TPTP Problem File: SLH0485^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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_00388_011108__16147476_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1367 ( 662 unt; 96 typ; 0 def)
% Number of atoms : 2960 ( 909 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9320 ( 210 ~; 39 |; 150 &;7803 @)
% ( 0 <=>;1118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 647 ( 647 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 89 usr; 14 con; 0-3 aty)
% Number of variables : 3404 ( 308 ^;3036 !; 60 ?;3404 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:48:23.426
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (89)
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OClique,type,
clique6749503327923060270Clique: set_nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
clique5786534781347292306Graphs: set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique134924887794942129at_nat: set_nat_nat > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
clique8906516429304539640et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique1181040904276305582et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC,type,
clique3210737319928189260st_ACC: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC__cf,type,
clique951075384711337423ACC_cf: nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OC,type,
clique5033774636164728462irst_C: nat > ( nat > nat ) > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
clique363107459185959606CLIQUE: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_ONEG,type,
clique3210737375870294875st_NEG: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060F_062,type,
clique2971579238625216137irst_F: nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oaccepts,type,
clique3686358387679108662ccepts: set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodot,type,
clique5469973757772500719t_odot: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite3586981331298542604at_nat: set_set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
finite5926941155766903689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inf_inf_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
sup_sup_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
sup_sup_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
sup_su5917979686466268903_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bot_bot_nat_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_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_OBex_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bex_nat_nat: set_nat_nat > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Nat__Onat,type,
bex_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Set__Oset_It__Nat__Onat_J,type,
bex_set_nat: set_set_nat > ( set_nat > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bex_set_set_nat: set_set_set_nat > ( set_set_nat > $o ) > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_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_D____,type,
d: set_set_nat ).
thf(sy_v_E____,type,
e: set_set_nat ).
thf(sy_v_G____,type,
g: 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,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1270)
thf(fact_0__C_K_C_I3_J,axiom,
member_set_set_nat @ g @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% "*"(3)
thf(fact_1__C_K_C_I5_J,axiom,
ord_le6893508408891458716et_nat @ e @ g ).
% "*"(5)
thf(fact_2__C_K_C_I4_J,axiom,
ord_le6893508408891458716et_nat @ d @ g ).
% "*"(4)
thf(fact_3__C_K_C_I2_J,axiom,
member_set_set_nat @ e @ y ).
% "*"(2)
thf(fact_4__C_K_C_I1_J,axiom,
member_set_set_nat @ d @ x ).
% "*"(1)
thf(fact_5_odot__def,axiom,
( clique5469973757772500719t_odot
= ( ^ [X: set_set_set_nat,Y: set_set_set_nat] :
( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D @ E ) )
& ( member_set_set_nat @ D @ X )
& ( member_set_set_nat @ E @ Y ) ) ) ) ) ).
% odot_def
thf(fact_6_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_7_UnCI,axiom,
! [C: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ A ) )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_8_UnCI,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ A ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_9_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_10_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_11_Un__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
| ( member_set_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_12_Un__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
| ( member_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_13_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_14_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_15_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_16_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_17_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_18_sup__idem,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_19_sup__idem,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_20_sup__idem,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_21_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_22_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_23_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_24_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_25_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_26_sup__left__idem,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_27_sup__left__idem,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_28_sup__left__idem,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_29_sup__left__idem,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_30_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_31_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_32_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_33_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_34__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_AG_A_092_060in_062_A_092_060G_062_059_AD_A_092_060subseteq_062_AG_059_AE_A_092_060subseteq_062_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D2: set_set_nat] :
( ( member_set_set_nat @ D2 @ x )
=> ! [E2: set_set_nat] :
( ( member_set_set_nat @ E2 @ y )
=> ( ( member_set_set_nat @ g @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ g )
=> ~ ( ord_le6893508408891458716et_nat @ E2 @ g ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>D E. \<lbrakk>D \<in> X; E \<in> Y; G \<in> \<G>; D \<subseteq> G; E \<subseteq> G\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_35_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
| ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_36_Un__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
| ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_37_Un__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
| ( member_nat_nat @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_38_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
| ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_39_Collect__disj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_40_Collect__disj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su4213647025997063966et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_41_Collect__disj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_42_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_43__092_060open_062G_A_092_060in_062_AACC_AX_A_092_060inter_062_AACC_AY_092_060close_062,axiom,
member_set_set_nat @ g @ ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ x ) @ ( clique3210737319928189260st_ACC @ k @ y ) ) ).
% \<open>G \<in> ACC X \<inter> ACC Y\<close>
thf(fact_44_ACC__union,axiom,
! [X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y3 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X4 ) @ ( clique3210737319928189260st_ACC @ k @ Y3 ) ) ) ).
% ACC_union
thf(fact_45_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( member_set_set_nat @ X5 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_46_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( member_set_nat @ X5 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_47_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( member_nat @ X5 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_48_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
=> ( member_nat_nat @ X5 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_49_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_50_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_51_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_52_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_53_inf__idem,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_54_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_55_inf__left__idem,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_56_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_57_inf__right__idem,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Y2 )
= ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_58_IntI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_59_IntI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_60_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_61_IntI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_62_Int__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_63_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_64_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_65_Int__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ( member_set_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_66_le__inf__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
& ( ord_le9131159989063066194et_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_67_le__inf__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
& ( ord_le6893508408891458716et_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_68_le__inf__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_69_le__inf__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
& ( ord_le9059583361652607317at_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_70_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_71_le__inf__iff,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat_nat @ X2 @ Y2 )
& ( ord_less_eq_nat_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_72_inf_Obounded__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_73_inf_Obounded__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_74_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_75_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_76_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_77_inf_Obounded__iff,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat_nat @ A2 @ B2 )
& ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_78_sup_Obounded__iff,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 )
= ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
& ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_79_sup_Obounded__iff,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_80_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_81_sup_Obounded__iff,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
= ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_82_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_83_sup_Obounded__iff,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat_nat @ B2 @ A2 )
& ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_84_le__sup__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Z )
& ( ord_le9131159989063066194et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_85_le__sup__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Z )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_86_le__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X2 @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_87_le__sup__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Z )
& ( ord_le9059583361652607317at_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_88_le__sup__iff,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_89_le__sup__iff,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat_nat @ X2 @ Z )
& ( ord_less_eq_nat_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_90_Int__subset__iff,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C2 @ A )
& ( ord_le9131159989063066194et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_91_Int__subset__iff,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A )
& ( ord_le6893508408891458716et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_92_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_93_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_94_Un__subset__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 )
= ( ( ord_le9131159989063066194et_nat @ A @ C2 )
& ( ord_le9131159989063066194et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_95_Un__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A @ C2 )
& ( ord_le6893508408891458716et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_96_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_97_Un__subset__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( ( ord_le9059583361652607317at_nat @ A @ C2 )
& ( ord_le9059583361652607317at_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_98_sup__inf__absorb,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_99_sup__inf__absorb,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_100_sup__inf__absorb,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_101_sup__inf__absorb,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_102_inf__sup__absorb,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_103_inf__sup__absorb,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_104_inf__sup__absorb,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_105_inf__sup__absorb,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_106_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_107_Un__Int__eq_I1_J,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_108_Un__Int__eq_I1_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_109_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_110_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_111_Un__Int__eq_I2_J,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_112_Un__Int__eq_I2_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_113_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_114_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_115_Un__Int__eq_I3_J,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S @ ( sup_su4213647025997063966et_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_116_Un__Int__eq_I3_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S @ ( sup_sup_set_nat_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_117_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_118_Un__Int__eq_I4_J,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T @ ( sup_sup_set_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_119_Un__Int__eq_I4_J,axiom,
! [T: set_set_set_nat,S: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T @ ( sup_su4213647025997063966et_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_120_Un__Int__eq_I4_J,axiom,
! [T: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T @ ( sup_sup_set_nat_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_121_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_122_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_123_Int__Un__eq_I1_J,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_124_Int__Un__eq_I1_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_125_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_126_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_127_Int__Un__eq_I2_J,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_128_Int__Un__eq_I2_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_129_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_130_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_131_Int__Un__eq_I3_J,axiom,
! [S: set_set_set_nat,T: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S @ ( inf_in5711780100303410308et_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_132_Int__Un__eq_I3_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S @ ( inf_inf_set_nat_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_133_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_134_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_135_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_136_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_137_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_138_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_142_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X5: set_set_nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_143_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_144_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X5: nat > nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_145_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X5: set_nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_146_Int__Un__eq_I4_J,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T @ ( inf_inf_set_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_147_Int__Un__eq_I4_J,axiom,
! [T: set_set_set_nat,S: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T @ ( inf_in5711780100303410308et_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_148_Int__Un__eq_I4_J,axiom,
! [T: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T @ ( inf_inf_set_nat_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_149_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_150_calculation,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ x @ y ) ) )
=> ( member_set_set_nat @ G @ ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ x ) @ ( clique3210737319928189260st_ACC @ k @ y ) ) ) ) ).
