TPTP Problem File: SLH0530^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_00166_004856__16110802_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1382 ( 438 unt; 116 typ; 0 def)
% Number of atoms : 4037 (1042 equ; 0 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 11935 ( 513 ~; 85 |; 388 &;8848 @)
% ( 0 <=>;2101 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 682 ( 682 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 109 usr; 20 con; 0-3 aty)
% Number of variables : 3851 ( 415 ^;3182 !; 254 ?;3851 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:48:02.559
%------------------------------------------------------------------------------
% 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_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_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__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (109)
thf(sy_c_Assumptions__and__Approximations_Oeps,type,
assumptions_and_eps: real ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OClique,type,
clique6749503327923060270Clique: set_nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
clique5786534781347292306Graphs: set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_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_O_092_060K_062,type,
clique3326749438856946062irst_K: 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_Euclidean__Space_Oeuclidean__space__class_OBasis_001t__Real__Oreal,type,
euclid1305858884100475807s_real: set_real ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
finite5926941155766903689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
minus_463385787819020154_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J,type,
minus_495547888894627908_nat_o: ( set_set_set_nat > $o ) > ( set_set_set_nat > $o ) > set_set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
minus_3113942175840221057et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > 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__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic7132588981422310769at_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
lattic6497381205983422413t_real: ( set_nat > real ) > set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
lattic82989400242555431at_nat: ( set_set_nat > nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Real__Oreal,type,
lattic4725577848813236611t_real: ( set_set_nat > real ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo6227097192321305471_nat_o: set_set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J,type,
bot_bo5536612546450143305_nat_o: set_set_set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
bot_bo193956671110832956et_nat: set_set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le466346588697744319_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $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__Real__Oreal,type,
ord_less_real: real > real > $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__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $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_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le3616423863276227763_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $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__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
orderi1027199981026551883et_nat: ( set_set_nat > set_set_nat > $o ) > ( set_set_nat > set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
orderi7669286751058938369et_nat: ( set_set_set_nat > set_set_set_nat > $o ) > ( set_set_set_nat > set_set_set_nat > $o ) > set_set_set_nat > $o ).
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_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
insert3687027775829606434et_nat: set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
is_sin6612384548583640136et_nat: set_set_set_nat > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
remove_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
remove_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
the_elem_set_set_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_v_G,type,
g: set_set_nat ).
thf(sy_v_edge____,type,
edge: set_nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_x____,type,
x: nat ).
thf(sy_v_y____,type,
y: nat ).
% Relevant facts (1265)
thf(fact_0__C_K_C_I2_J,axiom,
member_set_nat @ edge @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% "*"(2)
thf(fact_1__C_K_C_I1_J,axiom,
member_set_nat @ edge @ g ).
% "*"(1)
thf(fact_2_edge_I2_J,axiom,
x != y ).
% edge(2)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ay_O_A_092_060lbrakk_062edge_A_061_A_123x_M_Ay_125_059_Ax_A_092_060noteq_062_Ay_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: nat,Y: nat] :
( ( edge
= ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( X = Y ) ) ).
% \<open>\<And>thesis. (\<And>x y. \<lbrakk>edge = {x, y}; x \<noteq> y\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_edge_I1_J,axiom,
( edge
= ( insert_nat @ x @ ( insert_nat @ y @ bot_bot_set_nat ) ) ) ).
% edge(1)
thf(fact_5_assms_I2_J,axiom,
g != bot_bot_set_set_nat ).
% assms(2)
thf(fact_6_binprod__def,axiom,
( clique1181040904276305582et_nat
= ( ^ [X2: set_set_set_nat,Y2: set_set_set_nat] :
( collec7201453139178570183et_nat
@ ^ [Uu: set_set_set_nat] :
? [X3: set_set_nat,Y3: set_set_nat] :
( ( Uu
= ( insert_set_set_nat @ X3 @ ( insert_set_set_nat @ Y3 @ bot_bo7198184520161983622et_nat ) ) )
& ( member_set_set_nat @ X3 @ X2 )
& ( member_set_set_nat @ Y3 @ Y2 )
& ( X3 != Y3 ) ) ) ) ) ).
% binprod_def
thf(fact_7_binprod__def,axiom,
( clique8906516429304539640et_nat
= ( ^ [X2: set_set_nat,Y2: set_set_nat] :
( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [X3: set_nat,Y3: set_nat] :
( ( Uu
= ( insert_set_nat @ X3 @ ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) ) )
& ( member_set_nat @ X3 @ X2 )
& ( member_set_nat @ Y3 @ Y2 )
& ( X3 != Y3 ) ) ) ) ) ).
% binprod_def
thf(fact_8_binprod__def,axiom,
( clique6722202388162463298od_nat
= ( ^ [X2: set_nat,Y2: set_nat] :
( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat,Y3: nat] :
( ( Uu
= ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
& ( member_nat @ X3 @ X2 )
& ( member_nat @ Y3 @ Y2 )
& ( X3 != Y3 ) ) ) ) ) ).
% binprod_def
thf(fact_9_singleton__insert__inj__eq,axiom,
! [B: set_set_nat,A: set_set_nat,A2: set_set_set_nat] :
( ( ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_10_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_11_singleton__insert__inj__eq,axiom,
! [B: set_nat,A: set_nat,A2: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_12_singleton__insert__inj__eq_H,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_nat] :
( ( ( insert_set_set_nat @ A @ A2 )
= ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) )
= ( ( A = B )
& ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_13_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_14_singleton__insert__inj__eq_H,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_15_singleton__conv,axiom,
! [A: set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( X3 = A ) )
= ( insert3687027775829606434et_nat @ A @ bot_bo193956671110832956et_nat ) ) ).
% singleton_conv
thf(fact_16_singleton__conv,axiom,
! [A: set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( X3 = A ) )
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ).
% singleton_conv
thf(fact_17_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_18_singleton__conv,axiom,
! [A: set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( X3 = A ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singleton_conv
thf(fact_19_singleton__conv2,axiom,
! [A: set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z )
@ A ) )
= ( insert3687027775829606434et_nat @ A @ bot_bo193956671110832956et_nat ) ) ).
% singleton_conv2
thf(fact_20_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_21_singleton__conv2,axiom,
! [A: set_nat] :
( ( collect_set_nat
@ ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z )
@ A ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singleton_conv2
thf(fact_22_singleton__conv2,axiom,
! [A: set_set_nat] :
( ( collect_set_set_nat
@ ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z )
@ A ) )
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ).
% singleton_conv2
thf(fact_23_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_24_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_25_singletonI,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ).
% singletonI
thf(fact_26_insert__subset,axiom,
! [X4: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X4 @ A2 ) @ B2 )
= ( ( member_set_set_nat @ X4 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_27_insert__subset,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A2 ) @ B2 )
= ( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_28_insert__subset,axiom,
! [X4: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ A2 ) @ B2 )
= ( ( member_set_nat @ X4 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_29_subset__empty,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_30_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_31_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_32_empty__subsetI,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% empty_subsetI
thf(fact_33_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_34_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_35_v__def,axiom,
( clique5033774636164728513irst_v
= ( ^ [G: set_set_nat] :
( collect_nat
@ ^ [X3: nat] :
? [Y3: nat] : ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ G ) ) ) ) ).
% v_def
thf(fact_36_sameprod__mono,axiom,
! [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) @ ( clique6722202388162463298od_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_37_sameprod__mono,axiom,
! [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) @ ( clique8906516429304539640et_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_38_sameprod__mono,axiom,
! [X5: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) @ ( clique1181040904276305582et_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_39_subset__singletonD,axiom,
! [A2: set_set_set_nat,X4: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
=> ( ( A2 = bot_bo7198184520161983622et_nat )
| ( A2
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% subset_singletonD
thf(fact_40_subset__singletonD,axiom,
! [A2: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_41_subset__singletonD,axiom,
! [A2: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
=> ( ( A2 = bot_bot_set_set_nat )
| ( A2
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_42_v__mono,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G2 @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_43_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_44_subset__antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_45_subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_set_nat @ X @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_46_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_47_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_48_empty__Collect__eq,axiom,
! [P: set_set_set_nat > $o] :
( ( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat @ P ) )
= ( ! [X3: set_set_set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_49_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_50_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_51_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_52_Collect__empty__eq,axiom,
! [P: set_set_set_nat > $o] :
( ( ( collec7201453139178570183et_nat @ P )
= bot_bo193956671110832956et_nat )
= ( ! [X3: set_set_set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_53_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_54_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_55_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_56_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_57_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_58_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X3: set_set_nat] :
~ ( member_set_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_59_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_60_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_61_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_62_insert__absorb2,axiom,
! [X4: nat,A2: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ X4 @ A2 ) )
= ( insert_nat @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_63_insert__absorb2,axiom,
! [X4: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X4 @ ( insert_set_nat @ X4 @ A2 ) )
= ( insert_set_nat @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_64_insert__absorb2,axiom,
! [X4: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ X4 @ A2 ) )
= ( insert_set_set_nat @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_65_insert__iff,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_66_insert__iff,axiom,
! [A: set_set_nat,B: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_67_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_68_insertCI,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_69_insertCI,axiom,
! [A: set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( ~ ( member_set_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_70_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_71__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062edge_O_A_092_060lbrakk_062edge_A_092_060in_062_AG_059_Aedge_A_092_060in_062_A_091m_093_094_092_060two_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Edge: set_nat] :
( ( member_set_nat @ Edge @ g )
=> ~ ( member_set_nat @ Edge @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>edge. \<lbrakk>edge \<in> G; edge \<in> [m]^\<two>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_72__092_060open_062G_A_092_060in_062_A_123G_O_AG_A_092_060subseteq_062_A_091m_093_094_092_060two_062_125_092_060close_062,axiom,
( member_set_set_nat @ g
@ ( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<open>G \<in> {G. G \<subseteq> [m]^\<two>}\<close>
thf(fact_73_assms_I1_J,axiom,
member_set_set_nat @ g @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% assms(1)
thf(fact_74_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_75_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_76_Collect__mono__iff,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) )
= ( ! [X3: set_set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_77_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_78_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_79_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_80_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_81_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_82_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_83_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_84_Collect__mono,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X: set_set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_85_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_86_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_87_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_88_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_89_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_90_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_91_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_92_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_93_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_94_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_95_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_96_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_97_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_98_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_99_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_100_equalityE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_101_subsetD,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_102_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_103_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_104_in__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X4: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ X4 @ A2 )
=> ( member_set_set_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_105_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_106_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X4 @ A2 )
=> ( member_set_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_107_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_108_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_109_ex__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ? [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_110_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A @ ( collect_set_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
! [A: set_set_set_nat,P: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ A @ ( collec7201453139178570183et_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A2: set_set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_118_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_119_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_120_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_121_Collect__cong,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X: set_set_set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec7201453139178570183et_nat @ P )
= ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_122_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_123_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_124_equals0I,axiom,
! [A2: set_set_set_nat] :
( ! [Y: set_set_nat] :
~ ( member_set_set_nat @ Y @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_125_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_126_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_127_equals0D,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( A2 = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_128_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_129_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_130_emptyE,axiom,
! [A: set_set_nat] :
~ ( member_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_131_mk__disjoint__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ? [B4: set_set_nat] :
( ( A2
= ( insert_set_nat @ A @ B4 ) )
& ~ ( member_set_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_132_mk__disjoint__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ? [B4: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ A @ B4 ) )
& ~ ( member_set_set_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_133_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_134_insert__commute,axiom,
! [X4: nat,Y6: nat,A2: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ Y6 @ A2 ) )
= ( insert_nat @ Y6 @ ( insert_nat @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_135_insert__commute,axiom,
! [X4: set_nat,Y6: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X4 @ ( insert_set_nat @ Y6 @ A2 ) )
= ( insert_set_nat @ Y6 @ ( insert_set_nat @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_136_insert__commute,axiom,
! [X4: set_set_nat,Y6: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ Y6 @ A2 ) )
= ( insert_set_set_nat @ Y6 @ ( insert_set_set_nat @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_137_insert__eq__iff,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_nat] :
( ( A2
= ( insert_set_nat @ B @ C3 ) )
& ~ ( member_set_nat @ B @ C3 )
& ( B2
= ( insert_set_nat @ A @ C3 ) )
& ~ ( member_set_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_138_insert__eq__iff,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ A @ A2 )
=> ( ~ ( member_set_set_nat @ B @ B2 )
=> ( ( ( insert_set_set_nat @ A @ A2 )
= ( insert_set_set_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ B @ C3 ) )
& ~ ( member_set_set_nat @ B @ C3 )
& ( B2
= ( insert_set_set_nat @ A @ C3 ) )
& ~ ( member_set_set_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_139_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_140_insert__absorb,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_141_insert__absorb,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( insert_set_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_142_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_143_insert__ident,axiom,
! [X4: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A2 )
=> ( ~ ( member_set_nat @ X4 @ B2 )
=> ( ( ( insert_set_nat @ X4 @ A2 )
= ( insert_set_nat @ X4 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_144_insert__ident,axiom,
! [X4: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ~ ( member_set_set_nat @ X4 @ B2 )
=> ( ( ( insert_set_set_nat @ X4 @ A2 )
= ( insert_set_set_nat @ X4 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_145_insert__ident,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ~ ( member_nat @ X4 @ B2 )
=> ( ( ( insert_nat @ X4 @ A2 )
= ( insert_nat @ X4 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_146_Set_Oset__insert,axiom,
! [X4: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ~ ! [B4: set_set_nat] :
( ( A2
= ( insert_set_nat @ X4 @ B4 ) )
=> ( member_set_nat @ X4 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_147_Set_Oset__insert,axiom,
! [X4: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ~ ! [B4: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ X4 @ B4 ) )
=> ( member_set_set_nat @ X4 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_148_Set_Oset__insert,axiom,
! [X4: nat,A2: set_nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X4 @ B4 ) )
=> ( member_nat @ X4 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_149_insertI2,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B2 )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_150_insertI2,axiom,
! [A: set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ B2 )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_151_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_152_insertI1,axiom,
! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_153_insertI1,axiom,
! [A: set_set_nat,B2: set_set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_154_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_155_insertE,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_156_insertE,axiom,
! [A: set_set_nat,B: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_157_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_158_Collect__subset,axiom,
! [A2: 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 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_159_Collect__subset,axiom,
! [A2: set_set_set_set_nat,P: set_set_set_nat > $o] :
( ord_le572741076514265352et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_160_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_161_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_162_empty__def,axiom,
( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : $false ) ) ).
