Psi2 ) @ ( monotone_vars_a @ phi ) )
=> ~ ( ord_less_eq_nat @ ( monotone_cs_a @ Psi2 ) @ ( monotone_cs_a @ phi ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>\<psi>. \<lbrakk>\<psi> \<in> tf_mformula; \<forall>\<theta>. eval \<theta> \<phi> = eval \<theta> \<psi>; vars \<psi> \<subseteq> vars \<phi>; cs \<psi> \<le> cs \<phi>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_22_M0,axiom,
ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ k ) ).
% M0
thf(fact_23_M0_H,axiom,
ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ k ) ).
% M0'
thf(fact_24_forth__assumptions_OSET_Ocong,axiom,
clique6509092761774629891_SET_a = clique6509092761774629891_SET_a ).
% forth_assumptions.SET.cong
thf(fact_25_forth__assumptions_Oeval__gs_Ocong,axiom,
clique835570645587132141l_gs_a = clique835570645587132141l_gs_a ).
% forth_assumptions.eval_gs.cong
thf(fact_26_forth__assumptions_Oeval__g_Ocong,axiom,
clique5859573001277246426al_g_a = clique5859573001277246426al_g_a ).
% forth_assumptions.eval_g.cong
thf(fact_27_eval__ACC,axiom,
! [Phi: monotone_mformula_a,G: set_set_nat] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ G ) @ Phi )
= ( member_set_set_nat @ G @ ( clique4708818501384062891C_mf_a @ k @ pi @ Phi ) ) ) ) ) ) ).
% eval_ACC
thf(fact_28__092_060open_062_123_125_A_092_060in_062_A_092_060G_062_A_092_060Longrightarrow_062_A_I_123_125_A_092_060in_062_ACLIQUE_J_A_061_Aeval_A_I_092_060theta_062_092_060_094sub_062g_A_123_125_J_A_092_060phi_062_092_060close_062,axiom,
( ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) )
= ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ bot_bot_set_set_nat ) @ phi ) ) ) ).
% \<open>{} \<in> \<G> \<Longrightarrow> ({} \<in> CLIQUE) = eval (\<theta>\<^sub>g {}) \<phi>\<close>
thf(fact_29_SET___092_060G_062,axiom,
! [Phi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ord_le9131159989063066194et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% SET_\<G>
thf(fact_30__092_060open_062_092_060not_062_Aeval_A_I_092_060theta_062_092_060_094sub_062g_A_123_125_J_A_092_060phi_062_092_060close_062,axiom,
~ ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ bot_bot_set_set_nat ) @ phi ) ).
% \<open>\<not> eval (\<theta>\<^sub>g {}) \<phi>\<close>
thf(fact_31_to__tf__mformula,axiom,
! [Theta2: set_nat > $o,Phi: monoto7244996872745832772et_nat] :
( ~ ( monoto7255863275561024424et_nat @ Theta2 @ Phi )
=> ? [X5: monoto7244996872745832772et_nat] :
( ( member7623223977307079021et_nat @ X5 @ monoto1591516381014137079et_nat )
& ! [Theta: set_nat > $o] :
( ( monoto7255863275561024424et_nat @ Theta @ Phi )
= ( monoto7255863275561024424et_nat @ Theta @ X5 ) )
& ( ord_le6893508408891458716et_nat @ ( monoto8378391831928444664et_nat @ X5 ) @ ( monoto8378391831928444664et_nat @ Phi ) )
& ( ord_less_eq_nat @ ( monotone_cs_set_nat @ X5 ) @ ( monotone_cs_set_nat @ Phi ) ) ) ) ).
% to_tf_mformula
thf(fact_32_to__tf__mformula,axiom,
! [Theta2: nat > $o,Phi: monoto4181647612830706830la_nat] :
( ~ ( monotone_eval_nat @ Theta2 @ Phi )
=> ? [X5: monoto4181647612830706830la_nat] :
( ( member6605755813721784503la_nat @ X5 @ monoto1762321936899617089la_nat )
& ! [Theta: nat > $o] :
( ( monotone_eval_nat @ Theta @ Phi )
= ( monotone_eval_nat @ Theta @ X5 ) )
& ( ord_less_eq_set_nat @ ( monotone_vars_nat @ X5 ) @ ( monotone_vars_nat @ Phi ) )
& ( ord_less_eq_nat @ ( monotone_cs_nat @ X5 ) @ ( monotone_cs_nat @ Phi ) ) ) ) ).
% to_tf_mformula
thf(fact_33_to__tf__mformula,axiom,
! [Theta2: ( nat > nat ) > $o,Phi: monoto8276428299528460797at_nat] :
( ~ ( monoto2874428580947315297at_nat @ Theta2 @ Phi )
=> ? [X5: monoto8276428299528460797at_nat] :
( ( member435044527007775910at_nat @ X5 @ monoto6982705221701894448at_nat )
& ! [Theta: ( nat > nat ) > $o] :
( ( monoto2874428580947315297at_nat @ Theta @ Phi )
= ( monoto2874428580947315297at_nat @ Theta @ X5 ) )
& ( ord_le9059583361652607317at_nat @ ( monoto4799612099597528497at_nat @ X5 ) @ ( monoto4799612099597528497at_nat @ Phi ) )
& ( ord_less_eq_nat @ ( monotone_cs_nat_nat @ X5 ) @ ( monotone_cs_nat_nat @ Phi ) ) ) ) ).
% to_tf_mformula
thf(fact_34_to__tf__mformula,axiom,
! [Theta2: a > $o,Phi: monotone_mformula_a] :
( ~ ( monotone_eval_a @ Theta2 @ Phi )
=> ? [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ monoto4877036962378694605mula_a )
& ! [Theta: a > $o] :
( ( monotone_eval_a @ Theta @ Phi )
= ( monotone_eval_a @ Theta @ X5 ) )
& ( ord_less_eq_set_a @ ( monotone_vars_a @ X5 ) @ ( monotone_vars_a @ Phi ) )
& ( ord_less_eq_nat @ ( monotone_cs_a @ X5 ) @ ( monotone_cs_a @ Phi ) ) ) ) ).
% to_tf_mformula
thf(fact_35_SET_Osimps_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ).
% SET.simps(4)
thf(fact_36_tf__mformula_Ocases,axiom,
! [A: monotone_mformula_a] :
( ( member535913909593306477mula_a @ A @ monoto4877036962378694605mula_a )
=> ( ( A != monotone_FALSE_a )
=> ( ! [X5: a] :
( A
!= ( monotone_Var_a @ X5 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( A
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ( member535913909593306477mula_a @ Phi3 @ monoto4877036962378694605mula_a )
=> ~ ( member535913909593306477mula_a @ Psi2 @ monoto4877036962378694605mula_a ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( A
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( ( member535913909593306477mula_a @ Phi3 @ monoto4877036962378694605mula_a )
=> ~ ( member535913909593306477mula_a @ Psi2 @ monoto4877036962378694605mula_a ) ) ) ) ) ) ) ).
% tf_mformula.cases
thf(fact_37_tf__mformula_Osimps,axiom,
! [A: monotone_mformula_a] :
( ( member535913909593306477mula_a @ A @ monoto4877036962378694605mula_a )
= ( ( A = monotone_FALSE_a )
| ? [X2: a] :
( A
= ( monotone_Var_a @ X2 ) )
| ? [Phi2: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( A
= ( monotone_Disj_a @ Phi2 @ Psi3 ) )
& ( member535913909593306477mula_a @ Phi2 @ monoto4877036962378694605mula_a )
& ( member535913909593306477mula_a @ Psi3 @ monoto4877036962378694605mula_a ) )
| ? [Phi2: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( A
= ( monotone_Conj_a @ Phi2 @ Psi3 ) )
& ( member535913909593306477mula_a @ Phi2 @ monoto4877036962378694605mula_a )
& ( member535913909593306477mula_a @ Psi3 @ monoto4877036962378694605mula_a ) ) ) ) ).
% tf_mformula.simps
thf(fact_38_eval_Oelims_I3_J,axiom,
! [X: a > $o,Xa: monotone_mformula_a] :
( ~ ( monotone_eval_a @ X @ Xa )
=> ( ( Xa != monotone_FALSE_a )
=> ( ! [X5: a] :
( ( Xa
= ( monotone_Var_a @ X5 ) )
=> ( X @ X5 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( Xa
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ( monotone_eval_a @ X @ Phi3 )
| ( monotone_eval_a @ X @ Psi2 ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( Xa
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( ( monotone_eval_a @ X @ Phi3 )
& ( monotone_eval_a @ X @ Psi2 ) ) ) ) ) ) ) ).
% eval.elims(3)
thf(fact_39_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_40_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_41_mformula_Oinject_I2_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a,Y41: monotone_mformula_a,Y42: monotone_mformula_a] :
( ( ( monotone_Conj_a @ X41 @ X42 )
= ( monotone_Conj_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% mformula.inject(2)
thf(fact_42_mformula_Oinject_I3_J,axiom,
! [X51: monotone_mformula_a,X52: monotone_mformula_a,Y51: monotone_mformula_a,Y52: monotone_mformula_a] :
( ( ( monotone_Disj_a @ X51 @ X52 )
= ( monotone_Disj_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% mformula.inject(3)
thf(fact_43_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: monotone_mformula_a,P: monotone_mformula_a > $o] :
( ( member535913909593306477mula_a @ A @ ( collec4794253742848188331mula_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_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_46_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_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_49_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_Mo2626137824023173004mula_a] :
( ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] : ( member535913909593306477mula_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__cong,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ! [X5: monotone_mformula_a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collec4794253742848188331mula_a @ P )
= ( collec4794253742848188331mula_a @ Q ) ) ) ).
% Collect_cong
thf(fact_56_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X5: set_set_nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_57_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X5: nat > nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_59_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X5: set_nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_60_mformula_Oinject_I1_J,axiom,
! [X32: a,Y3: a] :
( ( ( monotone_Var_a @ X32 )
= ( monotone_Var_a @ Y3 ) )
= ( X32 = Y3 ) ) ).
% mformula.inject(1)
thf(fact_61_odot___092_060G_062,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_62_eval__gs__odot,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat,Theta2: a > $o] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ X4 )
& ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ Y ) ) ) ) ) ).
% eval_gs_odot
thf(fact_63_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_64_forth__assumptions_OACC__mf_Ocong,axiom,
clique4708818501384062891C_mf_a = clique4708818501384062891C_mf_a ).
% forth_assumptions.ACC_mf.cong
thf(fact_65_vars_Osimps_I4_J,axiom,
( ( monoto8378391831928444664et_nat @ monoto2388303931541111964et_nat )
= bot_bot_set_set_nat ) ).
% vars.simps(4)
thf(fact_66_vars_Osimps_I4_J,axiom,
( ( monoto4799612099597528497at_nat @ monoto3735746735589746005at_nat )
= bot_bot_set_nat_nat ) ).
% vars.simps(4)
thf(fact_67_vars_Osimps_I4_J,axiom,
( ( monoto3765512064276419502et_nat @ monoto6214072352461320530et_nat )
= bot_bo7198184520161983622et_nat ) ).
% vars.simps(4)
thf(fact_68_vars_Osimps_I4_J,axiom,
( ( monotone_vars_nat @ monotone_FALSE_nat )
= bot_bot_set_nat ) ).
% vars.simps(4)
thf(fact_69_vars_Osimps_I4_J,axiom,
( ( monotone_vars_a @ monotone_FALSE_a )
= bot_bot_set_a ) ).
% vars.simps(4)
thf(fact_70_mformula_Odistinct_I19_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a,X51: monotone_mformula_a,X52: monotone_mformula_a] :
( ( monotone_Conj_a @ X41 @ X42 )
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(19)
thf(fact_71_mformula_Odistinct_I15_J,axiom,
! [X32: a,X41: monotone_mformula_a,X42: monotone_mformula_a] :
( ( monotone_Var_a @ X32 )
!= ( monotone_Conj_a @ X41 @ X42 ) ) ).
% mformula.distinct(15)
thf(fact_72_mformula_Odistinct_I17_J,axiom,
! [X32: a,X51: monotone_mformula_a,X52: monotone_mformula_a] :
( ( monotone_Var_a @ X32 )
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(17)
thf(fact_73_mformula_Odistinct_I11_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a] :
( monotone_FALSE_a
!= ( monotone_Conj_a @ X41 @ X42 ) ) ).
% mformula.distinct(11)
thf(fact_74_mformula_Odistinct_I13_J,axiom,
! [X51: monotone_mformula_a,X52: monotone_mformula_a] :
( monotone_FALSE_a
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(13)
thf(fact_75_mformula_Odistinct_I9_J,axiom,
! [X32: a] :
( monotone_FALSE_a
!= ( monotone_Var_a @ X32 ) ) ).
% mformula.distinct(9)
thf(fact_76_eval_Osimps_I4_J,axiom,
! [Theta2: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_eval_a @ Theta2 @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta2 @ Phi )
| ( monotone_eval_a @ Theta2 @ Psi ) ) ) ).
% eval.simps(4)
thf(fact_77_eval_Osimps_I5_J,axiom,
! [Theta2: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_eval_a @ Theta2 @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta2 @ Phi )
& ( monotone_eval_a @ Theta2 @ Psi ) ) ) ).
% eval.simps(5)
thf(fact_78_eval_Osimps_I3_J,axiom,
! [Theta2: a > $o,X: a] :
( ( monotone_eval_a @ Theta2 @ ( monotone_Var_a @ X ) )
= ( Theta2 @ X ) ) ).
% eval.simps(3)
thf(fact_79_eval_Osimps_I1_J,axiom,
! [Theta2: a > $o] :
~ ( monotone_eval_a @ Theta2 @ monotone_FALSE_a ) ).
% eval.simps(1)
thf(fact_80_tf__mformula_Otf__Disj,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Psi @ monoto4877036962378694605mula_a )
=> ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ monoto4877036962378694605mula_a ) ) ) ).
% tf_mformula.tf_Disj
thf(fact_81_tf__mformula_Otf__Conj,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Psi @ monoto4877036962378694605mula_a )
=> ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ monoto4877036962378694605mula_a ) ) ) ).
% tf_mformula.tf_Conj
thf(fact_82_tf__mformula_Otf__Var,axiom,
! [X: a] : ( member535913909593306477mula_a @ ( monotone_Var_a @ X ) @ monoto4877036962378694605mula_a ) ).
% tf_mformula.tf_Var
thf(fact_83_tf__mformula_Otf__False,axiom,
member535913909593306477mula_a @ monotone_FALSE_a @ monoto4877036962378694605mula_a ).
% tf_mformula.tf_False
thf(fact_84_eval__vars,axiom,
! [Phi: monoto5348676645462821012mula_a,Theta_1: monotone_mformula_a > $o,Theta_2: monotone_mformula_a > $o] :
( ! [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ ( monoto6984925928961046984mula_a @ Phi ) )
=> ( ( Theta_1 @ X5 )
= ( Theta_2 @ X5 ) ) )
=> ( ( monoto2605587986704299640mula_a @ Theta_1 @ Phi )
= ( monoto2605587986704299640mula_a @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_85_eval__vars,axiom,
! [Phi: monoto7244996872745832772et_nat,Theta_1: set_nat > $o,Theta_2: set_nat > $o] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ ( monoto8378391831928444664et_nat @ Phi ) )
=> ( ( Theta_1 @ X5 )
= ( Theta_2 @ X5 ) ) )
=> ( ( monoto7255863275561024424et_nat @ Theta_1 @ Phi )
= ( monoto7255863275561024424et_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_86_eval__vars,axiom,
! [Phi: monoto5483634261523599098et_nat,Theta_1: set_set_nat > $o,Theta_2: set_set_nat > $o] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ ( monoto3765512064276419502et_nat @ Phi ) )
=> ( ( Theta_1 @ X5 )
= ( Theta_2 @ X5 ) ) )
=> ( ( monoto5763065145529399390et_nat @ Theta_1 @ Phi )
= ( monoto5763065145529399390et_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_87_eval__vars,axiom,
! [Phi: monoto8276428299528460797at_nat,Theta_1: ( nat > nat ) > $o,Theta_2: ( nat > nat ) > $o] :
( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ ( monoto4799612099597528497at_nat @ Phi ) )
=> ( ( Theta_1 @ X5 )
= ( Theta_2 @ X5 ) ) )
=> ( ( monoto2874428580947315297at_nat @ Theta_1 @ Phi )
= ( monoto2874428580947315297at_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_88_eval__vars,axiom,
! [Phi: monotone_mformula_a,Theta_1: a > $o,Theta_2: a > $o] :
( ! [X5: a] :
( ( member_a @ X5 @ ( monotone_vars_a @ Phi ) )
=> ( ( Theta_1 @ X5 )
= ( Theta_2 @ X5 ) ) )
=> ( ( monotone_eval_a @ Theta_1 @ Phi )
= ( monotone_eval_a @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_89_CLIQUE__solution__imp__POS__sub__ACC,axiom,
! [Phi: monotone_mformula_a] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ X5 @ ( clique363107459185959606CLIQUE @ k ) )
= ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ X5 ) @ Phi ) ) )
=> ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique4708818501384062891C_mf_a @ k @ pi @ Phi ) ) ) ) ) ).