% calculation
thf(fact_151_union___092_060G_062,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% union_\<G>
thf(fact_152_inf__sup__aci_I4_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_153_inf__sup__aci_I3_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_154_inf__sup__aci_I2_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Z )
= ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_155_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_156_inf__sup__ord_I2_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_157_inf__sup__ord_I2_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_158_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_159_inf__sup__ord_I2_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_160_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_161_inf__sup__ord_I2_J,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_162_inf__sup__ord_I1_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_163_inf__sup__ord_I1_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_164_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_165_inf__sup__ord_I1_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_166_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_167_inf__sup__ord_I1_J,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_168_inf__le1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_169_inf__le1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_170_inf__le1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_171_inf__le1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_172_inf__le1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_173_inf__le1,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_174_inf__le2,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_175_inf__le2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_176_inf__le2,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_177_inf__le2,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_178_inf__le2,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_179_inf__le2,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_180_le__infE,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ X2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_181_le__infE,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_182_le__infE,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_183_le__infE,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_184_le__infE,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_185_le__infE,axiom,
! [X2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_186_le__infI,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_187_le__infI,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_188_le__infI,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_189_le__infI,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_190_le__infI,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_191_le__infI,axiom,
! [X2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ B2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_192_inf__mono,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat,D3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D3 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_193_inf__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D3 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_194_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_195_inf__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_196_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_197_inf__mono,axiom,
! [A2: nat > nat,C: nat > nat,B2: nat > nat,D3: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ( ord_less_eq_nat_nat @ B2 @ D3 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ ( inf_inf_nat_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_198_le__infI1,axiom,
! [A2: set_set_set_nat,X2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_199_le__infI1,axiom,
! [A2: set_set_nat,X2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_200_le__infI1,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_201_le__infI1,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_202_le__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_203_le__infI1,axiom,
! [A2: nat > nat,X2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_204_le__infI2,axiom,
! [B2: set_set_set_nat,X2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_205_le__infI2,axiom,
! [B2: set_set_nat,X2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_206_le__infI2,axiom,
! [B2: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_207_le__infI2,axiom,
! [B2: set_nat_nat,X2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_208_le__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_209_le__infI2,axiom,
! [B2: nat > nat,X2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_210_inf_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_211_inf__assoc,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Z )
= ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_212_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_213_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_214_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_215_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_216_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_217_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_218_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_219_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_220_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_221_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_222_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_223_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_224_inf__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X5 @ ( F @ Y5 @ Z2 ) ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_225_inf__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ! [X5: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_set_nat,Y5: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X5 @ ( F @ Y5 @ Z2 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_226_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X5: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_nat,Y5: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_set_nat @ X5 @ Z2 )
=> ( ord_less_eq_set_nat @ X5 @ ( F @ Y5 @ Z2 ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_227_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y2: set_nat_nat] :
( ! [X5: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_nat_nat,Y5: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X5 @ Y5 )
=> ( ( ord_le9059583361652607317at_nat @ X5 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X5 @ ( F @ Y5 @ Z2 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_228_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ ( F @ Y5 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_229_inf__unique,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X2: nat > nat,Y2: nat > nat] :
( ! [X5: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: nat > nat,Y5: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat_nat @ X5 @ ( F @ Y5 @ Z2 ) ) ) )
=> ( ( inf_inf_nat_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_230_le__iff__inf,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_231_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_232_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_233_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_234_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_235_le__iff__inf,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y4: nat > nat] :
( ( inf_inf_nat_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_236_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_237_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_238_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_239_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_240_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_241_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_242_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_243_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_244_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_245_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_246_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_247_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_248_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_249_inf__absorb1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_250_inf__absorb1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_251_inf__absorb1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_252_inf__absorb1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_253_inf__absorb1,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_254_inf__absorb1,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_255_inf__absorb2,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_256_inf__absorb2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_257_inf__absorb2,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_258_inf__absorb2,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_259_inf__absorb2,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_260_inf__absorb2,axiom,
! [Y2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_261_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_262_inf_OboundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ~ ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_263_inf_OboundedE,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_264_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_265_inf_OboundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_266_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_267_inf_OboundedE,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_268_inf_OboundedI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_269_inf_OboundedI,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_270_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_271_inf_OboundedI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_272_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_273_inf_OboundedI,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_274_inf__greatest,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Z )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_275_inf__greatest,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Z )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_276_inf__greatest,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_277_inf__greatest,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Z )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_278_inf__greatest,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_279_inf__greatest,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Z )
=> ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_280_inf_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( A4
= ( inf_in5711780100303410308et_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_281_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( A4
= ( inf_inf_set_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_282_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_283_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( A4
= ( inf_inf_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_284_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_285_inf_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( A4
= ( inf_inf_nat_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_286_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_287_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_288_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_289_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_290_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_291_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_292_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_293_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_294_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_295_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_296_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_297_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_298_inf_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_299_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_300_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_301_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_302_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_303_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( inf_inf_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_304_inf_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_305_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_306_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_307_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_308_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_309_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( inf_inf_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_310_inf_OcoboundedI1,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_311_inf_OcoboundedI1,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_312_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_313_inf_OcoboundedI1,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_314_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_315_inf_OcoboundedI1,axiom,
! [A2: nat > nat,C: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_316_inf_OcoboundedI2,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_317_inf_OcoboundedI2,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_318_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_319_inf_OcoboundedI2,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_320_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_321_inf_OcoboundedI2,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_322_inf_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C ) )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_323_inf__left__commute,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_324_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_325_IntE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ~ ( member_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_326_IntE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ~ ( member_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_327_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_328_IntE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A )
=> ~ ( member_set_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_329_IntD1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ A ) ) ).