% empty_def
thf(fact_163_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_164_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X3: set_nat] : $false ) ) ).
% empty_def
thf(fact_165_empty__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat
@ ^ [X3: set_set_nat] : $false ) ) ).
% empty_def
thf(fact_166_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_167_insert__Collect,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( insert_set_set_nat @ A @ ( collect_set_set_nat @ P ) )
= ( collect_set_set_nat
@ ^ [U: set_set_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_168_insert__Collect,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
= ( collect_set_nat
@ ^ [U: set_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_169_insert__Collect,axiom,
! [A: set_set_set_nat,P: set_set_set_nat > $o] :
( ( insert3687027775829606434et_nat @ A @ ( collec7201453139178570183et_nat @ P ) )
= ( collec7201453139178570183et_nat
@ ^ [U: set_set_set_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_170_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A4 )
| ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_171_insert__compr,axiom,
( insert_set_set_nat
= ( ^ [A4: set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( X3 = A4 )
| ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_172_insert__compr,axiom,
( insert_set_nat
= ( ^ [A4: set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( X3 = A4 )
| ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_173_insert__compr,axiom,
( insert3687027775829606434et_nat
= ( ^ [A4: set_set_set_nat,B3: set_set_set_set_nat] :
( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( X3 = A4 )
| ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_174_subset__insertI2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_175_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_176_subset__insertI2,axiom,
! [A2: set_set_nat,B2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_177_subset__insertI,axiom,
! [B2: set_set_set_nat,A: set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ ( insert_set_set_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_178_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_179_subset__insertI,axiom,
! [B2: set_set_nat,A: set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( insert_set_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_180_subset__insert,axiom,
! [X4: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X4 @ B2 ) )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_181_subset__insert,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_182_subset__insert,axiom,
! [X4: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X4 @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_183_insert__mono,axiom,
! [C2: set_set_set_nat,D: set_set_set_nat,A: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ A @ C2 ) @ ( insert_set_set_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_184_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_185_insert__mono,axiom,
! [C2: set_set_nat,D: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A @ C2 ) @ ( insert_set_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_186_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_187_singleton__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ bot_bot_set_set_nat )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_188_singleton__inject,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_189_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_190_insert__not__empty,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ A2 )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_191_insert__not__empty,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ A @ A2 )
!= bot_bo7198184520161983622et_nat ) ).
% insert_not_empty
thf(fact_192_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_193_doubleton__eq__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ( insert_set_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C @ ( insert_set_nat @ D2 @ bot_bot_set_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_194_doubleton__eq__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D2: set_set_nat] :
( ( ( insert_set_set_nat @ A @ ( insert_set_set_nat @ B @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_set_nat @ C @ ( insert_set_set_nat @ D2 @ bot_bo7198184520161983622et_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_195_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_196_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_197_singleton__iff,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_198_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_199_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_200_singletonD,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_201_Collect__conv__if2,axiom,
! [P: set_set_set_nat > $o,A: set_set_set_nat] :
( ( ( P @ A )
=> ( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert3687027775829606434et_nat @ A @ bot_bo193956671110832956et_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo193956671110832956et_nat ) ) ) ).
% Collect_conv_if2
thf(fact_202_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_203_Collect__conv__if2,axiom,
! [P: set_nat > $o,A: set_nat] :
( ( ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_204_Collect__conv__if2,axiom,
! [P: set_set_nat > $o,A: set_set_nat] :
( ( ( P @ A )
=> ( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo7198184520161983622et_nat ) ) ) ).
% Collect_conv_if2
thf(fact_205_Collect__conv__if,axiom,
! [P: set_set_set_nat > $o,A: set_set_set_nat] :
( ( ( P @ A )
=> ( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert3687027775829606434et_nat @ A @ bot_bo193956671110832956et_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo193956671110832956et_nat ) ) ) ).
% Collect_conv_if
thf(fact_206_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_207_Collect__conv__if,axiom,
! [P: set_nat > $o,A: set_nat] :
( ( ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_208_Collect__conv__if,axiom,
! [P: set_set_nat > $o,A: set_set_nat] :
( ( ( P @ A )
=> ( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo7198184520161983622et_nat ) ) ) ).
% Collect_conv_if
thf(fact_209_subset__singleton__iff,axiom,
! [X5: set_set_set_nat,A: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
= ( ( X5 = bot_bo7198184520161983622et_nat )
| ( X5
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_210_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_211_subset__singleton__iff,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_set_nat )
| ( X5
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_212_v___092_060G_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_213_v___092_060G_062__2,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ).
% v_\<G>_2
thf(fact_214__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<G>_def
thf(fact_215_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_216_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_217_ball__insert,axiom,
! [A: nat,B2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( insert_nat @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_218_ball__insert,axiom,
! [A: set_nat,B2: set_set_nat,P: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( insert_set_nat @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_219_ball__insert,axiom,
! [A: set_set_nat,B2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( insert_set_set_nat @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_220_the__elem__eq,axiom,
! [X4: nat] :
( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_221_the__elem__eq,axiom,
! [X4: set_nat] :
( ( the_elem_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_222_the__elem__eq,axiom,
! [X4: set_set_nat] :
( ( the_elem_set_set_nat @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_223_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_224_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_225_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_226_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_227_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_228_order__refl,axiom,
! [X4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_229_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_230_order__refl,axiom,
! [X4: real] : ( ord_less_eq_real @ X4 @ X4 ) ).
% order_refl
thf(fact_231_is__singletonI,axiom,
! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_232_is__singletonI,axiom,
! [X4: set_nat] : ( is_singleton_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ).
% is_singletonI
thf(fact_233_is__singletonI,axiom,
! [X4: set_set_nat] : ( is_sin6612384548583640136et_nat @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ).
% is_singletonI
thf(fact_234_insert__subsetI,axiom,
! [X4: set_set_nat,A2: set_set_set_nat,X5: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X4 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_235_insert__subsetI,axiom,
! [X4: nat,A2: set_nat,X5: set_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( ord_less_eq_set_nat @ X5 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_236_insert__subsetI,axiom,
! [X4: set_nat,A2: set_set_nat,X5: set_set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_237_order__less__imp__not__less,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ~ ( ord_less_nat @ Y6 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_238_order__less__imp__not__less,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ~ ( ord_less_real @ Y6 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_239_order__less__imp__not__eq2,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( Y6 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_240_order__less__imp__not__eq2,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( Y6 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_241_order__less__imp__not__eq,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( X4 != Y6 ) ) ).
% order_less_imp_not_eq
thf(fact_242_order__less__imp__not__eq,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( X4 != Y6 ) ) ).
% order_less_imp_not_eq
thf(fact_243_linorder__less__linear,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
| ( X4 = Y6 )
| ( ord_less_nat @ Y6 @ X4 ) ) ).
% linorder_less_linear
thf(fact_244_linorder__less__linear,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
| ( X4 = Y6 )
| ( ord_less_real @ Y6 @ X4 ) ) ).
% linorder_less_linear
thf(fact_245_order__less__imp__triv,axiom,
! [X4: nat,Y6: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( ( ord_less_nat @ Y6 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_246_order__less__imp__triv,axiom,
! [X4: real,Y6: real,P: $o] :
( ( ord_less_real @ X4 @ Y6 )
=> ( ( ord_less_real @ Y6 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_247_order__less__not__sym,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ~ ( ord_less_nat @ Y6 @ X4 ) ) ).
% order_less_not_sym
thf(fact_248_order__less__not__sym,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ~ ( ord_less_real @ Y6 @ X4 ) ) ).
% order_less_not_sym
thf(fact_249_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_250_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_251_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_252_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_253_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_254_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_255_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_256_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_257_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_258_order__less__irrefl,axiom,
! [X4: real] :
~ ( ord_less_real @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_259_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_260_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_261_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_262_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_263_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_264_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_265_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_266_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_267_order__less__trans,axiom,
! [X4: nat,Y6: nat,Z2: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( ( ord_less_nat @ Y6 @ Z2 )
=> ( ord_less_nat @ X4 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_268_order__less__trans,axiom,
! [X4: real,Y6: real,Z2: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( ( ord_less_real @ Y6 @ Z2 )
=> ( ord_less_real @ X4 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_269_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_270_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_271_linorder__neq__iff,axiom,
! [X4: nat,Y6: nat] :
( ( X4 != Y6 )
= ( ( ord_less_nat @ X4 @ Y6 )
| ( ord_less_nat @ Y6 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_272_linorder__neq__iff,axiom,
! [X4: real,Y6: real] :
( ( X4 != Y6 )
= ( ( ord_less_real @ X4 @ Y6 )
| ( ord_less_real @ Y6 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_273_order__less__asym,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ~ ( ord_less_nat @ Y6 @ X4 ) ) ).
% order_less_asym
thf(fact_274_order__less__asym,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ~ ( ord_less_real @ Y6 @ X4 ) ) ).
% order_less_asym
thf(fact_275_linorder__neqE,axiom,
! [X4: nat,Y6: nat] :
( ( X4 != Y6 )
=> ( ~ ( ord_less_nat @ X4 @ Y6 )
=> ( ord_less_nat @ Y6 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_276_linorder__neqE,axiom,
! [X4: real,Y6: real] :
( ( X4 != Y6 )
=> ( ~ ( ord_less_real @ X4 @ Y6 )
=> ( ord_less_real @ Y6 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_277_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_278_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_279_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_280_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_281_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_282_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_283_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y6: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y6 ) )
= ( ( ord_less_nat @ Y6 @ X4 )
| ( X4 = Y6 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_284_not__less__iff__gr__or__eq,axiom,
! [X4: real,Y6: real] :
( ( ~ ( ord_less_real @ X4 @ Y6 ) )
= ( ( ord_less_real @ Y6 @ X4 )
| ( X4 = Y6 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_285_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_286_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_287_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_288_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_289_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_290_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_291_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_292_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_293_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_294_linorder__cases,axiom,
! [X4: nat,Y6: nat] :
( ~ ( ord_less_nat @ X4 @ Y6 )
=> ( ( X4 != Y6 )
=> ( ord_less_nat @ Y6 @ X4 ) ) ) ).
% linorder_cases
thf(fact_295_linorder__cases,axiom,
! [X4: real,Y6: real] :
( ~ ( ord_less_real @ X4 @ Y6 )
=> ( ( X4 != Y6 )
=> ( ord_less_real @ Y6 @ X4 ) ) ) ).
% linorder_cases
thf(fact_296_antisym__conv3,axiom,
! [Y6: nat,X4: nat] :
( ~ ( ord_less_nat @ Y6 @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y6 ) )
= ( X4 = Y6 ) ) ) ).
