TPTP Problem File: SLH0628^1.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_01172_044932__16311898_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1381 ( 632 unt; 107 typ; 0 def)
% Number of atoms : 2967 ( 841 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9822 ( 219 ~; 41 |; 221 &;8211 @)
% ( 0 <=>;1130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 620 ( 620 >; 0 *; 0 +; 0 <<)
% Number of symbols : 100 ( 99 usr; 14 con; 0-5 aty)
% Number of variables : 3348 ( 357 ^;2902 !; 89 ?;3348 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:59.856
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
produc4045820344675478307at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_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 (99)
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_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > 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_060G_062l,type,
clique7840962075309931874st_G_l: nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oaccepts,type,
clique3686358387679108662ccepts: set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodot,type,
clique5469973757772500719t_odot: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodotl,type,
clique7966186356931407165_odotl: nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov__gs,type,
clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__graph,type,
clique711371890332037011_graph: nat > nat > nat > set_set_set_nat > produc4045820344675478307at_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__rel,type,
clique8954521387433384062in_rel: nat > nat > nat > set_set_set_nat > set_set_set_nat > $o ).
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_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
inf_inf_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inf_inf_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
inf_in2551356467856225537_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_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_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_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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_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_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_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_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_fChoice_001t__Set__Oset_It__Nat__Onat_J,type,
fChoice_set_nat: ( set_nat > $o ) > 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_v_C____,type,
c: set_nat > set_nat ).
thf(sy_v_D____,type,
d: set_set_nat ).
thf(sy_v_E____,type,
e: set_set_nat ).
thf(sy_v_GS____,type,
gs: set_set_set_nat ).
thf(sy_v_G____,type,
g: set_set_nat ).
thf(sy_v_H____,type,
h: set_set_nat ).
thf(sy_v_K____,type,
k: set_nat > set_set_nat ).
thf(sy_v_Vs____,type,
vs: set_set_nat ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_k,type,
k2: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_merge____,type,
merge: set_nat > set_nat > set_set_nat ).
% Relevant facts (1272)
thf(fact_0_PLU__main__graph_Ocong,axiom,
clique711371890332037011_graph = clique711371890332037011_graph ).
% PLU_main_graph.cong
thf(fact_1_D,axiom,
member_set_set_nat @ d @ x ).
% D
thf(fact_2_E,axiom,
member_set_set_nat @ e @ y ).
% E
thf(fact_3_H_I1_J,axiom,
member_set_set_nat @ h @ ( clique5469973757772500719t_odot @ x @ y ) ).
% H(1)
thf(fact_4_lower,axiom,
ord_less_nat @ l @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ h ) ) ).
% lower
thf(fact_5_vm,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ h ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ).
% vm
thf(fact_6_H_I2_J,axiom,
~ ( member_set_set_nat @ h @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ).
% H(2)
thf(fact_7_km,axiom,
ord_less_nat @ k2 @ ( assump1710595444109740334irst_m @ k2 ) ).
% km
thf(fact_8_k,axiom,
ord_less_nat @ l @ k2 ).
% k
thf(fact_9_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_10_Vs__def,axiom,
( vs
= ( inf_inf_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) )
@ ( collect_set_nat
@ ^ [V: set_nat] :
( ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) )
& ( ord_less_eq_nat @ ( suc @ l ) @ ( finite_card_nat @ V ) ) ) ) ) ) ).
% Vs_def
thf(fact_11_PLU__main__rel_Ocong,axiom,
clique8954521387433384062in_rel = clique8954521387433384062in_rel ).
% PLU_main_rel.cong
thf(fact_12_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_13_HGl,axiom,
~ ( member_set_set_nat @ h @ ( clique7840962075309931874st_G_l @ l @ k2 ) ) ).
% HGl
thf(fact_14__092_060open_062card_A_Iv__gs_A_IX_A_092_060odot_062l_AY_J_J_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_092_060odot_062_AY_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ) ).
% \<open>card (v_gs (X \<odot>l Y)) \<le> card (v_gs (X \<odot> Y))\<close>
thf(fact_15_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_16_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_17_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_18_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_19_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_20_le__inf__iff,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( ( ord_le9131159989063066194et_nat @ X @ Y )
& ( ord_le9131159989063066194et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_21_le__inf__iff,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X @ Y )
& ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_22_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_23_le__inf__iff,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X @ Y )
& ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_24_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_25_le__inf__iff,axiom,
! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ ( inf_inf_nat_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat_nat @ X @ Y )
& ( ord_less_eq_nat_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_26_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_27_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_28_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_29_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_30_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_31_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_32_finvXY,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ).
% finvXY
thf(fact_33_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ).
% v_\<G>
thf(fact_34_HG0,axiom,
member_set_set_nat @ h @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% HG0
thf(fact_35_v__gs__mono,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_mono
thf(fact_36_v__mono,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_37_finV_I1_J,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% finV(1)
thf(fact_38_finV_I2_J,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ y ) ).
% finV(2)
thf(fact_39_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_40_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_41_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_42_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( member_set_set_nat @ X3 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_44_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_45_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_46_inf__right__idem,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_47_inf__right__idem,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ Y )
= ( inf_in5711780100303410308et_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_48_inf__right__idem,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_49_inf_Oright__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_50_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_51_inf_Oright__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_52_inf__left__idem,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y ) )
= ( inf_inf_set_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_53_inf__left__idem,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y ) )
= ( inf_in5711780100303410308et_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_54_inf__left__idem,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y ) )
= ( inf_inf_set_nat_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_55_inf_Oleft__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_56_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_57_inf_Oleft__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_58_inf__idem,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_59_inf__idem,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_60_inf__idem,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_61_inf_Oidem,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_62_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_63_inf_Oidem,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_64_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_65_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_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_74_joinl__join,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k2 @ X2 @ Y2 ) @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) ) ).
% joinl_join
thf(fact_75_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
@ ^ [X4: set_nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_76_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
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_77_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
@ ^ [X4: set_set_nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_78_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
@ ^ [X4: nat > nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_79_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_80_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_81_finite__Collect__disjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X4: set_set_nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
& ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_82_finite__Collect__disjI,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [X4: nat > nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite2115694454571419734at_nat @ ( collect_nat_nat @ P ) )
& ( finite2115694454571419734at_nat @ ( collect_nat_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_83_XY_I1_J,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% XY(1)
thf(fact_84_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_85_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_86_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_87_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_88_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X4: set_set_nat] : ( member_set_set_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X4: nat > nat] : ( member_nat_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_92_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_93_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_94_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_95_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_96_XY_I2_J,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% XY(2)
thf(fact_97_odot___092_060G_062,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_98_finite__members___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ).
% finite_members_\<G>
thf(fact_99_XY_I3_J,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% XY(3)
thf(fact_100_XY_I4_J,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% XY(4)
thf(fact_101_odotl__def,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k2 @ X2 @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) @ ( clique7840962075309931874st_G_l @ l @ k2 ) ) ) ).
% odotl_def
thf(fact_102_finite__v__gs__Gl,axiom,
! [X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) ) ).
% finite_v_gs_Gl
thf(fact_103_Dl_I1_J,axiom,
member_set_set_nat @ d @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% Dl(1)
thf(fact_104_Dl_I2_J,axiom,
member_set_set_nat @ e @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% Dl(2)
thf(fact_105_finite__v__gs,axiom,
! [X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) ) ).
% finite_v_gs
thf(fact_106_Ep,axiom,
member_set_set_nat @ e @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% Ep
thf(fact_107_Dp,axiom,
member_set_set_nat @ d @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% Dp
thf(fact_108_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_109_finite__Int,axiom,
! [F: set_set_nat,G: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F )
| ( finite1152437895449049373et_nat @ G ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_110_finite__Int,axiom,
! [F: set_set_set_nat,G: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F )
| ( finite6739761609112101331et_nat @ G ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_111_finite__Int,axiom,
! [F: set_nat_nat,G: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F )
| ( finite2115694454571419734at_nat @ G ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_112_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_113_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_114_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_115_sub,axiom,
ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% sub
thf(fact_116_XYD,axiom,
ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ x @ y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% XYD
thf(fact_117_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_118__092_060G_062l__def,axiom,
( ( clique7840962075309931874st_G_l @ l @ k2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ( ord_less_eq_nat @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G2 ) ) @ l ) ) ) ) ).
% \<G>l_def
thf(fact_119_HG,axiom,
ord_le6893508408891458716et_nat @ h @ g ).
% HG
thf(fact_120_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_121_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_122_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_123_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_124_HDE,axiom,
( h
= ( sup_sup_set_set_nat @ d @ e ) ) ).