% IntD1
thf(fact_330_IntD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_331_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_332_IntD1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_333_IntD2,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ B ) ) ).
% IntD2
thf(fact_334_IntD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( member_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_335_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_336_IntD2,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_337_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_338_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
& ( member_nat_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_339_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
& ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_340_Int__def,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
& ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_341_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X2 @ A )
=> ( member_set_set_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_342_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_343_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_344_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_345_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_346_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_347_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_348_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_349_Int__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D4 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ C2 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_350_Int__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D4 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C2 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_351_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D4: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D4 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_352_Int__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D4 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ C2 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_353_Int__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_354_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_355_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_356_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_357_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
=> ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_358_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_359_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_360_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_361_Int__absorb,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_371_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_372_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_373_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_374_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_375_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_376_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T2: set_set_nat] :
( ( member_set_set_nat @ T2 @ A3 )
=> ( member_set_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_377_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A3 )
=> ( member_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_378_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_379_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A3 )
=> ( member_nat_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_380_Int__Collect,axiom,
! [X2: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_381_Int__Collect,axiom,
! [X2: nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) )
= ( ( member_nat_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_382_Int__Collect,axiom,
! [X2: set_nat,A: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_383_Int__Collect,axiom,
! [X2: set_set_nat,A: set_set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ X2 @ ( inf_in5711780100303410308et_nat @ A @ ( collect_set_set_nat @ P ) ) )
= ( ( member_set_set_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_393_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_394_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_395_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_396_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X5: set_set_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_397_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X5: set_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_398_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_399_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X5: nat > nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_400_Int__greatest,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ B )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_401_Int__greatest,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_402_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_403_Int__greatest,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_404_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_405_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_406_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_407_set__eq__subset,axiom,
( ( ^ [Y6: set_set_nat,Z3: set_set_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_408_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_409_set__eq__subset,axiom,
( ( ^ [Y6: set_nat_nat,Z3: set_nat_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_410_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_411_Collect__conj__eq,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_412_Collect__conj__eq,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_413_Collect__conj__eq,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in5711780100303410308et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_414_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_415_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 ) )
= ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_416_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_417_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_418_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_419_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 )
=> ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( 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_420_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 )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( 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_421_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( 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_422_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 )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( 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_423_Int__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ B @ ( inf_in5711780100303410308et_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_424_distrib__sup__le,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) ) @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_425_distrib__sup__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) ) @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_426_distrib__sup__le,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_427_distrib__sup__le,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) ) @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_428_distrib__sup__le,axiom,
! [X2: nat,Y2: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_429_distrib__sup__le,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y2 @ Z ) ) @ ( inf_inf_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y2 ) @ ( sup_sup_nat_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_430_distrib__inf__le,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z ) ) @ ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_431_distrib__inf__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_432_distrib__inf__le,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_433_distrib__inf__le,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_434_distrib__inf__le,axiom,
! [X2: nat,Y2: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_435_distrib__inf__le,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ ( inf_inf_nat_nat @ X2 @ Z ) ) @ ( inf_inf_nat_nat @ X2 @ ( sup_sup_nat_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_436_Un__Int__assoc__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) )
= ( ord_le9131159989063066194et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_437_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_438_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_439_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_440_sup__inf__distrib2,axiom,
! [Y2: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_set_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_441_sup__inf__distrib2,axiom,
! [Y2: set_set_set_nat,Z: set_set_set_nat,X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) @ X2 )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ X2 ) @ ( sup_su4213647025997063966et_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_442_sup__inf__distrib2,axiom,
! [Y2: set_nat_nat,Z: set_nat_nat,X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y2 @ X2 ) @ ( sup_sup_set_nat_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_443_sup__inf__distrib2,axiom,
! [Y2: set_nat,Z: set_nat,X2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_444_sup__inf__distrib1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_445_sup__inf__distrib1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_446_sup__inf__distrib1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_447_sup__inf__distrib1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_448_inf__sup__distrib2,axiom,
! [Y2: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_set_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_449_inf__sup__distrib2,axiom,
! [Y2: set_set_set_nat,Z: set_set_set_nat,X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) @ X2 )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y2 @ X2 ) @ ( inf_in5711780100303410308et_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_450_inf__sup__distrib2,axiom,
! [Y2: set_nat_nat,Z: set_nat_nat,X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y2 @ X2 ) @ ( inf_inf_set_nat_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_451_inf__sup__distrib2,axiom,
! [Y2: set_nat,Z: set_nat,X2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_452_inf__sup__distrib1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_453_inf__sup__distrib1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_454_inf__sup__distrib1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_455_inf__sup__distrib1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_456_distrib__imp2,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ! [X5: set_set_nat,Y5: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X5 @ ( inf_inf_set_set_nat @ Y5 @ Z2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X5 @ Y5 ) @ ( sup_sup_set_set_nat @ X5 @ Z2 ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_457_distrib__imp2,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y5: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X5 @ ( inf_in5711780100303410308et_nat @ Y5 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) @ ( sup_su4213647025997063966et_nat @ X5 @ Z2 ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_458_distrib__imp2,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ! [X5: set_nat_nat,Y5: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X5 @ ( inf_inf_set_nat_nat @ Y5 @ Z2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X5 @ Y5 ) @ ( sup_sup_set_nat_nat @ X5 @ Z2 ) ) )
=> ( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_459_distrib__imp2,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ! [X5: set_nat,Y5: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X5 @ ( inf_inf_set_nat @ Y5 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X5 @ Y5 ) @ ( sup_sup_set_nat @ X5 @ Z2 ) ) )
=> ( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_460_distrib__imp1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ! [X5: set_set_nat,Y5: set_set_nat,Z2: set_set_nat] :
( ( inf_inf_set_set_nat @ X5 @ ( sup_sup_set_set_nat @ Y5 @ Z2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X5 @ Y5 ) @ ( inf_inf_set_set_nat @ X5 @ Z2 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_461_distrib__imp1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y5: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X5 @ ( sup_su4213647025997063966et_nat @ Y5 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X5 @ Y5 ) @ ( inf_in5711780100303410308et_nat @ X5 @ Z2 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_462_distrib__imp1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ! [X5: set_nat_nat,Y5: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X5 @ ( sup_sup_set_nat_nat @ Y5 @ Z2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X5 @ Y5 ) @ ( inf_inf_set_nat_nat @ X5 @ Z2 ) ) )
=> ( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_463_distrib__imp1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ! [X5: set_nat,Y5: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X5 @ ( sup_sup_set_nat @ Y5 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X5 @ Y5 ) @ ( inf_inf_set_nat @ X5 @ Z2 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_464_Collect__subset,axiom,
! [A: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_465_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_466_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_467_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_468_Un__Int__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ A ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_469_Un__Int__distrib2,axiom,
! [B: set_set_set_nat,C2: set_set_set_nat,A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ C2 ) @ A )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ A ) @ ( sup_su4213647025997063966et_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_470_Un__Int__distrib2,axiom,
! [B: set_nat_nat,C2: set_nat_nat,A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ C2 ) @ A )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ A ) @ ( sup_sup_set_nat_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_471_Un__Int__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_472_Int__Un__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ A ) @ ( inf_inf_set_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_473_Int__Un__distrib2,axiom,
! [B: set_set_set_nat,C2: set_set_set_nat,A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ C2 ) @ A )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ A ) @ ( inf_in5711780100303410308et_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_474_Int__Un__distrib2,axiom,
! [B: set_nat_nat,C2: set_nat_nat,A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ A ) @ ( inf_inf_set_nat_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_475_Int__Un__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_476_Un__Int__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_477_Un__Int__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_478_Un__Int__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_479_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_480_Int__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_481_Int__Un__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_482_Int__Un__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_483_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_484_Un__Int__crazy,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ B @ C2 ) ) @ ( inf_inf_set_set_nat @ C2 @ A ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ B @ C2 ) ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_485_Un__Int__crazy,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) @ ( inf_in5711780100303410308et_nat @ C2 @ A ) )
= ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) @ ( sup_su4213647025997063966et_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_486_Un__Int__crazy,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ B @ C2 ) ) @ ( inf_inf_set_nat_nat @ C2 @ A ) )
= ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ B @ C2 ) ) @ ( sup_sup_set_nat_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_487_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_488_sup__set__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ( sup_sup_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_489_sup__set__def,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ( sup_su5917979686466268903_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A3 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_490_sup__set__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( collect_nat_nat
@ ( sup_sup_nat_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A3 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_491_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_492_sup_OcoboundedI2,axiom,
! [C: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ B2 )
=> ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_493_sup_OcoboundedI2,axiom,
! [C: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_494_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_495_sup_OcoboundedI2,axiom,
! [C: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ B2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_496_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_497_sup_OcoboundedI2,axiom,
! [C: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ C @ B2 )
=> ( ord_less_eq_nat_nat @ C @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_498_sup_OcoboundedI1,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_499_sup_OcoboundedI1,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_500_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_501_sup_OcoboundedI1,axiom,
! [C: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_502_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_503_sup_OcoboundedI1,axiom,
! [C: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C @ A2 )
=> ( ord_less_eq_nat_nat @ C @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_504_sup_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_505_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_506_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_507_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_508_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_509_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( sup_sup_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_510_sup_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_511_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_512_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_513_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_514_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_515_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( sup_sup_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_516_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_517_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_518_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_519_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_520_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_521_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_522_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_523_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_524_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_525_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_526_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_527_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_528_sup_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( A4
= ( sup_su4213647025997063966et_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_529_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( A4
= ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_530_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_531_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( A4
= ( sup_sup_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_532_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_533_sup_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( A4
= ( sup_sup_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_534_sup_OboundedI,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_535_sup_OboundedI,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_536_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_537_sup_OboundedI,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_538_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_539_sup_OboundedI,axiom,
! [B2: nat > nat,A2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ C @ A2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_540_sup_OboundedE,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_541_sup_OboundedE,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_542_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_543_sup_OboundedE,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_544_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_545_sup_OboundedE,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_546_sup__absorb2,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_547_sup__absorb2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_548_sup__absorb2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_549_sup__absorb2,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_550_sup__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_551_sup__absorb2,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_552_sup__absorb1,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_553_sup__absorb1,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_554_sup__absorb1,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_555_sup__absorb1,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_556_sup__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_557_sup__absorb1,axiom,
! [Y2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_558_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_559_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_560_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_561_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_562_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_563_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_564_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_565_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_566_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_567_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_568_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_569_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_570_sup__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y5 @ X5 )
=> ( ( ord_le9131159989063066194et_nat @ Z2 @ X5 )
=> ( ord_le9131159989063066194et_nat @ ( F @ Y5 @ Z2 ) @ X5 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_571_sup__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ! [X5: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_set_nat,Y5: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y5 @ X5 )
=> ( ( ord_le6893508408891458716et_nat @ Z2 @ X5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ Y5 @ Z2 ) @ X5 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_572_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X5: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_nat,Y5: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y5 @ X5 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X5 )
=> ( ord_less_eq_set_nat @ ( F @ Y5 @ Z2 ) @ X5 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_573_sup__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y2: set_nat_nat] :
( ! [X5: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_nat_nat,Y5: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_nat_nat,Y5: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y5 @ X5 )
=> ( ( ord_le9059583361652607317at_nat @ Z2 @ X5 )
=> ( ord_le9059583361652607317at_nat @ ( F @ Y5 @ Z2 ) @ X5 ) ) )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_574_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z2 ) @ X5 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_575_sup__unique,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X2: nat > nat,Y2: nat > nat] :
( ! [X5: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: nat > nat,Y5: nat > nat] : ( ord_less_eq_nat_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: nat > nat,Y5: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y5 @ X5 )
=> ( ( ord_less_eq_nat_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat_nat @ ( F @ Y5 @ Z2 ) @ X5 ) ) )
=> ( ( sup_sup_nat_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_576_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_577_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_578_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_579_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_580_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_581_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_582_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_583_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_584_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_585_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_586_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_587_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_588_le__iff__sup,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_589_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_590_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_591_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_592_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_593_le__iff__sup,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y4: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_594_sup__least,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ Z @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_595_sup__least,axiom,
! [Y2: set_set_nat,X2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ Z @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_596_sup__least,axiom,
! [Y2: set_nat,X2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_597_sup__least,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ Z @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_598_sup__least,axiom,
! [Y2: nat,X2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_599_sup__least,axiom,
! [Y2: nat > nat,X2: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat_nat @ Z @ X2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_600_sup__mono,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat,D3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D3 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_601_sup__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D3 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_602_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_603_sup__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D3 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_604_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_605_sup__mono,axiom,
! [A2: nat > nat,C: nat > nat,B2: nat > nat,D3: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ( ord_less_eq_nat_nat @ B2 @ D3 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ ( sup_sup_nat_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_606_sup_Omono,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,D3: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ D3 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C @ D3 ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_607_sup_Omono,axiom,
! [C: set_set_nat,A2: set_set_nat,D3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D3 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C @ D3 ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_608_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D3 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D3 ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_609_sup_Omono,axiom,
! [C: set_nat_nat,A2: set_nat_nat,D3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C @ D3 ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_610_sup_Omono,axiom,
! [C: nat,A2: nat,D3: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D3 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D3 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_611_sup_Omono,axiom,
! [C: nat > nat,A2: nat > nat,D3: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C @ A2 )
=> ( ( ord_less_eq_nat_nat @ D3 @ B2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ C @ D3 ) @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_612_le__supI2,axiom,
! [X2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_613_le__supI2,axiom,
! [X2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_614_le__supI2,axiom,
! [X2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_615_le__supI2,axiom,
! [X2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_616_le__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_617_le__supI2,axiom,
! [X2: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ B2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_618_le__supI1,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_619_le__supI1,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_620_le__supI1,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_621_le__supI1,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_622_le__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_623_le__supI1,axiom,
! [X2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ A2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_624_sup__ge2,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_625_sup__ge2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_626_sup__ge2,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_627_sup__ge2,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_628_sup__ge2,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_629_sup__ge2,axiom,
! [Y2: nat > nat,X2: nat > nat] : ( ord_less_eq_nat_nat @ Y2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_630_sup__ge1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_631_sup__ge1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_632_sup__ge1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_633_sup__ge1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_634_sup__ge1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_635_sup__ge1,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_636_le__supI,axiom,
! [A2: set_set_set_nat,X2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_637_le__supI,axiom,
! [A2: set_set_nat,X2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_638_le__supI,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_639_le__supI,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_640_le__supI,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_641_le__supI,axiom,
! [A2: nat > nat,X2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_642_le__supE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ X2 )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_643_le__supE,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_644_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_645_le__supE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_646_le__supE,axiom,
! [A2: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_647_le__supE,axiom,