% antisym_conv3
thf(fact_297_antisym__conv3,axiom,
! [Y6: real,X4: real] :
( ~ ( ord_less_real @ Y6 @ X4 )
=> ( ( ~ ( ord_less_real @ X4 @ Y6 ) )
= ( X4 = Y6 ) ) ) ).
% antisym_conv3
thf(fact_298_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X )
=> ( P @ Y7 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_299_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_300_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_301_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_302_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_303_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_304_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_305_less__imp__neq,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( X4 != Y6 ) ) ).
% less_imp_neq
thf(fact_306_less__imp__neq,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( X4 != Y6 ) ) ).
% less_imp_neq
thf(fact_307_dense,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ? [Z3: real] :
( ( ord_less_real @ X4 @ Z3 )
& ( ord_less_real @ Z3 @ Y6 ) ) ) ).
% dense
thf(fact_308_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_309_gt__ex,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% gt_ex
thf(fact_310_lt__ex,axiom,
! [X4: real] :
? [Y: real] : ( ord_less_real @ Y @ X4 ) ).
% lt_ex
thf(fact_311_leD,axiom,
! [Y6: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y6 @ X4 )
=> ~ ( ord_less_set_nat @ X4 @ Y6 ) ) ).
% leD
thf(fact_312_leD,axiom,
! [Y6: set_set_nat,X4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y6 @ X4 )
=> ~ ( ord_less_set_set_nat @ X4 @ Y6 ) ) ).
% leD
thf(fact_313_leD,axiom,
! [Y6: nat,X4: nat] :
( ( ord_less_eq_nat @ Y6 @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y6 ) ) ).
% leD
thf(fact_314_leD,axiom,
! [Y6: real,X4: real] :
( ( ord_less_eq_real @ Y6 @ X4 )
=> ~ ( ord_less_real @ X4 @ Y6 ) ) ).
% leD
thf(fact_315_leI,axiom,
! [X4: nat,Y6: nat] :
( ~ ( ord_less_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ).
% leI
thf(fact_316_leI,axiom,
! [X4: real,Y6: real] :
( ~ ( ord_less_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ Y6 @ X4 ) ) ).
% leI
thf(fact_317_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_318_nless__le,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A @ B ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_319_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_320_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_321_antisym__conv1,axiom,
! [X4: set_nat,Y6: set_nat] :
( ~ ( ord_less_set_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% antisym_conv1
thf(fact_322_antisym__conv1,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X4 @ Y6 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% antisym_conv1
thf(fact_323_antisym__conv1,axiom,
! [X4: nat,Y6: nat] :
( ~ ( ord_less_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_nat @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% antisym_conv1
thf(fact_324_antisym__conv1,axiom,
! [X4: real,Y6: real] :
( ~ ( ord_less_real @ X4 @ Y6 )
=> ( ( ord_less_eq_real @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% antisym_conv1
thf(fact_325_antisym__conv2,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y6 )
=> ( ( ~ ( ord_less_set_nat @ X4 @ Y6 ) )
= ( X4 = Y6 ) ) ) ).
% antisym_conv2
thf(fact_326_antisym__conv2,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
=> ( ( ~ ( ord_less_set_set_nat @ X4 @ Y6 ) )
= ( X4 = Y6 ) ) ) ).
% antisym_conv2
thf(fact_327_antisym__conv2,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y6 ) )
= ( X4 = Y6 ) ) ) ).
% antisym_conv2
thf(fact_328_antisym__conv2,axiom,
! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ( ~ ( ord_less_real @ X4 @ Y6 ) )
= ( X4 = Y6 ) ) ) ).
% antisym_conv2
thf(fact_329_dense__ge,axiom,
! [Z2: real,Y6: real] :
( ! [X: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ord_less_eq_real @ Y6 @ X ) )
=> ( ord_less_eq_real @ Y6 @ Z2 ) ) ).
% dense_ge
thf(fact_330_dense__le,axiom,
! [Y6: real,Z2: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y6 )
=> ( ord_less_eq_real @ X @ Z2 ) )
=> ( ord_less_eq_real @ Y6 @ Z2 ) ) ).
% dense_le
thf(fact_331_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_332_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y3 )
& ~ ( ord_le6893508408891458716et_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_333_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_334_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_335_not__le__imp__less,axiom,
! [Y6: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y6 @ X4 )
=> ( ord_less_nat @ X4 @ Y6 ) ) ).
% not_le_imp_less
thf(fact_336_not__le__imp__less,axiom,
! [Y6: real,X4: real] :
( ~ ( ord_less_eq_real @ Y6 @ X4 )
=> ( ord_less_real @ X4 @ Y6 ) ) ).
% not_le_imp_less
thf(fact_337_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A4 @ B6 )
| ( A4 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_338_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B6: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B6 )
| ( A4 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_339_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_nat @ A4 @ B6 )
| ( A4 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_340_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B6: real] :
( ( ord_less_real @ A4 @ B6 )
| ( A4 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_341_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
& ( A4 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_342_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B6 )
& ( A4 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_343_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ A4 @ B6 )
& ( A4 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_344_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B6: real] :
( ( ord_less_eq_real @ A4 @ B6 )
& ( A4 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_345_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_346_order_Ostrict__trans1,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_347_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_348_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_349_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_350_order_Ostrict__trans2,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_351_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_352_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_353_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_354_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B6 )
& ~ ( ord_le6893508408891458716et_nat @ B6 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_355_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ A4 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_356_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B6: real] :
( ( ord_less_eq_real @ A4 @ B6 )
& ~ ( ord_less_eq_real @ B6 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_357_dense__ge__bounded,axiom,
! [Z2: real,X4: real,Y6: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X4 )
=> ( ord_less_eq_real @ Y6 @ W ) ) )
=> ( ord_less_eq_real @ Y6 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_358_dense__le__bounded,axiom,
! [X4: real,Y6: real,Z2: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( ! [W: real] :
( ( ord_less_real @ X4 @ W )
=> ( ( ord_less_real @ W @ Y6 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y6 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_359_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B6: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B6 @ A4 )
| ( A4 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_360_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B6: set_set_nat,A4: set_set_nat] :
( ( ord_less_set_set_nat @ B6 @ A4 )
| ( A4 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_361_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A4: nat] :
( ( ord_less_nat @ B6 @ A4 )
| ( A4 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_362_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B6: real,A4: real] :
( ( ord_less_real @ B6 @ A4 )
| ( A4 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_363_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B6: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
& ( A4 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_364_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B6: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ A4 )
& ( A4 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_365_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A4: nat] :
( ( ord_less_eq_nat @ B6 @ A4 )
& ( A4 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_366_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B6: real,A4: real] :
( ( ord_less_eq_real @ B6 @ A4 )
& ( A4 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_367_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_368_dual__order_Ostrict__trans1,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_369_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_370_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_371_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_372_dual__order_Ostrict__trans2,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_373_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_374_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_375_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B6: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_376_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B6: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ A4 )
& ~ ( ord_le6893508408891458716et_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_377_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A4: nat] :
( ( ord_less_eq_nat @ B6 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_378_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B6: real,A4: real] :
( ( ord_less_eq_real @ B6 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_379_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_380_order_Ostrict__implies__order,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_381_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_382_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_383_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_384_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_385_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_386_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_387_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_388_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_389_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_390_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_391_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_392_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_393_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_394_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_395_linorder__not__le,axiom,
! [X4: nat,Y6: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y6 ) )
= ( ord_less_nat @ Y6 @ X4 ) ) ).
% linorder_not_le
thf(fact_396_linorder__not__le,axiom,
! [X4: real,Y6: real] :
( ( ~ ( ord_less_eq_real @ X4 @ Y6 ) )
= ( ord_less_real @ Y6 @ X4 ) ) ).
% linorder_not_le
thf(fact_397_linorder__not__less,axiom,
! [X4: nat,Y6: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y6 ) )
= ( ord_less_eq_nat @ Y6 @ X4 ) ) ).
% linorder_not_less
thf(fact_398_linorder__not__less,axiom,
! [X4: real,Y6: real] :
( ( ~ ( ord_less_real @ X4 @ Y6 ) )
= ( ord_less_eq_real @ Y6 @ X4 ) ) ).
% linorder_not_less
thf(fact_399_order__less__imp__le,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ord_less_set_nat @ X4 @ Y6 )
=> ( ord_less_eq_set_nat @ X4 @ Y6 ) ) ).
% order_less_imp_le
thf(fact_400_order__less__imp__le,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y6 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Y6 ) ) ).
% order_less_imp_le
thf(fact_401_order__less__imp__le,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ X4 @ Y6 ) ) ).
% order_less_imp_le
thf(fact_402_order__less__imp__le,axiom,
! [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ X4 @ Y6 ) ) ).
% order_less_imp_le
thf(fact_403_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_404_order__le__neq__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_405_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_406_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_407_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_408_order__neq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A != B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_409_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_410_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_411_order__le__less__trans,axiom,
! [X4: set_nat,Y6: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y6 )
=> ( ( ord_less_set_nat @ Y6 @ Z2 )
=> ( ord_less_set_nat @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_412_order__le__less__trans,axiom,
! [X4: set_set_nat,Y6: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
=> ( ( ord_less_set_set_nat @ Y6 @ Z2 )
=> ( ord_less_set_set_nat @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_413_order__le__less__trans,axiom,
! [X4: nat,Y6: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ( ord_less_nat @ Y6 @ Z2 )
=> ( ord_less_nat @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_414_order__le__less__trans,axiom,
! [X4: real,Y6: real,Z2: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ( ord_less_real @ Y6 @ Z2 )
=> ( ord_less_real @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_415_order__less__le__trans,axiom,
! [X4: set_nat,Y6: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_set_nat @ Y6 @ Z2 )
=> ( ord_less_set_nat @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_416_order__less__le__trans,axiom,
! [X4: set_set_nat,Y6: set_set_nat,Z2: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y6 )
=> ( ( ord_le6893508408891458716et_nat @ Y6 @ Z2 )
=> ( ord_less_set_set_nat @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_417_order__less__le__trans,axiom,
! [X4: nat,Y6: nat,Z2: nat] :
( ( ord_less_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_nat @ Y6 @ Z2 )
=> ( ord_less_nat @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_418_order__less__le__trans,axiom,
! [X4: real,Y6: real,Z2: real] :
( ( ord_less_real @ X4 @ Y6 )
=> ( ( ord_less_eq_real @ Y6 @ Z2 )
=> ( ord_less_real @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_419_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_420_order__le__less__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_421_order__le__less__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_422_order__le__less__subst1,axiom,
! [A: set_set_nat,F: real > set_set_nat,B: real,C: real] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_423_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_424_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_425_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_426_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_427_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_428_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_429_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_430_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_431_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_432_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_433_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_434_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_435_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_436_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_437_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_438_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_439_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_440_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_441_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_442_order__less__le__subst1,axiom,
! [A: real,F: set_nat > real,B: set_nat,C: set_nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_443_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_444_order__less__le__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_445_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_446_order__less__le__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_447_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_448_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_449_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_450_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_set_nat,C: set_set_nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_set_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_451_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_452_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_453_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_454_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_455_linorder__le__less__linear,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
| ( ord_less_nat @ Y6 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_456_linorder__le__less__linear,axiom,
! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
| ( ord_less_real @ Y6 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_457_order__le__imp__less__or__eq,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y6 )
=> ( ( ord_less_set_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_458_order__le__imp__less__or__eq,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
=> ( ( ord_less_set_set_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_459_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ( ord_less_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_460_order__le__imp__less__or__eq,axiom,
! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ( ord_less_real @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_461_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_462_bot_Oextremum__strict,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% bot.extremum_strict
thf(fact_463_bot_Oextremum__strict,axiom,
! [A: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% bot.extremum_strict
thf(fact_464_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_465_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_466_bot_Onot__eq__extremum,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
= ( ord_less_set_set_nat @ bot_bot_set_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_467_bot_Onot__eq__extremum,axiom,
! [A: set_set_set_nat] :
( ( A != bot_bo7198184520161983622et_nat )
= ( ord_le152980574450754630et_nat @ bot_bo7198184520161983622et_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_468_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_469_bot__set__def,axiom,
( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat @ bot_bo5536612546450143305_nat_o ) ) ).
% bot_set_def
thf(fact_470_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_471_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_472_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_473_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A3 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_474_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_475_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_476_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_477_is__singleton__the__elem,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
( A3
= ( insert_set_nat @ ( the_elem_set_nat @ A3 ) @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_478_is__singleton__the__elem,axiom,
( is_sin6612384548583640136et_nat
= ( ^ [A3: set_set_set_nat] :
( A3
= ( insert_set_set_nat @ ( the_elem_set_set_nat @ A3 ) @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_479_Graphs__def,axiom,
( clique5786534781347292306Graphs
= ( ^ [V: set_nat] :
( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ V @ V ) ) ) ) ) ).