% HDE
thf(fact_125_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% NEG_\<G>
thf(fact_126_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_127_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_128_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_129_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_130_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_131_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_132_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_133_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_134_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_135_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_136_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_137_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_138_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_139_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_140_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_141_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_set_nat,R: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_142_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_set_nat,R: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_143_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 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_144_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat_nat,R: nat > ( nat > nat ) > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_145_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 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_146_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat,B: set_nat,R: ( nat > nat ) > nat > $o] :
( ~ ( finite2115694454571419734at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_147_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 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_148_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_nat_nat,R: set_nat > ( nat > nat ) > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_149_finite__has__minimal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( ord_le6893508408891458716et_nat @ X3 @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_150_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ord_less_eq_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_151_finite__has__minimal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A )
& ( ord_le9059583361652607317at_nat @ X3 @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_152_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_153_finite__has__minimal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( ord_less_eq_nat_nat @ X3 @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_154_finite__has__maximal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( ord_le6893508408891458716et_nat @ A2 @ X3 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_155_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ord_less_eq_set_nat @ A2 @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_156_finite__has__maximal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A )
& ( ord_le9059583361652607317at_nat @ A2 @ X3 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_157_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_158_finite__has__maximal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( ord_less_eq_nat_nat @ A2 @ X3 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_159_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_160_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_161_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_162_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_163_infinite__super,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S2 )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_164_infinite__super,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S2 )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_165_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_166_infinite__super,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S2 @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S2 )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_167_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_168_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_169_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_170_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_171_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_172_inf__set__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ( inf_inf_set_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 )
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_173_inf__set__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ( inf_in2551356467856225537_nat_o
@ ^ [X4: set_set_nat] : ( member_set_set_nat @ X4 @ A4 )
@ ^ [X4: set_set_nat] : ( member_set_set_nat @ X4 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_174_inf__set__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ( inf_inf_nat_nat_o
@ ^ [X4: nat > nat] : ( member_nat_nat @ X4 @ A4 )
@ ^ [X4: nat > nat] : ( member_nat_nat @ X4 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_175_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X4: set_set_nat] : ( member_set_set_nat @ X4 @ A4 )
@ ^ [X4: set_set_nat] : ( member_set_set_nat @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_176_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 )
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_177_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_178_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X4: nat > nat] : ( member_nat_nat @ X4 @ A4 )
@ ^ [X4: nat > nat] : ( member_nat_nat @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_179_lift__Suc__mono__less__iff,axiom,
! [F2: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_180_lift__Suc__mono__less__iff,axiom,
! [F2: nat > set_set_set_nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_le152980574450754630et_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_le152980574450754630et_nat @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_181_lift__Suc__mono__less,axiom,
! [F2: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_182_lift__Suc__mono__less,axiom,
! [F2: nat > set_set_set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le152980574450754630et_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_le152980574450754630et_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_183_infinite__arbitrarily__large,axiom,
! [A: set_set_set_nat,N: nat] :
( ~ ( finite6739761609112101331et_nat @ A )
=> ? [B4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B4 )
& ( ( finite1149291290879098388et_nat @ B4 )
= N )
& ( ord_le9131159989063066194et_nat @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_184_infinite__arbitrarily__large,axiom,
! [A: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ( finite_card_set_nat @ B4 )
= N )
& ( ord_le6893508408891458716et_nat @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_185_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ( finite_card_nat @ B4 )
= N )
& ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_186_infinite__arbitrarily__large,axiom,
! [A: set_nat_nat,N: nat] :
( ~ ( finite2115694454571419734at_nat @ A )
=> ? [B4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B4 )
& ( ( finite_card_nat_nat @ B4 )
= N )
& ( ord_le9059583361652607317at_nat @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_187_card__subset__eq,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ( finite1149291290879098388et_nat @ A )
= ( finite1149291290879098388et_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_188_card__subset__eq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( finite_card_set_nat @ A )
= ( finite_card_set_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_189_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_190_card__subset__eq,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( finite_card_nat_nat @ A )
= ( finite_card_nat_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_191_inf_Ostrict__coboundedI2,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_192_inf_Ostrict__coboundedI2,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ C2 )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_193_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_194_inf_Ostrict__coboundedI2,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_195_inf_Ostrict__coboundedI1,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ C2 )
=> ( ord_less_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_196_inf_Ostrict__coboundedI1,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ C2 )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_197_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_198_inf_Ostrict__coboundedI1,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ C2 )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_199_inf_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ( A3
= ( inf_inf_set_set_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_200_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] :
( ( A3
= ( inf_inf_set_nat_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_201_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_202_inf_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B5: set_set_set_nat] :
( ( A3
= ( inf_in5711780100303410308et_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_203_inf_Ostrict__boundedE,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_set_set_nat @ A2 @ B2 )
=> ~ ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_204_inf_Ostrict__boundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_205_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_206_inf_Ostrict__boundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_207_inf_Oabsorb4,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_208_inf_Oabsorb4,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_209_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_210_inf_Oabsorb4,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_211_inf_Oabsorb3,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_212_inf_Oabsorb3,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_213_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_214_inf_Oabsorb3,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_215_less__infI2,axiom,
! [B2: set_set_nat,X: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ X )
=> ( ord_less_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_216_less__infI2,axiom,
! [B2: set_nat_nat,X: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ X )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_217_less__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_218_less__infI2,axiom,
! [B2: set_set_set_nat,X: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ X )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_219_less__infI1,axiom,
! [A2: set_set_nat,X: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ X )
=> ( ord_less_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_220_less__infI1,axiom,
! [A2: set_nat_nat,X: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ X )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_221_less__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_222_less__infI1,axiom,
! [A2: set_set_set_nat,X: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ X )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_223_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_224_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_225_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I3 @ K ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_226_less__trans__Suc,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_227_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_228_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_229_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_230_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_231_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_232_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_233_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_234_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_235_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_236_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_237_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_238_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_239_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_240_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_241_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_242_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_243_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_244_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_245_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_246_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_247_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_set_nat,C: nat] :
( ! [G3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G3 @ F )
=> ( ( finite6739761609112101331et_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G3 ) @ C ) ) )
=> ( ( finite6739761609112101331et_nat @ F )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_248_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_nat,C: nat] :
( ! [G3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G3 @ F )
=> ( ( finite1152437895449049373et_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G3 ) @ C ) ) )
=> ( ( finite1152437895449049373et_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_249_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_250_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat_nat,C: nat] :
( ! [G3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ G3 @ F )
=> ( ( finite2115694454571419734at_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ G3 ) @ C ) ) )
=> ( ( finite2115694454571419734at_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_251_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ S2 ) )
=> ~ ! [T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T3 @ S2 )
=> ( ( ( finite1149291290879098388et_nat @ T3 )
= N )
=> ~ ( finite6739761609112101331et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_252_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S2 ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S2 )
=> ( ( ( finite_card_set_nat @ T3 )
= N )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_253_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_254_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ S2 ) )
=> ~ ! [T3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat_nat @ T3 )
= N )
=> ~ ( finite2115694454571419734at_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_255_exists__subset__between,axiom,
! [A: set_set_set_nat,N: nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ C ) )
=> ( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( finite6739761609112101331et_nat @ C )
=> ? [B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ C )
& ( ( finite1149291290879098388et_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_256_exists__subset__between,axiom,
! [A: set_set_nat,N: nat,C: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( finite1152437895449049373et_nat @ C )
=> ? [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ C )
& ( ( finite_card_set_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_257_exists__subset__between,axiom,
! [A: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
& ( ord_less_eq_set_nat @ B4 @ C )
& ( ( finite_card_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_258_exists__subset__between,axiom,
! [A: set_nat_nat,N: nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ C ) )
=> ( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( finite2115694454571419734at_nat @ C )
=> ? [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ C )
& ( ( finite_card_nat_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_259_card__seteq,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B ) @ ( finite1149291290879098388et_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_260_card__seteq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_set_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_261_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_262_card__seteq,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_nat_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_263_card__mono,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ).
% card_mono
thf(fact_264_card__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% card_mono
thf(fact_265_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_266_card__mono,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A ) @ ( finite_card_nat_nat @ B ) ) ) ) ).
% card_mono
thf(fact_267_Collect__mono__iff,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X4: set_set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_268_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_269_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_270_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X4: nat > nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_271_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_272_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_273_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_nat,Z2: set_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_274_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_275_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_276_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_277_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_278_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_279_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_280_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_281_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_282_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_283_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_284_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
! [T4: set_set_nat] :
( ( member_set_set_nat @ T4 @ A4 )
=> ( member_set_set_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_285_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
! [T4: set_nat] :
( ( member_set_nat @ T4 @ A4 )
=> ( member_set_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_286_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A4 )
=> ( member_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_287_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
! [T4: nat > nat] :
( ( member_nat_nat @ T4 @ A4 )
=> ( member_nat_nat @ T4 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_288_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_289_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_290_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_291_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_292_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_293_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_294_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A4 )
=> ( member_set_set_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_295_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( member_set_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_296_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_297_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A4 )
=> ( member_nat_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X @ A )
=> ( member_set_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_306_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_307_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_308_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X @ A )
=> ( member_nat_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_309_inf__left__commute,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ Y @ ( inf_inf_set_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_310_inf__left__commute,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y @ ( inf_in5711780100303410308et_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_311_inf__left__commute,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ Y @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_312_inf_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ B2 @ ( inf_inf_set_set_nat @ A2 @ C2 ) )
= ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_313_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_314_inf_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C2 ) )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_315_inf__commute,axiom,
( inf_inf_set_set_nat
= ( ^ [X4: set_set_nat,Y4: set_set_nat] : ( inf_inf_set_set_nat @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_316_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X4: set_set_set_nat,Y4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_317_inf__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [X4: set_nat_nat,Y4: set_nat_nat] : ( inf_inf_set_nat_nat @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_318_inf_Ocommute,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] : ( inf_inf_set_set_nat @ B5 @ A3 ) ) ) ).
% inf.commute
thf(fact_319_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B5 @ A3 ) ) ) ).
% inf.commute
thf(fact_320_inf_Ocommute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] : ( inf_inf_set_nat_nat @ B5 @ A3 ) ) ) ).