! [A2: nat > nat,B2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_648_inf__sup__ord_I3_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_649_inf__sup__ord_I3_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_650_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_651_inf__sup__ord_I3_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_652_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_653_inf__sup__ord_I3_J,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_654_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_655_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_656_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_657_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_658_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_659_inf__sup__ord_I4_J,axiom,
! [Y2: nat > nat,X2: nat > nat] : ( ord_less_eq_nat_nat @ Y2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_660_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_661_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_662_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_663_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_664_subset__UnE,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ~ ! [A5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ A )
=> ! [B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ B )
=> ( C2
!= ( sup_su4213647025997063966et_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_665_subset__UnE,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ~ ! [A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ A )
=> ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_666_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_667_subset__UnE,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ~ ! [A5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ A )
=> ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_nat_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_Un__least,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_685_Un__least,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_686_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_687_Un__least,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_688_Un__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D4 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C2 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_689_Un__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D4 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C2 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_690_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D4: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D4 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_691_Un__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D4 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C2 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_692_sup__left__commute,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_693_sup__left__commute,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_694_sup__left__commute,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_695_sup__left__commute,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_696_sup_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C ) )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_697_sup_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ C ) )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_698_sup_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ C ) )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_699_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_700_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_701_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_702_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X3: set_nat_nat,Y4: set_nat_nat] : ( sup_sup_set_nat_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_703_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_704_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_705_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_706_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_707_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_708_sup__assoc,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_709_sup__assoc,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ Z )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_710_sup__assoc,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_711_sup__assoc,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_712_sup_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_713_sup_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ C )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_714_sup_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_715_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_716_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_717_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_718_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X3: set_nat_nat,Y4: set_nat_nat] : ( sup_sup_set_nat_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_719_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_720_inf__sup__aci_I6_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_721_inf__sup__aci_I6_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ Z )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_722_inf__sup__aci_I6_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_723_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_724_inf__sup__aci_I7_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_725_inf__sup__aci_I7_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_726_inf__sup__aci_I7_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_727_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_728_inf__sup__aci_I8_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_729_inf__sup__aci_I8_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_730_inf__sup__aci_I8_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_731_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_732_Un__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_733_Un__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_734_Un__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ B @ ( sup_sup_set_nat_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_735_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_736_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_737_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_738_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_739_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_740_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_741_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_742_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_743_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_744_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_745_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_746_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_747_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_748_Un__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_749_Un__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 )
= ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_750_Un__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_751_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_752_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_753_ball__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_754_ball__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_755_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_756_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_757_bex__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_758_bex__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_759_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_760_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_761_UnI2,axiom,
! [C: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_762_UnI2,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_763_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_764_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_765_UnI1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_766_UnI1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_767_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_768_UnE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_769_UnE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( ~ ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_770_UnE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( ~ ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% UnE
thf(fact_771_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_772_ACC__def,axiom,
! [X4: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ X4 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( clique3686358387679108662ccepts @ X4 @ G2 ) ) ) ) ).
% ACC_def
thf(fact_773_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X: set_set_set_nat,G2: set_set_nat] :
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ).
% accepts_def
thf(fact_774__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<G>_def
thf(fact_775__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_O_AD_A_092_060subseteq_062_AG_125_A_092_060inter_062_A_123G_A_092_060in_062_A_092_060G_062_O_A_092_060exists_062D_092_060in_062Y_O_AD_A_092_060subseteq_062_AG_125_092_060close_062,axiom,
( member_set_set_nat @ g
@ ( inf_in5711780100303410308et_nat
@ ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ x )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) )
@ ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ y )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ) ) ).
% \<open>G \<in> {G \<in> \<G>. \<exists>D\<in>X. D \<subseteq> G} \<inter> {G \<in> \<G>. \<exists>D\<in>Y. D \<subseteq> G}\<close>
thf(fact_776_odot___092_060G_062,axiom,
! [X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y3 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_777_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_778_ACC__I,axiom,
! [G: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ X4 ) ) ) ) ).
% ACC_I
thf(fact_779_finite__members___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ).
% finite_members_\<G>
thf(fact_780_ACC__cf__union,axiom,
! [X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X4 @ Y3 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y3 ) ) ) ).
% ACC_cf_union
thf(fact_781_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_782_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_783_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_784_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_785_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_786_order__refl,axiom,
! [X2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_787_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_788_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_789_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_790_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_791_dual__order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_792_ACC__cf__mono,axiom,
! [X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique951075384711337423ACC_cf @ k @ Y3 ) ) ) ).
% ACC_cf_mono
thf(fact_793_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_794_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_795_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_796_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_797_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_798_all__not__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ! [X3: set_set_nat] :
~ ( member_set_set_nat @ X3 @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_799_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_800_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_801_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_802_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_803_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_804_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_805_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_806_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X3: set_set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_807_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_808_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_809_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_810_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_811_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_812_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_813_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_814_subset__empty,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_815_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_816_subset__empty,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_817_inf__bot__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_818_inf__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_819_inf__bot__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_820_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_821_inf__bot__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_822_inf__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_823_inf__bot__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_824_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_825_sup__bot__left,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_826_sup__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_827_sup__bot__left,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_828_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_829_sup__bot__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_830_sup__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= X2 ) ).
% sup_bot_right
thf(fact_831_sup__bot__right,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= X2 ) ).
% sup_bot_right
thf(fact_832_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_833_bot__eq__sup__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_834_bot__eq__sup__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_835_bot__eq__sup__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_836_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_837_sup__eq__bot__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= bot_bot_set_set_nat )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_838_sup__eq__bot__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ( X2 = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_839_sup__eq__bot__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= bot_bot_set_nat_nat )
= ( ( X2 = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_840_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_841_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_842_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_843_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_844_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_845_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_846_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_847_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_848_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_849_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_850_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_851_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_852_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_853_sup__bot_Oright__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_854_sup__bot_Oright__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_855_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_856_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_857_Un__empty,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_858_Un__empty,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ( A = bot_bo7198184520161983622et_nat )
& ( B = bot_bo7198184520161983622et_nat ) ) ) ).
