TPTP Problem File: SLH0630^1.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_00367_010517__16145678_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1380 ( 666 unt; 109 typ; 0 def)
% Number of atoms : 3071 ( 952 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 8946 ( 268 ~; 42 |; 258 &;7309 @)
% ( 0 <=>;1069 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 730 ( 730 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 102 usr; 13 con; 0-3 aty)
% Number of variables : 3277 ( 416 ^;2736 !; 125 ?;3277 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:48:23.233
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (102)
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_Binomial_Obinomial,type,
binomial: nat > 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_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_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_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_card_nat_nat: set_nat_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $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_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inf_inf_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_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__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_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__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__Nat__Onat_Mt__Nat__Onat_J,type,
sup_sup_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
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_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__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_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo6227097192321305471_nat_o: set_set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
bot_bo193956671110832956et_nat: set_set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le3616423863276227763_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Set_OBex_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bex_nat_nat: set_nat_nat > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Nat__Onat,type,
bex_nat: set_nat > ( nat > $o ) > $o ).
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_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1270)
thf(fact_0_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X: set_set_set_nat,G: set_set_nat] :
? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) ) ).
% accepts_def
thf(fact_1__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<G>_def
thf(fact_2_Un__subset__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
= ( ( ord_le9131159989063066194et_nat @ A @ C )
& ( ord_le9131159989063066194et_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_3_Un__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
= ( ( ord_le6893508408891458716et_nat @ A @ C )
& ( ord_le6893508408891458716et_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_4_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( ( ord_less_eq_set_nat @ A @ C )
& ( ord_less_eq_set_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_5_Un__subset__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
= ( ( ord_le9059583361652607317at_nat @ A @ C )
& ( ord_le9059583361652607317at_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_6_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_7_le__sup__iff,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) @ Z )
= ( ( ord_le9131159989063066194et_nat @ X3 @ Z )
& ( ord_le9131159989063066194et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_8_le__sup__iff,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X3 @ Y ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X3 @ Z )
& ( ord_le6893508408891458716et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_9_le__sup__iff,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X3 @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_10_le__sup__iff,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X3 @ Y ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X3 @ Z )
& ( ord_le9059583361652607317at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_11_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_12_le__sup__iff,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_nat_nat @ X3 @ Z )
& ( ord_less_eq_nat_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_13_sup_Obounded__iff,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
& ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_14_sup_Obounded__iff,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_15_sup_Obounded__iff,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_16_sup_Obounded__iff,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_17_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_18_sup_Obounded__iff,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat_nat @ B2 @ A2 )
& ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_19_finite__members___092_060G_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G2 ) ) ).
% finite_members_\<G>
thf(fact_20_CLIQUE__def,axiom,
( ( clique363107459185959606CLIQUE @ k )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( clique3326749438856946062irst_K @ k ) )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) ) ) ).
% CLIQUE_def
thf(fact_21_ACC__def,axiom,
! [X4: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ X4 )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ) ).
% ACC_def
thf(fact_22_UnCI,axiom,
! [C2: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C2 @ B )
=> ( member_set_set_nat @ C2 @ A ) )
=> ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_23_UnCI,axiom,
! [C2: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C2 @ B )
=> ( member_set_nat @ C2 @ A ) )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_24_UnCI,axiom,
! [C2: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C2 @ B )
=> ( member_nat_nat @ C2 @ A ) )
=> ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_25_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_26_Un__iff,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C2 @ A )
| ( member_set_set_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_27_Un__iff,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C2 @ A )
| ( member_set_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_28_Un__iff,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C2 @ A )
| ( member_nat_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_29_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_30_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_31_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_32_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_33_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_34_sup_Oidem,axiom,
! [A2: nat > nat] :
( ( sup_sup_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_35_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_36_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_37_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_38_subset__antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_39_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_40_subset__antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_41_subset__antisym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_42_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( member_set_nat @ X5 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( member_nat @ X5 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_44_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
=> ( member_nat_nat @ X5 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_45_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( member_set_set_nat @ X5 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_46_empty__iff,axiom,
! [C2: set_nat] :
~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_47_empty__iff,axiom,
! [C2: set_set_nat] :
~ ( member_set_set_nat @ C2 @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_48_empty__iff,axiom,
! [C2: nat > nat] :
~ ( member_nat_nat @ C2 @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_49_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_50_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ! [X2: set_set_nat] :
~ ( member_set_set_nat @ X2 @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_53_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_54_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_55_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_56_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_57_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_58_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_59_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_60_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_61_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_62_sup__apply,axiom,
( sup_sup_nat_nat
= ( ^ [F: nat > nat,G3: nat > nat,X2: nat] : ( sup_sup_nat @ ( F @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% sup_apply
thf(fact_63_sup_Oright__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ B2 )
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_64_sup_Oright__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_65_sup_Oright__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_66_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_67_sup_Oright__idem,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( sup_sup_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_nat_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_68_sup_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_69_sup__left__idem,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_70_sup__left__idem,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y ) )
= ( sup_sup_set_set_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_71_sup__left__idem,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ X3 @ Y ) )
= ( sup_sup_set_nat_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_72_sup__left__idem,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_73_sup__left__idem,axiom,
! [X3: nat > nat,Y: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ ( sup_sup_nat_nat @ X3 @ Y ) )
= ( sup_sup_nat_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_74_sup__left__idem,axiom,
! [X3: nat,Y: nat] :
( ( sup_sup_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) )
= ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_75_sup_Oleft__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_76_sup_Oleft__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_77_sup_Oleft__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_78_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_79_sup_Oleft__idem,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( sup_sup_nat_nat @ A2 @ ( sup_sup_nat_nat @ A2 @ B2 ) )
= ( sup_sup_nat_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_80_sup_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_81_sup__idem,axiom,
! [X3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_82_sup__idem,axiom,
! [X3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_83_sup__idem,axiom,
! [X3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_84_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_85_sup__idem,axiom,
! [X3: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_86_sup__idem,axiom,
! [X3: nat] :
( ( sup_sup_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_87__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_88_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_89_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_90_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_91_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_92_subset__empty,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_93_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_94_subset__empty,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_95_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_96_sup__bot_Oright__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_97_sup__bot_Oright__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_98_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_99_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_100_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_101_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_102_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_103_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_104_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_105_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_106_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_107_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_108_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_109_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_110_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_111_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_112_sup__eq__bot__iff,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X3 @ Y )
= bot_bot_set_set_nat )
= ( ( X3 = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_113_sup__eq__bot__iff,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X3 @ Y )
= bot_bo7198184520161983622et_nat )
= ( ( X3 = bot_bo7198184520161983622et_nat )
& ( Y = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_114_sup__eq__bot__iff,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X3 @ Y )
= bot_bot_set_nat_nat )
= ( ( X3 = bot_bot_set_nat_nat )
& ( Y = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_115_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_116_bot__eq__sup__iff,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X3 @ Y ) )
= ( ( X3 = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_117_bot__eq__sup__iff,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( ( X3 = bot_bo7198184520161983622et_nat )
& ( Y = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_118_bot__eq__sup__iff,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X3 @ Y ) )
= ( ( X3 = bot_bot_set_nat_nat )
& ( Y = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_119_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_120_sup__bot__right,axiom,
! [X3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ bot_bot_set_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_121_sup__bot__right,axiom,
! [X3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ bot_bo7198184520161983622et_nat )
= X3 ) ).
% sup_bot_right
thf(fact_122_sup__bot__right,axiom,
! [X3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ bot_bot_set_nat_nat )
= X3 ) ).
% sup_bot_right
thf(fact_123_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_124_sup__bot__left,axiom,
! [X3: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_125_sup__bot__left,axiom,
! [X3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_126_sup__bot__left,axiom,
! [X3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_127_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_128_Un__empty,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_129_Un__empty,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ( A = bot_bo7198184520161983622et_nat )
& ( B = bot_bo7198184520161983622et_nat ) ) ) ).
% Un_empty
thf(fact_130_Un__empty,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ( A = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_131_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_132_bex__empty,axiom,
! [P: set_nat > $o] :
~ ? [X6: set_nat] :
( ( member_set_nat @ X6 @ bot_bot_set_set_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_133_bex__empty,axiom,
! [P: set_set_nat > $o] :
~ ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ bot_bo7198184520161983622et_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_134_bex__empty,axiom,
! [P: ( nat > nat ) > $o] :
~ ? [X6: nat > nat] :
( ( member_nat_nat @ X6 @ bot_bot_set_nat_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_135_bex__empty,axiom,
! [P: nat > $o] :
~ ? [X6: nat] :
( ( member_nat @ X6 @ bot_bot_set_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_136_acceptsI,axiom,
! [D: set_set_nat,G2: set_set_nat,X4: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D @ G2 )
=> ( ( member_set_set_nat @ D @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G2 ) ) ) ).