% inf.commute
thf(fact_321_inf__assoc,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_322_inf__assoc,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ Z )
= ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_323_inf__assoc,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_324_inf_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_325_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_326_inf_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_327_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_nat
= ( ^ [X4: set_set_nat,Y4: set_set_nat] : ( inf_inf_set_set_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_328_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X4: set_set_set_nat,Y4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_329_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat_nat
= ( ^ [X4: set_nat_nat,Y4: set_nat_nat] : ( inf_inf_set_nat_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_330_inf__sup__aci_I2_J,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_331_inf__sup__aci_I2_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ Z )
= ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_332_inf__sup__aci_I2_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_333_inf__sup__aci_I3_J,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ Y @ ( inf_inf_set_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_334_inf__sup__aci_I3_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y @ ( inf_in5711780100303410308et_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_335_inf__sup__aci_I3_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ Y @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_336_inf__sup__aci_I4_J,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y ) )
= ( inf_inf_set_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_337_inf__sup__aci_I4_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y ) )
= ( inf_in5711780100303410308et_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_338_inf__sup__aci_I4_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y ) )
= ( inf_inf_set_nat_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_339_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_340_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_341_Int__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
= ( inf_inf_set_set_nat @ B @ ( inf_inf_set_set_nat @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_342_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_343_Int__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( inf_inf_set_nat_nat @ B @ ( inf_inf_set_nat_nat @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_344_Int__left__absorb,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( inf_inf_set_set_nat @ A @ B ) )
= ( inf_inf_set_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_345_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_346_Int__left__absorb,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_347_Int__commute,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] : ( inf_inf_set_set_nat @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_348_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_349_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] : ( inf_inf_set_nat_nat @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_350_Int__absorb,axiom,
! [A: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_351_Int__absorb,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_352_Int__absorb,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_353_Int__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C )
= ( inf_inf_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_354_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_355_Int__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_369_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_370_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_371_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_372_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_373_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_374_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_375_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_376_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_377_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_378_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_379_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_380_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_381_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_382_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_383_Collect__subset,axiom,
! [A: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_384_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_385_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_386_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_387_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_388_Collect__conj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X4: set_nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_389_Collect__conj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X4: set_set_nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_in5711780100303410308et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_390_Collect__conj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X4: nat > nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_391_Int__Collect,axiom,
! [X: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_392_Int__Collect,axiom,
! [X: set_nat,A: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_393_Int__Collect,axiom,
! [X: set_set_nat,A: set_set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ X @ ( inf_in5711780100303410308et_nat @ A @ ( collect_set_set_nat @ P ) ) )
= ( ( member_set_set_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_394_Int__Collect,axiom,
! [X: nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) )
= ( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_395_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( member_nat @ X4 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_396_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( member_set_nat @ X4 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_397_Int__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A4 )
& ( member_set_set_nat @ X4 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_398_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A4 )
& ( member_nat_nat @ X4 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_inf_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B5: set_set_set_nat,A3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_412_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A3: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_413_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_414_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_415_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_416_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [B5: nat > nat,A3: nat > nat] :
( ( inf_inf_nat_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_417_inf_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B5: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_418_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_419_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_420_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_421_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( inf_inf_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_422_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [A3: nat > nat,B5: nat > nat] :
( ( inf_inf_nat_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_423_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_424_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_425_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_426_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_427_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_428_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_429_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_430_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_431_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_432_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_433_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_434_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_435_inf_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B5: set_set_set_nat] :
( A3
= ( inf_in5711780100303410308et_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_436_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( A3
= ( inf_inf_set_set_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_437_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( A3
= ( inf_inf_set_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_438_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] :
( A3
= ( inf_inf_set_nat_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_439_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( A3
= ( inf_inf_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_440_inf_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [A3: nat > nat,B5: nat > nat] :
( A3
= ( inf_inf_nat_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_441_inf__greatest,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ( ord_le9131159989063066194et_nat @ X @ Z )
=> ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_442_inf__greatest,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X @ Z )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_443_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_444_inf__greatest,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X @ Z )
=> ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_445_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_446_inf__greatest,axiom,
! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ( ord_less_eq_nat_nat @ X @ Z )
=> ( ord_less_eq_nat_nat @ X @ ( inf_inf_nat_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_inf__absorb2,axiom,
! [Y: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X )
=> ( ( inf_in5711780100303410308et_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_460_inf__absorb2,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_461_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_462_inf__absorb2,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( inf_inf_set_nat_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_463_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_464_inf__absorb2,axiom,
! [Y: nat > nat,X: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X )
=> ( ( inf_inf_nat_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_465_inf__absorb1,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ( inf_in5711780100303410308et_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_466_inf__absorb1,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_467_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_468_inf__absorb1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( inf_inf_set_nat_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_469_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_470_inf__absorb1,axiom,
! [X: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ( inf_inf_nat_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_471_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_472_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_473_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_474_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_475_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_476_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_477_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_478_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_479_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_480_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_481_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_482_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_483_le__iff__inf,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X4: set_set_set_nat,Y4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_484_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X4: set_set_nat,Y4: set_set_nat] :
( ( inf_inf_set_set_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_485_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_486_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X4: set_nat_nat,Y4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_487_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( inf_inf_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_488_le__iff__inf,axiom,
( ord_less_eq_nat_nat
= ( ^ [X4: nat > nat,Y4: nat > nat] :
( ( inf_inf_nat_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_489_inf__unique,axiom,
! [F2: set_set_set_nat > set_set_set_nat > set_set_set_nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ! [X3: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F2 @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F2 @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_set_set_nat,Y5: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y5 )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ Z3 )
=> ( ord_le9131159989063066194et_nat @ X3 @ ( F2 @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_490_inf__unique,axiom,
! [F2: set_set_nat > set_set_nat > set_set_nat,X: set_set_nat,Y: set_set_nat] :
( ! [X3: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F2 @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F2 @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_set_nat,Y5: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y5 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( F2 @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_491_inf__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ ( F2 @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_492_inf__unique,axiom,
! [F2: set_nat_nat > set_nat_nat > set_nat_nat,X: set_nat_nat,Y: set_nat_nat] :
( ! [X3: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F2 @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F2 @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_nat_nat,Y5: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y5 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Z3 )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( F2 @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_493_inf__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F2 @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_494_inf__unique,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat > nat,X: nat > nat,Y: nat > nat] :
( ! [X3: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ ( F2 @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ ( F2 @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: nat > nat,Y5: nat > nat,Z3: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ Y5 )
=> ( ( ord_less_eq_nat_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat_nat @ X3 @ ( F2 @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_495_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_496_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_497_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_498_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_499_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_500_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_501_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_502_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_503_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_504_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_505_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_506_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_507_le__infI2,axiom,
! [B2: set_set_set_nat,X: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ X )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_508_le__infI2,axiom,
! [B2: set_set_nat,X: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_509_le__infI2,axiom,
! [B2: set_nat,X: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_510_le__infI2,axiom,
! [B2: set_nat_nat,X: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ X )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_511_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_512_le__infI2,axiom,
! [B2: nat > nat,X: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ X )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_513_le__infI1,axiom,
! [A2: set_set_set_nat,X: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_514_le__infI1,axiom,
! [A2: set_set_nat,X: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_515_le__infI1,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_516_le__infI1,axiom,
! [A2: set_nat_nat,X: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_517_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_518_le__infI1,axiom,
! [A2: nat > nat,X: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_519_inf__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_520_inf__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_521_inf__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_522_inf__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_523_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_524_inf__mono,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ D )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ ( inf_inf_nat_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_525_le__infI,axiom,
! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X @ B2 )
=> ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_526_le__infI,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ B2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_527_le__infI,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_528_le__infI,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ B2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_529_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_530_le__infI,axiom,
! [X: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ A2 )
=> ( ( ord_less_eq_nat_nat @ X @ B2 )
=> ( ord_less_eq_nat_nat @ X @ ( inf_inf_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_531_le__infE,axiom,
! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ X @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_532_le__infE,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_533_le__infE,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A2 )
=> ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_534_le__infE,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_535_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_536_le__infE,axiom,
! [X: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ ( inf_inf_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_537_inf__le2,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_538_inf__le2,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_539_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_540_inf__le2,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_541_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_542_inf__le2,axiom,
! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_543_inf__le1,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_544_inf__le1,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_545_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_546_inf__le1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_547_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_548_inf__le1,axiom,
! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_549_inf__sup__ord_I1_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_550_inf__sup__ord_I1_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_551_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_552_inf__sup__ord_I1_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_553_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_554_inf__sup__ord_I1_J,axiom,
! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_555_inf__sup__ord_I2_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_556_inf__sup__ord_I2_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_557_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_558_inf__sup__ord_I2_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_559_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_560_inf__sup__ord_I2_J,axiom,
! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_561_Int__Collect__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ ( collect_set_set_nat @ P ) ) @ ( inf_in5711780100303410308et_nat @ B @ ( collect_set_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_562_Int__Collect__mono,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_563_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_564_Int__Collect__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_565_Int__greatest,axiom,
! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A )
=> ( ( ord_le9131159989063066194et_nat @ C @ B )
=> ( ord_le9131159989063066194et_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_566_Int__greatest,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_567_Int__greatest,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_568_Int__greatest,axiom,
! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_569_Int__absorb2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_570_Int__absorb2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( inf_inf_set_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_571_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_572_Int__absorb2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_573_Int__absorb1,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_574_Int__absorb1,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( inf_inf_set_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_575_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_576_Int__absorb1,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_577_Int__lower2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_578_Int__lower2,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_579_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_580_Int__lower2,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_581_Int__lower1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_582_Int__lower1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_583_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_584_Int__lower1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_585_Int__mono,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_586_Int__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_587_Int__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_588_Int__mono,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_589_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( R @ X3 @ Y5 )
=> ( ( R @ Y5 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_590_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_591_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_592_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_593_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_594_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_595_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_596_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_597_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_598_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_599_lift__Suc__antimono__le,axiom,
! [F2: nat > set_set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le6893508408891458716et_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le6893508408891458716et_nat @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_600_lift__Suc__antimono__le,axiom,
! [F2: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_601_lift__Suc__antimono__le,axiom,
! [F2: nat > set_nat_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le9059583361652607317at_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_602_lift__Suc__antimono__le,axiom,
! [F2: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_603_lift__Suc__antimono__le,axiom,
! [F2: nat > nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat_nat @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_604_lift__Suc__mono__le,axiom,
! [F2: nat > set_set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le6893508408891458716et_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le6893508408891458716et_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_605_lift__Suc__mono__le,axiom,
! [F2: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_606_lift__Suc__mono__le,axiom,
! [F2: nat > set_nat_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le9059583361652607317at_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_607_lift__Suc__mono__le,axiom,
! [F2: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_608_lift__Suc__mono__le,axiom,
! [F2: nat > nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat_nat @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_609_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X5: set_set_set_nat,G2: set_set_nat] :
? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ X5 )
& ( ord_le6893508408891458716et_nat @ X4 @ G2 ) ) ) ) ).