% Un_empty
thf(fact_859_Un__empty,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ( A = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_860_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_861_bex__empty,axiom,
! [P: set_nat > $o] :
~ ? [X6: set_nat] :
( ( member_set_nat @ X6 @ bot_bot_set_set_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_862_bex__empty,axiom,
! [P: set_set_nat > $o] :
~ ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ bot_bo7198184520161983622et_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_863_bex__empty,axiom,
! [P: ( nat > nat ) > $o] :
~ ? [X6: nat > nat] :
( ( member_nat_nat @ X6 @ bot_bot_set_nat_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_864_bex__empty,axiom,
! [P: nat > $o] :
~ ? [X6: nat] :
( ( member_nat @ X6 @ bot_bot_set_nat )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_865_acceptsI,axiom,
! [D4: set_set_nat,G: set_set_nat,X4: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D4 @ G )
=> ( ( member_set_set_nat @ D4 @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ).
% acceptsI
thf(fact_866_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_867_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_868_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_869_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_870_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_871_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_872_Bex__def,axiom,
( bex_set_set_nat
= ( ^ [A3: set_set_set_nat,P2: set_set_nat > $o] :
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_873_Bex__def,axiom,
( bex_nat_nat
= ( ^ [A3: set_nat_nat,P2: ( nat > nat ) > $o] :
? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_874_Bex__def,axiom,
( bex_set_nat
= ( ^ [A3: set_set_nat,P2: set_nat > $o] :
? [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_875_Bex__def,axiom,
( bex_nat
= ( ^ [A3: set_nat,P2: nat > $o] :
? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_876_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_877_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_878_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_879_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_880_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y5: set_nat] :
~ ( member_set_nat @ Y5 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_881_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y5: set_set_nat] :
~ ( member_set_set_nat @ Y5 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_882_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y5: nat > nat] :
~ ( member_nat_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_883_equals0I,axiom,
! [A: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_884_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_885_ex__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ? [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
= ( A != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_886_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_887_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_888_sameprod__finite,axiom,
! [X4: set_set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_889_sameprod__finite,axiom,
! [X4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X4 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_890_sameprod__finite,axiom,
! [X4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X4 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_891_sameprod__finite,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_892_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_893_order__less__imp__not__less,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ~ ( ord_le152980574450754630et_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_894_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_895_order__less__imp__not__eq2,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_896_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_897_order__less__imp__not__eq,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_898_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_899_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_900_order__less__imp__triv,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,P: $o] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( ( ord_le152980574450754630et_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_901_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_902_order__less__not__sym,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ~ ( ord_le152980574450754630et_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_903_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_904_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_905_order__less__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_906_order__less__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_907_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_908_order__less__subst1,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_909_order__less__subst1,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_910_order__less__subst1,axiom,
! [A2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_911_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_912_order__less__irrefl,axiom,
! [X2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_913_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_914_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_915_ord__less__eq__subst,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_916_ord__less__eq__subst,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_917_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_918_ord__eq__less__subst,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_919_ord__eq__less__subst,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_920_ord__eq__less__subst,axiom,
! [A2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_921_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_922_order__less__trans,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( ( ord_le152980574450754630et_nat @ Y2 @ Z )
=> ( ord_le152980574450754630et_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_923_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_924_order__less__asym_H,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ord_le152980574450754630et_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_925_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_926_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_927_order__less__asym,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ~ ( ord_le152980574450754630et_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_928_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_929_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_930_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_931_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_932_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_933_order_Ostrict__implies__not__eq,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_934_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_935_dual__order_Ostrict__trans,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( ( ord_le152980574450754630et_nat @ C @ B2 )
=> ( ord_le152980574450754630et_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_936_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_937_bot_Onot__eq__extremum,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
= ( ord_less_set_set_nat @ bot_bot_set_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_938_bot_Onot__eq__extremum,axiom,
! [A2: set_nat_nat] :
( ( A2 != bot_bot_set_nat_nat )
= ( ord_less_set_nat_nat @ bot_bot_set_nat_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_939_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_940_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_941_bot_Onot__eq__extremum,axiom,
! [A2: set_set_set_nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
= ( ord_le152980574450754630et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_942_bot_Oextremum__strict,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% bot.extremum_strict
thf(fact_943_bot_Oextremum__strict,axiom,
! [A2: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% bot.extremum_strict
thf(fact_944_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_945_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_946_bot_Oextremum__strict,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% bot.extremum_strict
thf(fact_947_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_948_order_Ostrict__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_949_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_950_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P2 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_951_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_952_dual__order_Oirrefl,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_953_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_954_dual__order_Oasym,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ~ ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_955_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_956_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_957_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X5: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X5 )
=> ( P @ Y7 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_958_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_959_ord__less__eq__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_960_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_961_ord__eq__less__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2 = B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_962_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_963_order_Oasym,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ord_le152980574450754630et_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_964_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_965_less__imp__neq,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_966_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_967_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_968_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_969_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_970_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_971_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_972_bot_Oextremum__uniqueI,axiom,
! [A2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ bot_bot_nat_nat )
=> ( A2 = bot_bot_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_973_bot_Oextremum__unique,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_974_bot_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_975_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_976_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_977_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_978_bot_Oextremum__unique,axiom,
! [A2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ bot_bot_nat_nat )
= ( A2 = bot_bot_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_979_bot_Oextremum,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% bot.extremum
thf(fact_980_bot_Oextremum,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% bot.extremum
thf(fact_981_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_982_bot_Oextremum,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_983_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_984_bot_Oextremum,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ bot_bot_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_985_order__le__imp__less__or__eq,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_986_order__le__imp__less__or__eq,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_less_set_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_987_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_988_order__le__imp__less__or__eq,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_989_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_990_order__le__imp__less__or__eq,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_nat_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_991_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_992_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_993_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_994_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_set_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_995_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_996_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_997_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_set_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_998_order__less__le__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_999_order__less__le__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > set_nat,C: set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1000_order__less__le__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > set_set_nat,C: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_set_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1001_order__less__le__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat > nat,C: nat > nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1002_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1003_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1004_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1005_order__less__le__subst1,axiom,
! [A2: nat,F: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1006_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1007_order__less__le__subst1,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C: nat] :
( ( ord_less_set_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1008_order__less__le__subst1,axiom,
! [A2: nat > nat,F: nat > nat > nat,B2: nat,C: nat] :
( ( ord_less_nat_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1009_order__less__le__subst1,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X5: nat > nat,Y5: nat > nat] :
( ( ord_less_eq_nat_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1010_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_set_nat > set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1011_order__less__le__subst1,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1012_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1013_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1014_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1015_order__le__less__subst2,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1016_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1017_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1018_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1019_order__le__less__subst2,axiom,
! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat > nat,Y5: nat > nat] :
( ( ord_less_eq_nat_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1020_order__le__less__subst2,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1021_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1022_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1023_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1024_order__le__less__subst1,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_set_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1025_order__le__less__subst1,axiom,
! [A2: nat > nat,F: nat > nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1026_order__le__less__subst1,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_le152980574450754630et_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le152980574450754630et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1027_order__le__less__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_set_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1028_order__le__less__subst1,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1029_order__le__less__subst1,axiom,
! [A2: set_nat,F: set_set_set_nat > set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1030_order__le__less__subst1,axiom,
! [A2: set_set_nat,F: set_set_set_nat > set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_set_set_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1031_order__le__less__subst1,axiom,
! [A2: nat > nat,F: set_set_set_nat > nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X5 @ Y5 )
=> ( ord_less_nat_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1032_order__less__le__trans,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ Z )
=> ( ord_le152980574450754630et_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1033_order__less__le__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z )
=> ( ord_less_set_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1034_order__less__le__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1035_order__less__le__trans,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_less_set_nat_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1036_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1037_order__less__le__trans,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] :
( ( ord_less_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat_nat @ Y2 @ Z )
=> ( ord_less_nat_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1038_order__le__less__trans,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le152980574450754630et_nat @ Y2 @ Z )
=> ( ord_le152980574450754630et_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1039_order__le__less__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_less_set_set_nat @ Y2 @ Z )
=> ( ord_less_set_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1040_order__le__less__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1041_order__le__less__trans,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat_nat @ Y2 @ Z )
=> ( ord_less_set_nat_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1042_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1043_order__le__less__trans,axiom,
! [X2: nat > nat,Y2: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_nat_nat @ Y2 @ Z )
=> ( ord_less_nat_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1044_order__neq__le__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 != B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1045_order__neq__le__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 != B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1046_order__neq__le__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1047_order__neq__le__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 != B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1048_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1049_order__neq__le__trans,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ord_less_nat_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1050_order__le__neq__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1051_order__le__neq__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1052_order__le__neq__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1053_order__le__neq__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1054_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1055_order__le__neq__trans,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1056_order__less__imp__le,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1057_order__less__imp__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1058_order__less__imp__le,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1059_order__less__imp__le,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1060_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1061_order__less__imp__le,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_nat_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1062_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_1063_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_1064_order__less__le,axiom,
( ord_le152980574450754630et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1065_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1066_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1067_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1068_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1069_order__less__le,axiom,
( ord_less_nat_nat
= ( ^ [X3: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1070_order__le__less,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1071_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1072_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1073_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1074_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1075_order__le__less,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y4: nat > nat] :
( ( ord_less_nat_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1076_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1077_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1078_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1079_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1080_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1081_dual__order_Ostrict__implies__order,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_nat_nat @ B2 @ A2 )
=> ( ord_less_eq_nat_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1082_numbers2__mono,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X2 ) @ ( clique3652268606331196573umbers @ X2 ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y2 ) @ ( clique3652268606331196573umbers @ Y2 ) ) ) ) ).
% numbers2_mono
thf(fact_1083_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_1084_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_1085_ACC__cf___092_060F_062,axiom,
! [X4: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_1086__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_1087_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_1088_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_1089_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_1090_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_1091_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_1092_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_1093_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_1094_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1095_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_1096_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_1097_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_1098_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1099_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1100_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_1101_finite__vG,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ).
% finite_vG
thf(fact_1102_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_1103_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_1104_CLIQUE__def,axiom,
( ( clique363107459185959606CLIQUE @ k )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( clique3326749438856946062irst_K @ k ) )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ) ).
% CLIQUE_def
thf(fact_1105_finite__ACC,axiom,
! [X4: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X4 ) ) ).
% finite_ACC
thf(fact_1106_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_1107_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_1108_ACC__cf__def,axiom,
! [X4: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ X4 )
= ( collect_nat_nat
@ ^ [F2: nat > nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
& ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ k @ F2 ) ) ) ) ) ).
% ACC_cf_def
thf(fact_1109_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_1110_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_1111_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_1112_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_1113__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [K2: set_set_nat] :
( ( member_set_set_nat @ K2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K2 ) )
= k )
& ( K2
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K2 ) @ ( clique5033774636164728513irst_v @ K2 ) ) ) ) ) ) ).
% \<K>_def
thf(fact_1114_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1115_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1116__092_060K_062__altdef,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
& ( ( finite_card_nat @ V )
= k ) ) ) ) ).
% \<K>_altdef
thf(fact_1117_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_1118_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_1119_ACC__cf__I,axiom,
! [F3: nat > nat,X4: set_set_set_nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ k @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ k @ X4 ) ) ) ) ).
% ACC_cf_I
thf(fact_1120_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_1121_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1122_Clique__def,axiom,
( clique6749503327923060270Clique
= ( ^ [V: set_nat,K3: nat] :
( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ V ) )
& ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ C3 @ C3 ) @ G2 )
& ( ( finite_card_nat @ C3 )
= K3 ) ) ) ) ) ) ).
% Clique_def
thf(fact_1123_C__def,axiom,
! [F: nat > nat] :
( ( clique5033774636164728462irst_C @ k @ F )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat,Y4: nat] :
( ( Uu
= ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) ) ).
% C_def
thf(fact_1124_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1125_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1126_v__def,axiom,
( clique5033774636164728513irst_v
= ( ^ [G2: set_set_nat] :
( collect_nat
@ ^ [X3: nat] :
? [Y4: nat] : ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ G2 ) ) ) ) ).
% v_def
thf(fact_1127_first__assumptions_Ov__def,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique5033774636164728513irst_v @ G )
= ( collect_nat
@ ^ [X3: nat] :
? [Y4: nat] : ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ G ) ) ) ) ).
% first_assumptions.v_def
thf(fact_1128_first__assumptions_ONEG__def,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1129_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P4: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_1130_first__assumptions_OacceptsI,axiom,
! [L: nat,P4: nat,K: nat,D4: set_set_nat,G: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D4 @ G )
=> ( ( member_set_set_nat @ D4 @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1131_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X4 )
& ( ord_le6893508408891458716et_nat @ X3 @ G ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1132_first__assumptions_OACC__union,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y3 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X4 ) @ ( clique3210737319928189260st_ACC @ K @ Y3 ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_1133_first__assumptions_OACC__empty,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1134_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1135_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1136_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1137_first__assumptions_OC__def,axiom,
! [L: nat,P4: nat,K: nat,F: nat > nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique5033774636164728462irst_C @ K @ F )
= ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat,Y4: nat] :
( ( Uu
= ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) ) ) ).
% first_assumptions.C_def
thf(fact_1138_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1139_first__assumptions_Ov__mono,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1140_first__assumptions_Ov__empty,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_1141_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1142_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1143_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P4: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_1144_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1145_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y3 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1146_first__assumptions_Ov__union,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( 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_1147_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1148_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_1149_first__assumptions_Ounion___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.union_\<G>
thf(fact_1150_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y3 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y3 ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1151_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_1152_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_1153_first__assumptions_Oodot__def,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique5469973757772500719t_odot @ X4 @ Y3 )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D @ E ) )
& ( member_set_set_nat @ D @ X4 )
& ( member_set_set_nat @ E @ Y3 ) ) ) ) ) ).
% first_assumptions.odot_def
thf(fact_1154_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P4: nat,K: nat,F3: nat > nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1155_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_1156_first__assumptions_OACC__cf__def,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ X4 )
= ( collect_nat_nat
@ ^ [F2: nat > nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ K ) )
& ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K @ F2 ) ) ) ) ) ) ).