% Graphs_def
thf(fact_480_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_481_is__singletonI_H,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_singleton_set_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_482_is__singletonI_H,axiom,
! [A2: set_set_set_nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( member_set_set_nat @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_sin6612384548583640136et_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_483_numbers2__mono,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X4 ) @ ( clique3652268606331196573umbers @ X4 ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y6 ) @ ( clique3652268606331196573umbers @ Y6 ) ) ) ) ).
% numbers2_mono
thf(fact_484_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_485_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_486_le__cases3,axiom,
! [X4: nat,Y6: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y6 )
=> ~ ( ord_less_eq_nat @ Y6 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y6 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y6 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y6 )
=> ~ ( ord_less_eq_nat @ Y6 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y6 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y6 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_487_le__cases3,axiom,
! [X4: real,Y6: real,Z2: real] :
( ( ( ord_less_eq_real @ X4 @ Y6 )
=> ~ ( ord_less_eq_real @ Y6 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y6 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y6 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y6 )
=> ~ ( ord_less_eq_real @ Y6 @ X4 ) )
=> ( ( ( ord_less_eq_real @ Y6 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Y6 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_488_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_489_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y3 )
& ( ord_le6893508408891458716et_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_490_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_491_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_492_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_493_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_494_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_495_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_496_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_497_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_498_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_499_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_500_order__antisym,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_set_nat @ Y6 @ X4 )
=> ( X4 = Y6 ) ) ) ).
% order_antisym
thf(fact_501_order__antisym,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
=> ( ( ord_le6893508408891458716et_nat @ Y6 @ X4 )
=> ( X4 = Y6 ) ) ) ).
% order_antisym
thf(fact_502_order__antisym,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_nat @ Y6 @ X4 )
=> ( X4 = Y6 ) ) ) ).
% order_antisym
thf(fact_503_order__antisym,axiom,
! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ( ord_less_eq_real @ Y6 @ X4 )
=> ( X4 = Y6 ) ) ) ).
% order_antisym
thf(fact_504_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_505_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_506_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_507_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_508_order__trans,axiom,
! [X4: set_nat,Y6: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_set_nat @ Y6 @ Z2 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_509_order__trans,axiom,
! [X4: set_set_nat,Y6: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
=> ( ( ord_le6893508408891458716et_nat @ Y6 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_510_order__trans,axiom,
! [X4: nat,Y6: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_nat @ Y6 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_511_order__trans,axiom,
! [X4: real,Y6: real,Z2: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ( ord_less_eq_real @ Y6 @ Z2 )
=> ( ord_less_eq_real @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_512_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_513_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_514_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_515_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_516_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A4 )
& ( ord_less_eq_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_517_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A4: real,B6: real] :
( ( ord_less_eq_real @ B6 @ A4 )
& ( ord_less_eq_real @ A4 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_518_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_519_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_520_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_521_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_522_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_523_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_524_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_525_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_526_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_527_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_528_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_529_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_530_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_531_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_nat,B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_532_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ A4 @ B6 )
& ( ord_less_eq_nat @ B6 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_533_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A4: real,B6: real] :
( ( ord_less_eq_real @ A4 @ B6 )
& ( ord_less_eq_real @ B6 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_534_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_535_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_536_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_537_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_538_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_539_order__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_540_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_541_order__subst1,axiom,
! [A: real,F: set_nat > real,B: set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_542_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_543_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_544_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_545_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_546_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_547_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_548_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_549_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_550_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_551_order__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_552_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_553_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_554_order__eq__refl,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( X4 = Y6 )
=> ( ord_less_eq_set_nat @ X4 @ Y6 ) ) ).
% order_eq_refl
thf(fact_555_order__eq__refl,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ( X4 = Y6 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Y6 ) ) ).
% order_eq_refl
thf(fact_556_order__eq__refl,axiom,
! [X4: nat,Y6: nat] :
( ( X4 = Y6 )
=> ( ord_less_eq_nat @ X4 @ Y6 ) ) ).
% order_eq_refl
thf(fact_557_order__eq__refl,axiom,
! [X4: real,Y6: real] :
( ( X4 = Y6 )
=> ( ord_less_eq_real @ X4 @ Y6 ) ) ).
% order_eq_refl
thf(fact_558_linorder__linear,axiom,
! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
| ( ord_less_eq_nat @ Y6 @ X4 ) ) ).
% linorder_linear
thf(fact_559_linorder__linear,axiom,
! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
| ( ord_less_eq_real @ Y6 @ X4 ) ) ).
% linorder_linear
thf(fact_560_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_561_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_562_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_563_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_564_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_565_ord__eq__le__subst,axiom,
! [A: real,F: set_nat > real,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_566_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_567_ord__eq__le__subst,axiom,
! [A: set_nat,F: real > set_nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_568_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_569_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_570_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_571_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_572_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_573_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_574_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_575_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_576_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_577_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_578_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_579_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_580_linorder__le__cases,axiom,
! [X4: nat,Y6: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ).
% linorder_le_cases
thf(fact_581_linorder__le__cases,axiom,
! [X4: real,Y6: real] :
( ~ ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ Y6 @ X4 ) ) ).
% linorder_le_cases
thf(fact_582_order__antisym__conv,axiom,
! [Y6: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y6 @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% order_antisym_conv
thf(fact_583_order__antisym__conv,axiom,
! [Y6: set_set_nat,X4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y6 @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% order_antisym_conv
thf(fact_584_order__antisym__conv,axiom,
! [Y6: nat,X4: nat] :
( ( ord_less_eq_nat @ Y6 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% order_antisym_conv
thf(fact_585_order__antisym__conv,axiom,
! [Y6: real,X4: real] :
( ( ord_less_eq_real @ Y6 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ Y6 )
= ( X4 = Y6 ) ) ) ).
% order_antisym_conv
thf(fact_586_prop__restrict,axiom,
! [X4: set_set_nat,Z4: set_set_set_nat,X5: set_set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ X4 @ Z4 )
=> ( ( ord_le9131159989063066194et_nat @ Z4
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_587_prop__restrict,axiom,
! [X4: set_set_set_nat,Z4: set_set_set_set_nat,X5: set_set_set_set_nat,P: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ X4 @ Z4 )
=> ( ( ord_le572741076514265352et_nat @ Z4
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_588_prop__restrict,axiom,
! [X4: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X4 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_589_prop__restrict,axiom,
! [X4: set_nat,Z4: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X4 @ Z4 )
=> ( ( ord_le6893508408891458716et_nat @ Z4
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_590_Collect__restrict,axiom,
! [X5: 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 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_591_Collect__restrict,axiom,
! [X5: set_set_set_set_nat,P: set_set_set_nat > $o] :
( ord_le572741076514265352et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_592_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_593_Collect__restrict,axiom,
! [X5: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_594_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X: nat] :
( A2
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_595_is__singletonE,axiom,
! [A2: set_set_nat] :
( ( is_singleton_set_nat @ A2 )
=> ~ ! [X: set_nat] :
( A2
!= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% is_singletonE
thf(fact_596_is__singletonE,axiom,
! [A2: set_set_set_nat] :
( ( is_sin6612384548583640136et_nat @ A2 )
=> ~ ! [X: set_set_nat] :
( A2
!= ( insert_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) ) ) ).
% is_singletonE
thf(fact_597_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_598_is__singleton__def,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
? [X3: set_nat] :
( A3
= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_599_is__singleton__def,axiom,
( is_sin6612384548583640136et_nat
= ( ^ [A3: set_set_set_nat] :
? [X3: set_set_nat] :
( A3
= ( insert_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% is_singleton_def
thf(fact_600_bot_Oextremum,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% bot.extremum
thf(fact_601_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_602_bot_Oextremum,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% bot.extremum
thf(fact_603_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_604_bot_Oextremum__unique,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_605_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_606_bot_Oextremum__unique,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_607_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_608_bot_Oextremum__uniqueI,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_609_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_610_bot_Oextremum__uniqueI,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_611_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_612_subset__emptyI,axiom,
! [A2: set_set_set_nat] :
( ! [X: set_set_nat] :
~ ( member_set_set_nat @ X @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ).
% subset_emptyI
thf(fact_613_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_614_subset__emptyI,axiom,
! [A2: set_set_nat] :
( ! [X: set_nat] :
~ ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_615_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_616_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_617_finite__vG,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G2 ) ) ) ).
% finite_vG
thf(fact_618_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_619_finite__members___092_060G_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G2 ) ) ).
% finite_members_\<G>
thf(fact_620__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [K: set_set_nat] :
( ( member_set_set_nat @ K @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K ) )
= k )
& ( K
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K ) @ ( clique5033774636164728513irst_v @ K ) ) ) ) ) ) ).
% \<K>_def
thf(fact_621_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_622_first__assumptions_O_092_060G_062__def,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) )
= ( collect_set_set_nat
@ ^ [G: set_set_nat] : ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ) ) ).
% first_assumptions.\<G>_def
thf(fact_623_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_624_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_625_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_626_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_627_psubsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_628_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_629_finite__numbers,axiom,
! [N2: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N2 ) ) ).
% finite_numbers
thf(fact_630_card__numbers,axiom,
! [N2: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N2 ) )
= N2 ) ).
% card_numbers
thf(fact_631_finite__numbers2,axiom,
! [N2: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N2 ) @ ( clique3652268606331196573umbers @ N2 ) ) ) ).
% finite_numbers2
thf(fact_632_sameprod__finite,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_633_sameprod__finite,axiom,
! [X5: set_set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_634_sameprod__finite,axiom,
! [X5: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X5 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_635_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P4: nat,K2: nat,N2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N2 ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_636_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_637_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_638_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_639_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_640_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_641_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_642_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_643_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_644_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_645_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_646_psubset__imp__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_647_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_648_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_649_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_650_psubsetE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_651_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_652_not__psubset__empty,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_653_not__psubset__empty,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% not_psubset_empty
thf(fact_654_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P4: nat,K2: nat,N2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N2 ) @ ( clique3652268606331196573umbers @ N2 ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_655_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite1152437895449049373et_nat @ G2 ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_656_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_657_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_658_first__assumptions_O_092_060K_062__def,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( clique3326749438856946062irst_K @ K2 )
= ( collect_set_set_nat
@ ^ [K: set_set_nat] :
( ( member_set_set_nat @ K @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K ) )
= K2 )
& ( K
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K ) @ ( clique5033774636164728513irst_v @ K ) ) ) ) ) ) ) ).
% first_assumptions.\<K>_def
thf(fact_659_first__assumptions_Ov__mono,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( ord_le6893508408891458716et_nat @ G2 @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_660_first__assumptions_Ov__empty,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_661_first__assumptions_Ov__def,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( clique5033774636164728513irst_v @ G2 )
= ( collect_nat
@ ^ [X3: nat] :
? [Y3: nat] : ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ G2 ) ) ) ) ).
% first_assumptions.v_def
thf(fact_662_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_663_Clique__def,axiom,
( clique6749503327923060270Clique
= ( ^ [V: set_nat,K3: nat] :
( collect_set_set_nat
@ ^ [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ V ) )
& ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ V )
& ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ C3 @ C3 ) @ G )
& ( ( finite_card_nat @ C3 )
= K3 ) ) ) ) ) ) ).
% Clique_def
thf(fact_664_finite__Collect__subsets,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [B3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_665_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_666_finite__Collect__subsets,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_667_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_668_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_669_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_670_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
? [Y3: nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_671_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: set_nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
? [Y3: nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_672_finite__Collect__bounded__ex,axiom,
! [P: set_nat > $o,Q: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
? [Y3: set_nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_673_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: set_set_nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
? [Y3: nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_674_finite__Collect__bounded__ex,axiom,
! [P: set_nat > $o,Q: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
? [Y3: set_nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_675_finite__Collect__bounded__ex,axiom,
! [P: set_set_nat > $o,Q: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
? [Y3: set_set_nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_676_finite__Collect__bounded__ex,axiom,
! [P: set_set_set_nat > $o,Q: nat > set_set_set_nat > $o] :
( ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
? [Y3: set_set_set_nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: set_set_set_nat] :
( ( P @ Y3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_677_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: set_set_set_nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
? [Y3: nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_678_finite__Collect__bounded__ex,axiom,
! [P: set_nat > $o,Q: set_set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
? [Y3: set_nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_679_finite__Collect__bounded__ex,axiom,
! [P: set_set_nat > $o,Q: set_nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
? [Y3: set_set_nat] :
( ( P @ Y3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( Q @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_680_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_681_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_682_finite__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ).