% accepts_def
thf(fact_610_finK,axiom,
! [V2: set_nat] : ( finite1152437895449049373et_nat @ ( k @ V2 ) ) ).
% finK
thf(fact_611_Vs__C_I4_J,axiom,
! [V2: set_nat] :
( ( member_set_nat @ V2 @ vs )
=> ( ord_less_eq_set_nat @ ( c @ V2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ).
% Vs_C(4)
thf(fact_612_Vs__C_I2_J,axiom,
! [V2: set_nat] :
( ( member_set_nat @ V2 @ vs )
=> ( ( finite_card_nat @ ( c @ V2 ) )
= ( suc @ l ) ) ) ).
% Vs_C(2)
thf(fact_613_card__v__gs__join,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z4 @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z4 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_614__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% \<K>_\<G>
thf(fact_615_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_616__092_060open_062card_A_Iv__gs_A_IX_A_092_060odot_062_AY_J_J_A_092_060le_062_Acard_A_Iv__gs_AX_J_A_K_Acard_A_Iv__gs_AY_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) ) ) ).
% \<open>card (v_gs (X \<odot> Y)) \<le> card (v_gs X) * card (v_gs Y)\<close>
thf(fact_617__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062H_O_A_092_060lbrakk_062H_A_092_060in_062_AX_A_092_060odot_062_AY_059_AH_A_092_060notin_062_AX_A_092_060odot_062l_AY_059_AH_A_092_060subseteq_062_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [H2: set_set_nat] :
( ( member_set_set_nat @ H2 @ ( clique5469973757772500719t_odot @ x @ y ) )
=> ( ~ ( member_set_set_nat @ H2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) )
=> ~ ( ord_le6893508408891458716et_nat @ H2 @ g ) ) ) ).
% \<open>\<And>thesis. (\<And>H. \<lbrakk>H \<in> X \<odot> Y; H \<notin> X \<odot>l Y; H \<subseteq> G\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_618__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% \<G>_def
thf(fact_619_finite__vG,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ).
% finite_vG
thf(fact_620_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_621_vGk_I2_J,axiom,
( ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ g ) @ ( clique5033774636164728513irst_v @ g ) )
= g ) ).
% vGk(2)
thf(fact_622_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_623_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_624_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_625_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_626_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_627_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_628_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_629_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_630_sup__idem,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_631_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_632_sup__idem,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_633_sup__idem,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_634_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_635_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_636_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_637_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_638_sup__left__idem,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) )
= ( sup_sup_set_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_639_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_640_sup__left__idem,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_641_sup__left__idem,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) )
= ( sup_sup_set_nat_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_G_I1_J,axiom,
member_set_set_nat @ g @ ( clique3326749438856946062irst_K @ k2 ) ).
% G(1)
thf(fact_655_Vs__C_I3_J,axiom,
! [V2: set_nat] :
( ( member_set_nat @ V2 @ vs )
=> ( finite_finite_nat @ ( c @ V2 ) ) ) ).
% Vs_C(3)
thf(fact_656_Vs__C_I1_J,axiom,
! [V2: set_nat] :
( ( member_set_nat @ V2 @ vs )
=> ( ord_less_eq_set_nat @ ( c @ V2 ) @ V2 ) ) ).
% Vs_C(1)
thf(fact_657_vGk_I1_J,axiom,
( ( finite_card_nat @ ( clique5033774636164728513irst_v @ g ) )
= k2 ) ).
% vGk(1)
thf(fact_658_vHG,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ h ) @ ( clique5033774636164728513irst_v @ g ) ).
% vHG
thf(fact_659_G0,axiom,
member_set_set_nat @ g @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% G0
thf(fact_660__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062D_AE_O_A_092_060lbrakk_062D_A_092_060in_062_AX_059_AE_A_092_060in_062_AY_059_AH_A_061_AD_A_092_060union_062_AE_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D3: set_set_nat] :
( ( member_set_set_nat @ D3 @ x )
=> ! [E: set_set_nat] :
( ( member_set_set_nat @ E @ y )
=> ( h
!= ( sup_sup_set_set_nat @ D3 @ E ) ) ) ) ).
% \<open>\<And>thesis. (\<And>D E. \<lbrakk>D \<in> X; E \<in> Y; H = D \<union> E\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_661__092_060open_062card_A_Iv_AG_J_A_061_Ak_A_092_060and_062_Av_AG_094_092_060two_062_A_061_AG_092_060close_062,axiom,
( ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ g ) )
= k2 )
& ( ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ g ) @ ( clique5033774636164728513irst_v @ g ) )
= g ) ) ).
% \<open>card (v G) = k \<and> v G^\<two> = G\<close>
thf(fact_662_le__sup__iff,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ Z )
= ( ( ord_le9131159989063066194et_nat @ X @ Z )
& ( ord_le9131159989063066194et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_663_le__sup__iff,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X @ Z )
& ( ord_le6893508408891458716et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_664_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_665_le__sup__iff,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X @ Z )
& ( ord_le9059583361652607317at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_666_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_667_le__sup__iff,axiom,
! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat_nat @ X @ Z )
& ( ord_less_eq_nat_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_inf__sup__absorb,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_679_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_680_inf__sup__absorb,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_681_inf__sup__absorb,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_682_sup__inf__absorb,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_683_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_684_sup__inf__absorb,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_685_sup__inf__absorb,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_686_finite__Un,axiom,
! [F: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F @ G ) )
= ( ( finite1152437895449049373et_nat @ F )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_687_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_688_finite__Un,axiom,
! [F: set_set_set_nat,G: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F @ G ) )
= ( ( finite6739761609112101331et_nat @ F )
& ( finite6739761609112101331et_nat @ G ) ) ) ).
% finite_Un
thf(fact_689_finite__Un,axiom,
! [F: set_nat_nat,G: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F @ G ) )
= ( ( finite2115694454571419734at_nat @ F )
& ( finite2115694454571419734at_nat @ G ) ) ) ).
% finite_Un
thf(fact_690_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S2: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_691_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S2: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_692_Int__Un__eq_I4_J,axiom,
! [T2: set_set_set_nat,S2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T2 @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_693_Int__Un__eq_I4_J,axiom,
! [T2: set_nat_nat,S2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T2 @ ( inf_inf_set_nat_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_694_Int__Un__eq_I3_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S2 @ ( inf_inf_set_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_695_Int__Un__eq_I3_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S2 @ ( inf_inf_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_696_Int__Un__eq_I3_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S2 @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_697_Int__Un__eq_I3_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S2 @ ( inf_inf_set_nat_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_698_Int__Un__eq_I2_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_699_Int__Un__eq_I2_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_700_Int__Un__eq_I2_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_701_Int__Un__eq_I2_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_702_Int__Un__eq_I1_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_703_Int__Un__eq_I1_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_704_Int__Un__eq_I1_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_705_Int__Un__eq_I1_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_706_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S2: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_707_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S2: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_708_Un__Int__eq_I4_J,axiom,
! [T2: set_set_set_nat,S2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T2 @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_709_Un__Int__eq_I4_J,axiom,
! [T2: set_nat_nat,S2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T2 @ ( sup_sup_set_nat_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_710_Un__Int__eq_I3_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S2 @ ( sup_sup_set_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_711_Un__Int__eq_I3_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S2 @ ( sup_sup_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_712_Un__Int__eq_I3_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S2 @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_713_Un__Int__eq_I3_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S2 @ ( sup_sup_set_nat_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_714_Un__Int__eq_I2_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_715_Un__Int__eq_I2_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_716_Un__Int__eq_I2_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_717_Un__Int__eq_I2_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_718_Un__Int__eq_I1_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_719_Un__Int__eq_I1_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_720_Un__Int__eq_I1_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_721_Un__Int__eq_I1_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_722_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_723__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k2 )
= ( collect_set_set_nat
@ ^ [K3: set_set_nat] :
( ( member_set_set_nat @ K3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K3 ) )
= k2 )
& ( K3
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K3 ) @ ( clique5033774636164728513irst_v @ K3 ) ) ) ) ) ) ).
% \<K>_def
thf(fact_724_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_725_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_726_contra,axiom,
~ ( member_set_set_nat @ g @ gs ) ).
% contra
thf(fact_727_acceptsI,axiom,
! [D2: set_set_nat,G: set_set_nat,X2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G )
=> ( ( member_set_set_nat @ D2 @ X2 )
=> ( clique3686358387679108662ccepts @ X2 @ G ) ) ) ).