% first_assumptions.ACC_cf_def
thf(fact_1157_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_1158_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1159_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1160_first__assumptions_O_092_060K_062___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.\<K>_\<G>
thf(fact_1161_first__assumptions_OACC__I,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ K @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_1162_first__assumptions_ONEG___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.NEG_\<G>
thf(fact_1163_first__assumptions_O_092_060G_062__def,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] : ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.\<G>_def
thf(fact_1164_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat,Y3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y3 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_1165_first__assumptions_OACC__def,axiom,
! [L: nat,P4: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ X4 )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( clique3686358387679108662ccepts @ X4 @ G2 ) ) ) ) ) ).
% first_assumptions.ACC_def
thf(fact_1166_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P4: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_1167_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1168_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1169_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ N4 )
=> ( ord_less_nat @ X5 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1170_first__assumptions_O_092_060K_062__altdef,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3326749438856946062irst_K @ K )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [V: set_nat] :
( ( Uu
= ( clique6722202388162463298od_nat @ V @ V ) )
& ( ord_less_eq_set_nat @ V @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) )
& ( ( finite_card_nat @ V )
= K ) ) ) ) ) ).
% first_assumptions.\<K>_altdef
thf(fact_1171_first__assumptions_O_092_060K_062__def,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique3326749438856946062irst_K @ K )
= ( collect_set_set_nat
@ ^ [K2: set_set_nat] :
( ( member_set_set_nat @ K2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K2 ) )
= K )
& ( K2
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K2 ) @ ( clique5033774636164728513irst_v @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.\<K>_def
thf(fact_1172_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1173_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1174_first__assumptions_OCLIQUE__def,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique363107459185959606CLIQUE @ K )
= ( collect_set_set_nat
@ ^ [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( clique3326749438856946062irst_K @ K ) )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ) ) ).
% first_assumptions.CLIQUE_def
thf(fact_1175_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_1176_first__assumptions_Okml,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1177_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1178_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_1179_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_1180_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_1181_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1182_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1183_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1184_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1185_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ( P @ M5 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1186_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1187_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_1188_less__not__refl2,axiom,
! [N: nat,M4: nat] :
( ( ord_less_nat @ N @ M4 )
=> ( M4 != N ) ) ).
% less_not_refl2
thf(fact_1189_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1190_nat__neq__iff,axiom,
! [M4: nat,N: nat] :
( ( M4 != N )
= ( ( ord_less_nat @ M4 @ N )
| ( ord_less_nat @ N @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_1191_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1192_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1193_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1194_less__imp__le__nat,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_imp_le_nat
thf(fact_1195_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1196_less__or__eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( ( ord_less_nat @ M4 @ N )
| ( M4 = N ) )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1197_le__neq__implies__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( M4 != N )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1198_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1199_diff__le__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1200_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1201_diff__le__self,axiom,
! [M4: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N ) @ M4 ) ).
% diff_le_self
thf(fact_1202_diff__le__mono,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1203_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M4 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1204_le__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1205_eq__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M4 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1206_diff__less__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_1207_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1208_first__assumptions_Ok,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1209_first__assumptions_Okp,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_nat @ P4 @ K ) ) ).
% first_assumptions.kp
thf(fact_1210_first__assumptions_Opl,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_nat @ L @ P4 ) ) ).
% first_assumptions.pl
thf(fact_1211_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1212_less__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M4 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1213_first__assumptions_Okm,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_1214_first__assumptions_Omp,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_nat @ P4 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_1215_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat_nat )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S2 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X5 ) ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S2 )
=> ( ord_less_eq_nat_nat @ X5 @ Xa2 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1216_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat_nat )
=> ( ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S2 )
=> ( ( ord_less_eq_nat_nat @ X5 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X5 ) ) ) )
=> ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ S2 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S2 )
=> ( ord_less_eq_nat_nat @ Xa2 @ X5 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1217_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).
% card_POS
thf(fact_1218_first__assumptions_Ocard__POS,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K ) )
= ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) ) ) ).
% first_assumptions.card_POS
thf(fact_1219_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1220_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M3: nat] :
( ( ord_less_nat @ K @ M3 )
=> ? [N6: nat] :
( ( ord_less_nat @ M3 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1221_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1222_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_nat @ M @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1223_choose__mono,axiom,
! [N: nat,M4: nat,K: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ M4 @ K ) ) ) ).
% choose_mono
thf(fact_1224_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M4: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1225_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N5: nat] :
( ( infini8530281810654367211te_nat @ S @ N5 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_1226_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_1227_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1228_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1229__092_060F_062__def,axiom,
( ( clique2971579238625216137irst_F @ k )
= ( piE_nat_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) )
@ ^ [I: nat] : ( clique3652268606331196573umbers @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% \<F>_def
thf(fact_1230_nat__add__left__cancel__le,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1231_nat__add__left__cancel__less,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1232_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1233_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1234_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1235_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1236_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1237_trans__le__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_1238_trans__le__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1239_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1240_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1241_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1242_add__leD2,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1243_add__leD1,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% add_leD1
thf(fact_1244_le__add2,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M4 @ N ) ) ).
% le_add2
thf(fact_1245_le__add1,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M4 ) ) ).
% le_add1
thf(fact_1246_add__leE,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M4 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1247_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1248_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1249_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1250_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1251_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1252_mono__nat__linear__lb,axiom,
! [F: nat > nat,M4: nat,K: nat] :
( ! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M4 ) @ K ) @ ( F @ ( plus_plus_nat @ M4 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1253_Nat_Odiff__cancel,axiom,
! [K: nat,M4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M4 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1254_diff__cancel2,axiom,
! [M4: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M4 @ N ) ) ).
% diff_cancel2
thf(fact_1255_diff__add__inverse,axiom,
! [N: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M4 ) @ N )
= M4 ) ).
% diff_add_inverse
thf(fact_1256_diff__add__inverse2,axiom,
! [M4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N ) @ N )
= M4 ) ).
% diff_add_inverse2
thf(fact_1257_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1258_add__diff__inverse__nat,axiom,
! [M4: nat,N: nat] :
( ~ ( ord_less_nat @ M4 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M4 @ N ) )
= M4 ) ) ).
% add_diff_inverse_nat
thf(fact_1259_less__add__eq__less,axiom,
! [K: nat,L: nat,M4: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M4 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1260_trans__less__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_less_add2
thf(fact_1261_trans__less__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_less_add1
thf(fact_1262_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1263_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1264_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1265_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1266_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1267_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1268_first__assumptions_Olm,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1269_first__assumptions_O_092_060F_062__def,axiom,
! [L: nat,P4: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K )
=> ( ( clique2971579238625216137irst_F @ K )
= ( piE_nat_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) )
@ ^ [I: nat] : ( clique3652268606331196573umbers @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% first_assumptions.\<F>_def
% Conjectures (1)
thf(conj_0,conjecture,
( member_set_set_nat @ ( sup_sup_set_set_nat @ d @ e )
@ ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D @ E ) )
& ( member_set_set_nat @ D @ x )
& ( member_set_set_nat @ E @ y ) ) ) ) ).
%------------------------------------------------------------------------------