% acceptsI
thf(fact_137_odot___092_060G_062,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_138_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_139_union___092_060G_062,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G2 @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% union_\<G>
thf(fact_140_ACC__I,axiom,
! [G2: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G2 )
=> ( member_set_set_nat @ G2 @ ( clique3210737319928189260st_ACC @ k @ X4 ) ) ) ) ).
% ACC_I
thf(fact_141_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_142_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_143_mem__Collect__eq,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_144_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_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_152_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_153_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_154_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_155_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_156_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_157_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_158_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_159_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_160_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_161_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_162_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_163_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_164_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y3: set_set_nat] :
~ ( member_set_set_nat @ Y3 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_165_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y3: nat > nat] :
~ ( member_nat_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_166_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_167_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X2: set_nat] : $false ) ) ).
% empty_def
thf(fact_168_empty__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat
@ ^ [X2: set_set_nat] : $false ) ) ).
% empty_def
thf(fact_169_empty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X2: nat > nat] : $false ) ) ).
% empty_def
thf(fact_170_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_171_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_172_ex__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ? [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A ) )
= ( A != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_173_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_174_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_175_sameprod__finite,axiom,
! [X4: set_set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_176_sameprod__finite,axiom,
! [X4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X4 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_177_sameprod__finite,axiom,
! [X4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X4 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_178_sameprod__finite,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_179_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_180_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_181_Un__empty__right,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Un_empty_right
thf(fact_182_Un__empty__right,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Un_empty_right
thf(fact_183_Un__empty__right,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% Un_empty_right
thf(fact_184_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_185_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_186_Un__empty__left,axiom,
! [B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_187_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_188_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_189_sup__set__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: 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 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_190_sup__set__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: 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 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_191_sup__set__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: 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 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_192_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_193_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_194_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_195_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_196_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A3 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_197_numbers2__mono,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X3 ) @ ( clique3652268606331196573umbers @ X3 ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y ) @ ( clique3652268606331196573umbers @ Y ) ) ) ) ).
% numbers2_mono
thf(fact_198_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_199_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_200_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_201_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_202_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z2: set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_203_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_204_set__eq__subset,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_205_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_nat,Z2: set_set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_206_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_207_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_208_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_209_subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_210_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_211_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_212_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_213_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_214_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_215_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_216_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_217_subset__refl,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).
% subset_refl
thf(fact_218_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_219_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_220_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_221_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_222_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_223_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_224_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_225_equalityD2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_226_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_227_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_228_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_229_equalityD1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_230_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_231_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_232_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_233_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( member_set_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_234_equalityE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_235_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_236_equalityE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ~ ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_237_equalityE,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ~ ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_238_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_239_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_240_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_241_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_242_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_243_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_244_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X3: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_245_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X3: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X3 @ A )
=> ( member_set_set_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_246_sup__fun__def,axiom,
( sup_sup_nat_nat
= ( ^ [F: nat > nat,G3: nat > nat,X2: nat] : ( sup_sup_nat @ ( F @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% sup_fun_def
thf(fact_247_sup__left__commute,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_248_sup__left__commute,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_249_sup__left__commute,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_250_sup__left__commute,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_251_sup__left__commute,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ ( sup_sup_nat_nat @ Y @ Z ) )
= ( sup_sup_nat_nat @ Y @ ( sup_sup_nat_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_252_sup__left__commute,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X3 @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_253_sup_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_254_sup_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C2 ) )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_255_sup_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ C2 ) )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_256_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_257_sup_Oleft__commute,axiom,
! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
( ( sup_sup_nat_nat @ B2 @ ( sup_sup_nat_nat @ A2 @ C2 ) )
= ( sup_sup_nat_nat @ A2 @ ( sup_sup_nat_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_258_sup_Oleft__commute,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A2 @ C2 ) )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_259_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_260_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_261_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] : ( sup_sup_set_nat_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_262_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_263_sup__commute,axiom,
( sup_sup_nat_nat
= ( ^ [X2: nat > nat,Y5: nat > nat] : ( sup_sup_nat_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_264_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X2: nat,Y5: nat] : ( sup_sup_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_265_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_266_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_267_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_268_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_269_sup_Ocommute,axiom,
( sup_sup_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] : ( sup_sup_nat_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_270_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A4: nat,B4: nat] : ( sup_sup_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_271_sup__assoc,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) @ Z )
= ( sup_su4213647025997063966et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_272_sup__assoc,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_273_sup__assoc,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X3 @ Y ) @ Z )
= ( sup_sup_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_274_sup__assoc,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_275_sup__assoc,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( sup_sup_nat_nat @ ( sup_sup_nat_nat @ X3 @ Y ) @ Z )
= ( sup_sup_nat_nat @ X3 @ ( sup_sup_nat_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_276_sup__assoc,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z )
= ( sup_sup_nat @ X3 @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_277_sup_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ C2 )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_278_sup_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_279_sup_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_280_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_281_sup_Oassoc,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( sup_sup_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_nat_nat @ A2 @ ( sup_sup_nat_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_282_sup_Oassoc,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_283_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_284_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_285_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] : ( sup_sup_set_nat_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_286_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_287_inf__sup__aci_I5_J,axiom,
( sup_sup_nat_nat
= ( ^ [X2: nat > nat,Y5: nat > nat] : ( sup_sup_nat_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_288_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X2: nat,Y5: nat] : ( sup_sup_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_289_inf__sup__aci_I6_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) @ Z )
= ( sup_su4213647025997063966et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_290_inf__sup__aci_I6_J,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_291_inf__sup__aci_I6_J,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X3 @ Y ) @ Z )
= ( sup_sup_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_292_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_293_inf__sup__aci_I6_J,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( sup_sup_nat_nat @ ( sup_sup_nat_nat @ X3 @ Y ) @ Z )
= ( sup_sup_nat_nat @ X3 @ ( sup_sup_nat_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_294_inf__sup__aci_I6_J,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z )
= ( sup_sup_nat @ X3 @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_295_inf__sup__aci_I7_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_296_inf__sup__aci_I7_J,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_297_inf__sup__aci_I7_J,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_298_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_299_inf__sup__aci_I7_J,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ ( sup_sup_nat_nat @ Y @ Z ) )
= ( sup_sup_nat_nat @ Y @ ( sup_sup_nat_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_300_inf__sup__aci_I7_J,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X3 @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_301_inf__sup__aci_I8_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_302_inf__sup__aci_I8_J,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y ) )
= ( sup_sup_set_set_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_303_inf__sup__aci_I8_J,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ X3 @ Y ) )
= ( sup_sup_set_nat_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_304_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_305_inf__sup__aci_I8_J,axiom,
! [X3: nat > nat,Y: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ ( sup_sup_nat_nat @ X3 @ Y ) )
= ( sup_sup_nat_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_306_inf__sup__aci_I8_J,axiom,
! [X3: nat,Y: nat] :
( ( sup_sup_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) )
= ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_307_Un__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) )
= ( sup_su4213647025997063966et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_308_Un__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) )
= ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_309_Un__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) )
= ( sup_sup_set_nat_nat @ B @ ( sup_sup_set_nat_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_310_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_311_Un__left__absorb,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_312_Un__left__absorb,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_313_Un__left__absorb,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_314_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_315_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_316_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_317_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_318_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_319_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_320_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_321_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_322_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_323_Un__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
= ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_324_Un__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_325_Un__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
= ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_326_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_327_ball__Un,axiom,
! [A: 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 @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_328_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_329_ball__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_330_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_331_bex__Un,axiom,
! [A: 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 @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_332_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_333_bex__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_334_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_335_UnI2,axiom,
! [C2: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ B )
=> ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_336_UnI2,axiom,
! [C2: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C2 @ B )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_337_UnI2,axiom,
! [C2: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ C2 @ B )
=> ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_338_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_339_UnI1,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_340_UnI1,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_341_UnI1,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_342_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_343_UnE,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( ~ ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_344_UnE,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_345_UnE,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( ~ ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_346_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_347_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_348_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_349_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_350_Bex__def,axiom,
( bex_nat
= ( ^ [A3: set_nat,P2: nat > $o] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_351_Graphs__def,axiom,
( clique5786534781347292306Graphs
= ( ^ [V: set_nat] :
( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ V @ V ) ) ) ) ) ).