% CLIQUE_solution_imp_POS_sub_ACC
thf(fact_90_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G2: set_set_nat] :
? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X6 )
& ( ord_le6893508408891458716et_nat @ X2 @ G2 ) ) ) ) ).
% accepts_def
thf(fact_91__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_92__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<G>_def
thf(fact_93_inj__on__empty,axiom,
! [F: a > set_nat] : ( inj_on_a_set_nat @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_94_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_95_empty__subsetI,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% empty_subsetI
thf(fact_96_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_97_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_98_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_99_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_100_subset__empty,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_101_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_102_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_103_subset__empty,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_104_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_105_ACC__mf__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique4708818501384062891C_mf_a @ k @ pi @ Phi )
= ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ).
% ACC_mf_def
thf(fact_106_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_107_CLIQUE__solution__imp__ACC__cf__empty,axiom,
! [Phi: monotone_mformula_a] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ X5 @ ( clique363107459185959606CLIQUE @ k ) )
= ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ X5 ) @ Phi ) ) )
=> ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ( clique8961599393750669800f_mf_a @ k @ pi @ Phi )
= bot_bot_set_nat_nat ) ) ) ) ).
% CLIQUE_solution_imp_ACC_cf_empty
thf(fact_108_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_109_v__mono,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_110_SET_Osimps_I1_J,axiom,
( ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% SET.simps(1)
thf(fact_111_subsetI,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ! [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ A2 )
=> ( member535913909593306477mula_a @ X5 @ B ) )
=> ( ord_le5054881893329012716mula_a @ A2 @ B ) ) ).
% subsetI
thf(fact_112_subsetI,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
=> ( member_set_set_nat @ X5 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_113_subsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ( member_set_nat @ X5 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_114_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ X5 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_115_subsetI,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A2 )
=> ( member_nat_nat @ X5 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_116_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( member_a @ X5 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_117_subset__antisym,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_118_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_119_subset__antisym,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_120_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_121_empty__iff,axiom,
! [C: monotone_mformula_a] :
~ ( member535913909593306477mula_a @ C @ bot_bo3042613601904376864mula_a ) ).
% empty_iff
thf(fact_122_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_123_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_124_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_125_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_126_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_127_all__not__in__conv,axiom,
! [A2: set_Mo2626137824023173004mula_a] :
( ( ! [X2: monotone_mformula_a] :
~ ( member535913909593306477mula_a @ X2 @ A2 ) )
= ( A2 = bot_bo3042613601904376864mula_a ) ) ).
% all_not_in_conv
thf(fact_128_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_129_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_130_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_131_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X2: set_set_nat] :
~ ( member_set_set_nat @ X2 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_132_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_133_Collect__empty__eq,axiom,
! [P: monotone_mformula_a > $o] :
( ( ( collec4794253742848188331mula_a @ P )
= bot_bo3042613601904376864mula_a )
= ( ! [X2: monotone_mformula_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_134_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_135_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_136_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X2: set_set_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_137_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_138_empty__Collect__eq,axiom,
! [P: monotone_mformula_a > $o] :
( ( bot_bo3042613601904376864mula_a
= ( collec4794253742848188331mula_a @ P ) )
= ( ! [X2: monotone_mformula_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_139_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X2: set_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_140_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X2: nat > nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_141_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X2: set_set_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_142_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_143_ACC__def,axiom,
! [X4: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ X4 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( clique3686358387679108662ccepts @ X4 @ G2 ) ) ) ) ).
% ACC_def
thf(fact_144_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_145_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_146_acceptsI,axiom,
! [D: set_set_nat,G: set_set_nat,X4: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D @ G )
=> ( ( member_set_set_nat @ D @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ).
% acceptsI
thf(fact_147_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_148_ACC__SET_I2_J,axiom,
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
= bot_bo7198184520161983622et_nat ) ).
% ACC_SET(2)
thf(fact_149_ACC__I,axiom,
! [G: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ X4 ) ) ) ) ).
% ACC_I
thf(fact_150__092_060pi_062m2,axiom,
! [X: a] :
( ( member_a @ X @ v )
=> ( member_set_nat @ ( pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% \<pi>m2
thf(fact_151_ACC__SET_I1_J,axiom,
! [X: a] :
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( member_set_nat @ ( pi @ X ) @ G2 ) ) ) ) ).
% ACC_SET(1)
thf(fact_152_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_153_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_154_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_155_forth__assumptions_OACC__cf__mf_Ocong,axiom,
clique8961599393750669800f_mf_a = clique8961599393750669800f_mf_a ).
% forth_assumptions.ACC_cf_mf.cong
thf(fact_156_Graphs__def,axiom,
( clique5786534781347292306Graphs
= ( ^ [V: set_nat] :
( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ V @ V ) ) ) ) ) ).
% Graphs_def
thf(fact_157_numbers2__mono,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y2 ) @ ( clique3652268606331196573umbers @ Y2 ) ) ) ) ).
% numbers2_mono
thf(fact_158_in__mono,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,X: monotone_mformula_a] :
( ( ord_le5054881893329012716mula_a @ A2 @ B )
=> ( ( member535913909593306477mula_a @ X @ A2 )
=> ( member535913909593306477mula_a @ X @ B ) ) ) ).
% in_mono
thf(fact_159_in__mono,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,X: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_set_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_160_in__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_161_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_162_in__mono,axiom,
! [A2: set_nat_nat,B: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_163_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_164_subsetD,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,C: monotone_mformula_a] :
( ( ord_le5054881893329012716mula_a @ A2 @ B )
=> ( ( member535913909593306477mula_a @ C @ A2 )
=> ( member535913909593306477mula_a @ C @ B ) ) ) ).
% subsetD
thf(fact_165_subsetD,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_166_subsetD,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_167_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_168_subsetD,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_169_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_170_equalityE,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_171_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_172_equalityE,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 = B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ~ ( ord_le9059583361652607317at_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_173_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_174_subset__eq,axiom,
( ord_le5054881893329012716mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
=> ( member535913909593306477mula_a @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_175_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( member_set_set_nat @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_176_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( member_set_nat @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_177_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_178_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( member_nat_nat @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_179_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_180_equalityD1,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_181_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_182_equalityD1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 = B )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_183_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_184_equalityD2,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_185_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_186_equalityD2,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 = B )
=> ( ord_le9059583361652607317at_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_187_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_188_subset__iff,axiom,
( ord_le5054881893329012716mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
! [T: monotone_mformula_a] :
( ( member535913909593306477mula_a @ T @ A3 )
=> ( member535913909593306477mula_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_189_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_190_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_191_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_192_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_193_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_194_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_195_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_196_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_197_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_198_Collect__mono,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ! [X5: monotone_mformula_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le5054881893329012716mula_a @ ( collec4794253742848188331mula_a @ P ) @ ( collec4794253742848188331mula_a @ Q ) ) ) ).
% Collect_mono
thf(fact_199_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X5: set_set_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_200_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X5: set_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( 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] :
( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_202_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X5: nat > nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_203_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_204_subset__trans,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_205_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_206_subset__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_207_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_208_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_209_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_210_set__eq__subset,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_211_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_212_Collect__mono__iff,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ( ord_le5054881893329012716mula_a @ ( collec4794253742848188331mula_a @ P ) @ ( collec4794253742848188331mula_a @ Q ) )
= ( ! [X2: monotone_mformula_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_213_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 ) )
= ( ! [X2: set_set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_214_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_215_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_216_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_217_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_218_emptyE,axiom,
! [A: monotone_mformula_a] :
~ ( member535913909593306477mula_a @ A @ bot_bo3042613601904376864mula_a ) ).
% emptyE
thf(fact_219_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_220_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_221_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_222_emptyE,axiom,
! [A: set_set_nat] :
~ ( member_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_223_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_224_equals0D,axiom,
! [A2: set_Mo2626137824023173004mula_a,A: monotone_mformula_a] :
( ( A2 = bot_bo3042613601904376864mula_a )
=> ~ ( member535913909593306477mula_a @ A @ A2 ) ) ).
% equals0D
thf(fact_225_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_226_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_227_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_228_equals0D,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( A2 = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_229_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_230_equals0I,axiom,
! [A2: set_Mo2626137824023173004mula_a] :
( ! [Y5: monotone_mformula_a] :
~ ( member535913909593306477mula_a @ Y5 @ A2 )
=> ( A2 = bot_bo3042613601904376864mula_a ) ) ).
% equals0I
thf(fact_231_equals0I,axiom,
! [A2: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_232_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y5: set_nat] :
~ ( member_set_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_233_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y5: nat > nat] :
~ ( member_nat_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_234_equals0I,axiom,
! [A2: set_set_set_nat] :
( ! [Y5: set_set_nat] :
~ ( member_set_set_nat @ Y5 @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_235_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_236_ex__in__conv,axiom,
! [A2: set_Mo2626137824023173004mula_a] :
( ( ? [X2: monotone_mformula_a] : ( member535913909593306477mula_a @ X2 @ A2 ) )
= ( A2 != bot_bo3042613601904376864mula_a ) ) ).
% ex_in_conv
thf(fact_237_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_238_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_239_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_240_ex__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ? [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A2 ) )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_241_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_242_inj__onD,axiom,
! [F: a > set_nat,A2: set_a,X: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( X = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_243_inj__onI,axiom,
! [A2: set_a,F: a > set_nat] :
( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_a @ Y5 @ A2 )
=> ( ( ( F @ X5 )
= ( F @ Y5 ) )
=> ( X5 = Y5 ) ) ) )
=> ( inj_on_a_set_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_244_inj__on__def,axiom,
( inj_on_a_set_nat
= ( ^ [F2: a > set_nat,A3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ! [Y6: a] :
( ( member_a @ Y6 @ A3 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y6 ) )
=> ( X2 = Y6 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_245_inj__on__cong,axiom,
! [A2: set_a,F: a > set_nat,G3: a > set_nat] :
( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G3 @ A4 ) ) )
=> ( ( inj_on_a_set_nat @ F @ A2 )
= ( inj_on_a_set_nat @ G3 @ A2 ) ) ) ).
% inj_on_cong
thf(fact_246_inj__on__eq__iff,axiom,
! [F: a > set_nat,A2: set_a,X: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
= ( X = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_247_inj__on__contraD,axiom,
! [F: a > set_nat,A2: set_a,X: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A2 )
=> ( ( X != Y2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( ( F @ X )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_248_inj__on__inverseI,axiom,
! [A2: set_a,G3: set_nat > a,F: a > set_nat] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( G3 @ ( F @ X5 ) )
= X5 ) )
=> ( inj_on_a_set_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_249_Collect__subset,axiom,
! [A2: set_Mo2626137824023173004mula_a,P: monotone_mformula_a > $o] :
( ord_le5054881893329012716mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_250_Collect__subset,axiom,
! [A2: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_251_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_252_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_253_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_254_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_255_empty__def,axiom,
( bot_bo3042613601904376864mula_a
= ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] : $false ) ) ).
% empty_def
thf(fact_256_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X2: set_nat] : $false ) ) ).
% empty_def
thf(fact_257_empty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X2: nat > nat] : $false ) ) ).
% empty_def
thf(fact_258_empty__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat
@ ^ [X2: set_set_nat] : $false ) ) ).
% empty_def
thf(fact_259_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_260_sameprod__mono,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) @ ( clique8906516429304539640et_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_261_sameprod__mono,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) @ ( clique134924887794942129at_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_262_sameprod__mono,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( clique9072761800073521420prod_a @ X4 @ X4 ) @ ( clique9072761800073521420prod_a @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_263_sameprod__mono,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) @ ( clique6722202388162463298od_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_264_inj__on__subset,axiom,
! [F: a > set_nat,A2: set_a,B: set_a] :
( ( inj_on_a_set_nat @ F @ A2 )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( inj_on_a_set_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_265_subset__inj__on,axiom,
! [F: a > set_nat,B: set_a,A2: set_a] :
( ( inj_on_a_set_nat @ F @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( inj_on_a_set_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_266__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [K: set_set_nat] :
( ( member_set_set_nat @ K @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K ) )
= k )
& ( K
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K ) @ ( clique5033774636164728513irst_v @ K ) ) ) ) ) ) ).
% \<K>_def
thf(fact_267_CLIQUE__def,axiom,
( ( clique363107459185959606CLIQUE @ k )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( clique3326749438856946062irst_K @ k ) )
& ( ord_le6893508408891458716et_nat @ X2 @ G2 ) ) ) ) ) ).
% CLIQUE_def
thf(fact_268_finite__vG,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ).
% finite_vG
thf(fact_269_bij__betw___092_060pi_062,axiom,
bij_betw_a_set_nat @ pi @ v @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% bij_betw_\<pi>
thf(fact_270_ACC__cf__mf__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique8961599393750669800f_mf_a @ k @ pi @ Phi )
= ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ).
% ACC_cf_mf_def
thf(fact_271_finite__members___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ).
% finite_members_\<G>
thf(fact_272_eval__gs__union,axiom,
! [Theta2: a > $o,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ X4 )
| ( clique835570645587132141l_gs_a @ v @ pi @ Theta2 @ Y ) ) ) ).
% eval_gs_union
thf(fact_273_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_274_SET_Ocases,axiom,
! [X: monotone_mformula_a] :
( ( X != monotone_FALSE_a )
=> ( ! [X5: a] :
( X
!= ( monotone_Var_a @ X5 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( X = monotone_TRUE_a ) ) ) ) ) ).
% SET.cases
thf(fact_275_ACC__cf__mono,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_mono
thf(fact_276_ACC__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X4 ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_union
thf(fact_277_IntI,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ A2 )
=> ( ( member535913909593306477mula_a @ C @ B )
=> ( member535913909593306477mula_a @ C @ ( inf_in4741988911529734942mula_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_278_IntI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_279_IntI,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_280_IntI,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_281_IntI,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_282_Int__iff,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ ( inf_in4741988911529734942mula_a @ A2 @ B ) )
= ( ( member535913909593306477mula_a @ C @ A2 )
& ( member535913909593306477mula_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_283_Int__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_284_Int__iff,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
= ( ( member_set_nat @ C @ A2 )
& ( member_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_285_Int__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
= ( ( member_set_set_nat @ C @ A2 )
& ( member_set_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_286_Int__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
= ( ( member_nat_nat @ C @ A2 )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_287_UnCI,axiom,
! [C: monotone_mformula_a,B: set_Mo2626137824023173004mula_a,A2: set_Mo2626137824023173004mula_a] :
( ( ~ ( member535913909593306477mula_a @ C @ B )
=> ( member535913909593306477mula_a @ C @ A2 ) )
=> ( member535913909593306477mula_a @ C @ ( sup_su7438456061012554424mula_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_288_UnCI,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_289_UnCI,axiom,
! [C: set_set_nat,B: set_set_set_nat,A2: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ A2 ) )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_290_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A2: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A2 ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_291_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_292_UnCI,axiom,
! [C: nat > nat,B: set_nat_nat,A2: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ A2 ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_293_Un__iff,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ ( sup_su7438456061012554424mula_a @ A2 @ B ) )
= ( ( member535913909593306477mula_a @ C @ A2 )
| ( member535913909593306477mula_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_294_Un__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_295_Un__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
= ( ( member_set_set_nat @ C @ A2 )
| ( member_set_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_296_Un__iff,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( ( member_set_nat @ C @ A2 )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_297_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_298_Un__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) )
= ( ( member_nat_nat @ C @ A2 )
| ( member_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_299_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_300_ACC__odot,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X4 ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_odot
thf(fact_301_SET_Osimps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ).