% acceptsI
thf(fact_728__092_060open_062G_A_092_060in_062_A_123K_A_092_060in_062_A_092_060G_062_O_Acard_A_Iv_AK_J_A_061_Ak_A_092_060and_062_AK_A_061_Av_AK_094_092_060two_062_125_092_060close_062,axiom,
( member_set_set_nat @ g
@ ( collect_set_set_nat
@ ^ [K3: set_set_nat] :
( ( member_set_set_nat @ K3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K3 ) )
= k2 )
& ( K3
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K3 ) @ ( clique5033774636164728513irst_v @ K3 ) ) ) ) ) ) ).
% \<open>G \<in> {K \<in> \<G>. card (v K) = k \<and> K = v K^\<two>}\<close>
thf(fact_729_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_730_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% finite_\<G>
thf(fact_731_union___092_060G_062,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% union_\<G>
thf(fact_732__092_060open_062C_A_092_060equiv_062_A_092_060lambda_062V_O_ASOME_AC_O_AC_A_092_060subseteq_062_AV_A_092_060and_062_Acard_AC_A_061_ASuc_Al_092_060close_062,axiom,
( c
= ( ^ [V: set_nat] :
( fChoice_set_nat
@ ^ [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ( finite_card_nat @ C3 )
= ( suc @ l ) ) ) ) ) ) ).
% \<open>C \<equiv> \<lambda>V. SOME C. C \<subseteq> V \<and> card C = Suc l\<close>
thf(fact_733_inf__sup__aci_I8_J,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) )
= ( sup_sup_set_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_734_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_735_inf__sup__aci_I8_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_736_inf__sup__aci_I8_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) )
= ( sup_sup_set_nat_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_737_inf__sup__aci_I7_J,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_738_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_739_inf__sup__aci_I7_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_740_inf__sup__aci_I7_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_741_inf__sup__aci_I6_J,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_742_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_743_inf__sup__aci_I6_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ Z )
= ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_744_inf__sup__aci_I6_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_745_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X4: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_746_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_747_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X4: set_set_set_nat,Y4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_748_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X4: set_nat_nat,Y4: set_nat_nat] : ( sup_sup_set_nat_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_749_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_750_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_751_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_752_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_753_sup__assoc,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_754_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_755_sup__assoc,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ Z )
= ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_756_sup__assoc,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_757_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] : ( sup_sup_set_set_nat @ B5 @ A3 ) ) ) ).
% sup.commute
thf(fact_758_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A3 ) ) ) ).
% sup.commute
thf(fact_759_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B5 @ A3 ) ) ) ).
% sup.commute
thf(fact_760_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] : ( sup_sup_set_nat_nat @ B5 @ A3 ) ) ) ).
% sup.commute
thf(fact_761_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X4: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_762_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_763_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X4: set_set_set_nat,Y4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_764_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X4: set_nat_nat,Y4: set_nat_nat] : ( sup_sup_set_nat_nat @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_765_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_766_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_767_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_768_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_769_sup__left__commute,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_770_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_771_sup__left__commute,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_772_sup__left__commute,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_786_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_787_bex__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_788_bex__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_789_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_790_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_791_ball__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_792_ball__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_793_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_794_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_795_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_796_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_797_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_798_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_799_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_800_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_801_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_802_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_803_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_804_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_814_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_815_Collect__disj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X4: set_nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_816_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_817_Collect__disj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X4: set_set_nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_su4213647025997063966et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_818_Collect__disj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X4: nat > nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_819_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
| ( member_set_nat @ X4 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_820_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
| ( member_nat @ X4 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_821_Un__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A4 )
| ( member_set_set_nat @ X4 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_822_Un__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A4 )
| ( member_nat_nat @ X4 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_823_inf__sup__ord_I4_J,axiom,
! [Y: set_set_set_nat,X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_824_inf__sup__ord_I4_J,axiom,
! [Y: set_set_nat,X: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_825_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_826_inf__sup__ord_I4_J,axiom,
! [Y: set_nat_nat,X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_827_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_828_inf__sup__ord_I4_J,axiom,
! [Y: nat > nat,X: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_829_inf__sup__ord_I3_J,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_830_inf__sup__ord_I3_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_831_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_832_inf__sup__ord_I3_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_833_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_834_inf__sup__ord_I3_J,axiom,
! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_835_le__supE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ X )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_836_le__supE,axiom,
! [A2: set_set_nat,B2: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_837_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_838_le__supE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_839_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_840_le__supE,axiom,
! [A2: nat > nat,B2: nat > nat,X: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_841_le__supI,axiom,
! [A2: set_set_set_nat,X: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ X )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_842_le__supI,axiom,
! [A2: set_set_nat,X: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ X )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_843_le__supI,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_844_le__supI,axiom,
! [A2: set_nat_nat,X: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ X )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_845_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_846_le__supI,axiom,
! [A2: nat > nat,X: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X )
=> ( ( ord_less_eq_nat_nat @ B2 @ X )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_847_sup__ge1,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_848_sup__ge1,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_849_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_850_sup__ge1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_851_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_852_sup__ge1,axiom,
! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_853_sup__ge2,axiom,
! [Y: set_set_set_nat,X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_854_sup__ge2,axiom,
! [Y: set_set_nat,X: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_855_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_856_sup__ge2,axiom,
! [Y: set_nat_nat,X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_857_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_858_sup__ge2,axiom,
! [Y: nat > nat,X: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_859_le__supI1,axiom,
! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ A2 )
=> ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_860_le__supI1,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_861_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_862_le__supI1,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_863_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_864_le__supI1,axiom,
! [X: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ A2 )
=> ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_865_le__supI2,axiom,
! [X: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ B2 )
=> ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_866_le__supI2,axiom,
! [X: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ B2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_867_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_868_le__supI2,axiom,
! [X: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ B2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_869_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_870_le__supI2,axiom,
! [X: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ B2 )
=> ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_871_sup_Omono,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,D: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ D @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C2 @ D ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_872_sup_Omono,axiom,
! [C2: set_set_nat,A2: set_set_nat,D: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C2 @ D ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_873_sup_Omono,axiom,
! [C2: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_874_sup_Omono,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,D: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C2 @ D ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_875_sup_Omono,axiom,
! [C2: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_876_sup_Omono,axiom,
! [C2: nat > nat,A2: nat > nat,D: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ D @ B2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ C2 @ D ) @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_877_sup__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_878_sup__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_879_sup__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_880_sup__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_881_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_882_sup__mono,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ D )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ ( sup_sup_nat_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_883_sup__least,axiom,
! [Y: set_set_set_nat,X: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X )
=> ( ( ord_le9131159989063066194et_nat @ Z @ X )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_884_sup__least,axiom,
! [Y: set_set_nat,X: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ Z @ X )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_885_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_886_sup__least,axiom,
! [Y: set_nat_nat,X: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( ord_le9059583361652607317at_nat @ Z @ X )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_887_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_888_sup__least,axiom,
! [Y: nat > nat,X: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X )
=> ( ( ord_less_eq_nat_nat @ Z @ X )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_889_le__iff__sup,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X4: set_set_set_nat,Y4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_890_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X4: set_set_nat,Y4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_891_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_892_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X4: set_nat_nat,Y4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_893_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( sup_sup_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_894_le__iff__sup,axiom,
( ord_less_eq_nat_nat
= ( ^ [X4: nat > nat,Y4: nat > nat] :
( ( sup_sup_nat_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_895_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_896_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_897_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_898_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_899_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_900_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_901_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_902_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_903_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_904_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_905_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_906_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_907_sup__unique,axiom,
! [F2: set_set_set_nat > set_set_set_nat > set_set_set_nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ! [X3: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_set_set_nat,Y5: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y5 @ X3 )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ X3 )
=> ( ord_le9131159989063066194et_nat @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_908_sup__unique,axiom,
! [F2: set_set_nat > set_set_nat > set_set_nat,X: set_set_nat,Y: set_set_nat] :
( ! [X3: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_set_nat,Y5: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y5 @ X3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3 @ X3 )
=> ( ord_le6893508408891458716et_nat @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_set_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_909_sup__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y5 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X3 )
=> ( ord_less_eq_set_nat @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_910_sup__unique,axiom,
! [F2: set_nat_nat > set_nat_nat > set_nat_nat,X: set_nat_nat,Y: set_nat_nat] :
( ! [X3: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_nat_nat,Y5: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y5 @ X3 )
=> ( ( ord_le9059583361652607317at_nat @ Z3 @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_911_sup__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y5 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_912_sup__unique,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat > nat,X: nat > nat,Y: nat > nat] :
( ! [X3: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: nat > nat,Y5: nat > nat,Z3: nat > nat] :
( ( ord_less_eq_nat_nat @ Y5 @ X3 )
=> ( ( ord_less_eq_nat_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat_nat @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_913_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_914_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_915_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_916_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_917_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_918_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_919_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_920_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_921_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_922_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_923_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_924_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_925_sup__absorb1,axiom,
! [Y: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X )