% Graphs_def
thf(fact_352_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_353_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_354_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_355_Collect__subset,axiom,
! [A: 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 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_356_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_357_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_358_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_359_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_360_Un__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
| ( member_set_set_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_361_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
| ( member_set_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_362_Un__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
| ( member_nat_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_363_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_364_sameprod__mono,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) @ ( clique8906516429304539640et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_365_sameprod__mono,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) @ ( clique134924887794942129at_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_366_sameprod__mono,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y2 )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X4 @ X4 ) @ ( clique1181040904276305582et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_367_sameprod__mono,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) @ ( clique6722202388162463298od_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_368_sup_OcoboundedI2,axiom,
! [C2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_369_sup_OcoboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_370_sup_OcoboundedI2,axiom,
! [C2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_371_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_372_sup_OcoboundedI2,axiom,
! [C2: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ B2 )
=> ( ord_less_eq_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_373_sup_OcoboundedI2,axiom,
! [C2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_374_sup_OcoboundedI1,axiom,
! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_375_sup_OcoboundedI1,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_376_sup_OcoboundedI1,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_377_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_378_sup_OcoboundedI1,axiom,
! [C2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ord_less_eq_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_379_sup_OcoboundedI1,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_380_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_381_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_382_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_383_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_384_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( sup_sup_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_385_sup_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_386_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_387_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_388_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_389_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_390_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( sup_sup_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_391_sup_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_392_sup_Ocobounded2,axiom,
! [B2: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_393_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_394_sup_Ocobounded2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_395_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_396_sup_Ocobounded2,axiom,
! [B2: nat > nat,A2: nat > nat] : ( ord_less_eq_nat_nat @ B2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_397_sup_Ocobounded2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_398_sup_Ocobounded1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_399_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_400_sup_Ocobounded1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_401_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_402_sup_Ocobounded1,axiom,
! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_403_sup_Ocobounded1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_404_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( A4
= ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_405_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_406_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( A4
= ( sup_sup_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_407_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_408_sup_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( A4
= ( sup_sup_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_409_sup_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( A4
= ( sup_su4213647025997063966et_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_410_sup_OboundedI,axiom,
! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_411_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_412_sup_OboundedI,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_413_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_414_sup_OboundedI,axiom,
! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_415_sup_OboundedI,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_416_sup_OboundedE,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_417_sup_OboundedE,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_418_sup_OboundedE,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_419_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_420_sup_OboundedE,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_421_sup_OboundedE,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_422_sup__absorb2,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ( sup_sup_set_set_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_423_sup__absorb2,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_424_sup__absorb2,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
=> ( ( sup_sup_set_nat_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_425_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_426_sup__absorb2,axiom,
! [X3: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ Y )
=> ( ( sup_sup_nat_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_427_sup__absorb2,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
=> ( ( sup_su4213647025997063966et_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_428_sup__absorb1,axiom,
! [Y: set_set_nat,X3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X3 )
=> ( ( sup_sup_set_set_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_429_sup__absorb1,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_430_sup__absorb1,axiom,
! [Y: set_nat_nat,X3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X3 )
=> ( ( sup_sup_set_nat_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_431_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_432_sup__absorb1,axiom,
! [Y: nat > nat,X3: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X3 )
=> ( ( sup_sup_nat_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_433_sup__absorb1,axiom,
! [Y: set_set_set_nat,X3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X3 )
=> ( ( sup_su4213647025997063966et_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_434_sup_Oabsorb2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_435_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_436_sup_Oabsorb2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_437_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_438_sup_Oabsorb2,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( sup_sup_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_439_sup_Oabsorb2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_440_sup_Oabsorb1,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_441_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_442_sup_Oabsorb1,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_443_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_444_sup_Oabsorb1,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( sup_sup_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_445_sup_Oabsorb1,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_446_sup__unique,axiom,
! [F2: set_set_nat > set_set_nat > set_set_nat,X3: set_set_nat,Y: set_set_nat] :
( ! [X5: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_set_nat,Y3: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X5 )
=> ( ( ord_le6893508408891458716et_nat @ Z3 @ X5 )
=> ( ord_le6893508408891458716et_nat @ ( F2 @ Y3 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup_set_set_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_447_sup__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X5: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X5 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X5 )
=> ( ord_less_eq_set_nat @ ( F2 @ Y3 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_448_sup__unique,axiom,
! [F2: set_nat_nat > set_nat_nat > set_nat_nat,X3: set_nat_nat,Y: set_nat_nat] :
( ! [X5: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_nat_nat,Y3: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X5 )
=> ( ( ord_le9059583361652607317at_nat @ Z3 @ X5 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ Y3 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup_set_nat_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_449_sup__unique,axiom,
! [F2: nat > nat > nat,X3: nat,Y: nat] :
( ! [X5: nat,Y3: nat] : ( ord_less_eq_nat @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X5 )
=> ( ( ord_less_eq_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_450_sup__unique,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat > nat,X3: nat > nat,Y: nat > nat] :
( ! [X5: nat > nat,Y3: nat > nat] : ( ord_less_eq_nat_nat @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: nat > nat,Y3: nat > nat] : ( ord_less_eq_nat_nat @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: nat > nat,Y3: nat > nat,Z3: nat > nat] :
( ( ord_less_eq_nat_nat @ Y3 @ X5 )
=> ( ( ord_less_eq_nat_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat_nat @ ( F2 @ Y3 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup_nat_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_451_sup__unique,axiom,
! [F2: set_set_set_nat > set_set_set_nat > set_set_set_nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: set_set_set_nat,Y3: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y3 @ X5 )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ X5 )
=> ( ord_le9131159989063066194et_nat @ ( F2 @ Y3 @ Z3 ) @ X5 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_452_sup_OorderI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_453_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_454_sup_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_455_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_456_sup_OorderI,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( A2
= ( sup_sup_nat_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_457_sup_OorderI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_458_sup_OorderE,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_459_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_460_sup_OorderE,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_461_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_462_sup_OorderE,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_463_sup_OorderE,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_464_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_465_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_466_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_467_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( sup_sup_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_468_le__iff__sup,axiom,
( ord_less_eq_nat_nat
= ( ^ [X2: nat > nat,Y5: nat > nat] :
( ( sup_sup_nat_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_469_le__iff__sup,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_470_sup__least,axiom,
! [Y: set_set_nat,X3: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X3 )
=> ( ( ord_le6893508408891458716et_nat @ Z @ X3 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_471_sup__least,axiom,
! [Y: set_nat,X3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ Z @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_472_sup__least,axiom,
! [Y: set_nat_nat,X3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X3 )
=> ( ( ord_le9059583361652607317at_nat @ Z @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_473_sup__least,axiom,
! [Y: nat,X3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_474_sup__least,axiom,
! [Y: nat > nat,X3: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat_nat @ Z @ X3 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_475_sup__least,axiom,
! [Y: set_set_set_nat,X3: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X3 )
=> ( ( ord_le9131159989063066194et_nat @ Z @ X3 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_476_sup__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_477_sup__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_478_sup__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_479_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_480_sup__mono,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ D2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ ( sup_sup_nat_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_481_sup__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_482_sup_Omono,axiom,
! [C2: set_set_nat,A2: set_set_nat,D2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C2 @ D2 ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_483_sup_Omono,axiom,
! [C2: set_nat,A2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_484_sup_Omono,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,D2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C2 @ D2 ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_485_sup_Omono,axiom,