% SET.simps(3)
thf(fact_302_approx__pos_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ( X != monotone_TRUE_a )
=> ( ( X != monotone_FALSE_a )
=> ( ! [V2: a] :
( X
!= ( monotone_Var_a @ V2 ) )
=> ~ ! [V2: monotone_mformula_a,Va: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ V2 @ Va ) ) ) ) ) ) ).
% approx_pos.cases
thf(fact_303_approx__neg_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( ( X != monotone_TRUE_a )
=> ( ( X != monotone_FALSE_a )
=> ~ ! [V2: a] :
( X
!= ( monotone_Var_a @ V2 ) ) ) ) ) ) ).
% approx_neg.cases
thf(fact_304_Int__subset__iff,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
& ( ord_le9131159989063066194et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_305_Int__subset__iff,axiom,
! [C2: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_306_Int__subset__iff,axiom,
! [C2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_307_Int__subset__iff,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A2 @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_308_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_309_Un__subset__iff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ C2 )
= ( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
& ( ord_le9131159989063066194et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_310_Un__subset__iff,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
& ( ord_le6893508408891458716et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_311_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_312_Un__subset__iff,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ C2 )
= ( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
& ( ord_le9059583361652607317at_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_313_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_314_Un__empty,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_315_Un__empty,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_316_Un__empty,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B = bot_bo7198184520161983622et_nat ) ) ) ).
% Un_empty
thf(fact_317_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_318_Int__Un__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T2 @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_319_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_320_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_321_Int__Un__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T2 @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_322_Int__Un__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_323_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_324_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_325_Int__Un__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_326_Int__Un__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_327_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_328_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_329_Int__Un__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_330_Int__Un__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_331_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_332_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_333_Int__Un__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_334_Un__Int__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T2 @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_335_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_336_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_337_Un__Int__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T2 @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_338_Un__Int__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_339_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_340_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_341_Un__Int__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_342_Un__Int__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_343_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_344_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_345_Un__Int__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_346_Un__Int__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_347_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_348_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_349_Un__Int__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_350_bex__empty,axiom,
! [P: set_nat > $o] :
~ ? [X3: set_nat] :
( ( member_set_nat @ X3 @ bot_bot_set_set_nat )
& ( P @ X3 ) ) ).
% bex_empty
thf(fact_351_bex__empty,axiom,
! [P: ( nat > nat ) > $o] :
~ ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ bot_bot_set_nat_nat )
& ( P @ X3 ) ) ).
% bex_empty
thf(fact_352_bex__empty,axiom,
! [P: set_set_nat > $o] :
~ ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat )
& ( P @ X3 ) ) ).
% bex_empty
thf(fact_353_bex__empty,axiom,
! [P: nat > $o] :
~ ? [X3: nat] :
( ( member_nat @ X3 @ bot_bot_set_nat )
& ( P @ X3 ) ) ).
% bex_empty
thf(fact_354_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_355_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_356_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_357_ACC__cf__SET_I2_J,axiom,
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
= bot_bot_set_nat_nat ) ).
% ACC_cf_SET(2)
thf(fact_358_ACC__SET_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_SET(3)
thf(fact_359_ACC__SET_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_SET(4)
thf(fact_360_Un__Int__assoc__eq,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) )
= ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_361_Un__Int__assoc__eq,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_362_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_363_Un__Int__assoc__eq,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_364_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_365_bij__betw__disjoint__Un,axiom,
! [F: a > nat,A2: set_a,C2: set_nat,G3: a > nat,B: set_a,D: set_nat] :
( ( bij_betw_a_nat @ F @ A2 @ C2 )
=> ( ( bij_betw_a_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_nat @ C2 @ D )
= bot_bot_set_nat )
=> ( bij_betw_a_nat
@ ^ [X2: a] : ( if_nat @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B )
@ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_366_bij__betw__disjoint__Un,axiom,
! [F: nat > nat,A2: set_nat,C2: set_nat,G3: nat > nat,B: set_nat,D: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ C2 )
=> ( ( bij_betw_nat_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ C2 @ D )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat
@ ^ [X2: nat] : ( if_nat @ ( member_nat @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_nat @ A2 @ B )
@ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_367_bij__betw__disjoint__Un,axiom,
! [F: a > set_nat,A2: set_a,C2: set_set_nat,G3: a > set_nat,B: set_a,D: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A2 @ C2 )
=> ( ( bij_betw_a_set_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_set_nat @ C2 @ D )
= bot_bot_set_set_nat )
=> ( bij_betw_a_set_nat
@ ^ [X2: a] : ( if_set_nat @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B )
@ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_368_bij__betw__disjoint__Un,axiom,
! [F: monotone_mformula_a > nat,A2: set_Mo2626137824023173004mula_a,C2: set_nat,G3: monotone_mformula_a > nat,B: set_Mo2626137824023173004mula_a,D: set_nat] :
( ( bij_be4432079924810155166_a_nat @ F @ A2 @ C2 )
=> ( ( bij_be4432079924810155166_a_nat @ G3 @ B @ D )
=> ( ( ( inf_in4741988911529734942mula_a @ A2 @ B )
= bot_bo3042613601904376864mula_a )
=> ( ( ( inf_inf_set_nat @ C2 @ D )
= bot_bot_set_nat )
=> ( bij_be4432079924810155166_a_nat
@ ^ [X2: monotone_mformula_a] : ( if_nat @ ( member535913909593306477mula_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_su7438456061012554424mula_a @ A2 @ B )
@ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_369_bij__betw__disjoint__Un,axiom,
! [F: set_nat > nat,A2: set_set_nat,C2: set_nat,G3: set_nat > nat,B: set_set_nat,D: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ C2 )
=> ( ( bij_betw_set_nat_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
=> ( ( ( inf_inf_set_nat @ C2 @ D )
= bot_bot_set_nat )
=> ( bij_betw_set_nat_nat
@ ^ [X2: set_nat] : ( if_nat @ ( member_set_nat @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_set_nat @ A2 @ B )
@ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_370_bij__betw__disjoint__Un,axiom,
! [F: nat > set_nat,A2: set_nat,C2: set_set_nat,G3: nat > set_nat,B: set_nat,D: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ C2 )
=> ( ( bij_betw_nat_set_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_set_nat @ C2 @ D )
= bot_bot_set_set_nat )
=> ( bij_betw_nat_set_nat
@ ^ [X2: nat] : ( if_set_nat @ ( member_nat @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_nat @ A2 @ B )
@ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_371_bij__betw__disjoint__Un,axiom,
! [F: monotone_mformula_a > set_nat,A2: set_Mo2626137824023173004mula_a,C2: set_set_nat,G3: monotone_mformula_a > set_nat,B: set_Mo2626137824023173004mula_a,D: set_set_nat] :
( ( bij_be1996446448813045844et_nat @ F @ A2 @ C2 )
=> ( ( bij_be1996446448813045844et_nat @ G3 @ B @ D )
=> ( ( ( inf_in4741988911529734942mula_a @ A2 @ B )
= bot_bo3042613601904376864mula_a )
=> ( ( ( inf_inf_set_set_nat @ C2 @ D )
= bot_bot_set_set_nat )
=> ( bij_be1996446448813045844et_nat
@ ^ [X2: monotone_mformula_a] : ( if_set_nat @ ( member535913909593306477mula_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_su7438456061012554424mula_a @ A2 @ B )
@ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_372_bij__betw__disjoint__Un,axiom,
! [F: a > nat > nat,A2: set_a,C2: set_nat_nat,G3: a > nat > nat,B: set_a,D: set_nat_nat] :
( ( bij_betw_a_nat_nat @ F @ A2 @ C2 )
=> ( ( bij_betw_a_nat_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_nat_nat @ C2 @ D )
= bot_bot_set_nat_nat )
=> ( bij_betw_a_nat_nat
@ ^ [X2: a] : ( if_nat_nat @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B )
@ ( sup_sup_set_nat_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_373_bij__betw__disjoint__Un,axiom,
! [F: a > set_set_nat,A2: set_a,C2: set_set_set_nat,G3: a > set_set_nat,B: set_a,D: set_set_set_nat] :
( ( bij_be2639851105560558660et_nat @ F @ A2 @ C2 )
=> ( ( bij_be2639851105560558660et_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( ( inf_in5711780100303410308et_nat @ C2 @ D )
= bot_bo7198184520161983622et_nat )
=> ( bij_be2639851105560558660et_nat
@ ^ [X2: a] : ( if_set_set_nat @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B )
@ ( sup_su4213647025997063966et_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_374_bij__betw__disjoint__Un,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,C2: set_set_nat,G3: set_nat > set_nat,B: set_set_nat,D: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ C2 )
=> ( ( bij_be3438014552859920132et_nat @ G3 @ B @ D )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
=> ( ( ( inf_inf_set_set_nat @ C2 @ D )
= bot_bot_set_set_nat )
=> ( bij_be3438014552859920132et_nat
@ ^ [X2: set_nat] : ( if_set_nat @ ( member_set_nat @ X2 @ A2 ) @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_set_nat @ A2 @ B )
@ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_375_bij__betw__partition,axiom,
! [F: nat > nat,A2: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( bij_betw_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) )
=> ( ( bij_betw_nat_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ A2 @ C2 )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_376_bij__betw__partition,axiom,
! [F: a > set_nat,A2: set_a,C2: set_a,B: set_set_nat,D: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ ( sup_sup_set_a @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) )
=> ( ( bij_betw_a_set_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_a @ A2 @ C2 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_betw_a_set_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_377_bij__betw__partition,axiom,
! [F: set_nat > nat,A2: set_set_nat,C2: set_set_nat,B: set_nat,D: set_nat] :
( ( bij_betw_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) )
=> ( ( bij_betw_set_nat_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ A2 @ C2 )
= bot_bot_set_set_nat )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_betw_set_nat_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_378_bij__betw__partition,axiom,
! [F: nat > set_nat,A2: set_nat,C2: set_nat,B: set_set_nat,D: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) )
=> ( ( bij_betw_nat_set_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ A2 @ C2 )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_betw_nat_set_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_379_bij__betw__partition,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,C2: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) )
=> ( ( bij_be3438014552859920132et_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ A2 @ C2 )
= bot_bot_set_set_nat )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_be3438014552859920132et_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_380_bij__betw__partition,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,C2: set_nat_nat,B: set_nat,D: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ ( sup_sup_set_nat_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) )
=> ( ( bij_betw_nat_nat_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat_nat @ A2 @ C2 )
= bot_bot_set_nat_nat )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_381_bij__betw__partition,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,C2: set_set_set_nat,B: set_nat,D: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) )
=> ( ( bij_be6199415091885040644at_nat @ F @ C2 @ D )
=> ( ( ( inf_in5711780100303410308et_nat @ A2 @ C2 )
= bot_bo7198184520161983622et_nat )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_be6199415091885040644at_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_382_bij__betw__partition,axiom,
! [F: nat > nat > nat,A2: set_nat,C2: set_nat,B: set_nat_nat,D: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat_nat @ B @ D ) )
=> ( ( bij_betw_nat_nat_nat2 @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ A2 @ C2 )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat_nat @ B @ D )
= bot_bot_set_nat_nat )
=> ( bij_betw_nat_nat_nat2 @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_383_bij__betw__partition,axiom,
! [F: nat > set_set_nat,A2: set_nat,C2: set_nat,B: set_set_set_nat,D: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_su4213647025997063966et_nat @ B @ D ) )
=> ( ( bij_be6938610931847138308et_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ A2 @ C2 )
= bot_bot_set_nat )
=> ( ( ( inf_in5711780100303410308et_nat @ B @ D )
= bot_bo7198184520161983622et_nat )
=> ( bij_be6938610931847138308et_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_384_bij__betw__partition,axiom,
! [F: set_nat > nat > nat,A2: set_set_nat,C2: set_set_nat,B: set_nat_nat,D: set_nat_nat] :
( ( bij_be3458689793592806333at_nat @ F @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat_nat @ B @ D ) )
=> ( ( bij_be3458689793592806333at_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ A2 @ C2 )
= bot_bot_set_set_nat )
=> ( ( ( inf_inf_set_nat_nat @ B @ D )
= bot_bot_set_nat_nat )
=> ( bij_be3458689793592806333at_nat @ F @ A2 @ B ) ) ) ) ) ).
% bij_betw_partition
thf(fact_385_bij__betw__combine,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( bij_betw_nat_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_386_bij__betw__combine,axiom,
! [F: a > set_nat,A2: set_a,B: set_set_nat,C2: set_a,D: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A2 @ B )
=> ( ( bij_betw_a_set_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_betw_a_set_nat @ F @ ( sup_sup_set_a @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_387_bij__betw__combine,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_set_nat,C2: set_nat,D: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B )
=> ( ( bij_betw_nat_set_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_betw_nat_set_nat @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_388_bij__betw__combine,axiom,
! [F: set_nat > nat,A2: set_set_nat,B: set_nat,C2: set_set_nat,D: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B )
=> ( ( bij_betw_set_nat_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_betw_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_389_bij__betw__combine,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B: set_set_nat,C2: set_set_nat,D: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B )
=> ( ( bij_be3438014552859920132et_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_be3438014552859920132et_nat @ F @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_390_bij__betw__combine,axiom,
! [F: nat > nat > nat,A2: set_nat,B: set_nat_nat,C2: set_nat,D: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B )
=> ( ( bij_betw_nat_nat_nat2 @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat_nat @ B @ D )
= bot_bot_set_nat_nat )
=> ( bij_betw_nat_nat_nat2 @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_391_bij__betw__combine,axiom,
! [F: nat > set_set_nat,A2: set_nat,B: set_set_set_nat,C2: set_nat,D: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B )
=> ( ( bij_be6938610931847138308et_nat @ F @ C2 @ D )
=> ( ( ( inf_in5711780100303410308et_nat @ B @ D )
= bot_bo7198184520161983622et_nat )
=> ( bij_be6938610931847138308et_nat @ F @ ( sup_sup_set_nat @ A2 @ C2 ) @ ( sup_su4213647025997063966et_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_392_bij__betw__combine,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B: set_nat,C2: set_set_set_nat,D: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B )
=> ( ( bij_be6199415091885040644at_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_be6199415091885040644at_nat @ F @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_393_bij__betw__combine,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B: set_nat,C2: set_nat_nat,D: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B )
=> ( ( bij_betw_nat_nat_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_nat @ B @ D )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat_nat @ F @ ( sup_sup_set_nat_nat @ A2 @ C2 ) @ ( sup_sup_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_394_bij__betw__combine,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B: set_set_nat,C2: set_set_set_nat,D: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B )
=> ( ( bij_be4885122793727115194et_nat @ F @ C2 @ D )
=> ( ( ( inf_inf_set_set_nat @ B @ D )
= bot_bot_set_set_nat )
=> ( bij_be4885122793727115194et_nat @ F @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B @ D ) ) ) ) ) ).
% bij_betw_combine
thf(fact_395_UnE,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ ( sup_su7438456061012554424mula_a @ A2 @ B ) )
=> ( ~ ( member535913909593306477mula_a @ C @ A2 )
=> ( member535913909593306477mula_a @ C @ B ) ) ) ).
% UnE
thf(fact_396_UnE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_397_UnE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
=> ( ~ ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_398_UnE,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) )
=> ( ~ ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_399_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_400_UnE,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) )
=> ( ~ ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B ) ) ) ).
% UnE
thf(fact_401_IntE,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ ( inf_in4741988911529734942mula_a @ A2 @ B ) )
=> ~ ( ( member535913909593306477mula_a @ C @ A2 )
=> ~ ( member535913909593306477mula_a @ C @ B ) ) ) ).