% finite_insert
thf(fact_683_finite__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( insert_set_set_nat @ A @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ).
% finite_insert
thf(fact_684_finite__Collect__disjI,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
& ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_685_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_686_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_687_finite__Collect__disjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
& ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_688_finite__Collect__conjI,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
| ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ Q ) ) )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_689_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_690_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_691_finite__Collect__conjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
| ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X3: set_set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_692_psubsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_693_psubsetD,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_694_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_695_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_696_less__set__def,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ord_le466346588697744319_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A3 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_697_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_698_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_699_finite__psubset__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ! [B7: set_set_nat] :
( ( ord_less_set_set_nat @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_700_finite__psubset__induct,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [A6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A6 )
=> ( ! [B7: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_701_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_702_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_set_nat,R: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_703_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_704_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_set_set_nat,R: nat > set_set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_705_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_set_nat,R: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_706_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_nat,B2: set_nat,R: set_set_nat > nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_707_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_set_nat,B2: set_nat,R: set_set_set_nat > nat > $o] :
( ~ ( finite5926941155766903689et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [A4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_708_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,R: set_nat > set_set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_709_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,R: set_set_nat > set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_710_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_set_nat,B2: set_set_nat,R: set_set_set_nat > set_nat > $o] :
( ~ ( finite5926941155766903689et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ~ ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [A4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A4 @ A2 )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_711_not__finite__existsD,axiom,
! [P: set_set_set_nat > $o] :
( ~ ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
=> ? [X_1: set_set_set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_712_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_713_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_714_not__finite__existsD,axiom,
! [P: set_set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ? [X_1: set_set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_715_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ A @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_716_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_717_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_718_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X: real] :
( ( member_real @ X @ A2 )
& ( ord_less_eq_real @ A @ X )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_719_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_720_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ X @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_721_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_722_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X: real] :
( ( member_real @ X @ A2 )
& ( ord_less_eq_real @ X @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_723_finite__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_724_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_725_finite__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_726_infinite__super,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_727_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_728_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_729_rev__finite__subset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_730_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_731_rev__finite__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_732_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_733_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_734_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_735_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_736_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_737_infinite__imp__nonempty,axiom,
! [S: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ( S != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_738_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_739_finite_OinsertI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( insert_set_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_740_finite_OinsertI,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite6739761609112101331et_nat @ ( insert_set_set_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_741_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_742_finite__image__set,axiom,
! [P: nat > $o,F: nat > set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_743_finite__image__set,axiom,
! [P: set_nat > $o,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: set_nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_744_finite__image__set,axiom,
! [P: nat > $o,F: nat > set_set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_745_finite__image__set,axiom,
! [P: set_nat > $o,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: set_nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_746_finite__image__set,axiom,
! [P: set_set_nat > $o,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: set_set_nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_747_finite__image__set,axiom,
! [P: set_set_set_nat > $o,F: set_set_set_nat > nat] :
( ( finite5926941155766903689et_nat @ ( collec7201453139178570183et_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: set_set_set_nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_748_finite__image__set,axiom,
! [P: nat > $o,F: nat > set_set_set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [Uu: set_set_set_nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_749_finite__image__set,axiom,
! [P: set_nat > $o,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [X3: set_nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_750_finite__image__set,axiom,
! [P: set_set_nat > $o,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: set_set_nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_751_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat,Y3: nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_752_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat,Y3: nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_753_finite__image__set2,axiom,
! [P: nat > $o,Q: set_nat > $o,F: nat > set_nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat,Y3: set_nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_754_finite__image__set2,axiom,
! [P: set_nat > $o,Q: nat > $o,F: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: set_nat,Y3: nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_755_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > set_set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [X3: nat,Y3: nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_756_finite__image__set2,axiom,
! [P: nat > $o,Q: set_nat > $o,F: nat > set_nat > set_nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: nat,Y3: set_nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_757_finite__image__set2,axiom,
! [P: nat > $o,Q: set_set_nat > $o,F: nat > set_set_nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat,Y3: set_set_nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_758_finite__image__set2,axiom,
! [P: set_nat > $o,Q: nat > $o,F: set_nat > nat > set_nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [X3: set_nat,Y3: nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_759_finite__image__set2,axiom,
! [P: set_nat > $o,Q: set_nat > $o,F: set_nat > set_nat > nat] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: set_nat,Y3: set_nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_760_finite__image__set2,axiom,
! [P: set_set_nat > $o,Q: nat > $o,F: set_set_nat > nat > nat] :
( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X3: set_set_nat,Y3: nat] :
( ( Uu
= ( F @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_761_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_762_finite__has__maximal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_763_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_764_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X: real] :
( ( member_real @ X @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_765_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_766_finite__has__minimal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_767_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_768_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X: real] :
( ( member_real @ X @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_769_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A
= ( insert_nat @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_770_finite_Ocases,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ~ ! [A6: set_set_nat] :
( ? [A5: set_nat] :
( A
= ( insert_set_nat @ A5 @ A6 ) )
=> ~ ( finite1152437895449049373et_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_771_finite_Ocases,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ~ ! [A6: set_set_set_nat] :
( ? [A5: set_set_nat] :
( A
= ( insert_set_set_nat @ A5 @ A6 ) )
=> ~ ( finite6739761609112101331et_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_772_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B6: nat] :
( ( A4
= ( insert_nat @ B6 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_773_finite_Osimps,axiom,
( finite1152437895449049373et_nat
= ( ^ [A4: set_set_nat] :
( ( A4 = bot_bot_set_set_nat )
| ? [A3: set_set_nat,B6: set_nat] :
( ( A4
= ( insert_set_nat @ B6 @ A3 ) )
& ( finite1152437895449049373et_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_774_finite_Osimps,axiom,
( finite6739761609112101331et_nat
= ( ^ [A4: set_set_set_nat] :
( ( A4 = bot_bo7198184520161983622et_nat )
| ? [A3: set_set_set_nat,B6: set_set_nat] :
( ( A4
= ( insert_set_set_nat @ B6 @ A3 ) )
& ( finite6739761609112101331et_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_775_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_776_finite__induct,axiom,
! [F2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ~ ( member_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_777_finite__induct,axiom,
! [F2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ~ ( member_set_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_778_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_779_finite__ne__induct,axiom,
! [F2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( F2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] : ( P @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( ! [X: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( F3 != bot_bot_set_set_nat )
=> ( ~ ( member_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_780_finite__ne__induct,axiom,
! [F2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( F2 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat] : ( P @ ( insert_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) )
=> ( ! [X: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( F3 != bot_bo7198184520161983622et_nat )
=> ( ~ ( member_set_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_781_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_782_infinite__finite__induct,axiom,
! [P: set_set_nat > $o,A2: set_set_nat] :
( ! [A6: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ~ ( member_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_783_infinite__finite__induct,axiom,
! [P: set_set_set_nat > $o,A2: set_set_set_nat] :
( ! [A6: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ~ ( member_set_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_784_infinite__arbitrarily__large,axiom,
! [A2: set_real,N2: nat] :
( ~ ( finite_finite_real @ A2 )
=> ? [B4: set_real] :
( ( finite_finite_real @ B4 )
& ( ( finite_card_real @ B4 )
= N2 )
& ( ord_less_eq_set_real @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_785_infinite__arbitrarily__large,axiom,
! [A2: set_set_set_nat,N2: nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ? [B4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B4 )
& ( ( finite1149291290879098388et_nat @ B4 )
= N2 )
& ( ord_le9131159989063066194et_nat @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_786_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ( finite_card_nat @ B4 )
= N2 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_787_infinite__arbitrarily__large,axiom,
! [A2: set_set_nat,N2: nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ( finite_card_set_nat @ B4 )
= N2 )
& ( ord_le6893508408891458716et_nat @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_788_card__subset__eq,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ( finite_card_real @ A2 )
= ( finite_card_real @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_789_card__subset__eq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_790_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_791_card__subset__eq,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_792_psubset__card__mono,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_set_real @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_793_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_794_psubset__card__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_795_psubset__card__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_796_card__insert__le,axiom,
! [A2: set_set_nat,X4: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_797_card__insert__le,axiom,
! [A2: set_set_set_nat,X4: set_set_nat] : ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ ( insert_set_set_nat @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_798_card__insert__le,axiom,
! [A2: set_nat,X4: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_799_card__insert__le,axiom,
! [A2: set_real,X4: real] : ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ ( insert_real @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_800_finite__subset__induct,axiom,
! [F2: set_set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( ord_le9131159989063066194et_nat @ F2 @ A2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [A5: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( member_set_set_nat @ A5 @ A2 )
=> ( ~ ( member_set_set_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_801_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_802_finite__subset__induct,axiom,
! [F2: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A5: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( member_set_nat @ A5 @ A2 )
=> ( ~ ( member_set_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_803_finite__subset__induct_H,axiom,
! [F2: set_set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( ord_le9131159989063066194et_nat @ F2 @ A2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [A5: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( member_set_set_nat @ A5 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ F3 @ A2 )
=> ( ~ ( member_set_set_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_804_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_805_finite__subset__induct_H,axiom,
! [F2: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A5: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( member_set_nat @ A5 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
=> ( ~ ( member_set_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_806_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_real,C2: nat] :
( ! [G3: set_real] :
( ( ord_less_eq_set_real @ G3 @ F2 )
=> ( ( finite_finite_real @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_real @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_real @ F2 )
& ( ord_less_eq_nat @ ( finite_card_real @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_807_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_set_nat,C2: nat] :
( ! [G3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G3 @ F2 )
=> ( ( finite6739761609112101331et_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G3 ) @ C2 ) ) )
=> ( ( finite6739761609112101331et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_808_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F2 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_809_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_nat,C2: nat] :
( ! [G3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G3 @ F2 )
=> ( ( finite1152437895449049373et_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G3 ) @ C2 ) ) )
=> ( ( finite1152437895449049373et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_810_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_real] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_real @ S ) )
=> ~ ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S )
=> ( ( ( finite_card_real @ T3 )
= N2 )
=> ~ ( finite_finite_real @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_811_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_set_set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite1149291290879098388et_nat @ S ) )
=> ~ ! [T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T3 @ S )
=> ( ( ( finite1149291290879098388et_nat @ T3 )
= N2 )
=> ~ ( finite6739761609112101331et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_812_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_813_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ S ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S )
=> ( ( ( finite_card_set_nat @ T3 )
= N2 )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_814_exists__subset__between,axiom,
! [A2: set_real,N2: nat,C2: set_real] :
( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_real @ C2 ) )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( finite_finite_real @ C2 )
=> ? [B4: set_real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
& ( ord_less_eq_set_real @ B4 @ C2 )
& ( ( finite_card_real @ B4 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_815_exists__subset__between,axiom,
! [A2: set_set_set_nat,N2: nat,C2: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite1149291290879098388et_nat @ C2 ) )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( finite6739761609112101331et_nat @ C2 )
=> ? [B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ C2 )
& ( ( finite1149291290879098388et_nat @ B4 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_816_exists__subset__between,axiom,
! [A2: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ C2 )
& ( ( finite_card_nat @ B4 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_817_exists__subset__between,axiom,
! [A2: set_set_nat,N2: nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ C2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( finite1152437895449049373et_nat @ C2 )
=> ? [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ C2 )
& ( ( finite_card_set_nat @ B4 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_818_card__seteq,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ B2 ) @ ( finite_card_real @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_819_card__seteq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_820_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_821_card__seteq,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_822_card__mono,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ).