! [C2: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_486_sup_Omono,axiom,
! [C2: nat > nat,A2: nat > nat,D2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ D2 @ B2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ C2 @ D2 ) @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_487_sup_Omono,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,D2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ D2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C2 @ D2 ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_488_le__supI2,axiom,
! [X3: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ B2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_489_le__supI2,axiom,
! [X3: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_490_le__supI2,axiom,
! [X3: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_491_le__supI2,axiom,
! [X3: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_492_le__supI2,axiom,
! [X3: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ B2 )
=> ( ord_less_eq_nat_nat @ X3 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_493_le__supI2,axiom,
! [X3: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ B2 )
=> ( ord_le9131159989063066194et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_494_le__supI1,axiom,
! [X3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_495_le__supI1,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_496_le__supI1,axiom,
! [X3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_497_le__supI1,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_498_le__supI1,axiom,
! [X3: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_nat_nat @ X3 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_499_le__supI1,axiom,
! [X3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_500_sup__ge2,axiom,
! [Y: set_set_nat,X3: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_501_sup__ge2,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_502_sup__ge2,axiom,
! [Y: set_nat_nat,X3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_503_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_504_sup__ge2,axiom,
! [Y: nat > nat,X3: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_505_sup__ge2,axiom,
! [Y: set_set_set_nat,X3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_506_sup__ge1,axiom,
! [X3: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_507_sup__ge1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_508_sup__ge1,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ ( sup_sup_set_nat_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_509_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_510_sup__ge1,axiom,
! [X3: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X3 @ ( sup_sup_nat_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_511_sup__ge1,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_512_le__supI,axiom,
! [A2: set_set_nat,X3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X3 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ X3 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_513_le__supI,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_514_le__supI,axiom,
! [A2: set_nat_nat,X3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X3 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_515_le__supI,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_516_le__supI,axiom,
! [A2: nat > nat,X3: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat_nat @ B2 @ X3 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_517_le__supI,axiom,
! [A2: set_set_set_nat,X3: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X3 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ X3 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_518_le__supE,axiom,
! [A2: set_set_nat,B2: set_set_nat,X3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X3 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_519_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_520_le__supE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X3 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_521_le__supE,axiom,
! [A2: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_522_le__supE,axiom,
! [A2: nat > nat,B2: nat > nat,X3: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_523_le__supE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ X3 )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_524_inf__sup__ord_I3_J,axiom,
! [X3: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_525_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_526_inf__sup__ord_I3_J,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ ( sup_sup_set_nat_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_527_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_528_inf__sup__ord_I3_J,axiom,
! [X3: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X3 @ ( sup_sup_nat_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_529_inf__sup__ord_I3_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_530_inf__sup__ord_I4_J,axiom,
! [Y: set_set_nat,X3: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_531_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_532_inf__sup__ord_I4_J,axiom,
! [Y: set_nat_nat,X3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_533_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_534_inf__sup__ord_I4_J,axiom,
! [Y: nat > nat,X3: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_535_inf__sup__ord_I4_J,axiom,
! [Y: set_set_set_nat,X3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_536_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_537_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_538_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_539_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_540_subset__UnE,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ~ ! [A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ A )
=> ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_541_subset__UnE,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_542_subset__UnE,axiom,
! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ~ ! [A5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ A )
=> ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_nat_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_543_subset__UnE,axiom,
! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ~ ! [A5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ A )
=> ! [B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ B )
=> ( C
!= ( sup_su4213647025997063966et_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_544_Un__absorb2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_545_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_546_Un__absorb2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_547_Un__absorb2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( sup_su4213647025997063966et_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_548_Un__absorb1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_549_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_550_Un__absorb1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_551_Un__absorb1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( sup_su4213647025997063966et_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_552_Un__upper2,axiom,
! [B: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_553_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_554_Un__upper2,axiom,
! [B: set_nat_nat,A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_555_Un__upper2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_556_Un__upper1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_557_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_558_Un__upper1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_559_Un__upper1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_560_Un__least,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_561_Un__least,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_562_Un__least,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_563_Un__least,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_564_Un__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_565_Un__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_566_Un__mono,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_567_Un__mono,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ D )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_568_v___092_060G_062__2,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ).
% v_\<G>_2
thf(fact_569_finite__Collect__subsets,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_570_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_571_finite__Collect__subsets,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( finite3586981331298542604at_nat
@ ( collect_set_nat_nat
@ ^ [B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_572_finite__Collect__subsets,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [B3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_573_ACC__cf__union,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_union
thf(fact_574_finite__Collect__bex,axiom,
! [A: set_nat,Q: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y5: nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_575_finite__Collect__bex,axiom,
! [A: set_set_nat,Q: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y5: nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_576_finite__Collect__bex,axiom,
! [A: set_nat,Q: set_nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Y5: set_nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_577_finite__Collect__bex,axiom,
! [A: set_set_nat,Q: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Y5: set_nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_578_finite__Collect__bex,axiom,
! [A: set_nat,Q: set_set_nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Y5: set_set_nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_579_finite__Collect__bex,axiom,
! [A: set_nat,Q: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [Y5: nat > nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_580_finite__Collect__bex,axiom,
! [A: set_set_set_nat,Q: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y5: nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_581_finite__Collect__bex,axiom,
! [A: set_nat_nat,Q: nat > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y5: nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_582_finite__Collect__bex,axiom,
! [A: set_set_nat,Q: set_set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Y5: set_set_nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_583_finite__Collect__bex,axiom,
! [A: set_set_nat,Q: ( nat > nat ) > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A )
& ( Q @ X2 @ Y5 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [Y5: nat > nat] : ( Q @ Y5 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_584_v___092_060G_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_585_finite__Un,axiom,
! [F3: set_set_set_nat,G2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F3 @ G2 ) )
= ( ( finite6739761609112101331et_nat @ F3 )
& ( finite6739761609112101331et_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_586_finite__Un,axiom,
! [F3: set_set_nat,G2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F3 @ G2 ) )
= ( ( finite1152437895449049373et_nat @ F3 )
& ( finite1152437895449049373et_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_587_finite__Un,axiom,
! [F3: set_nat_nat,G2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F3 @ G2 ) )
= ( ( finite2115694454571419734at_nat @ F3 )
& ( finite2115694454571419734at_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_588_finite__Un,axiom,
! [F3: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G2 ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_589_ACC__cf__mono,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_mono
thf(fact_590_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_finite__has__maximal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_600_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_601_finite__has__maximal,axiom,
! [A: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ? [X5: set_nat_nat] :
( ( member_set_nat_nat @ X5 @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_602_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_603_finite__has__maximal,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_604_finite__has__maximal,axiom,
! [A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( A != bot_bo193956671110832956et_nat )
=> ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_605_finite__has__minimal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_606_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_607_finite__has__minimal,axiom,
! [A: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ? [X5: set_nat_nat] :
( ( member_set_nat_nat @ X5 @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_608_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_609_finite__has__minimal,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_610_finite__has__minimal,axiom,
! [A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( A != bot_bo193956671110832956et_nat )
=> ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_611_v__union,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_612_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_613_v__mono,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G2 @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_614_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_615_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_616_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_617_finite__vG,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G2 ) ) ) ).
% finite_vG
thf(fact_618_psubsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_619_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_620_psubsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_621_psubsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_622_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_623_finite__ACC,axiom,
! [X4: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) ) ).