% IntE
thf(fact_402_IntE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_403_IntE,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ~ ( member_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_404_IntE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A2 )
=> ~ ( member_set_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_405_IntE,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ~ ( member_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_406_UnI1,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ A2 )
=> ( member535913909593306477mula_a @ C @ ( sup_su7438456061012554424mula_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_407_UnI1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_408_UnI1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_409_UnI1,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_410_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_411_UnI1,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_412_UnI2,axiom,
! [C: monotone_mformula_a,B: set_Mo2626137824023173004mula_a,A2: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ B )
=> ( member535913909593306477mula_a @ C @ ( sup_su7438456061012554424mula_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_413_UnI2,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_414_UnI2,axiom,
! [C: set_set_nat,B: set_set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_415_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_416_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_417_UnI2,axiom,
! [C: nat > nat,B: set_nat_nat,A2: set_nat_nat] :
( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_418_IntD1,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ ( inf_in4741988911529734942mula_a @ A2 @ B ) )
=> ( member535913909593306477mula_a @ C @ A2 ) ) ).
% IntD1
thf(fact_419_IntD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_420_IntD1,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_421_IntD1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
=> ( member_set_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_422_IntD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_423_IntD2,axiom,
! [C: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( member535913909593306477mula_a @ C @ ( inf_in4741988911529734942mula_a @ A2 @ B ) )
=> ( member535913909593306477mula_a @ C @ B ) ) ).
% IntD2
thf(fact_424_IntD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_425_IntD2,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
=> ( member_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_426_IntD2,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
=> ( member_set_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_427_IntD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ( member_nat_nat @ C @ B ) ) ).
% IntD2
thf(fact_428_bex__Un,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_429_bex__Un,axiom,
! [A2: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_430_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_431_bex__Un,axiom,
! [A2: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_432_Bex__def,axiom,
( bex_Mo775092102567092726mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,P2: monotone_mformula_a > $o] :
? [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_433_Bex__def,axiom,
( bex_a
= ( ^ [A3: set_a,P2: a > $o] :
? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_434_Bex__def,axiom,
( bex_set_nat
= ( ^ [A3: set_set_nat,P2: set_nat > $o] :
? [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_435_Bex__def,axiom,
( bex_set_set_nat
= ( ^ [A3: set_set_set_nat,P2: set_set_nat > $o] :
? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_436_Bex__def,axiom,
( bex_nat_nat
= ( ^ [A3: set_nat_nat,P2: ( nat > nat ) > $o] :
? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_437_ball__Un,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_438_ball__Un,axiom,
! [A2: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_439_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_440_ball__Un,axiom,
! [A2: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_441_Un__assoc,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ C2 )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_442_Un__assoc,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ C2 )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_443_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_444_Un__assoc,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ C2 )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_445_bij__betwE,axiom,
! [F: a > set_nat,A2: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A2 @ B )
=> ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B ) ) ) ).
% bij_betwE
thf(fact_446_Int__assoc,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_447_Int__assoc,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_448_Un__absorb,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_449_Un__absorb,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_450_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_451_Un__absorb,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_452_Int__absorb,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_453_Int__absorb,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_454_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_455_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] : ( sup_sup_set_set_nat @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_456_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] : ( sup_sup_set_nat @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_457_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] : ( sup_sup_set_nat_nat @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_458_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B2 @ A3 ) ) ) ).
% Int_commute
thf(fact_459_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] : ( inf_inf_set_nat_nat @ B2 @ A3 ) ) ) ).
% Int_commute
thf(fact_460_bij__betw__inv,axiom,
! [F: set_nat > a,A2: set_set_nat,B: set_a] :
( ( bij_betw_set_nat_a @ F @ A2 @ B )
=> ? [G4: a > set_nat] : ( bij_betw_a_set_nat @ G4 @ B @ A2 ) ) ).
% bij_betw_inv
thf(fact_461_bij__betw__inv,axiom,
! [F: a > set_nat,A2: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A2 @ B )
=> ? [G4: set_nat > a] : ( bij_betw_set_nat_a @ G4 @ B @ A2 ) ) ).
% bij_betw_inv
thf(fact_462_Un__Int__crazy,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) @ ( inf_in5711780100303410308et_nat @ C2 @ A2 ) )
= ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) @ ( sup_su4213647025997063966et_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_463_Un__Int__crazy,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ ( inf_inf_set_set_nat @ B @ C2 ) ) @ ( inf_inf_set_set_nat @ C2 @ A2 ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ ( sup_sup_set_set_nat @ B @ C2 ) ) @ ( sup_sup_set_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_464_Un__Int__crazy,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A2 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_465_Un__Int__crazy,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ ( inf_inf_set_nat_nat @ B @ C2 ) ) @ ( inf_inf_set_nat_nat @ C2 @ A2 ) )
= ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ ( sup_sup_set_nat_nat @ B @ C2 ) ) @ ( sup_sup_set_nat_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_466_bij__betw__ball,axiom,
! [F: a > set_nat,A2: set_a,B: set_set_nat,Phi5: set_nat > $o] :
( ( bij_betw_a_set_nat @ F @ A2 @ B )
=> ( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( Phi5 @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( Phi5 @ ( F @ X2 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_467_bij__betw__cong,axiom,
! [A2: set_a,F: a > set_nat,G3: a > set_nat,A5: set_set_nat] :
( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G3 @ A4 ) ) )
=> ( ( bij_betw_a_set_nat @ F @ A2 @ A5 )
= ( bij_betw_a_set_nat @ G3 @ A2 @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_468_bij__betw__apply,axiom,
! [F: a > a,A2: set_a,B: set_a,A: a] :
( ( bij_betw_a_a @ F @ A2 @ B )
=> ( ( member_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_469_bij__betw__apply,axiom,
! [F: monotone_mformula_a > a,A2: set_Mo2626137824023173004mula_a,B: set_a,A: monotone_mformula_a] :
( ( bij_be7751601365550188976la_a_a @ F @ A2 @ B )
=> ( ( member535913909593306477mula_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_470_bij__betw__apply,axiom,
! [F: a > monotone_mformula_a,A2: set_a,B: set_Mo2626137824023173004mula_a,A: a] :
( ( bij_be1655973440275287390mula_a @ F @ A2 @ B )
=> ( ( member_a @ A @ A2 )
=> ( member535913909593306477mula_a @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_471_bij__betw__apply,axiom,
! [F: set_nat > a,A2: set_set_nat,B: set_a,A: set_nat] :
( ( bij_betw_set_nat_a @ F @ A2 @ B )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_a @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_472_bij__betw__apply,axiom,
! [F: a > set_nat,A2: set_a,B: set_set_nat,A: a] :
( ( bij_betw_a_set_nat @ F @ A2 @ B )
=> ( ( member_a @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_473_bij__betw__apply,axiom,
! [F: monotone_mformula_a > monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,A: monotone_mformula_a] :
( ( bij_be6841836389319334180mula_a @ F @ A2 @ B )
=> ( ( member535913909593306477mula_a @ A @ A2 )
=> ( member535913909593306477mula_a @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_474_bij__betw__apply,axiom,
! [F: monotone_mformula_a > set_nat,A2: set_Mo2626137824023173004mula_a,B: set_set_nat,A: monotone_mformula_a] :
( ( bij_be1996446448813045844et_nat @ F @ A2 @ B )
=> ( ( member535913909593306477mula_a @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_475_bij__betw__apply,axiom,
! [F: a > set_set_nat,A2: set_a,B: set_set_set_nat,A: a] :
( ( bij_be2639851105560558660et_nat @ F @ A2 @ B )
=> ( ( member_a @ A @ A2 )
=> ( member_set_set_nat @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_476_bij__betw__apply,axiom,
! [F: a > nat > nat,A2: set_a,B: set_nat_nat,A: a] :
( ( bij_betw_a_nat_nat @ F @ A2 @ B )
=> ( ( member_a @ A @ A2 )
=> ( member_nat_nat @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_477_bij__betw__apply,axiom,
! [F: set_nat > monotone_mformula_a,A2: set_set_nat,B: set_Mo2626137824023173004mula_a,A: set_nat] :
( ( bij_be3858092191235522516mula_a @ F @ A2 @ B )
=> ( ( member_set_nat @ A @ A2 )
=> ( member535913909593306477mula_a @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_478_Int__Un__distrib,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_479_Int__Un__distrib,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ ( inf_inf_set_set_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_480_Int__Un__distrib,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_481_Int__Un__distrib,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ ( inf_inf_set_nat_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_482_Un__Int__distrib,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_483_Un__Int__distrib,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ ( sup_sup_set_set_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_484_Un__Int__distrib,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_485_Un__Int__distrib,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ ( sup_sup_set_nat_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_486_Un__left__absorb,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
= ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_487_Un__left__absorb,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_488_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_489_Un__left__absorb,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B ) )
= ( sup_sup_set_nat_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_490_Int__Un__distrib2,axiom,
! [B: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ C2 ) @ A2 )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ A2 ) @ ( inf_in5711780100303410308et_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_491_Int__Un__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ A2 ) @ ( inf_inf_set_set_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_492_Int__Un__distrib2,axiom,
! [B: set_nat,C2: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A2 ) @ ( inf_inf_set_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_493_Int__Un__distrib2,axiom,
! [B: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ C2 ) @ A2 )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ A2 ) @ ( inf_inf_set_nat_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_494_Int__left__absorb,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_495_Int__left__absorb,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A2 @ B ) )
= ( inf_inf_set_nat_nat @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_496_Un__Int__distrib2,axiom,
! [B: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ C2 ) @ A2 )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ A2 ) @ ( sup_su4213647025997063966et_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_497_Un__Int__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ C2 ) @ A2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ A2 ) @ ( sup_sup_set_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_498_Un__Int__distrib2,axiom,
! [B: set_nat,C2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C2 ) @ A2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A2 ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_499_Un__Int__distrib2,axiom,
! [B: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ C2 ) @ A2 )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ A2 ) @ ( sup_sup_set_nat_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_500_Un__left__commute,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ B @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_501_Un__left__commute,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_502_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_503_Un__left__commute,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ B @ ( sup_sup_set_nat_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_504_Int__left__commute,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ B @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_505_Int__left__commute,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ B @ ( inf_inf_set_nat_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_506_bij__betw__iff__bijections,axiom,
( bij_betw_a_a
= ( ^ [F2: a > a,A3: set_a,B2: set_a] :
? [G5: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( member_a @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_507_bij__betw__iff__bijections,axiom,
( bij_be1655973440275287390mula_a
= ( ^ [F2: a > monotone_mformula_a,A3: set_a,B2: set_Mo2626137824023173004mula_a] :
? [G5: monotone_mformula_a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member535913909593306477mula_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ B2 )
=> ( ( member_a @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_508_bij__betw__iff__bijections,axiom,
( bij_be7751601365550188976la_a_a
= ( ^ [F2: monotone_mformula_a > a,A3: set_Mo2626137824023173004mula_a,B2: set_a] :
? [G5: a > monotone_mformula_a] :
( ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
=> ( ( member_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( member535913909593306477mula_a @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_509_bij__betw__iff__bijections,axiom,
( bij_betw_set_nat_a
= ( ^ [F2: set_nat > a,A3: set_set_nat,B2: set_a] :
? [G5: a > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( member_set_nat @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_510_bij__betw__iff__bijections,axiom,
( bij_betw_a_set_nat
= ( ^ [F2: a > set_nat,A3: set_a,B2: set_set_nat] :
? [G5: set_nat > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_set_nat @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B2 )
=> ( ( member_a @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_511_bij__betw__iff__bijections,axiom,
( bij_be6841836389319334180mula_a
= ( ^ [F2: monotone_mformula_a > monotone_mformula_a,A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
? [G5: monotone_mformula_a > monotone_mformula_a] :
( ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
=> ( ( member535913909593306477mula_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ B2 )
=> ( ( member535913909593306477mula_a @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_512_bij__betw__iff__bijections,axiom,
( bij_be3858092191235522516mula_a
= ( ^ [F2: set_nat > monotone_mformula_a,A3: set_set_nat,B2: set_Mo2626137824023173004mula_a] :
? [G5: monotone_mformula_a > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member535913909593306477mula_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ B2 )
=> ( ( member_set_nat @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_513_bij__betw__iff__bijections,axiom,
( bij_be3032674665972365258_nat_a
= ( ^ [F2: set_set_nat > a,A3: set_set_set_nat,B2: set_a] :
? [G5: a > set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( ( member_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( member_set_set_nat @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_514_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat_a
= ( ^ [F2: ( nat > nat ) > a,A3: set_nat_nat,B2: set_a] :
? [G5: a > nat > nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( ( member_a @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( member_nat_nat @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_515_bij__betw__iff__bijections,axiom,
( bij_be1996446448813045844et_nat
= ( ^ [F2: monotone_mformula_a > set_nat,A3: set_Mo2626137824023173004mula_a,B2: set_set_nat] :
? [G5: set_nat > monotone_mformula_a] :
( ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
=> ( ( member_set_nat @ ( F2 @ X2 ) @ B2 )
& ( ( G5 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B2 )
=> ( ( member535913909593306477mula_a @ ( G5 @ X2 ) @ A3 )
& ( ( F2 @ ( G5 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_516_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_517_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B2: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
| ( member_a @ X2 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_518_Un__def,axiom,
( sup_su7438456061012554424mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
| ( member535913909593306477mula_a @ X2 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_519_Un__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
| ( member_set_set_nat @ X2 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_520_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
| ( member_set_nat @ X2 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_521_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
| ( member_nat @ X2 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_522_Un__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
| ( member_nat_nat @ X2 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_523_Collect__disj__eq,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su7438456061012554424mula_a @ ( collec4794253742848188331mula_a @ P ) @ ( collec4794253742848188331mula_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_524_Collect__disj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su4213647025997063966et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_525_Collect__disj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_526_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_527_Collect__disj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_528_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B2: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( member_a @ X2 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_529_Int__def,axiom,
( inf_in4741988911529734942mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A3 )
& ( member535913909593306477mula_a @ X2 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_530_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( member_nat @ X2 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_531_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
& ( member_set_nat @ X2 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_532_Int__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
& ( member_set_set_nat @ X2 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_533_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
& ( member_nat_nat @ X2 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_534_Int__Collect,axiom,
! [X: a,A2: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_535_Int__Collect,axiom,
! [X: monotone_mformula_a,A2: set_Mo2626137824023173004mula_a,P: monotone_mformula_a > $o] :
( ( member535913909593306477mula_a @ X @ ( inf_in4741988911529734942mula_a @ A2 @ ( collec4794253742848188331mula_a @ P ) ) )
= ( ( member535913909593306477mula_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_536_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_537_Int__Collect,axiom,
! [X: set_nat,A2: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_538_Int__Collect,axiom,
! [X: set_set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ X @ ( inf_in5711780100303410308et_nat @ A2 @ ( collect_set_set_nat @ P ) ) )
= ( ( member_set_set_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_539_Int__Collect,axiom,
! [X: nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X @ ( inf_inf_set_nat_nat @ A2 @ ( collect_nat_nat @ P ) ) )
= ( ( member_nat_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_540_Collect__conj__eq,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_in4741988911529734942mula_a @ ( collec4794253742848188331mula_a @ P ) @ ( collec4794253742848188331mula_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_541_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_542_Collect__conj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_543_Collect__conj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_in5711780100303410308et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_544_Collect__conj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_545_sameprod__finite,axiom,
! [X4: set_set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_546_sameprod__finite,axiom,
! [X4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X4 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_547_sameprod__finite,axiom,
! [X4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X4 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_548_sameprod__finite,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_549_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_550_bij__betw__empty2,axiom,
! [F: a > set_nat,A2: set_a] :
( ( bij_betw_a_set_nat @ F @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_a ) ) ).