=> ( ( sup_su4213647025997063966et_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_926_sup__absorb1,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( sup_sup_set_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_927_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_928_sup__absorb1,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( sup_sup_set_nat_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_929_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_930_sup__absorb1,axiom,
! [Y: nat > nat,X: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X )
=> ( ( sup_sup_nat_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_931_sup__absorb2,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ( sup_su4213647025997063966et_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_932_sup__absorb2,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( sup_sup_set_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_933_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_934_sup__absorb2,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( sup_sup_set_nat_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_935_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_936_sup__absorb2,axiom,
! [X: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y )
=> ( ( sup_sup_nat_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_sup_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B5: set_set_set_nat,A3: set_set_set_nat] :
( A3
= ( sup_su4213647025997063966et_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_950_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A3: set_set_nat] :
( A3
= ( sup_sup_set_set_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_951_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_952_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A3: set_nat_nat] :
( A3
= ( sup_sup_set_nat_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_953_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_954_sup_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [B5: nat > nat,A3: nat > nat] :
( A3
= ( sup_sup_nat_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_955_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_956_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_957_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_958_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_959_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_960_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_961_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_962_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_963_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_964_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_965_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_966_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_967_sup_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B5: set_set_set_nat,A3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_968_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_969_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_970_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_971_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_972_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [B5: nat > nat,A3: nat > nat] :
( ( sup_sup_nat_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_973_sup_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B5: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_974_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_975_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_976_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_977_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( sup_sup_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_978_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [A3: nat > nat,B5: nat > nat] :
( ( sup_sup_nat_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [B5: set_set_nat,A3: set_set_nat] :
( ( A3
= ( sup_sup_set_set_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1002_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1003_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [B5: set_nat_nat,A3: set_nat_nat] :
( ( A3
= ( sup_sup_set_nat_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1004_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1005_sup_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B5: set_set_set_nat,A3: set_set_set_nat] :
( ( A3
= ( sup_su4213647025997063966et_nat @ A3 @ B5 ) )
& ( A3 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1015_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_1016_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_1017_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_1018_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_1019_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1020_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_1021_less__supI2,axiom,
! [X: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ X @ B2 )
=> ( ord_less_set_set_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1022_less__supI2,axiom,
! [X: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ X @ B2 )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1023_less__supI2,axiom,
! [X: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ B2 )
=> ( ord_less_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1024_less__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1025_less__supI2,axiom,
! [X: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ B2 )
=> ( ord_le152980574450754630et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1026_less__supI1,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ X @ A2 )
=> ( ord_less_set_set_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1027_less__supI1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ X @ A2 )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1028_less__supI1,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ A2 )
=> ( ord_less_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1029_less__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1030_less__supI1,axiom,
! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X @ A2 )
=> ( ord_le152980574450754630et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1031_distrib__imp1,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ! [X3: set_set_nat,Y5: set_set_nat,Z3: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y5 @ Z3 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X3 @ Y5 ) @ ( inf_inf_set_set_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1032_distrib__imp1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X3: set_nat,Y5: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y5 @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y5 ) @ ( inf_inf_set_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1033_distrib__imp1,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ! [X3: set_set_set_nat,Y5: set_set_set_nat,Z3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ ( sup_su4213647025997063966et_nat @ Y5 @ Z3 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X3 @ Y5 ) @ ( inf_in5711780100303410308et_nat @ X3 @ Z3 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1034_distrib__imp1,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ! [X3: set_nat_nat,Y5: set_nat_nat,Z3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ Y5 @ Z3 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X3 @ Y5 ) @ ( inf_inf_set_nat_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1035_distrib__imp2,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ! [X3: set_set_nat,Y5: set_set_nat,Z3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( inf_inf_set_set_nat @ Y5 @ Z3 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y5 ) @ ( sup_sup_set_set_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ ( inf_inf_set_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1036_distrib__imp2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X3: set_nat,Y5: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y5 @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y5 ) @ ( sup_sup_set_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1037_distrib__imp2,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ! [X3: set_set_set_nat,Y5: set_set_set_nat,Z3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ ( inf_in5711780100303410308et_nat @ Y5 @ Z3 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X3 @ Y5 ) @ ( sup_su4213647025997063966et_nat @ X3 @ Z3 ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ ( inf_in5711780100303410308et_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1038_distrib__imp2,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ! [X3: set_nat_nat,Y5: set_nat_nat,Z3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ ( inf_inf_set_nat_nat @ Y5 @ Z3 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X3 @ Y5 ) @ ( sup_sup_set_nat_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1039_inf__sup__distrib1,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ ( inf_inf_set_set_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1040_inf__sup__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1041_inf__sup__distrib1,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y ) @ ( inf_in5711780100303410308et_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1042_inf__sup__distrib1,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1043_inf__sup__distrib2,axiom,
! [Y: set_set_nat,Z: set_set_nat,X: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ X ) @ ( inf_inf_set_set_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1044_inf__sup__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1045_inf__sup__distrib2,axiom,
! [Y: set_set_set_nat,Z: set_set_set_nat,X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z ) @ X )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y @ X ) @ ( inf_in5711780100303410308et_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1046_inf__sup__distrib2,axiom,
! [Y: set_nat_nat,Z: set_nat_nat,X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y @ X ) @ ( inf_inf_set_nat_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1047_sup__inf__distrib1,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1048_sup__inf__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1049_sup__inf__distrib1,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1050_sup__inf__distrib1,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1051_sup__inf__distrib2,axiom,
! [Y: set_set_nat,Z: set_set_nat,X: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ X ) @ ( sup_sup_set_set_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1052_sup__inf__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1053_sup__inf__distrib2,axiom,
! [Y: set_set_set_nat,Z: set_set_set_nat,X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y @ Z ) @ X )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y @ X ) @ ( sup_su4213647025997063966et_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1054_sup__inf__distrib2,axiom,
! [Y: set_nat_nat,Z: set_nat_nat,X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y @ X ) @ ( sup_sup_set_nat_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1055_Un__mono,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1056_Un__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1057_Un__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1058_Un__mono,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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 )
=> ! [B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B6 @ B )
=> ( C
!= ( sup_su4213647025997063966et_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1080_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 )
=> ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ B )
=> ( C
!= ( sup_sup_set_set_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1081_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 )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B )
=> ( C
!= ( sup_sup_set_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1082_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 )
=> ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ B )
=> ( C
!= ( sup_sup_set_nat_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1083_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1084_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1085_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1086_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1087_infinite__Un,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S2 @ T2 ) ) )
= ( ~ ( finite1152437895449049373et_nat @ S2 )
| ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1088_infinite__Un,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S2 )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1089_infinite__Un,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) ) )
= ( ~ ( finite6739761609112101331et_nat @ S2 )
| ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1090_infinite__Un,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S2 @ T2 ) ) )
= ( ~ ( finite2115694454571419734at_nat @ S2 )
| ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1091_Un__infinite,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S2 )
=> ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1092_Un__infinite,axiom,
! [S2: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1093_Un__infinite,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S2 )
=> ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1094_Un__infinite,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S2 )
=> ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1095_finite__UnI,axiom,
! [F: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ F )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_1096_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_1097_finite__UnI,axiom,
! [F: set_set_set_nat,G: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F )
=> ( ( finite6739761609112101331et_nat @ G )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_1098_finite__UnI,axiom,
! [F: set_nat_nat,G: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F )
=> ( ( finite2115694454571419734at_nat @ G )
=> ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_1099_Un__Int__crazy,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ B @ C ) ) @ ( inf_inf_set_set_nat @ C @ A ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ B @ C ) ) @ ( sup_sup_set_set_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1100_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C ) ) @ ( inf_inf_set_nat @ C @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C ) ) @ ( sup_sup_set_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1101_Un__Int__crazy,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ B @ C ) ) @ ( inf_in5711780100303410308et_nat @ C @ A ) )
= ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ B @ C ) ) @ ( sup_su4213647025997063966et_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1102_Un__Int__crazy,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ B @ C ) ) @ ( inf_inf_set_nat_nat @ C @ A ) )
= ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ B @ C ) ) @ ( sup_sup_set_nat_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1103_Int__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_1104_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_1105_Int__Un__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_1106_Int__Un__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_1107_Un__Int__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_1108_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_1109_Un__Int__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_1110_Un__Int__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_1111_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1112_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1113_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1114_mult__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1115_mult__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1116_mult__le__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1117_Graphs__def,axiom,
( clique5786534781347292306Graphs
= ( ^ [V: set_nat] :
( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ V @ V ) ) ) ) ) ).
% Graphs_def
thf(fact_1118_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1119_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1120_numbers2__mono,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y ) @ ( clique3652268606331196573umbers @ Y ) ) ) ) ).
% numbers2_mono
thf(fact_1121_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique363107459185959606CLIQUE @ k2 ) ).
% POS_sub_CLIQUE
thf(fact_1122_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique363107459185959606CLIQUE @ k2 ) ).
% POS_CLIQUE
thf(fact_1123_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k2 ) ).
% finite_\<F>
thf(fact_1124_G_I3_J,axiom,
~ ( member_set_set_nat @ g @ ( clique3210737319928189260st_ACC @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) ).
% G(3)
thf(fact_1125__092_060open_062G_A_092_060notin_062_A_123G_A_092_060in_062_A_092_060G_062_O_A_092_060exists_062D_092_060in_062X_A_092_060odot_062l_AY_O_AD_A_092_060subseteq_062_AG_125_092_060close_062,axiom,
~ ( member_set_set_nat @ g
@ ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) )
& ( ord_le6893508408891458716et_nat @ X4 @ G2 ) ) ) ) ) ).
% \<open>G \<notin> {G \<in> \<G>. \<exists>D\<in>X \<odot>l Y. D \<subseteq> G}\<close>
thf(fact_1126__092_060open_062G_A_092_060in_062_A_123G_A_092_060in_062_A_092_060G_062_O_A_092_060exists_062D_092_060in_062X_A_092_060odot_062_AY_O_AD_A_092_060subseteq_062_AG_125_092_060close_062,axiom,
( member_set_set_nat @ g
@ ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( clique5469973757772500719t_odot @ x @ y ) )
& ( ord_le6893508408891458716et_nat @ X4 @ G2 ) ) ) ) ) ).