% finite_ACC
thf(fact_624_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_625_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_626_psubsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: nat > nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_627_psubsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_628_psubsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_629_psubsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_630_psubset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% psubset_trans
thf(fact_631_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_632_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_633_finite__psubset__induct,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ! [B6: set_set_nat] :
( ( ord_less_set_set_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_634_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_635_finite__psubset__induct,axiom,
! [A: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ A )
=> ( ! [A6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A6 )
=> ( ! [B6: set_nat_nat] :
( ( ord_less_set_nat_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_636_finite__psubset__induct,axiom,
! [A: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [A6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A6 )
=> ( ! [B6: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_637_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_638_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_639_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_640_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_641_psubsetE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_642_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_643_psubsetE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_644_psubsetE,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_645_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_646_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_647_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_648_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_649_psubset__imp__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_650_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_651_psubset__imp__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_652_psubset__imp__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_653_psubset__subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_654_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_655_psubset__subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_656_psubset__subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_657_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_658_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_659_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_660_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ~ ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_661_subset__psubset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_662_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_663_subset__psubset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_664_subset__psubset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_665_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_666_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_667_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_668_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_669_not__psubset__empty,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_670_not__psubset__empty,axiom,
! [A: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_671_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_672_not__psubset__empty,axiom,
! [A: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% not_psubset_empty
thf(fact_673_less__supI1,axiom,
! [X3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ A2 )
=> ( ord_less_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_674_less__supI1,axiom,
! [X3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ A2 )
=> ( ord_less_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_675_less__supI1,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ X3 @ A2 )
=> ( ord_less_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_676_less__supI1,axiom,
! [X3: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_nat_nat @ X3 @ A2 )
=> ( ord_less_nat_nat @ X3 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_677_less__supI1,axiom,
! [X3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X3 @ A2 )
=> ( ord_le152980574450754630et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_678_less__supI1,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X3 @ A2 )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_679_less__supI2,axiom,
! [X3: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ B2 )
=> ( ord_less_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_680_less__supI2,axiom,
! [X3: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ B2 )
=> ( ord_less_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_681_less__supI2,axiom,
! [X3: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ X3 @ B2 )
=> ( ord_less_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_682_less__supI2,axiom,
! [X3: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_nat_nat @ X3 @ B2 )
=> ( ord_less_nat_nat @ X3 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_683_less__supI2,axiom,
! [X3: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X3 @ B2 )
=> ( ord_le152980574450754630et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_684_less__supI2,axiom,
! [X3: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X3 @ B2 )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_685_sup_Oabsorb3,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_686_sup_Oabsorb3,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_687_sup_Oabsorb3,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_688_sup_Oabsorb3,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_nat_nat @ B2 @ A2 )
=> ( ( sup_sup_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_689_sup_Oabsorb3,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_690_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_691_sup_Oabsorb4,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_692_sup_Oabsorb4,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_693_sup_Oabsorb4,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_694_sup_Oabsorb4,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_nat_nat @ A2 @ B2 )
=> ( ( sup_sup_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_695_sup_Oabsorb4,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_696_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_697_sup_Ostrict__boundedE,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_set_nat @ B2 @ A2 )
=> ~ ( ord_less_set_set_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_698_sup_Ostrict__boundedE,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_set_nat_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_699_sup_Ostrict__boundedE,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_nat @ B2 @ A2 )
=> ~ ( ord_less_set_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_700_sup_Ostrict__boundedE,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_nat_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_701_sup_Ostrict__boundedE,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ~ ( ord_le152980574450754630et_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_702_sup_Ostrict__boundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_703_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( A4
= ( sup_sup_set_set_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_704_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( A4
= ( sup_sup_set_nat_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_705_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( A4
= ( sup_sup_set_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_706_sup_Ostrict__order__iff,axiom,
( ord_less_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( A4
= ( sup_sup_nat_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_707_sup_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( A4
= ( sup_su4213647025997063966et_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_708_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_709_sup_Ostrict__coboundedI1,axiom,
! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ C2 @ A2 )
=> ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_710_sup_Ostrict__coboundedI1,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ C2 @ A2 )
=> ( ord_less_set_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_711_sup_Ostrict__coboundedI1,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ C2 @ A2 )
=> ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_712_sup_Ostrict__coboundedI1,axiom,
! [C2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_nat_nat @ C2 @ A2 )
=> ( ord_less_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_713_sup_Ostrict__coboundedI1,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ C2 @ A2 )
=> ( ord_le152980574450754630et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_714_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C2 @ A2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_715_sup_Ostrict__coboundedI2,axiom,
! [C2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ C2 @ B2 )
=> ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_716_sup_Ostrict__coboundedI2,axiom,
! [C2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ C2 @ B2 )
=> ( ord_less_set_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_717_sup_Ostrict__coboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ C2 @ B2 )
=> ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_718_sup_Ostrict__coboundedI2,axiom,
! [C2: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_nat_nat @ C2 @ B2 )
=> ( ord_less_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_719_sup_Ostrict__coboundedI2,axiom,
! [C2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ C2 @ B2 )
=> ( ord_le152980574450754630et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_720_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_721_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_722_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_723_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_set_nat,R: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_724_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_set_nat,R: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_725_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_set_set_nat,R: nat > set_set_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_726_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat_nat,R: nat > ( nat > nat ) > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_727_pigeonhole__infinite__rel,axiom,
! [A: set_set_set_nat,B: set_nat,R: set_set_nat > nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_728_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat,B: set_nat,R: ( nat > nat ) > nat > $o] :
( ~ ( finite2115694454571419734at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A4: nat > nat] :
( ( member_nat_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_729_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_set_set_nat,R: set_nat > set_set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_730_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_nat_nat,R: set_nat > ( nat > nat ) > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_731_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_732_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_733_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_734_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_735_finite__has__maximal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
& ( ord_le6893508408891458716et_nat @ A2 @ X5 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_736_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ( ord_less_eq_set_nat @ A2 @ X5 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_737_finite__has__maximal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X5: set_nat_nat] :
( ( member_set_nat_nat @ X5 @ A )
& ( ord_le9059583361652607317at_nat @ A2 @ X5 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_738_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_nat @ A2 @ X5 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_739_finite__has__maximal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
& ( ord_less_eq_nat_nat @ A2 @ X5 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_740_finite__has__maximal2,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ A )
& ( ord_le9131159989063066194et_nat @ A2 @ X5 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_741_finite__has__minimal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
& ( ord_le6893508408891458716et_nat @ X5 @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_742_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ( ord_less_eq_set_nat @ X5 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_743_finite__has__minimal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X5: set_nat_nat] :
( ( member_set_nat_nat @ X5 @ A )
& ( ord_le9059583361652607317at_nat @ X5 @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_744_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_nat @ X5 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_745_finite__has__minimal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
& ( ord_less_eq_nat_nat @ X5 @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_746_finite__has__minimal2,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ A )
& ( ord_le9131159989063066194et_nat @ X5 @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_747_finite__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_subset
thf(fact_748_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_749_finite__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( finite2115694454571419734at_nat @ A ) ) ) ).
% finite_subset
thf(fact_750_finite__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( finite6739761609112101331et_nat @ A ) ) ) ).
% finite_subset
thf(fact_751_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_752_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_753_infinite__super,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_754_infinite__super,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_755_rev__finite__subset,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_756_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_757_rev__finite__subset,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( finite2115694454571419734at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_758_rev__finite__subset,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( finite6739761609112101331et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_759_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_760_infinite__imp__nonempty,axiom,
! [S: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ( S != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_761_infinite__imp__nonempty,axiom,
! [S: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ( S != bot_bot_set_nat_nat ) ) ).
% infinite_imp_nonempty
thf(fact_762_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_763_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_764_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_765_finite_OemptyI,axiom,
finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).
% finite.emptyI
thf(fact_766_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_767_finite__UnI,axiom,
! [F3: set_set_set_nat,G2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( finite6739761609112101331et_nat @ G2 )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F3 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_768_finite__UnI,axiom,
! [F3: set_set_nat,G2: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( finite1152437895449049373et_nat @ G2 )
=> ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F3 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_769_finite__UnI,axiom,
! [F3: set_nat_nat,G2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( ( finite2115694454571419734at_nat @ G2 )
=> ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F3 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_770_finite__UnI,axiom,
! [F3: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_771_Un__infinite,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_772_Un__infinite,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_773_Un__infinite,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_774_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_775_infinite__Un,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) ) )
= ( ~ ( finite6739761609112101331et_nat @ S )
| ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_776_infinite__Un,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) )
= ( ~ ( finite1152437895449049373et_nat @ S )
| ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_777_infinite__Un,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) ) )
= ( ~ ( finite2115694454571419734at_nat @ S )
| ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_778_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_779_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_780_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_781__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [K2: set_set_nat] :
( ( member_set_set_nat @ K2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K2 ) )
= k )
& ( K2
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K2 ) @ ( clique5033774636164728513irst_v @ K2 ) ) ) ) ) ) ).