% bij_betw_empty2
thf(fact_551_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_552_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_553_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_554_bij__betw__empty2,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ bot_bot_set_nat_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_555_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_556_bij__betw__empty2,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bij_betw_empty2
thf(fact_557_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_558_bij__betw__empty2,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( bij_be2321430536510320189et_nat @ F @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bij_betw_empty2
thf(fact_559_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_560_bij__betw__empty1,axiom,
! [F: a > set_nat,A2: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ bot_bot_set_a @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_561_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_562_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_563_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_564_bij__betw__empty1,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ bot_bot_set_nat_nat @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_565_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_566_bij__betw__empty1,axiom,
! [F: nat > nat > nat,A2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ bot_bot_set_nat @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bij_betw_empty1
thf(fact_567_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_568_bij__betw__empty1,axiom,
! [F: set_nat > nat > nat,A2: set_nat_nat] :
( ( bij_be3458689793592806333at_nat @ F @ bot_bot_set_set_nat @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bij_betw_empty1
thf(fact_569_bij__betw__imp__inj__on,axiom,
! [F: a > set_nat,A2: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A2 @ B )
=> ( inj_on_a_set_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_570_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_571_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_572_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_573_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_574_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_575_subset__UnE,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
=> ~ ! [A6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A6 @ A2 )
=> ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ B )
=> ( C2
!= ( sup_su4213647025997063966et_nat @ A6 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_576_subset__UnE,axiom,
! [C2: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B ) )
=> ~ ! [A6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ A2 )
=> ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ B )
=> ( C2
!= ( sup_sup_set_set_nat @ A6 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_577_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A2 )
=> ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_578_subset__UnE,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B ) )
=> ~ ! [A6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ A2 )
=> ! [B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ B )
=> ( C2
!= ( sup_sup_set_nat_nat @ A6 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_579_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A2 )
=> ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ B )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_580_Un__absorb2,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_581_Un__absorb2,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_582_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_583_Un__absorb2,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_584_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_585_Un__absorb1,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_586_Un__absorb1,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( sup_sup_set_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_587_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_588_Un__absorb1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( sup_sup_set_nat_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_589_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_590_Un__upper2,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_591_Un__upper2,axiom,
! [B: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_592_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_593_Un__upper2,axiom,
! [B: set_nat_nat,A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_594_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_595_Un__upper1,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_596_Un__upper1,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_597_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_598_Un__upper1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_599_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_600_Un__least,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_601_Un__least,axiom,
! [A2: set_set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_602_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_603_Un__least,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_604_Un__least,axiom,
! [A2: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_605_Un__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ ( sup_su4213647025997063966et_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_606_Un__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_607_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_608_Un__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ ( sup_sup_set_nat_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_609_Un__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_610_Un__empty__right,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_611_Un__empty__right,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% Un_empty_right
thf(fact_612_Un__empty__right,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% Un_empty_right
thf(fact_613_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_614_Un__empty__left,axiom,
! [B: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_615_Un__empty__left,axiom,
! [B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_616_Un__empty__left,axiom,
! [B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_617_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_618_Int__Collect__mono,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ( ord_le5054881893329012716mula_a @ A2 @ B )
=> ( ! [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le5054881893329012716mula_a @ ( inf_in4741988911529734942mula_a @ A2 @ ( collec4794253742848188331mula_a @ P ) ) @ ( inf_in4741988911529734942mula_a @ B @ ( collec4794253742848188331mula_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_619_Int__Collect__mono,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ ( collect_set_set_nat @ P ) ) @ ( inf_in5711780100303410308et_nat @ B @ ( collect_set_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_620_Int__Collect__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_621_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_622_Int__Collect__mono,axiom,
! [A2: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_623_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_624_Int__greatest,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ B )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_625_Int__greatest,axiom,
! [C2: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_626_Int__greatest,axiom,
! [C2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_627_Int__greatest,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_628_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_629_Int__absorb2,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_630_Int__absorb2,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( inf_inf_set_set_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_631_Int__absorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_632_Int__absorb2,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_633_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_634_Int__absorb1,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_635_Int__absorb1,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_636_Int__absorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_637_Int__absorb1,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_638_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_639_Int__lower2,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_640_Int__lower2,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_641_Int__lower2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_642_Int__lower2,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_643_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_644_Int__lower1,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_645_Int__lower1,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_646_Int__lower1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_647_Int__lower1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_648_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_649_Int__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ ( inf_in5711780100303410308et_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_650_Int__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ ( inf_inf_set_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_651_Int__mono,axiom,
! [A2: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_652_Int__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ ( inf_inf_set_nat_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_653_Int__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_654_disjoint__iff__not__equal,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ! [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B )
=> ( X2 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_655_disjoint__iff__not__equal,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A2 @ B )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ! [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ B )
=> ( X2 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_656_disjoint__iff__not__equal,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ! [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ B )
=> ( X2 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_657_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ B )
=> ( X2 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_658_Int__empty__right,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_659_Int__empty__right,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% Int_empty_right
thf(fact_660_Int__empty__right,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_right
thf(fact_661_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_662_Int__empty__left,axiom,
! [B: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_663_Int__empty__left,axiom,
! [B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ B )
= bot_bot_set_nat_nat ) ).
% Int_empty_left
thf(fact_664_Int__empty__left,axiom,
! [B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ B )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_left
thf(fact_665_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_666_disjoint__iff,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ( ( inf_in4741988911529734942mula_a @ A2 @ B )
= bot_bo3042613601904376864mula_a )
= ( ! [X2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X2 @ A2 )
=> ~ ( member535913909593306477mula_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_667_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_668_disjoint__iff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ( member_set_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_669_disjoint__iff,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A2 @ B )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ~ ( member_nat_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_670_disjoint__iff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ~ ( member_set_set_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_671_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_672_Int__emptyI,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
( ! [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ A2 )
=> ~ ( member535913909593306477mula_a @ X5 @ B ) )
=> ( ( inf_in4741988911529734942mula_a @ A2 @ B )
= bot_bo3042613601904376864mula_a ) ) ).
% Int_emptyI
thf(fact_673_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ~ ( member_a @ X5 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_674_Int__emptyI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ~ ( member_set_nat @ X5 @ B ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_675_Int__emptyI,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A2 )
=> ~ ( member_nat_nat @ X5 @ B ) )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_676_Int__emptyI,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
=> ~ ( member_set_set_nat @ X5 @ B ) )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat ) ) ).
% Int_emptyI
thf(fact_677_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ~ ( member_nat @ X5 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_678_mformula_Odistinct_I5_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a] :
( monotone_TRUE_a
!= ( monotone_Conj_a @ X41 @ X42 ) ) ).
% mformula.distinct(5)
thf(fact_679_mformula_Odistinct_I7_J,axiom,
! [X51: monotone_mformula_a,X52: monotone_mformula_a] :
( monotone_TRUE_a
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(7)
thf(fact_680_mformula_Odistinct_I3_J,axiom,
! [X32: a] :
( monotone_TRUE_a
!= ( monotone_Var_a @ X32 ) ) ).
% mformula.distinct(3)
thf(fact_681_mformula_Odistinct_I1_J,axiom,
monotone_TRUE_a != monotone_FALSE_a ).
% mformula.distinct(1)
thf(fact_682_inj__on__Int,axiom,
! [F: a > set_nat,A2: set_a,B: set_a] :
( ( ( inj_on_a_set_nat @ F @ A2 )
| ( inj_on_a_set_nat @ F @ B ) )
=> ( inj_on_a_set_nat @ F @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% inj_on_Int
thf(fact_683_eval_Osimps_I2_J,axiom,
! [Theta2: a > $o] : ( monotone_eval_a @ Theta2 @ monotone_TRUE_a ) ).
% eval.simps(2)
thf(fact_684_vars_Osimps_I5_J,axiom,
( ( monoto8378391831928444664et_nat @ monoto7549873196617247779et_nat )
= bot_bot_set_set_nat ) ).
% vars.simps(5)
thf(fact_685_vars_Osimps_I5_J,axiom,
( ( monoto4799612099597528497at_nat @ monoto581402444256252508at_nat )
= bot_bot_set_nat_nat ) ).
% vars.simps(5)
thf(fact_686_vars_Osimps_I5_J,axiom,
( ( monoto3765512064276419502et_nat @ monoto5104785069271071961et_nat )
= bot_bo7198184520161983622et_nat ) ).
% vars.simps(5)
thf(fact_687_vars_Osimps_I5_J,axiom,
( ( monotone_vars_nat @ monotone_TRUE_nat )
= bot_bot_set_nat ) ).
% vars.simps(5)
thf(fact_688_vars_Osimps_I5_J,axiom,
( ( monotone_vars_a @ monotone_TRUE_a )
= bot_bot_set_a ) ).
% vars.simps(5)
thf(fact_689_vars_Osimps_I2_J,axiom,
! [Phi: monoto5483634261523599098et_nat,Psi: monoto5483634261523599098et_nat] :
( ( monoto3765512064276419502et_nat @ ( monoto78699555928797203et_nat @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi ) @ ( monoto3765512064276419502et_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_690_vars_Osimps_I2_J,axiom,
! [Phi: monoto7244996872745832772et_nat,Psi: monoto7244996872745832772et_nat] :
( ( monoto8378391831928444664et_nat @ ( monoto3675431328128845661et_nat @ Phi @ Psi ) )
= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi ) @ ( monoto8378391831928444664et_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_691_vars_Osimps_I2_J,axiom,
! [Phi: monoto4181647612830706830la_nat,Psi: monoto4181647612830706830la_nat] :
( ( monotone_vars_nat @ ( monotone_Conj_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi ) @ ( monotone_vars_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_692_vars_Osimps_I2_J,axiom,
! [Phi: monoto8276428299528460797at_nat,Psi: monoto8276428299528460797at_nat] :
( ( monoto4799612099597528497at_nat @ ( monoto3849437543655978646at_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi ) @ ( monoto4799612099597528497at_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_693_vars_Osimps_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_vars_a @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( sup_sup_set_a @ ( monotone_vars_a @ Phi ) @ ( monotone_vars_a @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_694_vars_Osimps_I3_J,axiom,
! [Phi: monoto5483634261523599098et_nat,Psi: monoto5483634261523599098et_nat] :
( ( monoto3765512064276419502et_nat @ ( monoto5378078033476056327et_nat @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi ) @ ( monoto3765512064276419502et_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_695_vars_Osimps_I3_J,axiom,
! [Phi: monoto7244996872745832772et_nat,Psi: monoto7244996872745832772et_nat] :
( ( monoto8378391831928444664et_nat @ ( monoto2996447309290675281et_nat @ Phi @ Psi ) )
= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi ) @ ( monoto8378391831928444664et_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_696_vars_Osimps_I3_J,axiom,
! [Phi: monoto4181647612830706830la_nat,Psi: monoto4181647612830706830la_nat] :
( ( monotone_vars_nat @ ( monotone_Disj_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi ) @ ( monotone_vars_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_697_vars_Osimps_I3_J,axiom,
! [Phi: monoto8276428299528460797at_nat,Psi: monoto8276428299528460797at_nat] :
( ( monoto4799612099597528497at_nat @ ( monoto6252637820927860106at_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi ) @ ( monoto4799612099597528497at_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_698_vars_Osimps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_vars_a @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( sup_sup_set_a @ ( monotone_vars_a @ Phi ) @ ( monotone_vars_a @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_699_to__tf__formula_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ( X != monotone_TRUE_a )
=> ( ( X != monotone_FALSE_a )
=> ~ ! [V2: a] :
( X
!= ( monotone_Var_a @ V2 ) ) ) ) ) ) ).
% to_tf_formula.cases
thf(fact_700_mformula_Oexhaust,axiom,
! [Y2: monotone_mformula_a] :
( ( Y2 != monotone_TRUE_a )
=> ( ( Y2 != monotone_FALSE_a )
=> ( ! [X33: a] :
( Y2
!= ( monotone_Var_a @ X33 ) )
=> ( ! [X412: monotone_mformula_a,X422: monotone_mformula_a] :
( Y2
!= ( monotone_Conj_a @ X412 @ X422 ) )
=> ~ ! [X512: monotone_mformula_a,X522: monotone_mformula_a] :
( Y2
!= ( monotone_Disj_a @ X512 @ X522 ) ) ) ) ) ) ).
% mformula.exhaust
thf(fact_701_vars_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [X5: a] :
( X
!= ( monotone_Var_a @ X5 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ( X != monotone_FALSE_a )
=> ( X = monotone_TRUE_a ) ) ) ) ) ).
% vars.cases
thf(fact_702_SUB_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ! [X5: a] :
( X
!= ( monotone_Var_a @ X5 ) )
=> ( ( X != monotone_FALSE_a )
=> ( X = monotone_TRUE_a ) ) ) ) ) ).
% SUB.cases
thf(fact_703_eval_Oelims_I2_J,axiom,
! [X: a > $o,Xa: monotone_mformula_a] :
( ( monotone_eval_a @ X @ Xa )
=> ( ( Xa != monotone_TRUE_a )
=> ( ! [X5: a] :
( ( Xa
= ( monotone_Var_a @ X5 ) )
=> ~ ( X @ X5 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( Xa
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ~ ( ( monotone_eval_a @ X @ Phi3 )
| ( monotone_eval_a @ X @ Psi2 ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( Xa
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ~ ( ( monotone_eval_a @ X @ Phi3 )
& ( monotone_eval_a @ X @ Psi2 ) ) ) ) ) ) ) ).
% eval.elims(2)
thf(fact_704_eval_Oelims_I1_J,axiom,
! [X: a > $o,Xa: monotone_mformula_a,Y2: $o] :
( ( ( monotone_eval_a @ X @ Xa )
= Y2 )
=> ( ( ( Xa = monotone_FALSE_a )
=> Y2 )
=> ( ( ( Xa = monotone_TRUE_a )
=> ~ Y2 )
=> ( ! [X5: a] :
( ( Xa
= ( monotone_Var_a @ X5 ) )
=> ( Y2
= ( ~ ( X @ X5 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( Xa
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( Y2
= ( ~ ( ( monotone_eval_a @ X @ Phi3 )
| ( monotone_eval_a @ X @ Psi2 ) ) ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( Xa
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( Y2
= ( ~ ( ( monotone_eval_a @ X @ Phi3 )
& ( monotone_eval_a @ X @ Psi2 ) ) ) ) ) ) ) ) ) ) ).
% eval.elims(1)
thf(fact_705_ACC__cf___092_060F_062,axiom,
! [X4: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_706_finite__Collect__subsets,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_707_finite__Collect__subsets,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_708_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_709_finite__Collect__subsets,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( finite3586981331298542604at_nat
@ ( collect_set_nat_nat
@ ^ [B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_710_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_711_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_712__092_060K_062__altdef,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
& ( ( finite_card_nat @ V )
= k ) ) ) ) ).
% \<K>_altdef
thf(fact_713_finite__v__gs,axiom,
! [X4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ).
% finite_v_gs
thf(fact_714_finite__Collect__bex,axiom,
! [A2: set_nat,Q: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y6: nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_715_finite__Collect__bex,axiom,
! [A2: set_nat,Q: monotone_mformula_a > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Y6: monotone_mformula_a] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_716_finite__Collect__bex,axiom,
! [A2: set_nat,Q: set_nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Y6: set_nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_717_finite__Collect__bex,axiom,
! [A2: set_set_nat,Q: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y6: nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_718_finite__Collect__bex,axiom,
! [A2: set_nat,Q: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [Y6: nat > nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_719_finite__Collect__bex,axiom,
! [A2: set_nat,Q: set_set_nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Y6: set_set_nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_720_finite__Collect__bex,axiom,
! [A2: set_set_nat,Q: monotone_mformula_a > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Y6: monotone_mformula_a] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_721_finite__Collect__bex,axiom,
! [A2: set_set_nat,Q: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Y6: set_nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_722_finite__Collect__bex,axiom,
! [A2: set_nat_nat,Q: nat > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y6: nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_723_finite__Collect__bex,axiom,
! [A2: set_set_set_nat,Q: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ A2 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y6: nat] : ( Q @ Y6 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_724_finite__Un,axiom,
! [F3: set_set_set_nat,G: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F3 @ G ) )
= ( ( finite6739761609112101331et_nat @ F3 )
& ( finite6739761609112101331et_nat @ G ) ) ) ).