% \<open>G \<in> {G \<in> \<G>. \<exists>D\<in>X \<odot> Y. D \<subseteq> G}\<close>
thf(fact_1127_G_I2_J,axiom,
member_set_set_nat @ g @ ( clique3210737319928189260st_ACC @ k2 @ ( clique5469973757772500719t_odot @ x @ y ) ) ).
% G(2)
thf(fact_1128__092_060open_062merge_A_092_060equiv_062_A_092_060lambda_062C_AV_O_AC_A_092_060union_062_AV_094_092_060two_062_092_060close_062,axiom,
( merge
= ( ^ [C3: set_nat,V: set_nat] : ( clique6722202388162463298od_nat @ ( sup_sup_set_nat @ C3 @ V ) @ ( sup_sup_set_nat @ C3 @ V ) ) ) ) ).
% \<open>merge \<equiv> \<lambda>C V. C \<union> V^\<two>\<close>
thf(fact_1129_v__gs__union,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_union
thf(fact_1130_v__union,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_1131_ACC__union,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k2 @ X2 ) @ ( clique3210737319928189260st_ACC @ k2 @ Y2 ) ) ) ).
% ACC_union
thf(fact_1132_ACC__odot,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k2 @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k2 @ X2 ) @ ( clique3210737319928189260st_ACC @ k2 @ Y2 ) ) ) ).
% ACC_odot
thf(fact_1133_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737375870294875st_NEG @ k2 ) ) ).
% finite_POS_NEG
thf(fact_1134__092_060open_062G_A_092_060notin_062_AACC_A_IX_A_092_060odot_062l_AY_J_A_092_060union_062_AGS_092_060close_062,axiom,
~ ( member_set_set_nat @ g @ ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) @ gs ) ) ).
% \<open>G \<notin> ACC (X \<odot>l Y) \<union> GS\<close>
thf(fact_1135_ACC__def,axiom,
! [X2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k2 @ X2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ( clique3686358387679108662ccepts @ X2 @ G2 ) ) ) ) ).
% ACC_def
thf(fact_1136_local_Omerge__def,axiom,
! [C: set_nat,V2: set_nat] :
( ( merge @ C @ V2 )
= ( clique6722202388162463298od_nat @ ( sup_sup_set_nat @ C @ V2 ) @ ( sup_sup_set_nat @ C @ V2 ) ) ) ).
% local.merge_def
thf(fact_1137_C__def,axiom,
! [V2: set_nat] :
( ( c @ V2 )
= ( fChoice_set_nat
@ ^ [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V2 )
& ( ( finite_card_nat @ C3 )
= ( suc @ l ) ) ) ) ) ).
% C_def
thf(fact_1138_CLIQUE__def,axiom,
( ( clique363107459185959606CLIQUE @ k2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
& ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( clique3326749438856946062irst_K @ k2 ) )
& ( ord_le6893508408891458716et_nat @ X4 @ G2 ) ) ) ) ) ).
% CLIQUE_def
thf(fact_1139_ACC__I,axiom,
! [G: set_set_nat,X2: set_set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X2 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k2 @ X2 ) ) ) ) ).
% ACC_I
thf(fact_1140_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_1141_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_1142_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_1143__092_060K_062__altdef,axiom,
( ( clique3326749438856946062irst_K @ k2 )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) )
& ( ( finite_card_nat @ V )
= k2 ) ) ) ) ).
% \<K>_altdef
thf(fact_1144_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k2 ) @ ( clique3210737375870294875st_NEG @ k2 ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_1145__092_060open_062H_A_092_060in_062_A_123D_A_092_060union_062_AE_A_124D_AE_O_AD_A_092_060in_062_AX_A_092_060and_062_AE_A_092_060in_062_AY_125_092_060close_062,axiom,
( member_set_set_nat @ h
@ ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D4: set_set_nat,E2: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D4 @ E2 ) )
& ( member_set_set_nat @ D4 @ x )
& ( member_set_set_nat @ E2 @ y ) ) ) ) ).
% \<open>H \<in> {D \<union> E |D E. D \<in> X \<and> E \<in> Y}\<close>
thf(fact_1146_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% empty_\<G>
thf(fact_1147_kml,axiom,
ord_less_eq_nat @ k2 @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k2 ) @ l ) ).
% kml
thf(fact_1148_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k2 ) ) ).
% empty_CLIQUE
thf(fact_1149_odot__def,axiom,
( clique5469973757772500719t_odot
= ( ^ [X5: set_set_set_nat,Y7: set_set_set_nat] :
( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D4: set_set_nat,E2: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D4 @ E2 ) )
& ( member_set_set_nat @ D4 @ X5 )
& ( member_set_set_nat @ E2 @ Y7 ) ) ) ) ) ).
% odot_def
thf(fact_1150_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1151_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1152_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_1153_GS__def,axiom,
( gs
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat,W: set_nat] :
( ( Uu
= ( merge @ ( c @ V ) @ W ) )
& ( member_set_nat @ V @ vs )
& ( member_set_nat @ W @ ( k @ ( c @ V ) ) ) ) ) ) ).
% GS_def
thf(fact_1154_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_1155_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_1156__092_060open_062K_A_092_060equiv_062_A_092_060lambda_062C_O_A_123W_O_AW_A_092_060subseteq_062_A_091m_093_A_N_AC_A_092_060and_062_Acard_AW_A_061_Ak_A_N_ASuc_Al_125_092_060close_062,axiom,
( k
= ( ^ [C3: set_nat] :
( collect_set_nat
@ ^ [W: set_nat] :
( ( ord_less_eq_set_nat @ W @ ( minus_minus_set_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) @ C3 ) )
& ( ( finite_card_nat @ W )
= ( minus_minus_nat @ k2 @ ( suc @ l ) ) ) ) ) ) ) ).
% \<open>K \<equiv> \<lambda>C. {W. W \<subseteq> [m] - C \<and> card W = k - Suc l}\<close>
thf(fact_1157_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1158_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1159_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1160_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1161_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1162_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1163_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1164_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_1165_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1166_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1167_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1168_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1169_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1170_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1171_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1172_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1173_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1174_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_1175_Clique__def,axiom,
( clique6749503327923060270Clique
= ( ^ [V: set_nat,K4: nat] :
( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ V ) )
& ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ C3 @ C3 ) @ G2 )
& ( ( finite_card_nat @ C3 )
= K4 ) ) ) ) ) ) ).
% Clique_def
thf(fact_1176_K__def,axiom,
! [C: set_nat] :
( ( k @ C )
= ( collect_set_nat
@ ^ [W: set_nat] :
( ( ord_less_eq_set_nat @ W @ ( minus_minus_set_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) @ C ) )
& ( ( finite_card_nat @ W )
= ( minus_minus_nat @ k2 @ ( suc @ l ) ) ) ) ) ) ).
% K_def
thf(fact_1177_Vs__C_I5_J,axiom,
! [V2: set_nat] :
( ( member_set_nat @ V2 @ vs )
=> ( ( finite_card_set_nat @ ( k @ ( c @ V2 ) ) )
= ( binomial @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k2 ) @ ( suc @ l ) ) @ ( minus_minus_nat @ k2 @ ( suc @ l ) ) ) ) ) ).
% Vs_C(5)
thf(fact_1178_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k2 ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k2 ) @ k2 ) ) ).
% card_POS
thf(fact_1179_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_1180_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_1181_choose__mult,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K2 ) )
= ( times_times_nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) ) ) ) ) ) ).
% choose_mult
thf(fact_1182_binomial__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% binomial_symmetric
thf(fact_1183_Suc__times__binomial__eq,axiom,
! [N: nat,K2: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1184_Suc__times__binomial,axiom,
! [K2: nat,N: nat] :
( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).
% Suc_times_binomial
thf(fact_1185_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1186_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1187_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1188_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1189_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F2 @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F2 @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1190_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less_nat @ K4 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1191_ACC__cf___092_060F_062,axiom,
! [X2: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k2 @ X2 ) @ ( clique2971579238625216137irst_F @ k2 ) ) ).
% ACC_cf_\<F>
thf(fact_1192_first__assumptions_Ocard__v__gs__join,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat,Z4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z4 @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z4 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ) ) ) ) ).
% first_assumptions.card_v_gs_join
thf(fact_1193_ACC__cf__odot,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k2 @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k2 @ X2 ) @ ( clique951075384711337423ACC_cf @ k2 @ Y2 ) ) ) ).
% ACC_cf_odot
thf(fact_1194_ACC__cf__union,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k2 @ X2 ) @ ( clique951075384711337423ACC_cf @ k2 @ Y2 ) ) ) ).
% ACC_cf_union
thf(fact_1195_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k2 @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_1196_ACC__cf__mono,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k2 @ X2 ) @ ( clique951075384711337423ACC_cf @ k2 @ Y2 ) ) ) ).
% ACC_cf_mono
thf(fact_1197_finite__ACC,axiom,
! [X2: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k2 @ X2 ) ) ).
% finite_ACC
thf(fact_1198_ACC__cf__def,axiom,
! [X2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k2 @ X2 )
= ( collect_nat_nat
@ ^ [F3: nat > nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k2 ) )
& ( clique3686358387679108662ccepts @ X2 @ ( clique5033774636164728462irst_C @ k2 @ F3 ) ) ) ) ) ).