% card_mono
thf(fact_823_card__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_824_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_825_card__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_826_card__psubset,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) )
=> ( ord_less_set_real @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_827_card__psubset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_828_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_829_card__psubset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_830_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_831_first__assumptions_Okml,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_eq_nat @ K2 @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K2 ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_832_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A6 )
=> ( ord_less_nat @ X7 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_833_finite__linorder__max__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B5: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X7: real] :
( ( member_real @ X7 @ A6 )
=> ( ord_less_real @ X7 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_real @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_834_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A6 )
=> ( ord_less_nat @ B5 @ X7 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_835_finite__linorder__min__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B5: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X7: real] :
( ( member_real @ X7 @ A6 )
=> ( ord_less_real @ B5 @ X7 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_real @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_836_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_837_finite__ranking__induct,axiom,
! [S: set_set_nat,P: set_set_nat > $o,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,S2: set_set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [Y7: set_nat] :
( ( member_set_nat @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_838_finite__ranking__induct,axiom,
! [S: set_set_set_nat,P: set_set_set_nat > $o,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,S2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ S2 )
=> ( ! [Y7: set_set_nat] :
( ( member_set_set_nat @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_set_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_839_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_840_finite__ranking__induct,axiom,
! [S: set_set_nat,P: set_set_nat > $o,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,S2: set_set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [Y7: set_nat] :
( ( member_set_nat @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_841_finite__ranking__induct,axiom,
! [S: set_set_set_nat,P: set_set_set_nat > $o,F: set_set_nat > real] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,S2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ S2 )
=> ( ! [Y7: set_set_nat] :
( ( member_set_set_nat @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_set_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_842_first__assumptions_Okm,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_nat @ K2 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.km
thf(fact_843_first__assumptions_Omp,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_nat @ P4 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.mp
thf(fact_844_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_845_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_846_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_847_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_848_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_849_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_850_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_851_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_852_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_853_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less_nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_854_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_855_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_856_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_857_linorder__neqE__nat,axiom,
! [X4: nat,Y6: nat] :
( ( X4 != Y6 )
=> ( ~ ( ord_less_nat @ X4 @ Y6 )
=> ( ord_less_nat @ Y6 @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_858_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_859_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_860_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_861_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_862_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_863_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_864_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_865_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_866_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_867_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_868_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_869_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_870_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_871_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_872_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_873_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_874_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_875_first__assumptions_Ok,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_nat @ L @ K2 ) ) ).
% first_assumptions.k
thf(fact_876_first__assumptions_Okp,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_nat @ P4 @ K2 ) ) ).
% first_assumptions.kp
thf(fact_877_first__assumptions_Opl,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_nat @ L @ P4 ) ) ).
% first_assumptions.pl
thf(fact_878_ex__min__if__finite,axiom,
! [S: set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ S )
& ~ ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ S )
& ( ord_less_set_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_879_ex__min__if__finite,axiom,
! [S: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( S != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ S )
& ~ ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ S )
& ( ord_less_set_set_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_880_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_881_ex__min__if__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ? [X: real] :
( ( member_real @ X @ S )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S )
& ( ord_less_real @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_882_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_883_infinite__growing,axiom,
! [X5: set_real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X5 )
& ( ord_less_real @ X @ Xa ) ) )
=> ~ ( finite_finite_real @ X5 ) ) ) ).
% infinite_growing
thf(fact_884_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_885_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_886_card__le__if__inj__on__rel,axiom,
! [B2: set_real,A2: set_nat,R2: nat > real > $o] :
( ( finite_finite_real @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B8: real] :
( ( member_real @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: real] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_real @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_887_card__le__if__inj__on__rel,axiom,
! [B2: set_real,A2: set_real,R2: real > real > $o] :
( ( finite_finite_real @ B2 )
=> ( ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ? [B8: real] :
( ( member_real @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: real,A22: real,B5: real] :
( ( member_real @ A1 @ A2 )
=> ( ( member_real @ A22 @ A2 )
=> ( ( member_real @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_888_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_889_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_real,R2: real > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: real,A22: real,B5: nat] :
( ( member_real @ A1 @ A2 )
=> ( ( member_real @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_890_card__le__if__inj__on__rel,axiom,
! [B2: set_real,A2: set_set_nat,R2: set_nat > real > $o] :
( ( finite_finite_real @ B2 )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [B8: real] :
( ( member_real @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B5: real] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_real @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_891_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_nat,R2: set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B5: nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_892_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_nat,R2: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B8: set_nat] :
( ( member_set_nat @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_893_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_real,R2: real > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ? [B8: set_nat] :
( ( member_set_nat @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: real,A22: real,B5: set_nat] :
( ( member_real @ A1 @ A2 )
=> ( ( member_real @ A22 @ A2 )
=> ( ( member_set_nat @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_894_card__le__if__inj__on__rel,axiom,
! [B2: set_real,A2: set_set_set_nat,R2: set_set_nat > real > $o] :
( ( finite_finite_real @ B2 )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
=> ? [B8: real] :
( ( member_real @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B5: real] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_real @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_895_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_set_nat,R2: set_set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A5 @ B8 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B5: nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_896_ex__card,axiom,
! [N2: nat,A2: set_real] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_real @ A2 ) )
=> ? [S2: set_real] :
( ( ord_less_eq_set_real @ S2 @ A2 )
& ( ( finite_card_real @ S2 )
= N2 ) ) ) ).
% ex_card
thf(fact_897_ex__card,axiom,
! [N2: nat,A2: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ A2 ) )
=> ? [S2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ A2 )
& ( ( finite_card_nat @ S2 )
= N2 ) ) ) ).
% ex_card
thf(fact_898_ex__card,axiom,
! [N2: nat,A2: set_set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ A2 ) )
=> ? [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ A2 )
& ( ( finite_card_set_nat @ S2 )
= N2 ) ) ) ).
% ex_card
thf(fact_899_diff__shunt__var,axiom,
! [X4: set_set_set_nat,Y6: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ X4 @ Y6 )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ X4 @ Y6 ) ) ).
% diff_shunt_var
thf(fact_900_diff__shunt__var,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ( minus_minus_set_nat @ X4 @ Y6 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ Y6 ) ) ).
% diff_shunt_var
thf(fact_901_diff__shunt__var,axiom,
! [X4: set_set_nat,Y6: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X4 @ Y6 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X4 @ Y6 ) ) ).
% diff_shunt_var
thf(fact_902_pred__subset__eq,axiom,
! [R: set_set_set_nat,S: set_set_set_nat] :
( ( ord_le3616423863276227763_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ R )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ S ) )
= ( ord_le9131159989063066194et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_903_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_904_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_905_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_906_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_907_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_908_Diff__empty,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_909_Diff__empty,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% Diff_empty
thf(fact_910_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_911_empty__Diff,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A2 )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_912_empty__Diff,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% empty_Diff
thf(fact_913_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_914_Diff__cancel,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ A2 )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_915_Diff__cancel,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% Diff_cancel
thf(fact_916_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_917_finite__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_918_finite__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_919_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_920_finite__Diff2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_921_finite__Diff2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_922_Diff__insert0,axiom,
! [X4: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ B2 ) )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_923_Diff__insert0,axiom,
! [X4: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ B2 ) )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_924_Diff__insert0,axiom,
! [X4: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_925_insert__Diff1,axiom,
! [X4: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_926_insert__Diff1,axiom,
! [X4: set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ B2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X4 @ A2 ) @ B2 )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_927_insert__Diff1,axiom,
! [X4: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_928_Diff__eq__empty__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_929_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_930_Diff__eq__empty__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_931_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_932_insert__Diff__single,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_933_insert__Diff__single,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ A @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
= ( insert_set_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_934_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_935_finite__Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) ) )
= ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_936_finite__Diff__insert,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B2 ) ) )
= ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_937_Diff__mono,axiom,
! [A2: set_nat,C2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_938_Diff__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,D: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_939_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_940_Diff__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_941_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_942_double__diff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ( minus_2163939370556025621et_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_943_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_944_Diff__infinite__finite,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( finite1152437895449049373et_nat @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_945_Diff__infinite__finite,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_946_insert__Diff__if,axiom,
! [X4: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X4 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
& ( ~ ( member_set_nat @ X4 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A2 ) @ B2 )
= ( insert_set_nat @ X4 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_947_insert__Diff__if,axiom,
! [X4: set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ( member_set_set_nat @ X4 @ B2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X4 @ A2 ) @ B2 )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) )
& ( ~ ( member_set_set_nat @ X4 @ B2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X4 @ A2 ) @ B2 )
= ( insert_set_set_nat @ X4 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_948_insert__Diff__if,axiom,
! [X4: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X4 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X4 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ B2 )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_949_psubset__imp__ex__mem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ? [B5: set_nat] : ( member_set_nat @ B5 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_950_psubset__imp__ex__mem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ? [B5: set_set_nat] : ( member_set_set_nat @ B5 @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_951_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_952_subset__Diff__insert,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X4: set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B2 @ ( insert_set_set_nat @ X4 @ C2 ) ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B2 @ C2 ) )
& ~ ( member_set_set_nat @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_953_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X4: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_954_subset__Diff__insert,axiom,
! [A2: set_set_nat,B2: set_set_nat,X4: set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X4 @ C2 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
& ~ ( member_set_nat @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_955_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_956_Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ).
% Diff_insert
thf(fact_957_Diff__insert,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ).
% Diff_insert
thf(fact_958_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_959_insert__Diff,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_960_insert__Diff,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( insert_set_set_nat @ A @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_961_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_962_Diff__insert2,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_963_Diff__insert2,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_964_Diff__insert__absorb,axiom,
! [X4: nat,A2: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_965_Diff__insert__absorb,axiom,
! [X4: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A2 ) @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_966_Diff__insert__absorb,axiom,
! [X4: set_set_nat,A2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X4 @ A2 ) @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_967_subset__insert__iff,axiom,
! [A2: set_set_set_nat,X4: set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X4 @ B2 ) )
= ( ( ( member_set_set_nat @ X4 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) @ B2 ) )
& ( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_968_subset__insert__iff,axiom,
! [A2: set_nat,X4: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ B2 ) )
= ( ( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_969_subset__insert__iff,axiom,
! [A2: set_set_nat,X4: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X4 @ B2 ) )
= ( ( ( member_set_nat @ X4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_970_Diff__single__insert,axiom,
! [A2: set_set_set_nat,X4: set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) @ B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_971_Diff__single__insert,axiom,
! [A2: set_nat,X4: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_972_Diff__single__insert,axiom,
! [A2: set_set_nat,X4: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_973_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_974_infinite__remove,axiom,
! [S: set_set_nat,A: set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_975_infinite__remove,axiom,
! [S: set_set_set_nat,A: set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% infinite_remove
thf(fact_976_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_977_infinite__coinduct,axiom,
! [X5: set_set_nat > $o,A2: set_set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_set_nat] :
( ( X5 @ A6 )
=> ? [X7: set_nat] :
( ( member_set_nat @ X7 @ A6 )
& ( ( X5 @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X7 @ bot_bot_set_set_nat ) ) )
| ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X7 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_978_infinite__coinduct,axiom,
! [X5: set_set_set_nat > $o,A2: set_set_set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_set_set_nat] :
( ( X5 @ A6 )
=> ? [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A6 )
& ( ( X5 @ ( minus_2447799839930672331et_nat @ A6 @ ( insert_set_set_nat @ X7 @ bot_bo7198184520161983622et_nat ) ) )
| ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A6 @ ( insert_set_set_nat @ X7 @ bot_bo7198184520161983622et_nat ) ) ) ) ) )
=> ~ ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_979_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_980_finite__empty__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: set_nat,A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ( member_set_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ A5 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_981_finite__empty__induct,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: set_set_nat,A6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A6 )
=> ( ( member_set_set_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_2447799839930672331et_nat @ A6 @ ( insert_set_set_nat @ A5 @ bot_bo7198184520161983622et_nat ) ) ) ) ) )
=> ( P @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_982_card__le__sym__Diff,axiom,
! [A2: set_real,B2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ ( finite_card_real @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_983_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_984_card__le__sym__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_985_card__le__sym__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_986_card__less__sym__Diff,axiom,
! [A2: set_real,B2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B2 )
=> ( ( ord_less_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ ( finite_card_real @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_987_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_988_card__less__sym__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_989_card__less__sym__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_990_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M4: nat] :
( ( P @ X4 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_991_remove__induct,axiom,
! [P: set_set_set_nat > $o,B2: set_set_set_nat] :
( ( P @ bot_bo7198184520161983622et_nat )
=> ( ( ~ ( finite6739761609112101331et_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A6 )
=> ( ( A6 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le9131159989063066194et_nat @ A6 @ B2 )
=> ( ! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A6 )
=> ( P @ ( minus_2447799839930672331et_nat @ A6 @ ( insert_set_set_nat @ X7 @ bot_bo7198184520161983622et_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_992_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_993_remove__induct,axiom,
! [P: set_set_nat > $o,B2: set_set_nat] :
( ( P @ bot_bot_set_set_nat )
=> ( ( ~ ( finite1152437895449049373et_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ( A6 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A6 @ B2 )
=> ( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A6 )
=> ( P @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X7 @ bot_bot_set_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_994_finite__remove__induct,axiom,
! [B2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [A6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A6 )
=> ( ( A6 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le9131159989063066194et_nat @ A6 @ B2 )
=> ( ! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A6 )
=> ( P @ ( minus_2447799839930672331et_nat @ A6 @ ( insert_set_set_nat @ X7 @ bot_bo7198184520161983622et_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_995_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_996_finite__remove__induct,axiom,
! [B2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ( A6 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A6 @ B2 )
=> ( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A6 )
=> ( P @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X7 @ bot_bot_set_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_997_card__Diff1__le,axiom,
! [A2: set_real,X4: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ).