% finite_Un
thf(fact_725_finite__Un,axiom,
! [F3: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F3 @ G ) )
= ( ( finite1152437895449049373et_nat @ F3 )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_726_finite__Un,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_727_finite__Un,axiom,
! [F3: set_nat_nat,G: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F3 @ G ) )
= ( ( finite2115694454571419734at_nat @ F3 )
& ( finite2115694454571419734at_nat @ G ) ) ) ).
% finite_Un
thf(fact_728_sup__inf__absorb,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_729_sup__inf__absorb,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_730_sup__inf__absorb,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_731_sup__inf__absorb,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_732_inf__sup__absorb,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_733_inf__sup__absorb,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_734_inf__sup__absorb,axiom,
! [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_735_inf__sup__absorb,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_736_v__union,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_737_v__gs__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_union
thf(fact_738_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_739_v__gs__mono,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_mono
thf(fact_740_ACC__cf__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_union
thf(fact_741_ACC__cf__odot,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_odot
thf(fact_742_inf_Oidem,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_743_inf_Oidem,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_744_inf__idem,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_745_inf__idem,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_746_inf_Oleft__idem,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ A @ B4 ) )
= ( inf_in5711780100303410308et_nat @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_747_inf_Oleft__idem,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ A @ B4 ) )
= ( inf_inf_set_nat_nat @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_748_inf__left__idem,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_749_inf__left__idem,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y2 ) )
= ( inf_inf_set_nat_nat @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_750_inf_Oright__idem,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ B4 )
= ( inf_in5711780100303410308et_nat @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_751_inf_Oright__idem,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ B4 )
= ( inf_inf_set_nat_nat @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_752_inf__right__idem,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Y2 )
= ( inf_in5711780100303410308et_nat @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_753_inf__right__idem,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_nat_nat @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_754_sup_Oidem,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_755_sup_Oidem,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_756_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_757_sup_Oidem,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_758_sup__idem,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_759_sup__idem,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_760_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_761_sup__idem,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_762_sup_Oleft__idem,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B4 ) )
= ( sup_su4213647025997063966et_nat @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_763_sup_Oleft__idem,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B4 ) )
= ( sup_sup_set_set_nat @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_764_sup_Oleft__idem,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_765_sup_Oleft__idem,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ A @ B4 ) )
= ( sup_sup_set_nat_nat @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_766_sup__left__idem,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_767_sup__left__idem,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) )
= ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_768_sup__left__idem,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_769_sup__left__idem,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) )
= ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_770_sup_Oright__idem,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B4 ) @ B4 )
= ( sup_su4213647025997063966et_nat @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_771_sup_Oright__idem,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B4 ) @ B4 )
= ( sup_sup_set_set_nat @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_772_sup_Oright__idem,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ B4 )
= ( sup_sup_set_nat @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_773_sup_Oright__idem,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B4 ) @ B4 )
= ( sup_sup_set_nat_nat @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_774_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_775_finite__Collect__disjI,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
& ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_776_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_777_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_778_finite__Collect__disjI,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
& ( finite2115694454571419734at_nat @ ( collect_nat_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_779_finite__Collect__disjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
& ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_780_finite__Collect__conjI,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o] :
( ( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
| ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ Q ) ) )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_781_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_782_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_783_finite__Collect__conjI,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
| ( finite2115694454571419734at_nat @ ( collect_nat_nat @ Q ) ) )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_784_finite__Collect__conjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
| ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_785_le__inf__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( ( ord_le9131159989063066194et_nat @ X @ Y2 )
& ( ord_le9131159989063066194et_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_786_le__inf__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) )
= ( ( ord_le6893508408891458716et_nat @ X @ Y2 )
& ( ord_le6893508408891458716et_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_787_le__inf__iff,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X @ Y2 )
& ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_788_le__inf__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( ( ord_le9059583361652607317at_nat @ X @ Y2 )
& ( ord_le9059583361652607317at_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_789_le__inf__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y2 )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_790_le__inf__iff,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_791_inf_Obounded__iff,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( inf_in5711780100303410308et_nat @ B4 @ C ) )
= ( ( ord_le9131159989063066194et_nat @ A @ B4 )
& ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_792_inf_Obounded__iff,axiom,
! [A: set_set_nat,B4: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B4 @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A @ B4 )
& ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_793_inf_Obounded__iff,axiom,
! [A: set_nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B4 )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_794_inf_Obounded__iff,axiom,
! [A: set_nat_nat,B4: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( inf_inf_set_nat_nat @ B4 @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A @ B4 )
& ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_795_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_796_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_797_le__sup__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_le9131159989063066194et_nat @ X @ Z2 )
& ( ord_le9131159989063066194et_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_798_le__sup__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_le6893508408891458716et_nat @ X @ Z2 )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_799_le__sup__iff,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_800_le__sup__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_le9059583361652607317at_nat @ X @ Z2 )
& ( ord_le9059583361652607317at_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_801_le__sup__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_802_le__sup__iff,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_803_sup_Obounded__iff,axiom,
! [B4: set_set_set_nat,C: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B4 @ C ) @ A )
= ( ( ord_le9131159989063066194et_nat @ B4 @ A )
& ( ord_le9131159989063066194et_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_804_sup_Obounded__iff,axiom,
! [B4: set_set_nat,C: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B4 @ C ) @ A )
= ( ( ord_le6893508408891458716et_nat @ B4 @ A )
& ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_805_sup_Obounded__iff,axiom,
! [B4: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B4 @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B4 @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_806_sup_Obounded__iff,axiom,
! [B4: set_nat_nat,C: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B4 @ C ) @ A )
= ( ( ord_le9059583361652607317at_nat @ B4 @ A )
& ( ord_le9059583361652607317at_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_807_sup_Obounded__iff,axiom,
! [B4: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_808_sup_Obounded__iff,axiom,
! [B4: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_809_inf__bot__left,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_810_inf__bot__left,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_811_inf__bot__left,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_812_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_813_inf__bot__right,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_814_inf__bot__right,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_815_inf__bot__right,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_816_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_817_sup__bot__left,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_818_sup__bot__left,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_819_sup__bot__left,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_820_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_821_sup__bot__right,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ bot_bot_set_set_nat )
= X ) ).
% sup_bot_right
thf(fact_822_sup__bot__right,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ bot_bot_set_nat_nat )
= X ) ).
% sup_bot_right
thf(fact_823_sup__bot__right,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ bot_bo7198184520161983622et_nat )
= X ) ).
% sup_bot_right
thf(fact_824_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_825_bot__eq__sup__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_826_bot__eq__sup__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_827_bot__eq__sup__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( ( X = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_828_bot__eq__sup__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_829_sup__eq__bot__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X @ Y2 )
= bot_bot_set_set_nat )
= ( ( X = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_830_sup__eq__bot__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X @ Y2 )
= bot_bot_set_nat_nat )
= ( ( X = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_831_sup__eq__bot__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ( X = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_832_sup__eq__bot__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_833_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B4 )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B4 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_834_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A @ B4 )
= bot_bot_set_nat_nat )
= ( ( A = bot_bot_set_nat_nat )
& ( B4 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_835_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A @ B4 )
= bot_bo7198184520161983622et_nat )
= ( ( A = bot_bo7198184520161983622et_nat )
& ( B4 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_836_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_837_sup__bot_Oleft__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_838_sup__bot_Oleft__neutral,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_839_sup__bot_Oleft__neutral,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_840_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_841_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A @ B4 ) )
= ( ( A = bot_bot_set_set_nat )
& ( B4 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_842_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ A @ B4 ) )
= ( ( A = bot_bot_set_nat_nat )
& ( B4 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_843_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ A @ B4 ) )
= ( ( A = bot_bo7198184520161983622et_nat )
& ( B4 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_844_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B4 ) )
= ( ( A = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_845_sup__bot_Oright__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_846_sup__bot_Oright__neutral,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_847_sup__bot_Oright__neutral,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_848_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_849_finite__Int,axiom,
! [F3: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_850_finite__Int,axiom,
! [F3: set_set_nat,G: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F3 )
| ( finite1152437895449049373et_nat @ G ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_851_finite__Int,axiom,
! [F3: set_set_set_nat,G: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F3 )
| ( finite6739761609112101331et_nat @ G ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_852_finite__Int,axiom,
! [F3: set_nat_nat,G: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F3 )
| ( finite2115694454571419734at_nat @ G ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_853_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_854_finite__Collect__bounded__ex,axiom,
! [P: monotone_mformula_a > $o,Q: nat > monotone_mformula_a > $o] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: monotone_mformula_a] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: monotone_mformula_a] :
( ( P @ Y6 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_855_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: monotone_mformula_a > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
? [Y6: nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_856_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: set_nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
? [Y6: nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_857_finite__Collect__bounded__ex,axiom,
! [P: set_nat > $o,Q: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y6: set_nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: set_nat] :
( ( P @ Y6 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_858_finite__Collect__bounded__ex,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > monotone_mformula_a > $o] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
? [Y6: monotone_mformula_a] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: monotone_mformula_a] :
( ( P @ Y6 )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_859_finite__Collect__bounded__ex,axiom,
! [P: monotone_mformula_a > $o,Q: set_nat > monotone_mformula_a > $o] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
? [Y6: monotone_mformula_a] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: monotone_mformula_a] :
( ( P @ Y6 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_860_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
? [Y6: nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_861_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: set_set_nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
? [Y6: nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_862_finite__Collect__bounded__ex,axiom,
! [P: set_nat > $o,Q: monotone_mformula_a > set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] :
? [Y6: set_nat] :
( ( P @ Y6 )
& ( Q @ X2 @ Y6 ) ) ) )
= ( ! [Y6: set_nat] :
( ( P @ Y6 )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [X2: monotone_mformula_a] : ( Q @ X2 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_863_finite__ACC,axiom,
! [X4: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) ) ).
% finite_ACC
thf(fact_864_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_865_union___092_060G_062,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% union_\<G>
thf(fact_866_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_867_ACC__cf__SET_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_cf_SET(3)
thf(fact_868_ACC__cf__SET_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_cf_SET(4)
thf(fact_869_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_870_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: nat,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_871_finite__image__set2,axiom,
! [P: monotone_mformula_a > $o,Q: nat > $o,F: monotone_mformula_a > nat > nat] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: monotone_mformula_a,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_872_finite__image__set2,axiom,
! [P: nat > $o,Q: monotone_mformula_a > $o,F: nat > monotone_mformula_a > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: nat,Y6: monotone_mformula_a] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_873_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > monotone_mformula_a] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Uu: monotone_mformula_a] :
? [X2: nat,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_874_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: nat,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_875_finite__image__set2,axiom,
! [P: nat > $o,Q: set_nat > $o,F: nat > set_nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: nat,Y6: set_nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_876_finite__image__set2,axiom,
! [P: set_nat > $o,Q: nat > $o,F: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: set_nat,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_877_finite__image__set2,axiom,
! [P: monotone_mformula_a > $o,Q: monotone_mformula_a > $o,F: monotone_mformula_a > monotone_mformula_a > nat] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: monotone_mformula_a,Y6: monotone_mformula_a] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_878_finite__image__set2,axiom,
! [P: monotone_mformula_a > $o,Q: nat > $o,F: monotone_mformula_a > nat > monotone_mformula_a] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Uu: monotone_mformula_a] :
? [X2: monotone_mformula_a,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_879_finite__image__set2,axiom,
! [P: monotone_mformula_a > $o,Q: nat > $o,F: monotone_mformula_a > nat > set_nat] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: monotone_mformula_a,Y6: nat] :
( ( Uu
= ( F @ X2 @ Y6 ) )
& ( P @ X2 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_880_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_881_finite__image__set,axiom,
! [P: monotone_mformula_a > $o,F: monotone_mformula_a > nat] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: monotone_mformula_a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_882_finite__image__set,axiom,
! [P: nat > $o,F: nat > monotone_mformula_a] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Uu: monotone_mformula_a] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_883_finite__image__set,axiom,
! [P: nat > $o,F: nat > set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_884_finite__image__set,axiom,
! [P: set_nat > $o,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X2: set_nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_885_finite__image__set,axiom,
! [P: monotone_mformula_a > $o,F: monotone_mformula_a > monotone_mformula_a] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Uu: monotone_mformula_a] :
? [X2: monotone_mformula_a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_886_finite__image__set,axiom,
! [P: monotone_mformula_a > $o,F: monotone_mformula_a > set_nat] :
( ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: monotone_mformula_a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_887_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_888_finite__image__set,axiom,
! [P: nat > $o,F: nat > set_set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_889_finite__image__set,axiom,
! [P: set_nat > $o,F: set_nat > monotone_mformula_a] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [Uu: monotone_mformula_a] :
? [X2: set_nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_890_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B2: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_891_inf__set__def,axiom,
( inf_in4741988911529734942mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
( collec4794253742848188331mula_a
@ ( inf_in6543652950005596391la_a_o
@ ^ [X2: monotone_mformula_a] : ( member535913909593306477mula_a @ X2 @ A3 )
@ ^ [X2: monotone_mformula_a] : ( member535913909593306477mula_a @ X2 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_892_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_893_inf__set__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( collect_set_nat
@ ( inf_inf_set_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_894_inf__set__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( collect_set_set_nat
@ ( inf_in2551356467856225537_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A3 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_895_inf__set__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( collect_nat_nat
@ ( inf_inf_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_896_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B2: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_897_sup__set__def,axiom,
( sup_su7438456061012554424mula_a
= ( ^ [A3: set_Mo2626137824023173004mula_a,B2: set_Mo2626137824023173004mula_a] :
( collec4794253742848188331mula_a
@ ( sup_su6423506137380736461la_a_o
@ ^ [X2: monotone_mformula_a] : ( member535913909593306477mula_a @ X2 @ A3 )
@ ^ [X2: monotone_mformula_a] : ( member535913909593306477mula_a @ X2 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_898_sup__set__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( collect_set_set_nat
@ ( sup_su5917979686466268903_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A3 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_899_sup__set__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( collect_set_nat
@ ( sup_sup_set_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_900_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_901_sup__set__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( collect_nat_nat
@ ( sup_sup_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_902_inf__sup__aci_I4_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_903_inf__sup__aci_I4_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y2 ) )
= ( inf_inf_set_nat_nat @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_904_inf__sup__aci_I3_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_905_inf__sup__aci_I3_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat_nat @ Y2 @ ( inf_inf_set_nat_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_906_inf__sup__aci_I2_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Z2 )
= ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_907_inf__sup__aci_I2_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Z2 )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_908_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X2: set_set_set_nat,Y6: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y6 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_909_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat_nat
= ( ^ [X2: set_nat_nat,Y6: set_nat_nat] : ( inf_inf_set_nat_nat @ Y6 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_910_inf_Oassoc,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ C )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B4 @ C ) ) ) ).
% inf.assoc
thf(fact_911_inf_Oassoc,axiom,
! [A: set_nat_nat,B4: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ C )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B4 @ C ) ) ) ).
% inf.assoc
thf(fact_912_inf__assoc,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Z2 )
= ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_913_inf__assoc,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Z2 )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_914_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A7: set_set_set_nat,B5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B5 @ A7 ) ) ) ).
% inf.commute
thf(fact_915_inf_Ocommute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A7: set_nat_nat,B5: set_nat_nat] : ( inf_inf_set_nat_nat @ B5 @ A7 ) ) ) ).
% inf.commute
thf(fact_916_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X2: set_set_set_nat,Y6: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y6 @ X2 ) ) ) ).
% inf_commute
thf(fact_917_inf__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [X2: set_nat_nat,Y6: set_nat_nat] : ( inf_inf_set_nat_nat @ Y6 @ X2 ) ) ) ).
% inf_commute
thf(fact_918_inf_Oleft__commute,axiom,
! [B4: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ B4 @ ( inf_in5711780100303410308et_nat @ A @ C ) )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B4 @ C ) ) ) ).
% inf.left_commute
thf(fact_919_inf_Oleft__commute,axiom,
! [B4: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ B4 @ ( inf_inf_set_nat_nat @ A @ C ) )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B4 @ C ) ) ) ).