% \<K>_def
thf(fact_782_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_783_ex__min__if__finite,axiom,
! [S: set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ S )
& ~ ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ S )
& ( ord_less_set_nat @ Xa @ X5 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_784_ex__min__if__finite,axiom,
! [S: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( S != bot_bo7198184520161983622et_nat )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ S )
& ~ ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ S )
& ( ord_less_set_set_nat @ Xa @ X5 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_785_ex__min__if__finite,axiom,
! [S: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S )
=> ( ( S != bot_bot_set_nat_nat )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S )
& ~ ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S )
& ( ord_less_nat_nat @ Xa @ X5 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_786_ex__min__if__finite,axiom,
! [S: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ S )
=> ( ( S != bot_bo193956671110832956et_nat )
=> ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ S )
& ~ ? [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ S )
& ( ord_le152980574450754630et_nat @ Xa @ X5 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_787_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_788_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X5 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_789_sup__Un__eq,axiom,
! [R: set_set_set_nat,S: set_set_set_nat] :
( ( sup_su5917979686466268903_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ R )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ S ) )
= ( ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ ( sup_su4213647025997063966et_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_790_sup__Un__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ R )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ S ) )
= ( ^ [X2: set_nat] : ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_791_sup__Un__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ R )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ S ) )
= ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_792_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_793_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_794_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_795_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_796_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_797_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_798_dual__order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_799_dual__order_Orefl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_800_order__refl,axiom,
! [X3: set_set_nat] : ( ord_le6893508408891458716et_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_801_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_802_order__refl,axiom,
! [X3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_803_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_804_order__refl,axiom,
! [X3: nat > nat] : ( ord_less_eq_nat_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_805_order__refl,axiom,
! [X3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_806_inf__right__idem,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) @ Y )
= ( inf_in5711780100303410308et_nat @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_807_inf_Oright__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ B2 )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_808_inf__left__idem,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) )
= ( inf_in5711780100303410308et_nat @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_809_inf_Oleft__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_810_inf__idem,axiom,
! [X3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_811_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_812_Int__iff,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C2 @ A )
& ( member_nat_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_813_Int__iff,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C2 @ A )
& ( member_set_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_814_Int__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_815_Int__iff,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C2 @ A )
& ( member_set_set_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_816_IntI,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ A )
=> ( ( member_nat_nat @ C2 @ B )
=> ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_817_IntI,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ A )
=> ( ( member_set_nat @ C2 @ B )
=> ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_818_IntI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_819_IntI,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ A )
=> ( ( member_set_set_nat @ C2 @ B )
=> ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_820_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_821_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_822_inf_Obounded__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_823_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_824_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_825_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_826_inf_Obounded__iff,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat_nat @ A2 @ B2 )
& ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_827_inf_Obounded__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_828_le__inf__iff,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X3 @ Y )
& ( ord_le6893508408891458716et_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_829_le__inf__iff,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y )
& ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_830_le__inf__iff,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X3 @ Y )
& ( ord_le9059583361652607317at_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_831_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_832_le__inf__iff,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ ( inf_inf_nat_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat_nat @ X3 @ Y )
& ( ord_less_eq_nat_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_833_le__inf__iff,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( ( ord_le9131159989063066194et_nat @ X3 @ Y )
& ( ord_le9131159989063066194et_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_834_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X3 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_835_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X3 )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_836_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X3 )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_837_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_838_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_839_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_840_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_841_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_842_inf__bot__left,axiom,
! [X3: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X3 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_843_inf__bot__left,axiom,
! [X3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X3 )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_844_inf__bot__left,axiom,
! [X3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X3 )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_845_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_846_inf__bot__right,axiom,
! [X3: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_847_inf__bot__right,axiom,
! [X3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_848_inf__bot__right,axiom,
! [X3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_849_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_850_Int__subset__iff,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_le6893508408891458716et_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_851_Int__subset__iff,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_852_Int__subset__iff,axiom,
! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C @ A )
& ( ord_le9059583361652607317at_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_853_Int__subset__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C @ A )
& ( ord_le9131159989063066194et_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_854_finite__Int,axiom,
! [F3: set_set_nat,G2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F3 )
| ( finite1152437895449049373et_nat @ G2 ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_855_finite__Int,axiom,
! [F3: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_856_finite__Int,axiom,
! [F3: set_nat_nat,G2: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F3 )
| ( finite2115694454571419734at_nat @ G2 ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_857_finite__Int,axiom,
! [F3: set_set_set_nat,G2: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F3 )
| ( finite6739761609112101331et_nat @ G2 ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_858_sup__inf__absorb,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_859_sup__inf__absorb,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( inf_inf_set_set_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_860_sup__inf__absorb,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( inf_inf_set_nat_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_861_sup__inf__absorb,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_862_sup__inf__absorb,axiom,
! [X3: nat > nat,Y: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ ( inf_inf_nat_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_863_sup__inf__absorb,axiom,
! [X3: nat,Y: nat] :
( ( sup_sup_nat @ X3 @ ( inf_inf_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_864_inf__sup__absorb,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_865_inf__sup__absorb,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_866_inf__sup__absorb,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_867_inf__sup__absorb,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_868_inf__sup__absorb,axiom,
! [X3: nat > nat,Y: nat > nat] :
( ( inf_inf_nat_nat @ X3 @ ( sup_sup_nat_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_869_inf__sup__absorb,axiom,
! [X3: nat,Y: nat] :
( ( inf_inf_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_903_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_904_less__fun__def,axiom,
( ord_less_nat_nat
= ( ^ [F: nat > nat,G3: nat > nat] :
( ( ord_less_eq_nat_nat @ F @ G3 )
& ~ ( ord_less_eq_nat_nat @ G3 @ F ) ) ) ) ).
% less_fun_def
thf(fact_905_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_906_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) @ ( inf_in5711780100303410308et_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_907_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X3 @ Y ) @ ( inf_inf_set_set_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_908_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X3 @ Y ) @ ( inf_inf_set_nat_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_909_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_910_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) @ ( sup_su4213647025997063966et_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_911_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y ) @ ( sup_sup_set_set_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_912_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X3 @ Y ) @ ( sup_sup_set_nat_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_913_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_914_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_set_nat,Z: set_set_set_nat,X3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z ) @ X3 )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y @ X3 ) @ ( inf_in5711780100303410308et_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_915_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_nat,Z: set_set_nat,X3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ Z ) @ X3 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ X3 ) @ ( inf_inf_set_set_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_916_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat_nat,Z: set_nat_nat,X3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y @ Z ) @ X3 )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y @ X3 ) @ ( inf_inf_set_nat_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_917_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X3 ) @ ( inf_inf_set_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_918_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_set_nat,Z: set_set_set_nat,X3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y @ Z ) @ X3 )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y @ X3 ) @ ( sup_su4213647025997063966et_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_919_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_nat,Z: set_set_nat,X3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ Z ) @ X3 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ X3 ) @ ( sup_sup_set_set_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_920_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat_nat,Z: set_nat_nat,X3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y @ Z ) @ X3 )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y @ X3 ) @ ( sup_sup_set_nat_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_921_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X3 ) @ ( sup_sup_set_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_922_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_923_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_924_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_925_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_926_Int__Collect,axiom,
! [X3: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_927_Int__Collect,axiom,
! [X3: nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X3 @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) )
= ( ( member_nat_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_928_Int__Collect,axiom,
! [X3: set_nat,A: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X3 @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_929_Int__Collect,axiom,
! [X3: set_set_nat,A: set_set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ X3 @ ( inf_in5711780100303410308et_nat @ A @ ( collect_set_set_nat @ P ) ) )
= ( ( member_set_set_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_930_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_931_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
& ( member_nat_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_932_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
& ( member_set_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_933_Int__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
& ( member_set_set_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_934_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_set_set_nat,K: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A
= ( inf_in5711780100303410308et_nat @ K @ A2 ) )
=> ( ( inf_in5711780100303410308et_nat @ A @ B2 )
= ( inf_in5711780100303410308et_nat @ K @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_935_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_set_nat,K: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( B
= ( inf_in5711780100303410308et_nat @ K @ B2 ) )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= ( inf_in5711780100303410308et_nat @ K @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_936_Int__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) )
= ( inf_in5711780100303410308et_nat @ B @ ( inf_in5711780100303410308et_nat @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_937_Int__left__absorb,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_938_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_939_Int__absorb,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_940_Int__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_941_IntD2,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_942_IntD2,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( member_set_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_943_IntD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_944_IntD2,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_945_IntD1,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_946_IntD1,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( member_set_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_947_IntD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_948_IntD1,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_949_IntE,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C2 @ A )
=> ~ ( member_nat_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_950_IntE,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C2 @ A )
=> ~ ( member_set_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_951_IntE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_952_IntE,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C2 @ A )
=> ~ ( member_set_set_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_953_inf__left__commute,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y @ ( inf_in5711780100303410308et_nat @ X3 @ Z ) ) ) ).
% inf_left_commute
thf(fact_954_inf_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_955_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y5 @ X2 ) ) ) ).