% card_Diff1_le
thf(fact_998_card__Diff1__le,axiom,
! [A2: set_nat,X4: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_999_card__Diff1__le,axiom,
! [A2: set_set_nat,X4: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1000_card__Diff1__le,axiom,
! [A2: set_set_set_nat,X4: set_set_nat] : ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1001_card__Diff__subset,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1002_card__Diff__subset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1003_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1004_card__Diff__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1005_psubset__insert__iff,axiom,
! [A2: set_set_set_nat,X4: set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( insert_set_set_nat @ X4 @ B2 ) )
= ( ( ( member_set_set_nat @ X4 @ B2 )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) )
& ( ~ ( member_set_set_nat @ X4 @ B2 )
=> ( ( ( member_set_set_nat @ X4 @ A2 )
=> ( ord_le152980574450754630et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) @ B2 ) )
& ( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1006_psubset__insert__iff,axiom,
! [A2: set_nat,X4: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X4 @ B2 ) )
= ( ( ( member_nat @ X4 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X4 @ B2 )
=> ( ( ( member_nat @ X4 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1007_psubset__insert__iff,axiom,
! [A2: set_set_nat,X4: set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X4 @ B2 ) )
= ( ( ( member_set_nat @ X4 @ B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) )
& ( ~ ( member_set_nat @ X4 @ B2 )
=> ( ( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1008_diff__card__le__card__Diff,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1009_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1010_diff__card__le__card__Diff,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1011_diff__card__le__card__Diff,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1012_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X7 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1013_finite__induct__select,axiom,
! [S: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [T3: set_set_nat] :
( ( ord_less_set_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X7: set_nat] :
( ( member_set_nat @ X7 @ ( minus_2163939370556025621et_nat @ S @ T3 ) )
& ( P @ ( insert_set_nat @ X7 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1014_finite__induct__select,axiom,
! [S: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [T3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ ( minus_2447799839930672331et_nat @ S @ T3 ) )
& ( P @ ( insert_set_set_nat @ X7 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1015_card__Diff1__less,axiom,
! [A2: set_real,X4: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X4 @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1016_card__Diff1__less,axiom,
! [A2: set_nat,X4: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X4 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1017_card__Diff1__less,axiom,
! [A2: set_set_nat,X4: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1018_card__Diff1__less,axiom,
! [A2: set_set_set_nat,X4: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ X4 @ A2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1019_card__Diff2__less,axiom,
! [A2: set_real,X4: real,Y6: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X4 @ A2 )
=> ( ( member_real @ Y6 @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) @ ( insert_real @ Y6 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1020_card__Diff2__less,axiom,
! [A2: set_nat,X4: nat,Y6: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y6 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1021_card__Diff2__less,axiom,
! [A2: set_set_nat,X4: set_nat,Y6: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X4 @ A2 )
=> ( ( member_set_nat @ Y6 @ A2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1022_card__Diff2__less,axiom,
! [A2: set_set_set_nat,X4: set_set_nat,Y6: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ X4 @ A2 )
=> ( ( member_set_set_nat @ Y6 @ A2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) @ ( insert_set_set_nat @ Y6 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1023_card__Diff1__less__iff,axiom,
! [A2: set_real,X4: real] :
( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) )
= ( ( finite_finite_real @ A2 )
& ( member_real @ X4 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1024_card__Diff1__less__iff,axiom,
! [A2: set_nat,X4: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X4 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1025_card__Diff1__less__iff,axiom,
! [A2: set_set_nat,X4: set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) )
= ( ( finite1152437895449049373et_nat @ A2 )
& ( member_set_nat @ X4 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1026_card__Diff1__less__iff,axiom,
! [A2: set_set_set_nat,X4: set_set_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) )
= ( ( finite6739761609112101331et_nat @ A2 )
& ( member_set_set_nat @ X4 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1027_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1028_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1029_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_nat @ X @ N2 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1030_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1031_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1032_bot__empty__eq,axiom,
( bot_bo6227097192321305471_nat_o
= ( ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat ) ) ) ).
% bot_empty_eq
thf(fact_1033_Collect__empty__eq__bot,axiom,
! [P: set_set_set_nat > $o] :
( ( ( collec7201453139178570183et_nat @ P )
= bot_bo193956671110832956et_nat )
= ( P = bot_bo5536612546450143305_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1034_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1035_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1036_Collect__empty__eq__bot,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( P = bot_bo6227097192321305471_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1037_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1038_unbounded__k__infinite,axiom,
! [K2: nat,S: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K2 @ M5 )
=> ? [N6: nat] :
( ( ord_less_nat @ M5 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1039_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1040_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X7: nat] :
( ( member_nat @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1041_arg__min__if__finite_I2_J,axiom,
! [S: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ~ ? [X7: set_nat] :
( ( member_set_nat @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic7132588981422310769at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1042_arg__min__if__finite_I2_J,axiom,
! [S: set_set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( S != bot_bo7198184520161983622et_nat )
=> ~ ? [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic82989400242555431at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1043_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X7: nat] :
( ( member_nat @ X7 @ S )
& ( ord_less_real @ ( F @ X7 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1044_arg__min__if__finite_I2_J,axiom,
! [S: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ~ ? [X7: set_nat] :
( ( member_set_nat @ X7 @ S )
& ( ord_less_real @ ( F @ X7 ) @ ( F @ ( lattic6497381205983422413t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1045_arg__min__if__finite_I2_J,axiom,
! [S: set_set_set_nat,F: set_set_nat > real] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( S != bot_bo7198184520161983622et_nat )
=> ~ ? [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ S )
& ( ord_less_real @ ( F @ X7 ) @ ( F @ ( lattic4725577848813236611t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1046_Diff__iff,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C @ A2 )
& ~ ( member_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1047_Diff__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( ( member_set_set_nat @ C @ A2 )
& ~ ( member_set_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1048_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1049_DiffI,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1050_DiffI,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( ~ ( member_set_set_nat @ C @ B2 )
=> ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1051_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1052_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1053_minus__set__def,axiom,
( minus_2447799839930672331et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ( minus_463385787819020154_nat_o
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A3 )
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1054_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1055_minus__set__def,axiom,
( minus_3113942175840221057et_nat
= ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
( collec7201453139178570183et_nat
@ ( minus_495547888894627908_nat_o
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A3 )
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1056_DiffD2,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( member_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1057_DiffD2,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ~ ( member_set_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1058_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1059_DiffD1,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1060_DiffD1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ( member_set_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1061_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1062_DiffE,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1063_DiffE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1064_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1065_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ~ ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1066_set__diff__eq,axiom,
( minus_2447799839930672331et_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 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1067_set__diff__eq,axiom,
( minus_2163939370556025621et_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 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1068_set__diff__eq,axiom,
( minus_3113942175840221057et_nat
= ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A3 )
& ~ ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1069_finite__transitivity__chain,axiom,
! [A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X: nat] :
~ ( R @ X @ X )
=> ( ! [X: nat,Y: nat,Z3: nat] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Y7: nat] :
( ( member_nat @ Y7 @ A2 )
& ( R @ X @ Y7 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1070_finite__transitivity__chain,axiom,
! [A2: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [X: set_nat] :
~ ( R @ X @ X )
=> ( ! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Y7: set_nat] :
( ( member_set_nat @ Y7 @ A2 )
& ( R @ X @ Y7 ) ) )
=> ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1071_finite__transitivity__chain,axiom,
! [A2: set_set_set_nat,R: set_set_nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [X: set_set_nat] :
~ ( R @ X @ X )
=> ( ! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Y7: set_set_nat] :
( ( member_set_set_nat @ Y7 @ A2 )
& ( R @ X @ Y7 ) ) )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1072_arg__min__least,axiom,
! [S: set_nat,Y6: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y6 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y6 ) ) ) ) ) ).
% arg_min_least
thf(fact_1073_arg__min__least,axiom,
! [S: set_set_nat,Y6: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ( ( member_set_nat @ Y6 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7132588981422310769at_nat @ F @ S ) ) @ ( F @ Y6 ) ) ) ) ) ).
% arg_min_least
thf(fact_1074_arg__min__least,axiom,
! [S: set_set_set_nat,Y6: set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( S != bot_bo7198184520161983622et_nat )
=> ( ( member_set_set_nat @ Y6 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic82989400242555431at_nat @ F @ S ) ) @ ( F @ Y6 ) ) ) ) ) ).
% arg_min_least
thf(fact_1075_arg__min__least,axiom,
! [S: set_nat,Y6: nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y6 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) @ ( F @ Y6 ) ) ) ) ) ).
% arg_min_least
thf(fact_1076_arg__min__least,axiom,
! [S: set_set_nat,Y6: set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ( ( member_set_nat @ Y6 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic6497381205983422413t_real @ F @ S ) ) @ ( F @ Y6 ) ) ) ) ) ).
% arg_min_least
thf(fact_1077_arg__min__least,axiom,
! [S: set_set_set_nat,Y6: set_set_nat,F: set_set_nat > real] :
( ( finite6739761609112101331et_nat @ S )
=> ( ( S != bot_bo7198184520161983622et_nat )
=> ( ( member_set_set_nat @ Y6 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic4725577848813236611t_real @ F @ S ) ) @ ( F @ Y6 ) ) ) ) ) ).
% arg_min_least
thf(fact_1078_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N2 @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1079_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [X_12: nat] : ( P @ X @ X_12 ) )
=> ? [M5: nat] :
! [X7: nat] :
( ( member_nat @ X7 @ A2 )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ M5 )
& ( P @ X7 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1080_finite__indexed__bound,axiom,
! [A2: set_set_nat,P: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [X_12: nat] : ( P @ X @ X_12 ) )
=> ? [M5: nat] :
! [X7: set_nat] :
( ( member_set_nat @ X7 @ A2 )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ M5 )
& ( P @ X7 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1081_finite__indexed__bound,axiom,
! [A2: set_set_set_nat,P: set_set_nat > nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [X_12: nat] : ( P @ X @ X_12 ) )
=> ? [M5: nat] :
! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A2 )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ M5 )
& ( P @ X7 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1082_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [X_12: real] : ( P @ X @ X_12 ) )
=> ? [M5: real] :
! [X7: nat] :
( ( member_nat @ X7 @ A2 )
=> ? [K4: real] :
( ( ord_less_eq_real @ K4 @ M5 )
& ( P @ X7 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1083_finite__indexed__bound,axiom,
! [A2: set_set_nat,P: set_nat > real > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [X_12: real] : ( P @ X @ X_12 ) )
=> ? [M5: real] :
! [X7: set_nat] :
( ( member_set_nat @ X7 @ A2 )
=> ? [K4: real] :
( ( ord_less_eq_real @ K4 @ M5 )
& ( P @ X7 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1084_finite__indexed__bound,axiom,
! [A2: set_set_set_nat,P: set_set_nat > real > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [X_12: real] : ( P @ X @ X_12 ) )
=> ? [M5: real] :
! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A2 )
=> ? [K4: real] :
( ( ord_less_eq_real @ K4 @ M5 )
& ( P @ X7 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1085_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1086_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1087_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_1088_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1089_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1090_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1091_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1092_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1093_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1094_diff__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1095_diff__strict__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_1096_minf_I8_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_eq_nat @ T4 @ X7 ) ) ).
% minf(8)
thf(fact_1097_minf_I8_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ~ ( ord_less_eq_real @ T4 @ X7 ) ) ).
% minf(8)
thf(fact_1098_minf_I6_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_eq_nat @ X7 @ T4 ) ) ).
% minf(6)
thf(fact_1099_minf_I6_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ( ord_less_eq_real @ X7 @ T4 ) ) ).
% minf(6)
thf(fact_1100_pinf_I8_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_eq_nat @ T4 @ X7 ) ) ).
% pinf(8)
thf(fact_1101_pinf_I8_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ( ord_less_eq_real @ T4 @ X7 ) ) ).
% pinf(8)
thf(fact_1102_pinf_I6_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T4 ) ) ).
% pinf(6)
thf(fact_1103_pinf_I6_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ~ ( ord_less_eq_real @ X7 @ T4 ) ) ).
% pinf(6)
thf(fact_1104_verit__comp__simplify1_I3_J,axiom,
! [B9: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B9 @ A7 ) )
= ( ord_less_nat @ A7 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1105_verit__comp__simplify1_I3_J,axiom,
! [B9: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B9 @ A7 ) )
= ( ord_less_real @ A7 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1106_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1107_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1108_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1109_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1110_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1111_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1112_minf_I7_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_nat @ T4 @ X7 ) ) ).