% inf.left_commute
thf(fact_920_inf__left__commute,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_921_inf__left__commute,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat_nat @ Y2 @ ( inf_inf_set_nat_nat @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_922_inf__sup__aci_I8_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_923_inf__sup__aci_I8_J,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) )
= ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_924_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_925_inf__sup__aci_I8_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) )
= ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_926_inf__sup__aci_I7_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_927_inf__sup__aci_I7_J,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_928_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_929_inf__sup__aci_I7_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_930_inf__sup__aci_I6_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) @ Z2 )
= ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_931_inf__sup__aci_I6_J,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_932_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_933_inf__sup__aci_I6_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_934_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X2: set_set_set_nat,Y6: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y6 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_935_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y6: set_set_nat] : ( sup_sup_set_set_nat @ Y6 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_936_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y6: set_nat] : ( sup_sup_set_nat @ Y6 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_937_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X2: set_nat_nat,Y6: set_nat_nat] : ( sup_sup_set_nat_nat @ Y6 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_938_sup_Oassoc,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B4 ) @ C )
= ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B4 @ C ) ) ) ).
% sup.assoc
thf(fact_939_sup_Oassoc,axiom,
! [A: set_set_nat,B4: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B4 ) @ C )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B4 @ C ) ) ) ).
% sup.assoc
thf(fact_940_sup_Oassoc,axiom,
! [A: set_nat,B4: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C ) ) ) ).
% sup.assoc
thf(fact_941_sup_Oassoc,axiom,
! [A: set_nat_nat,B4: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B4 ) @ C )
= ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B4 @ C ) ) ) ).
% sup.assoc
thf(fact_942_sup__assoc,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) @ Z2 )
= ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_943_sup__assoc,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_944_sup__assoc,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_945_sup__assoc,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_946_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A7: set_set_set_nat,B5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B5 @ A7 ) ) ) ).
% sup.commute
thf(fact_947_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A7: set_set_nat,B5: set_set_nat] : ( sup_sup_set_set_nat @ B5 @ A7 ) ) ) ).
% sup.commute
thf(fact_948_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A7: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A7 ) ) ) ).
% sup.commute
thf(fact_949_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A7: set_nat_nat,B5: set_nat_nat] : ( sup_sup_set_nat_nat @ B5 @ A7 ) ) ) ).
% sup.commute
thf(fact_950_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X2: set_set_set_nat,Y6: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y6 @ X2 ) ) ) ).
% sup_commute
thf(fact_951_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y6: set_set_nat] : ( sup_sup_set_set_nat @ Y6 @ X2 ) ) ) ).
% sup_commute
thf(fact_952_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y6: set_nat] : ( sup_sup_set_nat @ Y6 @ X2 ) ) ) ).
% sup_commute
thf(fact_953_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X2: set_nat_nat,Y6: set_nat_nat] : ( sup_sup_set_nat_nat @ Y6 @ X2 ) ) ) ).
% sup_commute
thf(fact_954_sup_Oleft__commute,axiom,
! [B4: set_set_set_nat,A: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ B4 @ ( sup_su4213647025997063966et_nat @ A @ C ) )
= ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B4 @ C ) ) ) ).
% sup.left_commute
thf(fact_955_sup_Oleft__commute,axiom,
! [B4: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ B4 @ ( sup_sup_set_set_nat @ A @ C ) )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B4 @ C ) ) ) ).
% sup.left_commute
thf(fact_956_sup_Oleft__commute,axiom,
! [B4: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C ) ) ) ).
% sup.left_commute
thf(fact_957_sup_Oleft__commute,axiom,
! [B4: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ B4 @ ( sup_sup_set_nat_nat @ A @ C ) )
= ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B4 @ C ) ) ) ).
% sup.left_commute
thf(fact_958_sup__left__commute,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_959_sup__left__commute,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_960_sup__left__commute,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_961_sup__left__commute,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_962_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A7: a] :
( ( member_a @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_963_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A7: nat] :
( ( member_nat @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_964_pigeonhole__infinite__rel,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_nat,R: monotone_mformula_a > nat > $o] :
( ~ ( finite694752133686741613mula_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [A7: monotone_mformula_a] :
( ( member535913909593306477mula_a @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_965_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_set_nat,R: a > set_nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A7: a] :
( ( member_a @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_966_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_set_nat,R: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A7: nat] :
( ( member_nat @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_967_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A7: set_nat] :
( ( member_set_nat @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_968_pigeonhole__infinite__rel,axiom,
! [A2: set_Mo2626137824023173004mula_a,B: set_set_nat,R: monotone_mformula_a > set_nat > $o] :
( ~ ( finite694752133686741613mula_a @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X5: monotone_mformula_a] :
( ( member535913909593306477mula_a @ X5 @ A2 )
=> ? [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ~ ( finite694752133686741613mula_a
@ ( collec4794253742848188331mula_a
@ ^ [A7: monotone_mformula_a] :
( ( member535913909593306477mula_a @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_969_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat_nat,R: a > ( nat > nat ) > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A7: a] :
( ( member_a @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_970_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_set_set_nat,R: a > set_set_nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [Xa2: set_set_nat] :
( ( member_set_set_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A7: a] :
( ( member_a @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_971_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat_nat,R: nat > ( nat > nat ) > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ? [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ B )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A7: nat] :
( ( member_nat @ A7 @ A2 )
& ( R @ A7 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_972_not__finite__existsD,axiom,
! [P: monotone_mformula_a > $o] :
( ~ ( finite694752133686741613mula_a @ ( collec4794253742848188331mula_a @ P ) )
=> ? [X_1: monotone_mformula_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_973_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_974_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_975_not__finite__existsD,axiom,
! [P: ( nat > nat ) > $o] :
( ~ ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
=> ? [X_1: nat > nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_976_not__finite__existsD,axiom,
! [P: set_set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ? [X_1: set_set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_977_finite__has__maximal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A2 )
& ( ord_less_eq_nat_nat @ A @ X5 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_978_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X5 )
& ! [Xa2: set_set_nat] :
( ( member_set_set_nat @ Xa2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_979_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ A @ X5 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_980_finite__has__maximal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X5: set_nat_nat] :
( ( member_set_nat_nat @ X5 @ A2 )
& ( ord_le9059583361652607317at_nat @ A @ X5 )
& ! [Xa2: set_nat_nat] :
( ( member_set_nat_nat @ Xa2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_981_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ A @ X5 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_982_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_nat @ A @ X5 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_983_finite__has__minimal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A2 )
& ( ord_less_eq_nat_nat @ X5 @ A )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_984_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A2 )
& ( ord_le6893508408891458716et_nat @ X5 @ A )
& ! [Xa2: set_set_nat] :
( ( member_set_set_nat @ Xa2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_985_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ X5 @ A )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_986_finite__has__minimal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X5: set_nat_nat] :
( ( member_set_nat_nat @ X5 @ A2 )
& ( ord_le9059583361652607317at_nat @ X5 @ A )
& ! [Xa2: set_nat_nat] :
( ( member_set_nat_nat @ Xa2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_987_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ X5 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_988_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_nat @ X5 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_989_inf__sup__ord_I2_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_990_inf__sup__ord_I2_J,axiom,
! [X: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_991_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_992_inf__sup__ord_I2_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_993_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_994_inf__sup__ord_I2_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_995_inf__sup__ord_I1_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_996_inf__sup__ord_I1_J,axiom,
! [X: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_997_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_998_inf__sup__ord_I1_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_999_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1000_inf__sup__ord_I1_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1001_inf__le1,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_1002_inf__le1,axiom,
! [X: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_1003_inf__le1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_1004_inf__le1,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_1005_inf__le1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_1006_inf__le1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_1007_inf__le2,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1008_inf__le2,axiom,
! [X: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1009_inf__le2,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1010_inf__le2,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1011_inf__le2,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1012_inf__le2,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1013_le__infE,axiom,
! [X: set_set_set_nat,A: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ A @ B4 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ X @ A )
=> ~ ( ord_le9131159989063066194et_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_1014_le__infE,axiom,
! [X: set_set_nat,A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A @ B4 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X @ A )
=> ~ ( ord_le6893508408891458716et_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_1015_le__infE,axiom,
! [X: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A )
=> ~ ( ord_less_eq_set_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_1016_le__infE,axiom,
! [X: set_nat_nat,A: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ A @ B4 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X @ A )
=> ~ ( ord_le9059583361652607317at_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_1017_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_1018_le__infE,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_1019_le__infI,axiom,
! [X: set_set_set_nat,A: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ A )
=> ( ( ord_le9131159989063066194et_nat @ X @ B4 )
=> ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_1020_le__infI,axiom,
! [X: set_set_nat,A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ A )
=> ( ( ord_le6893508408891458716et_nat @ X @ B4 )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_1021_le__infI,axiom,
! [X: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X @ B4 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_1022_le__infI,axiom,
! [X: set_nat_nat,A: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ A )
=> ( ( ord_le9059583361652607317at_nat @ X @ B4 )
=> ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_1023_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_1024_le__infI,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_1025_inf__mono,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B4: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B4 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ ( inf_in5711780100303410308et_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1026_inf__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B4: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) @ ( inf_inf_set_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1027_inf__mono,axiom,
! [A: set_nat,C: set_nat,B4: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B4 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1028_inf__mono,axiom,
! [A: set_nat_nat,C: set_nat_nat,B4: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B4 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ ( inf_inf_set_nat_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1029_inf__mono,axiom,
! [A: set_a,C: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1030_inf__mono,axiom,
! [A: nat,C: nat,B4: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B4 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1031_le__infI1,axiom,
! [A: set_set_set_nat,X: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ X )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_1032_le__infI1,axiom,
! [A: set_set_nat,X: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_1033_le__infI1,axiom,
! [A: set_nat,X: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_1034_le__infI1,axiom,
! [A: set_nat_nat,X: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ X )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_1035_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_1036_le__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_1037_le__infI2,axiom,
! [B4: set_set_set_nat,X: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ X )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_1038_le__infI2,axiom,
! [B4: set_set_nat,X: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_1039_le__infI2,axiom,
! [B4: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_1040_le__infI2,axiom,
! [B4: set_nat_nat,X: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ X )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_1041_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_1042_le__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_1043_inf_OorderE,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B4 )
=> ( A
= ( inf_in5711780100303410308et_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_1044_inf_OorderE,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( A
= ( inf_inf_set_set_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_1045_inf_OorderE,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( A
= ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_1046_inf_OorderE,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B4 )
=> ( A
= ( inf_inf_set_nat_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_1047_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_1048_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_1049_inf_OorderI,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( A
= ( inf_in5711780100303410308et_nat @ A @ B4 ) )
=> ( ord_le9131159989063066194et_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_1050_inf_OorderI,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A
= ( inf_inf_set_set_nat @ A @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_1051_inf_OorderI,axiom,
! [A: set_nat,B4: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_1052_inf_OorderI,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( A
= ( inf_inf_set_nat_nat @ A @ B4 ) )
=> ( ord_le9059583361652607317at_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_1053_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_1054_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_1055_inf__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Z3 )
=> ( ord_le9131159989063066194et_nat @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1056_inf__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X: set_set_nat,Y2: set_set_nat] :
( ! [X5: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_set_nat,Y5: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1057_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X5: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_set_nat @ X5 @ Z3 )
=> ( ord_less_eq_set_nat @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1058_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X: set_nat_nat,Y2: set_nat_nat] :
( ! [X5: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_nat_nat,Y5: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X5 @ Y5 )
=> ( ( ord_le9059583361652607317at_nat @ X5 @ Z3 )
=> ( ord_le9059583361652607317at_nat @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1059_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_a,Y5: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( ord_less_eq_set_a @ X5 @ Z3 )
=> ( ord_less_eq_set_a @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1060_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y2: nat] :
( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1061_le__iff__inf,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X2: set_set_set_nat,Y6: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ Y6 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1062_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X2: set_set_nat,Y6: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ Y6 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1063_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y6: set_nat] :
( ( inf_inf_set_nat @ X2 @ Y6 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1064_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y6: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ Y6 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1065_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y6: set_a] :
( ( inf_inf_set_a @ X2 @ Y6 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1066_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y6: nat] :
( ( inf_inf_nat @ X2 @ Y6 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1067_inf_Oabsorb1,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B4 )
=> ( ( inf_in5711780100303410308et_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1068_inf_Oabsorb1,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( inf_inf_set_set_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1069_inf_Oabsorb1,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( inf_inf_set_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1070_inf_Oabsorb1,axiom,
! [A: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B4 )
=> ( ( inf_inf_set_nat_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1071_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1072_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1073_inf_Oabsorb2,axiom,
! [B4: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1074_inf_Oabsorb2,axiom,
! [B4: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ( inf_inf_set_set_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1075_inf_Oabsorb2,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( inf_inf_set_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1076_inf_Oabsorb2,axiom,
! [B4: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1077_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1078_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1079_inf__absorb1,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ( inf_in5711780100303410308et_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_1080_inf__absorb1,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y2 )
=> ( ( inf_inf_set_set_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_1081_inf__absorb1,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( inf_inf_set_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_1082_inf__absorb1,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( inf_inf_set_nat_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_1083_inf__absorb1,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( inf_inf_set_a @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_1084_inf__absorb1,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( inf_inf_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_1085_inf__absorb2,axiom,
! [Y2: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X )
=> ( ( inf_in5711780100303410308et_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1086_inf__absorb2,axiom,
! [Y2: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X )
=> ( ( inf_inf_set_set_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1087_inf__absorb2,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( inf_inf_set_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1088_inf__absorb2,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( ( inf_inf_set_nat_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1089_inf__absorb2,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( inf_inf_set_a @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1090_inf__absorb2,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( inf_inf_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1091_inf_OboundedE,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( inf_in5711780100303410308et_nat @ B4 @ C ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B4 )
=> ~ ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1092_inf_OboundedE,axiom,
! [A: set_set_nat,B4: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B4 @ C ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ~ ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1093_inf_OboundedE,axiom,
! [A: set_nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1094_inf_OboundedE,axiom,
! [A: set_nat_nat,B4: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( inf_inf_set_nat_nat @ B4 @ C ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B4 )
=> ~ ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1095_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1096_inf_OboundedE,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1097_inf_OboundedI,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B4 )
=> ( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ord_le9131159989063066194et_nat @ A @ ( inf_in5711780100303410308et_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1098_inf_OboundedI,axiom,
! [A: set_set_nat,B4: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1099_inf_OboundedI,axiom,
! [A: set_nat,B4: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1100_inf_OboundedI,axiom,
! [A: set_nat_nat,B4: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B4 )
=> ( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ord_le9059583361652607317at_nat @ A @ ( inf_inf_set_nat_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1101_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1102_inf_OboundedI,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1103_inf__greatest,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ X @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1104_inf__greatest,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1105_inf__greatest,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ X @ Z2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1106_inf__greatest,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1107_inf__greatest,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1108_inf__greatest,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1109_inf_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A7: set_set_set_nat,B5: set_set_set_nat] :
( A7
= ( inf_in5711780100303410308et_nat @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1110_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A7: set_set_nat,B5: set_set_nat] :
( A7
= ( inf_inf_set_set_nat @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1111_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B5: set_nat] :
( A7
= ( inf_inf_set_nat @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1112_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A7: set_nat_nat,B5: set_nat_nat] :
( A7
= ( inf_inf_set_nat_nat @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1113_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
( A7
= ( inf_inf_set_a @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1114_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B5: nat] :
( A7
= ( inf_inf_nat @ A7 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1115_inf_Ocobounded1,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1116_inf_Ocobounded1,axiom,
! [A: set_set_nat,B4: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1117_inf_Ocobounded1,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1118_inf_Ocobounded1,axiom,
! [A: set_nat_nat,B4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1119_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1120_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1121_inf_Ocobounded2,axiom,
! [A: set_set_set_nat,B4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1122_inf_Ocobounded2,axiom,
! [A: set_set_nat,B4: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1123_inf_Ocobounded2,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1124_inf_Ocobounded2,axiom,
! [A: set_nat_nat,B4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1125_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1126_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1127_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
( ( inf_inf_set_a @ A7 @ B5 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_1128_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B5: nat] :
( ( inf_inf_nat @ A7 @ B5 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_1129_ACC__cf__def,axiom,
! [X4: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ X4 )
= ( collect_nat_nat
@ ^ [F4: nat > nat] :
( ( member_nat_nat @ F4 @ ( clique2971579238625216137irst_F @ k ) )
& ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ k @ F4 ) ) ) ) ) ).