% ACC_cf_def
thf(fact_1199_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K2 @ X2 ) @ ( clique2971579238625216137irst_F @ K2 ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1200_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K2 @ X2 ) @ ( clique951075384711337423ACC_cf @ K2 @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1201_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K2 @ X2 ) @ ( clique951075384711337423ACC_cf @ K2 @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1202_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K2 @ X2 ) @ ( clique951075384711337423ACC_cf @ K2 @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1203_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_1204_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K2 @ X2 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1205_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1206_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P2: nat,K2: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_1207_first__assumptions_OACC__union,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K2 @ X2 ) @ ( clique3210737319928189260st_ACC @ K2 @ Y2 ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_1208_first__assumptions_OACC__empty,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1209_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3686358387679108662ccepts @ X2 @ G )
= ( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ X2 )
& ( ord_le6893508408891458716et_nat @ X4 @ G ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1210_first__assumptions_OacceptsI,axiom,
! [L: nat,P2: nat,K2: nat,D2: set_set_nat,G: set_set_nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ G )
=> ( ( member_set_set_nat @ D2 @ X2 )
=> ( clique3686358387679108662ccepts @ X2 @ G ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1211_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K2 ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1212_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K2 ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1213_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1214_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K2 @ X2 @ Y2 ) @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1215_first__assumptions_Ov__mono,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1216_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P2: nat,K2: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_1217_first__assumptions_OACC__odot,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K2 @ X2 ) @ ( clique3210737319928189260st_ACC @ K2 @ Y2 ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1218_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique363107459185959606CLIQUE @ K2 ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1219_first__assumptions_Ov__empty,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_1220_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1221_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_1222_first__assumptions_Ov__union,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1223_first__assumptions_Oodot__def,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique5469973757772500719t_odot @ X2 @ Y2 )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D4: set_set_nat,E2: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D4 @ E2 ) )
& ( member_set_set_nat @ D4 @ X2 )
& ( member_set_set_nat @ E2 @ Y2 ) ) ) ) ) ).
% first_assumptions.odot_def
thf(fact_1224_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_1225_first__assumptions_Ounion___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.union_\<G>
thf(fact_1226_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_1227_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique363107459185959606CLIQUE @ K2 ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1228_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ L @ K2 ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1229_first__assumptions_Oodotl__def,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique7966186356931407165_odotl @ L @ K2 @ X2 @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) @ ( clique7840962075309931874st_G_l @ L @ K2 ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1230_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_1231_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K2 ) @ ( clique3210737375870294875st_NEG @ K2 ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1232_first__assumptions_Ocard__POS,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K2 ) )
= ( binomial @ ( assump1710595444109740334irst_m @ K2 ) @ K2 ) ) ) ).
% first_assumptions.card_POS
thf(fact_1233_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_1234_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique3210737375870294875st_NEG @ K2 ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1235_first__assumptions_O_092_060K_062___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.\<K>_\<G>
thf(fact_1236_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_1237_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X2 @ Y2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_1238_first__assumptions_ONEG___092_060G_062,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ K2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.NEG_\<G>
thf(fact_1239_first__assumptions_OACC__I,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X2 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ K2 @ X2 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_1240_first__assumptions_O_092_060G_062__def,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.\<G>_def
thf(fact_1241_first__assumptions_OACC__def,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3210737319928189260st_ACC @ K2 @ X2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( clique3686358387679108662ccepts @ X2 @ G2 ) ) ) ) ) ).
% first_assumptions.ACC_def
thf(fact_1242_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_1243_first__assumptions_Ofinite__v__gs,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) ) ) ).
% first_assumptions.finite_v_gs
thf(fact_1244_first__assumptions_O_092_060G_062l__def,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique7840962075309931874st_G_l @ L @ K2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ord_less_eq_nat @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G2 ) ) @ L ) ) ) ) ) ).
% first_assumptions.\<G>l_def
thf(fact_1245_first__assumptions_O_092_060K_062__def,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3326749438856946062irst_K @ K2 )
= ( collect_set_set_nat
@ ^ [K3: set_set_nat] :
( ( member_set_set_nat @ K3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K3 ) )
= K2 )
& ( K3
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K3 ) @ ( clique5033774636164728513irst_v @ K3 ) ) ) ) ) ) ) ).
% first_assumptions.\<K>_def
thf(fact_1246_first__assumptions_O_092_060K_062__altdef,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3326749438856946062irst_K @ K2 )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) )
& ( ( finite_card_nat @ V )
= K2 ) ) ) ) ) ).
% first_assumptions.\<K>_altdef
thf(fact_1247_first__assumptions_OCLIQUE__def,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique363107459185959606CLIQUE @ K2 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( clique3326749438856946062irst_K @ K2 ) )
& ( ord_le6893508408891458716et_nat @ X4 @ G2 ) ) ) ) ) ) ).
% first_assumptions.CLIQUE_def
thf(fact_1248_first__assumptions_Okml,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_less_eq_nat @ K2 @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K2 ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1249_ACC__cf__I,axiom,
! [F: nat > nat,X2: set_set_set_nat] :
( ( member_nat_nat @ F @ ( clique2971579238625216137irst_F @ k2 ) )
=> ( ( clique3686358387679108662ccepts @ X2 @ ( clique5033774636164728462irst_C @ k2 @ F ) )
=> ( member_nat_nat @ F @ ( clique951075384711337423ACC_cf @ k2 @ X2 ) ) ) ) ).
% ACC_cf_I
thf(fact_1250_first__assumptions_Omp,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.mp
thf(fact_1251_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1252_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1253_first__assumptions_Opl,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_less_nat @ L @ P2 ) ) ).
% first_assumptions.pl
thf(fact_1254_first__assumptions_Okp,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_less_nat @ P2 @ K2 ) ) ).
% first_assumptions.kp
thf(fact_1255_first__assumptions_Ok,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_less_nat @ L @ K2 ) ) ).
% first_assumptions.k
thf(fact_1256_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P2: nat,K2: nat,F: nat > nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( member_nat_nat @ F @ ( clique2971579238625216137irst_F @ K2 ) )
=> ( ( clique3686358387679108662ccepts @ X2 @ ( clique5033774636164728462irst_C @ K2 @ F ) )
=> ( member_nat_nat @ F @ ( clique951075384711337423ACC_cf @ K2 @ X2 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1257_first__assumptions_OACC__cf__def,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique951075384711337423ACC_cf @ K2 @ X2 )
= ( collect_nat_nat
@ ^ [F3: nat > nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K2 ) )
& ( clique3686358387679108662ccepts @ X2 @ ( clique5033774636164728462irst_C @ K2 @ F3 ) ) ) ) ) ) ).
% first_assumptions.ACC_cf_def
thf(fact_1258_first__assumptions_Okm,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ord_less_nat @ K2 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.km
thf(fact_1259_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k2 )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k2 ) @ ( clique2971579238625216137irst_F @ k2 ) ) ) ).
% local.NEG_def
thf(fact_1260_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S )
=> ( ( S != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ S )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S )
=> ( ( ord_less_eq_nat_nat @ X3 @ Xa2 )
| ( ord_less_eq_nat_nat @ Xa2 @ X3 ) ) ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ S )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S )
=> ( ord_less_eq_nat_nat @ X3 @ Xa ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1261_first__assumptions_ONEG__def,axiom,
! [L: nat,P2: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique3210737375870294875st_NEG @ K2 )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K2 ) @ ( clique2971579238625216137irst_F @ K2 ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1262_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S )
=> ( ( S != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ S )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S )
=> ( ( ord_less_eq_nat_nat @ X3 @ Xa2 )
| ( ord_less_eq_nat_nat @ Xa2 @ X3 ) ) ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ S )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S )
=> ( ord_less_eq_nat_nat @ Xa @ X3 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1263_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_1264_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P2: nat,K2: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique8462013130872731469t_v_gs @ X2 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X2 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_1265_v__def,axiom,
( clique5033774636164728513irst_v
= ( ^ [G2: set_set_nat] :
( collect_nat
@ ^ [X4: nat] :
? [Y4: nat] : ( member_set_nat @ ( insert_nat @ X4 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ G2 ) ) ) ) ).
% v_def
thf(fact_1266_first__assumptions_Ov__def,axiom,
! [L: nat,P2: nat,K2: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique5033774636164728513irst_v @ G )
= ( collect_nat
@ ^ [X4: nat] :
? [Y4: nat] : ( member_set_nat @ ( insert_nat @ X4 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ G ) ) ) ) ).
% first_assumptions.v_def
thf(fact_1267_first__assumptions_OC__def,axiom,
! [L: nat,P2: nat,K2: nat,F2: nat > nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K2 )
=> ( ( clique5033774636164728462irst_C @ K2 @ F2 )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X4: nat,Y4: nat] :
( ( Uu
= ( insert_nat @ X4 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X4 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ( F2 @ X4 )
!= ( F2 @ Y4 ) ) ) ) ) ) ).
% first_assumptions.C_def
thf(fact_1268_unbounded__k__infinite,axiom,
! [K2: nat,S2: set_nat] :
( ! [M6: nat] :
( ( ord_less_nat @ K2 @ M6 )
=> ? [N7: nat] :
( ( ord_less_nat @ M6 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1269_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M4: nat] :
? [N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1270_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M4: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1271_choose__mono,axiom,
! [N: nat,M: nat,K2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ M @ K2 ) ) ) ).
% choose_mono
% Helper facts (1)
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: set_nat > $o] :
( ( P @ ( fChoice_set_nat @ P ) )
= ( ? [X5: set_nat] : ( P @ X5 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_set_nat @ ( clique5033774636164728513irst_v @ h )
@ ( inf_inf_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) )
@ ( collect_set_nat
@ ^ [V: set_nat] :
( ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) )
& ( ord_less_eq_nat @ ( suc @ l ) @ ( finite_card_nat @ V ) ) ) ) ) ) ).
%------------------------------------------------------------------------------