% minf(7)
thf(fact_1113_minf_I7_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ~ ( ord_less_real @ T4 @ X7 ) ) ).
% minf(7)
thf(fact_1114_minf_I5_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_nat @ X7 @ T4 ) ) ).
% minf(5)
thf(fact_1115_minf_I5_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ( ord_less_real @ X7 @ T4 ) ) ).
% minf(5)
thf(fact_1116_minf_I4_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T4 ) ) ).
% minf(4)
thf(fact_1117_minf_I4_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ( X7 != T4 ) ) ).
% minf(4)
thf(fact_1118_minf_I3_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T4 ) ) ).
% minf(3)
thf(fact_1119_minf_I3_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ( X7 != T4 ) ) ).
% minf(3)
thf(fact_1120_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1121_minf_I2_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ X @ Z5 )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1122_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1123_minf_I1_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ X @ Z5 )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1124_pinf_I7_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_nat @ T4 @ X7 ) ) ).
% pinf(7)
thf(fact_1125_pinf_I7_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ( ord_less_real @ T4 @ X7 ) ) ).
% pinf(7)
thf(fact_1126_pinf_I5_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T4 ) ) ).
% pinf(5)
thf(fact_1127_pinf_I5_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ~ ( ord_less_real @ X7 @ T4 ) ) ).
% pinf(5)
thf(fact_1128_pinf_I4_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T4 ) ) ).
% pinf(4)
thf(fact_1129_pinf_I4_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ( X7 != T4 ) ) ).
% pinf(4)
thf(fact_1130_pinf_I3_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T4 ) ) ).
% pinf(3)
thf(fact_1131_pinf_I3_J,axiom,
! [T4: real] :
? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ( X7 != T4 ) ) ).
% pinf(3)
thf(fact_1132_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1133_pinf_I2_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ Z5 @ X )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1134_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1135_pinf_I1_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ Z5 @ X )
=> ( ( P @ X )
= ( P5 @ X ) ) )
=> ( ? [Z5: real] :
! [X: real] :
( ( ord_less_real @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X7: real] :
( ( ord_less_real @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1136_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1137_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1138_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1139_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: real] :
( ( ord_less_eq_real @ A @ C4 )
& ( ord_less_eq_real @ C4 @ B )
& ! [X7: real] :
( ( ( ord_less_eq_real @ A @ X7 )
& ( ord_less_real @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D3: real] :
( ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_real @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_real @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1140_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_1141_remove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_1142_remove__def,axiom,
( remove_set_nat
= ( ^ [X3: set_nat,A3: set_set_nat] : ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% remove_def
thf(fact_1143_remove__def,axiom,
( remove_set_set_nat
= ( ^ [X3: set_set_nat,A3: set_set_set_nat] : ( minus_2447799839930672331et_nat @ A3 @ ( insert_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% remove_def
thf(fact_1144_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M2: nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_1145_finite__enum__subset,axiom,
! [X5: set_nat,Y5: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X5 ) )
=> ( ( infini8530281810654367211te_nat @ X5 @ I3 )
= ( infini8530281810654367211te_nat @ Y5 @ I3 ) ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X5 ) @ ( finite_card_nat @ Y5 ) )
=> ( ord_less_eq_set_nat @ X5 @ Y5 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_1146_member__remove,axiom,
! [X4: set_nat,Y6: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X4 @ ( remove_set_nat @ Y6 @ A2 ) )
= ( ( member_set_nat @ X4 @ A2 )
& ( X4 != Y6 ) ) ) ).
% member_remove
thf(fact_1147_member__remove,axiom,
! [X4: set_set_nat,Y6: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ ( remove_set_set_nat @ Y6 @ A2 ) )
= ( ( member_set_set_nat @ X4 @ A2 )
& ( X4 != Y6 ) ) ) ).
% member_remove
thf(fact_1148_member__remove,axiom,
! [X4: nat,Y6: nat,A2: set_nat] :
( ( member_nat @ X4 @ ( remove_nat @ Y6 @ A2 ) )
= ( ( member_nat @ X4 @ A2 )
& ( X4 != Y6 ) ) ) ).
% member_remove
thf(fact_1149_enumerate__mono__le__iff,axiom,
! [S: set_nat,M2: nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_1150_enumerate__mono__iff,axiom,
! [S: set_nat,M2: nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_iff
thf(fact_1151_enumerate__in__set,axiom,
! [S: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N2 ) @ S ) ) ).
% enumerate_in_set
thf(fact_1152_enumerate__Ex,axiom,
! [S: set_nat,S3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S3 @ S )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S @ N3 )
= S3 ) ) ) ).
% enumerate_Ex
thf(fact_1153_le__enumerate,axiom,
! [S: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ).
% le_enumerate
thf(fact_1154_ex__gt__or__lt,axiom,
! [A: real] :
? [B5: real] :
( ( ord_less_real @ A @ B5 )
| ( ord_less_real @ B5 @ A ) ) ).
% ex_gt_or_lt
thf(fact_1155_enumerate__mono,axiom,
! [M2: nat,N2: nat,S: set_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ).
% enumerate_mono
thf(fact_1156_finite__enum__ext,axiom,
! [X5: set_nat,Y5: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X5 ) )
=> ( ( infini8530281810654367211te_nat @ X5 @ I3 )
= ( infini8530281810654367211te_nat @ Y5 @ I3 ) ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y5 )
=> ( ( ( finite_card_nat @ X5 )
= ( finite_card_nat @ Y5 ) )
=> ( X5 = Y5 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_1157_finite__enumerate__Ex,axiom,
! [S: set_nat,S3: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat @ S3 @ S )
=> ? [N3: nat] :
( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N3 )
= S3 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_1158_finite__enumerate__in__set,axiom,
! [S: set_nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N2 ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_1159_finite__enumerate__mono,axiom,
! [M2: nat,N2: nat,S: set_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_1160_finite__le__enumerate,axiom,
! [S: set_nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_1161_bot_Oordering__top__axioms,axiom,
( orderi7669286751058938369et_nat
@ ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y3 @ X3 )
@ ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( ord_le152980574450754630et_nat @ Y3 @ X3 )
@ bot_bo7198184520161983622et_nat ) ).
% bot.ordering_top_axioms
thf(fact_1162_bot_Oordering__top__axioms,axiom,
( ordering_top_set_nat
@ ^ [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ X3 )
@ ^ [X3: set_nat,Y3: set_nat] : ( ord_less_set_nat @ Y3 @ X3 )
@ bot_bot_set_nat ) ).
% bot.ordering_top_axioms
thf(fact_1163_bot_Oordering__top__axioms,axiom,
( orderi1027199981026551883et_nat
@ ^ [X3: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y3 @ X3 )
@ ^ [X3: set_set_nat,Y3: set_set_nat] : ( ord_less_set_set_nat @ Y3 @ X3 )
@ bot_bot_set_set_nat ) ).
% bot.ordering_top_axioms
thf(fact_1164_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_1165_card__Diff__singleton,axiom,
! [X4: real,A2: set_real] :
( ( member_real @ X4 @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1166_card__Diff__singleton,axiom,
! [X4: nat,A2: set_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1167_card__Diff__singleton,axiom,
! [X4: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1168_card__Diff__singleton,axiom,
! [X4: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1169_card__Diff__singleton__if,axiom,
! [X4: real,A2: set_real] :
( ( ( member_real @ X4 @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_real @ X4 @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
= ( finite_card_real @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1170_card__Diff__singleton__if,axiom,
! [X4: nat,A2: set_nat] :
( ( ( member_nat @ X4 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X4 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1171_card__Diff__singleton__if,axiom,
! [X4: set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X4 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_nat @ X4 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1172_card__Diff__singleton__if,axiom,
! [X4: set_set_nat,A2: set_set_set_nat] :
( ( ( member_set_set_nat @ X4 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_set_nat @ X4 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1173_card__Diff__insert,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ A @ B2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1174_card__Diff__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ~ ( member_set_set_nat @ A @ B2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1175_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1176_card__Diff__insert,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( member_real @ A @ A2 )
=> ( ~ ( member_real @ A @ B2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1177_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1178_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1179_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1180_diff__diff__left,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1181_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1182_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1183_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1184_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1185_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1186_diff__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1187_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1188_add__lessD1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
=> ( ord_less_nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_1189_add__less__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1190_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1191_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1192_add__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1193_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1194_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1195_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1196_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1197_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1198_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1199_add__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1200_add__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1201_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1202_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1203_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1204_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1205_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1206_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1207_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1208_le__diff__conv,axiom,
! [J: nat,K2: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1209_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1210_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1211_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1212_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1213_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1214_less__diff__conv,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1215_less__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1216_first__assumptions_Olm,axiom,
! [L: nat,P4: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K2 ) ) ) ).
% first_assumptions.lm
thf(fact_1217_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1218_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1219_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1220_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1221_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1222_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1223_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1224_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1225_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1226_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1227_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1228_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1229_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1230_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1231_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1232_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1233_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1234_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1235_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1236_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1237_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1238_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1239_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1240_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1241_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1242_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1243_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1244_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1245_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1246_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1247_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1248_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1249_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1250_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1251_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1252_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1253_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1254_kuhn__lemma,axiom,
! [P4: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P4 )
=> ( ! [X: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X @ I4 ) @ P4 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( Label @ X @ I3 )
= zero_zero_nat )
| ( ( Label @ X @ I3 )
= one_one_nat ) ) ) )
=> ( ! [X: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X @ I4 ) @ P4 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X @ I3 )
= zero_zero_nat )
=> ( ( Label @ X @ I3 )
= zero_zero_nat ) ) ) )
=> ( ! [X: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X @ I4 ) @ P4 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X @ I3 )
= P4 )
=> ( ( Label @ X @ I3 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_nat @ ( Q3 @ I4 ) @ P4 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ? [R3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J3 ) @ ( R3 @ J3 ) )
& ( ord_less_eq_nat @ ( R3 @ J3 ) @ ( plus_plus_nat @ ( Q3 @ J3 ) @ one_one_nat ) ) ) )
& ? [S4: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J3 ) @ ( S4 @ J3 ) )
& ( ord_less_eq_nat @ ( S4 @ J3 ) @ ( plus_plus_nat @ ( Q3 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I4 )
!= ( Label @ S4 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1255_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X: nat > real] :
( ( P @ X )
=> ( P @ ( F @ X ) ) )
=> ( ! [X: nat > real] :
( ( P @ X )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) )
& ( ord_less_eq_real @ ( X @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X7: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X7 @ I4 ) @ one_one_nat )
& ! [X7: nat > real,I4: nat] :
( ( ( P @ X7 )
& ( Q @ I4 )
& ( ( X7 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X7 @ I4 )
= zero_zero_nat ) )
& ! [X7: nat > real,I4: nat] :
( ( ( P @ X7 )
& ( Q @ I4 )
& ( ( X7 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X7 @ I4 )
= one_one_nat ) )
& ! [X7: nat > real,I4: nat] :
( ( ( P @ X7 )
& ( Q @ I4 )
& ( ( L2 @ X7 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X7 @ I4 ) @ ( F @ X7 @ I4 ) ) )
& ! [X7: nat > real,I4: nat] :
( ( ( P @ X7 )
& ( Q @ I4 )
& ( ( L2 @ X7 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X7 @ I4 ) @ ( X7 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1256_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M2 @ N6 )
=> ( ord_less_real @ ( D2 @ N6 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1257_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A5: real,B5: real,C4: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C4 )
=> ( ( ord_less_eq_real @ A5 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C4 )
=> ( P @ A5 @ C4 ) ) ) ) )
=> ( ! [X: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq_real @ A5 @ X )
& ( ord_less_eq_real @ X @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D3 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1258_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1259_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I3_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(3)
thf(fact_1260_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(1)
thf(fact_1261_eps,axiom,
ord_less_real @ zero_zero_real @ assumptions_and_eps ).
% eps
thf(fact_1262_v__union,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_1263_first__assumptions_Ov__union,axiom,
! [L: nat,P4: nat,K2: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P4 @ K2 )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1264_DIM__real,axiom,
( ( finite_card_real @ euclid1305858884100475807s_real )
= one_one_nat ) ).
% DIM_real
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_set_nat @ ( insert_nat @ x @ ( insert_nat @ y @ bot_bot_set_nat ) )
@ ( collect_nat
@ ^ [X3: nat] :
? [Y3: nat] : ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ g ) ) ) ).
%------------------------------------------------------------------------------