% ACC_cf_def
thf(fact_1130_Clique__def,axiom,
( clique6749503327923060270Clique
= ( ^ [V: set_nat,K3: nat] :
( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ V ) )
& ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ C3 @ C3 ) @ G2 )
& ( ( finite_card_nat @ C3 )
= K3 ) ) ) ) ) ) ).
% Clique_def
thf(fact_1131_ACC__cf__SET_I1_J,axiom,
! [X: a] :
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X ) ) )
= ( collect_nat_nat
@ ^ [F2: nat > nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
& ( member_set_nat @ ( pi @ X ) @ ( clique5033774636164728462irst_C @ k @ F2 ) ) ) ) ) ).
% ACC_cf_SET(1)
thf(fact_1132__092_060pi_062__singleton_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ v )
=> ( member_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% \<pi>_singleton(1)
thf(fact_1133_ACC__cf__I,axiom,
! [F3: nat > nat,X4: set_set_set_nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ k @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ k @ X4 ) ) ) ) ).
% ACC_cf_I
thf(fact_1134_odot__def,axiom,
( clique5469973757772500719t_odot
= ( ^ [X6: set_set_set_nat,Y7: set_set_set_nat] :
( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D3: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D3 @ E ) )
& ( member_set_set_nat @ D3 @ X6 )
& ( member_set_set_nat @ E @ Y7 ) ) ) ) ) ).
% odot_def
thf(fact_1135_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1136_card__v__gs__join,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat,Z4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z4 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z4 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_1137_C__def,axiom,
! [F: nat > nat] :
( ( clique5033774636164728462irst_C @ k @ F )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: nat,Y6: nat] :
( ( Uu
= ( insert_nat @ X2 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( F @ X2 )
!= ( F @ Y6 ) ) ) ) ) ).
% C_def
thf(fact_1138_SET_Osimps_I2_J,axiom,
! [X: a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% SET.simps(2)
thf(fact_1139_v__def,axiom,
( clique5033774636164728513irst_v
= ( ^ [G2: set_set_nat] :
( collect_nat
@ ^ [X2: nat] :
? [Y6: nat] : ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) @ G2 ) ) ) ) ).
% v_def
thf(fact_1140_SET_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X )
= Y2 )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X5: a] :
( ( X
= ( monotone_Var_a @ X5 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X5 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( Y2
!= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi2 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( Y2
!= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi2 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% SET.elims
thf(fact_1141_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1142_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1143_SET_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ X )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X5: a] :
( ( X
= ( monotone_Var_a @ X5 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X5 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Var_a @ X5 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ( Y2
= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi2 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( ( Y2
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi2 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% SET.pelims
thf(fact_1144_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_1145_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_1146_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1147_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1148_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1149_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ N3 )
=> ( ord_less_nat @ X5 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1150_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1151_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1152_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1153_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1154_first__assumptions_Ocard__v__gs__join,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat,Z4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z4 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z4 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ) ) ) ) ).
% first_assumptions.card_v_gs_join
thf(fact_1155_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P3: nat,K2: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_1156_first__assumptions_OACC__empty,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1157_first__assumptions_OACC__union,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K2 @ X4 ) @ ( clique3210737319928189260st_ACC @ K2 @ Y ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_1158_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K2 @ X4 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1159_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K2 ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1160_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K2 ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1161_first__assumptions_OacceptsI,axiom,
! [L: nat,P3: nat,K2: nat,D: set_set_nat,G: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ G )
=> ( ( member_set_set_nat @ D @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1162_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X4 )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1163_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1164_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_1165_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1166_first__assumptions_Ov__empty,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_1167_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique363107459185959606CLIQUE @ K2 ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1168_first__assumptions_OACC__odot,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K2 @ X4 ) @ ( clique3210737319928189260st_ACC @ K2 @ Y ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1169_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K2 @ X4 ) @ ( clique951075384711337423ACC_cf @ K2 @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1170_first__assumptions_Ov__mono,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1171_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K2 @ X4 ) @ ( clique951075384711337423ACC_cf @ K2 @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1172_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P3: nat,K2: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_1173_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1174_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K2 @ X4 ) @ ( clique2971579238625216137irst_F @ K2 ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1175_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K2 @ X4 ) @ ( clique951075384711337423ACC_cf @ K2 @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1176_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_1177_first__assumptions_Ov__union,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1178_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique8462013130872731469t_v_gs @ X4 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X4 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_1179_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique363107459185959606CLIQUE @ K2 ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1180_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_1181_first__assumptions_Ounion___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.union_\<G>
thf(fact_1182_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_1183_first__assumptions_Oodot__def,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique5469973757772500719t_odot @ X4 @ Y )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D3: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D3 @ E ) )
& ( member_set_set_nat @ D3 @ X4 )
& ( member_set_set_nat @ E @ Y ) ) ) ) ) ).
% first_assumptions.odot_def
thf(fact_1184_first__assumptions_Ov__def,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique5033774636164728513irst_v @ G )
= ( collect_nat
@ ^ [X2: nat] :
? [Y6: nat] : ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) @ G ) ) ) ) ).
% first_assumptions.v_def
thf(fact_1185_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K2 ) @ ( clique3210737375870294875st_NEG @ K2 ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1186_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique3210737375870294875st_NEG @ K2 ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1187_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P3: nat,K2: nat,F3: nat > nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K2 ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K2 @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ K2 @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1188_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_1189_first__assumptions_OACC__cf__def,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ X4 )
= ( collect_nat_nat
@ ^ [F4: nat > nat] :
( ( member_nat_nat @ F4 @ ( clique2971579238625216137irst_F @ K2 ) )
& ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K2 @ F4 ) ) ) ) ) ) ).
% first_assumptions.ACC_cf_def
thf(fact_1190_first__assumptions_O_092_060K_062___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.\<K>_\<G>
thf(fact_1191_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_1192_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_1193_first__assumptions_OACC__I,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ K2 @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_1194_first__assumptions_ONEG___092_060G_062,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ K2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.NEG_\<G>
thf(fact_1195_first__assumptions_O_092_060G_062__def,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.\<G>_def
thf(fact_1196_first__assumptions_OACC__def,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ X4 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( clique3686358387679108662ccepts @ X4 @ G2 ) ) ) ) ) ).
% first_assumptions.ACC_def
thf(fact_1197_first__assumptions_ONEG__def,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3210737375870294875st_NEG @ K2 )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K2 ) @ ( clique2971579238625216137irst_F @ K2 ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1198_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P3: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_1199_first__assumptions_Ofinite__v__gs,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ) ).
% first_assumptions.finite_v_gs
thf(fact_1200_first__assumptions_O_092_060K_062__def,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3326749438856946062irst_K @ K2 )
= ( collect_set_set_nat
@ ^ [K: set_set_nat] :
( ( member_set_set_nat @ K @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K ) )
= K2 )
& ( K
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K ) @ ( clique5033774636164728513irst_v @ K ) ) ) ) ) ) ) ).
% first_assumptions.\<K>_def
thf(fact_1201_first__assumptions_OC__def,axiom,
! [L: nat,P3: nat,K2: nat,F: nat > nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique5033774636164728462irst_C @ K2 @ F )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: nat,Y6: nat] :
( ( Uu
= ( insert_nat @ X2 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ( F @ X2 )
!= ( F @ Y6 ) ) ) ) ) ) ).
% first_assumptions.C_def
thf(fact_1202_first__assumptions_OCLIQUE__def,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique363107459185959606CLIQUE @ K2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( clique3326749438856946062irst_K @ K2 ) )
& ( ord_le6893508408891458716et_nat @ X2 @ G2 ) ) ) ) ) ) ).
% first_assumptions.CLIQUE_def
thf(fact_1203_first__assumptions_O_092_060K_062__altdef,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique3326749438856946062irst_K @ K2 )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) )
& ( ( finite_card_nat @ V )
= K2 ) ) ) ) ) ).
% first_assumptions.\<K>_altdef
thf(fact_1204_first__assumptions_Okm,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_less_nat @ K2 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.km
thf(fact_1205_first__assumptions_Omp,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_less_nat @ P3 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.mp
thf(fact_1206_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1207_first__assumptions_O_092_060G_062l__def,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique7840962075309931874st_G_l @ L @ K2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ord_less_eq_nat @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G2 ) ) @ L ) ) ) ) ) ).
% first_assumptions.\<G>l_def
thf(fact_1208_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_1209_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K2 ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1210__092_060G_062l__def,axiom,
( ( clique7840962075309931874st_G_l @ l @ k )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ord_less_eq_nat @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G2 ) ) @ l ) ) ) ) ).
% \<G>l_def
thf(fact_1211_first__assumptions_Oodotl__def,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ( clique7966186356931407165_odotl @ L @ K2 @ X4 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique7840962075309931874st_G_l @ L @ K2 ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1212_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1213_joinl__join,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X4 @ Y ) @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ).
% joinl_join
thf(fact_1214_finite__v__gs__Gl,axiom,
! [X4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ).
% finite_v_gs_Gl
thf(fact_1215_odotl__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X4 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1216_L0,axiom,
ord_less_eq_nat @ assumptions_and_L0 @ l ).
% L0
thf(fact_1217_L0_H,axiom,
ord_less_eq_nat @ assumptions_and_L02 @ l ).
% L0'
thf(fact_1218_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_1219_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P3: nat,K2: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K2 @ X4 @ Y ) @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1220_forth__assumptions__axioms,axiom,
clique8563529963003110213ions_a @ l @ p @ k @ v @ pi ).
% forth_assumptions_axioms
thf(fact_1221_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1222_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_1223_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1224_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_1225_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_1226_deviate__finite_I3_J,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( finite6739761609112101331et_nat @ ( clique3314026705536850673os_cup @ l @ p @ k @ A2 @ B ) ) ).
% deviate_finite(3)
thf(fact_1227_first__assumptions_Okml,axiom,
! [L: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K2 )
=> ( ord_less_eq_nat @ K2 @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K2 ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1228_deviate__finite_I5_J,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( finite6739761609112101331et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ A2 @ B ) ) ).
% deviate_finite(5)
thf(fact_1229_deviate__finite_I6_J,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k @ A2 @ B ) ) ).
% deviate_finite(6)
thf(fact_1230_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_1231_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_1232_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_1233_deviate__pos__cap__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k @ X4 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X4 @ Y ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1234_deviate__neg__cap__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k @ X4 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X4 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_1235_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_1236_sqcap__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k @ X4 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X4 @ Y ) ) ) ).
% sqcap_def
thf(fact_1237_deviate__pos__cup__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k @ X4 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X4 @ Y ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1238_sqcup__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k @ X4 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) ).
% sqcup_def
thf(fact_1239_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_1240_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_1241_deviate__neg__cup__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k @ X4 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X4 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_1242_deviate__finite_I4_J,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k @ A2 @ B ) ) ).
% deviate_finite(4)
thf(fact_1243_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_1244_approx__neg_Osimps_I3_J,axiom,
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ monotone_TRUE_a )
= bot_bot_set_nat_nat ) ).
% approx_neg.simps(3)
thf(fact_1245_approx__pos_Osimps_I2_J,axiom,
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ monotone_TRUE_a )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(2)
thf(fact_1246_finite__approx__neg,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_neg
thf(fact_1247_finite__approx__pos,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_pos
thf(fact_1248_approx__pos_Osimps_I5_J,axiom,
! [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ V3 @ Va2 ) )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(5)
thf(fact_1249_approx__pos_Osimps_I4_J,axiom,
! [V3: a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ V3 ) )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(4)
thf(fact_1250_approx__pos_Osimps_I3_J,axiom,
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(3)
thf(fact_1251_approx__neg_Osimps_I5_J,axiom,
! [V3: a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ V3 ) )
= bot_bot_set_nat_nat ) ).
% approx_neg.simps(5)
thf(fact_1252_approx__neg_Osimps_I4_J,axiom,
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ).
% approx_neg.simps(4)
thf(fact_1253_APR_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X5: a] :
( ( X
= ( monotone_Var_a @ X5 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X5 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( Y2
!= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( Y2
!= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% APR.elims
thf(fact_1254_deviate__finite_I1_J,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(1)
thf(fact_1255_APR_Osimps_I1_J,axiom,
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% APR.simps(1)
thf(fact_1256_APR_Osimps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% APR.simps(3)
thf(fact_1257_APR_Osimps_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% APR.simps(4)
thf(fact_1258_approx__neg_Osimps_I2_J,axiom,
! [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi5 @ Psi5 ) )
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi5 ) ) ) ).
% approx_neg.simps(2)
thf(fact_1259_approx__neg_Osimps_I1_J,axiom,
! [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi5 @ Psi5 ) )
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi5 ) ) ) ).
% approx_neg.simps(1)
thf(fact_1260_approx__pos_Osimps_I1_J,axiom,
! [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi5 @ Psi5 ) )
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi5 ) ) ) ).
% approx_pos.simps(1)
thf(fact_1261_APR_Osimps_I2_J,axiom,
! [X: a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% APR.simps(2)
thf(fact_1262_deviate__subset__Conj_I1_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Conj(1)
thf(fact_1263_deviate__subset__Disj_I1_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705536850673os_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Disj(1)
thf(fact_1264_deviate__pos__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique4708818501384062891C_mf_a @ k @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) ) ) ) ).
% deviate_pos_def
thf(fact_1265_approx__pos_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ? [V2: a] :
( X
= ( monotone_Var_a @ V2 ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ~ ( ? [V2: monotone_mformula_a,Va: monotone_mformula_a] :
( X
= ( monotone_Disj_a @ V2 @ Va ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).
% approx_pos.elims
thf(fact_1266_approx__neg_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ~ ( ? [V2: a] :
( X
= ( monotone_Var_a @ V2 ) )
=> ( Y2 != bot_bot_set_nat_nat ) ) ) ) ) ) ) ).
% approx_neg.elims
thf(fact_1267_no__deviation_I3_J,axiom,
! [X: a] :
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(3)
thf(fact_1268_no__deviation_I1_J,axiom,
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(1)
thf(fact_1269_APR_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ X )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X5: a] :
( ( X
= ( monotone_Var_a @ X5 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X5 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Var_a @ X5 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi3 @ Psi2 ) )
=> ( ( Y2
= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi2 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi2 ) )
=> ( ( Y2
= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi2 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% APR.pelims
thf(fact_1270_approx__neg_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ X )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Conj_a @ Phi4 @ Psi4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Disj_a @ Phi4 @ Psi4 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_FALSE_a ) ) )
=> ~ ! [V2: a] :
( ( X
= ( monotone_Var_a @ V2 ) )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Var_a @ V2 ) ) ) ) ) ) ) ) ) ) ).
% approx_neg.pelims
thf(fact_1271_approx__pos_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ X )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Conj_a @ Phi4 @ Psi4 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [V2: a] :
( ( X
= ( monotone_Var_a @ V2 ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Var_a @ V2 ) ) ) )
=> ~ ! [V2: monotone_mformula_a,Va: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ V2 @ Va ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Disj_a @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% approx_pos.pelims
thf(fact_1272_deviate__subset__Conj_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Conj(2)
thf(fact_1273_deviate__finite_I2_J,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(2)
thf(fact_1274_deviate__neg__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique8961599393750669800f_mf_a @ k @ pi @ Phi ) ) ) ).
% deviate_neg_def
thf(fact_1275_deviate__subset__Disj_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Disj(2)
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y2: set_nat] :
( ( if_set_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y2: set_nat] :
( ( if_set_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: nat > nat,Y2: nat > nat] :
( ( if_nat_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: nat > nat,Y2: nat > nat] :
( ( if_nat_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( if_set_set_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( if_set_set_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member535913909593306477mula_a @ psi @ ( clique5987991184601036204th_A_a @ v ) ).
%------------------------------------------------------------------------------