% inf_commute
thf(fact_956_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_957_inf__assoc,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) @ Z )
= ( inf_in5711780100303410308et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_958_inf_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_959_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_960_inf__sup__aci_I2_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) @ Z )
= ( inf_in5711780100303410308et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_961_inf__sup__aci_I3_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y @ ( inf_in5711780100303410308et_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_962_inf__sup__aci_I4_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) )
= ( inf_in5711780100303410308et_nat @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_963_psubset__card__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_less_set_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_964_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_965_psubset__card__mono,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat_nat @ A ) @ ( finite_card_nat_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_966_psubset__card__mono,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ A ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_967_inf_OcoboundedI2,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_968_inf_OcoboundedI2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_969_inf_OcoboundedI2,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_970_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_971_inf_OcoboundedI2,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ C2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_972_inf_OcoboundedI2,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_973_inf_OcoboundedI1,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_974_inf_OcoboundedI1,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_975_inf_OcoboundedI1,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_976_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_977_inf_OcoboundedI1,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_978_inf_OcoboundedI1,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_979_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_980_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_981_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_982_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_983_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( inf_inf_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_984_inf_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_985_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_986_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_987_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_988_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_989_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( inf_inf_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_990_inf_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_991_inf_Ocobounded2,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_992_inf_Ocobounded2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_993_inf_Ocobounded2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_994_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_995_inf_Ocobounded2,axiom,
! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_996_inf_Ocobounded2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_997_inf_Ocobounded1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_998_inf_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_999_inf_Ocobounded1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1000_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1001_inf_Ocobounded1,axiom,
! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1002_inf_Ocobounded1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1003_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( A4
= ( inf_inf_set_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1004_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1005_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( A4
= ( inf_inf_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1006_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1007_inf_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( A4
= ( inf_inf_nat_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1008_inf_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( A4
= ( inf_in5711780100303410308et_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1009_inf__greatest,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Z )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1010_inf__greatest,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ X3 @ Z )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1011_inf__greatest,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Z )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1012_inf__greatest,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1013_inf__greatest,axiom,
! [X3: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat_nat @ X3 @ Z )
=> ( ord_less_eq_nat_nat @ X3 @ ( inf_inf_nat_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1014_inf__greatest,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ Z )
=> ( ord_le9131159989063066194et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1015_inf_OboundedI,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1016_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1017_inf_OboundedI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1018_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1019_inf_OboundedI,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1020_inf_OboundedI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1021_inf_OboundedE,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1022_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1023_inf_OboundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1024_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1025_inf_OboundedE,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1026_inf_OboundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ~ ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1027_inf__absorb2,axiom,
! [Y: set_set_nat,X3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X3 )
=> ( ( inf_inf_set_set_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1028_inf__absorb2,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1029_inf__absorb2,axiom,
! [Y: set_nat_nat,X3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X3 )
=> ( ( inf_inf_set_nat_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1030_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1031_inf__absorb2,axiom,
! [Y: nat > nat,X3: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X3 )
=> ( ( inf_inf_nat_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1032_inf__absorb2,axiom,
! [Y: set_set_set_nat,X3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X3 )
=> ( ( inf_in5711780100303410308et_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1033_inf__absorb1,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ( inf_inf_set_set_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1034_inf__absorb1,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1035_inf__absorb1,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
=> ( ( inf_inf_set_nat_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1036_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1037_inf__absorb1,axiom,
! [X3: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ Y )
=> ( ( inf_inf_nat_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1038_inf__absorb1,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
=> ( ( inf_in5711780100303410308et_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1039_inf_Oabsorb2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1040_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1041_inf_Oabsorb2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1042_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1043_inf_Oabsorb2,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( inf_inf_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1044_inf_Oabsorb2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1045_inf_Oabsorb1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1046_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1047_inf_Oabsorb1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1048_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1049_inf_Oabsorb1,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( inf_inf_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1050_inf_Oabsorb1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1051_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1052_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( inf_inf_set_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1053_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1054_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( inf_inf_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1055_le__iff__inf,axiom,
( ord_less_eq_nat_nat
= ( ^ [X2: nat > nat,Y5: nat > nat] :
( ( inf_inf_nat_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1056_le__iff__inf,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1057_inf__unique,axiom,
! [F2: set_set_nat > set_set_nat > set_set_nat,X3: set_set_nat,Y: set_set_nat] :
( ! [X5: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: set_set_nat,Y3: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X5 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1058_inf__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X5: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X5 @ Z3 )
=> ( ord_less_eq_set_nat @ X5 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1059_inf__unique,axiom,
! [F2: set_nat_nat > set_nat_nat > set_nat_nat,X3: set_nat_nat,Y: set_nat_nat] :
( ! [X5: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: set_nat_nat,Y3: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X5 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ X5 @ Z3 )
=> ( ord_le9059583361652607317at_nat @ X5 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1060_inf__unique,axiom,
! [F2: nat > nat > nat,X3: nat,Y: nat] :
( ! [X5: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
=> ( ( ord_less_eq_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1061_inf__unique,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat > nat,X3: nat > nat,Y: nat > nat] :
( ! [X5: nat > nat,Y3: nat > nat] : ( ord_less_eq_nat_nat @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: nat > nat,Y3: nat > nat] : ( ord_less_eq_nat_nat @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: nat > nat,Y3: nat > nat,Z3: nat > nat] :
( ( ord_less_eq_nat_nat @ X5 @ Y3 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat_nat @ X5 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1062_inf__unique,axiom,
! [F2: set_set_set_nat > set_set_set_nat > set_set_set_nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: set_set_set_nat,Y3: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y3 )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Z3 )
=> ( ord_le9131159989063066194et_nat @ X5 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1063_inf_OorderI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2
= ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1064_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1065_inf_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1066_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1067_inf_OorderI,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( A2
= ( inf_inf_nat_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1068_inf_OorderI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1069_inf_OorderE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1070_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1071_inf_OorderE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1072_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1073_inf_OorderE,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1074_inf_OorderE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( A2
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1075_le__infI2,axiom,
! [B2: set_set_nat,X3: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ X3 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1076_le__infI2,axiom,
! [B2: set_nat,X3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1077_le__infI2,axiom,
! [B2: set_nat_nat,X3: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1078_le__infI2,axiom,
! [B2: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1079_le__infI2,axiom,
! [B2: nat > nat,X3: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ X3 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1080_le__infI2,axiom,
! [B2: set_set_set_nat,X3: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ X3 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1081_le__infI1,axiom,
! [A2: set_set_nat,X3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X3 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1082_le__infI1,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1083_le__infI1,axiom,
! [A2: set_nat_nat,X3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1084_le__infI1,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1085_le__infI1,axiom,
! [A2: nat > nat,X3: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X3 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1086_le__infI1,axiom,
! [A2: set_set_set_nat,X3: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X3 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1087_inf__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1088_inf__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1089_inf__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1090_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1091_inf__mono,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ D2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ ( inf_inf_nat_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1092_inf__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1093_le__infI,axiom,
! [X3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ B2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1094_le__infI,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1095_le__infI,axiom,
! [X3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1096_le__infI,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1097_le__infI,axiom,
! [X3: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat_nat @ X3 @ B2 )
=> ( ord_less_eq_nat_nat @ X3 @ ( inf_inf_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1098_le__infI,axiom,
! [X3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ B2 )
=> ( ord_le9131159989063066194et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1099_le__infE,axiom,
! [X3: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X3 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1100_le__infE,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1101_le__infE,axiom,
! [X3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X3 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1102_le__infE,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1103_le__infE,axiom,
! [X3: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ ( inf_inf_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1104_le__infE,axiom,
! [X3: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ X3 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1105_inf__le2,axiom,
! [X3: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1106_inf__le2,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1107_inf__le2,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1108_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1109_inf__le2,axiom,
! [X3: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1110_inf__le2,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1111_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1112_inf__le1,axiom,
! [X3: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1113_inf__le1,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1114_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_1115__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_1116_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_1117_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_1118_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1119_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat_nat )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S2 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa2 )
| ( ord_less_eq_nat_nat @ Xa2 @ X5 ) ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S2 )
=> ( ord_less_eq_nat_nat @ Xa @ X5 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1120_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1121_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_1122_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1123_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M: nat] :
( ( P @ X3 )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1124_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1125_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ N4 )
=> ( ord_less_nat @ X5 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1126_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_1127_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1128_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F2 @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1129_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat_nat )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S2 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa2 )
| ( ord_less_eq_nat_nat @ Xa2 @ X5 ) ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S2 )
=> ( ord_less_eq_nat_nat @ X5 @ Xa ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1130_Clique__def,axiom,
( clique6749503327923060270Clique
= ( ^ [V: set_nat,K3: nat] :
( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ V ) )
& ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ C3 @ C3 ) @ G )
& ( ( finite_card_nat @ C3 )
= K3 ) ) ) ) ) ) ).
% Clique_def
thf(fact_1131_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1132_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1133_odot__def,axiom,
( clique5469973757772500719t_odot
= ( ^ [X: set_set_set_nat,Y6: 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 @ X )
& ( member_set_set_nat @ E @ Y6 ) ) ) ) ) ).
% odot_def
thf(fact_1134_first__assumptions_Oodot__def,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5469973757772500719t_odot @ X4 @ Y2 )
= ( 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 @ Y2 ) ) ) ) ) ).
% first_assumptions.odot_def
thf(fact_1135_first__assumptions_ONEG__def,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1136_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P3: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_1137_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3686358387679108662ccepts @ X4 @ G2 )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X4 )
& ( ord_le6893508408891458716et_nat @ X2 @ G2 ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1138_first__assumptions_OacceptsI,axiom,
! [L: nat,P3: nat,K: nat,D: set_set_nat,G2: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D @ G2 )
=> ( ( member_set_set_nat @ D @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G2 ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1139_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1140_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1141_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1142_first__assumptions_OACC__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1143_first__assumptions_Ov__mono,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G2 @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1144_first__assumptions_Ov__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_1145_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1146_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1147_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P3: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_1148_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1149_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1150_first__assumptions_Ov__union,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1151_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1152_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_1153_first__assumptions_Ounion___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G2 @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.union_\<G>
thf(fact_1154_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_1155_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ G2 ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_1156_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1157_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P3: nat,K: nat,F3: nat > nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1158_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_1159_first__assumptions_OACC__cf__def,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ X4 )
= ( collect_nat_nat
@ ^ [F4: nat > nat] :
( ( member_nat_nat @ F4 @ ( clique2971579238625216137irst_F @ K ) )
& ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K @ F4 ) ) ) ) ) ) ).
% first_assumptions.ACC_cf_def
thf(fact_1160_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_1161_first__assumptions_O_092_060K_062___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.\<K>_\<G>
thf(fact_1162_first__assumptions_ONEG___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.NEG_\<G>
thf(fact_1163_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1164_first__assumptions_OACC__I,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G2 )
=> ( member_set_set_nat @ G2 @ ( clique3210737319928189260st_ACC @ K @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_1165_first__assumptions_O_092_060G_062__def,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.\<G>_def
thf(fact_1166_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1167_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_1168_first__assumptions_OACC__def,axiom,
! [L: nat,P3: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ X4 )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ) ) ).
% first_assumptions.ACC_def
thf(fact_1169_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_1170_first__assumptions_O_092_060K_062__def,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3326749438856946062irst_K @ K )
= ( collect_set_set_nat
@ ^ [K2: set_set_nat] :
( ( member_set_set_nat @ K2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K2 ) )
= K )
& ( K2
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K2 ) @ ( clique5033774636164728513irst_v @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.\<K>_def
thf(fact_1171_first__assumptions_O_092_060K_062__altdef,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( 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 ) ) ) ) ) ).
% first_assumptions.\<K>_altdef
thf(fact_1172_first__assumptions_OCLIQUE__def,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique363107459185959606CLIQUE @ K )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( clique3326749438856946062irst_K @ K ) )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) ) ) ) ).
% first_assumptions.CLIQUE_def
thf(fact_1173_C__def,axiom,
! [F2: nat > nat] :
( ( clique5033774636164728462irst_C @ k @ F2 )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: nat,Y5: nat] :
( ( Uu
= ( insert_nat @ X2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( F2 @ X2 )
!= ( F2 @ Y5 ) ) ) ) ) ).
% C_def
thf(fact_1174_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1175_first__assumptions_Okml,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1176_v__def,axiom,
( clique5033774636164728513irst_v
= ( ^ [G: set_set_nat] :
( collect_nat
@ ^ [X2: nat] :
? [Y5: nat] : ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) @ G ) ) ) ) ).
% v_def
thf(fact_1177_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1178_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1179_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_1180_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_1181_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_1182_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1183_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1184_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1185_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ( P @ M5 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1186_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1187_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_1188_less__not__refl2,axiom,
! [N: nat,M4: nat] :
( ( ord_less_nat @ N @ M4 )
=> ( M4 != N ) ) ).
% less_not_refl2
thf(fact_1189_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1190_nat__neq__iff,axiom,
! [M4: nat,N: nat] :
( ( M4 != N )
= ( ( ord_less_nat @ M4 @ N )
| ( ord_less_nat @ N @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_1191_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1192_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1193_first__assumptions_Ov__def,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728513irst_v @ G2 )
= ( collect_nat
@ ^ [X2: nat] :
? [Y5: nat] : ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) @ G2 ) ) ) ) ).
% first_assumptions.v_def
thf(fact_1194_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F2 @ I3 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1195_le__neq__implies__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( M4 != N )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1196_less__or__eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( ( ord_less_nat @ M4 @ N )
| ( M4 = N ) )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1197_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1198_less__imp__le__nat,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_imp_le_nat
thf(fact_1199_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1200_eq__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M4 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1201_le__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1202_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M4 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1203_diff__le__mono,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1204_diff__le__self,axiom,
! [M4: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N ) @ M4 ) ).
% diff_le_self
thf(fact_1205_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1206_diff__le__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1207_diff__less__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_1208_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1209_first__assumptions_Ok,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1210_first__assumptions_Okp,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ P3 @ K ) ) ).
% first_assumptions.kp
thf(fact_1211_first__assumptions_Opl,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ L @ P3 ) ) ).
% first_assumptions.pl
thf(fact_1212_first__assumptions_OC__def,axiom,
! [L: nat,P3: nat,K: nat,F2: nat > nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728462irst_C @ K @ F2 )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X2: nat,Y5: nat] :
( ( Uu
= ( insert_nat @ X2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( ( F2 @ X2 )
!= ( F2 @ Y5 ) ) ) ) ) ) ).
% first_assumptions.C_def
thf(fact_1213_less__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M4 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1214_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1215_first__assumptions_Omp,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ P3 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_1216_first__assumptions_Okm,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_1217_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).
% card_POS
thf(fact_1218_first__assumptions_Ocard__POS,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K ) )
= ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) ) ) ).
% first_assumptions.card_POS
thf(fact_1219_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1220_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M2: nat] :
( ( ord_less_nat @ K @ M2 )
=> ? [N6: nat] :
( ( ord_less_nat @ M2 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1221_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1222_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1223_choose__mono,axiom,
! [N: nat,M4: nat,K: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ M4 @ K ) ) ) ).
% choose_mono
thf(fact_1224_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M4: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1225_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N5: nat] :
( ( infini8530281810654367211te_nat @ S @ N5 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_1226_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_1227_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1228_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1229__092_060F_062__def,axiom,
( ( clique2971579238625216137irst_F @ k )
= ( piE_nat_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) )
@ ^ [I: nat] : ( clique3652268606331196573umbers @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% \<F>_def
thf(fact_1230_nat__add__left__cancel__le,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1231_nat__add__left__cancel__less,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1232_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1233_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1234_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1235_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1236_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1237_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1238_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1239_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1240_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1241_add__leE,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M4 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1242_le__add1,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M4 ) ) ).
% le_add1
thf(fact_1243_le__add2,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M4 @ N ) ) ).
% le_add2
thf(fact_1244_add__leD1,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% add_leD1
thf(fact_1245_add__leD2,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1246_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1247_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1248_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1249_trans__le__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1250_trans__le__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_1251_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1252_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M4: nat,K: nat] :
( ! [M2: nat,N5: nat] :
( ( ord_less_nat @ M2 @ N5 )
=> ( ord_less_nat @ ( F2 @ M2 ) @ ( F2 @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M4 ) @ K ) @ ( F2 @ ( plus_plus_nat @ M4 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1253_less__add__eq__less,axiom,
! [K: nat,L: nat,M4: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M4 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1254_trans__less__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_less_add2
thf(fact_1255_trans__less__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_less_add1
thf(fact_1256_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1257_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1258_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1259_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1260_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1261_add__diff__inverse__nat,axiom,
! [M4: nat,N: nat] :
( ~ ( ord_less_nat @ M4 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M4 @ N ) )
= M4 ) ) ).
% add_diff_inverse_nat
thf(fact_1262_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1263_diff__add__inverse2,axiom,
! [M4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N ) @ N )
= M4 ) ).
% diff_add_inverse2
thf(fact_1264_diff__add__inverse,axiom,
! [N: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M4 ) @ N )
= M4 ) ).
% diff_add_inverse
thf(fact_1265_diff__cancel2,axiom,
! [M4: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M4 @ N ) ) ).
% diff_cancel2
thf(fact_1266_Nat_Odiff__cancel,axiom,
! [K: nat,M4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M4 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1267_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1268_first__assumptions_Olm,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1269_first__assumptions_O_092_060F_062__def,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique2971579238625216137irst_F @ K )
= ( piE_nat_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) )
@ ^ [I: nat] : ( clique3652268606331196573umbers @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% first_assumptions.\<F>_def
% Conjectures (1)
thf(conj_0,conjecture,
( ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( sup_su4213647025997063966et_nat @ x @ y ) )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) )
= ( sup_su4213647025997063966et_nat
@ ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ x )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) )
@ ( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ y )
& ( ord_le6893508408891458716et_nat @ X2 @ G ) ) ) ) ) ) ).
%------------------------------------------------------------------------------