%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP093^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:29 AM UTC 2026
% Result : Theorem 3.17s 1.77s
% Output : Refutation 3.17s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 4
% Syntax : Number of formulae : 24 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 89 ( 25 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 133 ( 15 ~; 9 |; 0 &; 103 @)
% ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 4 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 108 usr; 8 con; 0-3 aty)
% Number of variables : 23 ( 11 ^; 12 !; 0 ?; 23 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_nat: $tType ).
thf(type_def_6,type,
set_set_nat: $tType ).
thf(type_def_8,type,
set_Pr1986765409at_nat: $tType ).
thf(type_def_9,type,
set_se1612935105at_nat: $tType ).
thf(type_def_10,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_11,type,
set_se958357159od_a_a: $tType ).
thf(type_def_12,type,
set_Product_prod_a_a: $tType ).
thf(type_def_13,type,
set_se1596668135od_a_a: $tType ).
thf(type_def_14,type,
set_a: $tType ).
thf(type_def_15,type,
set_set_a: $tType ).
thf(type_def_16,type,
nat: $tType ).
thf(type_def_17,type,
set_Pr1746169692at_nat: $tType ).
thf(type_def_18,type,
set_Pr339609346od_a_a: $tType ).
thf(type_def_19,type,
set_Pr967348953_nat_a: $tType ).
thf(type_def_20,type,
set_Pr1490359111at_nat: $tType ).
thf(type_def_21,type,
set_Pr1295299783od_a_a: $tType ).
thf(type_def_22,type,
set_Pr894832732_a_nat: $tType ).
thf(type_def_23,type,
set_Pr1689873822_a_a_a: $tType ).
thf(type_def_24,type,
set_Pr548851891_a_nat: $tType ).
thf(type_def_25,type,
set_Pr7688842at_nat: $tType ).
thf(type_def_26,type,
set_Pr681306928od_a_a: $tType ).
thf(type_def_27,type,
product_prod_a_a: $tType ).
thf(type_def_28,type,
a: $tType ).
thf(type_def_29,type,
pair_p1914262621t_unit: $tType ).
thf(type_def_30,type,
pair_p2041852168t_unit: $tType ).
thf(type_def_31,type,
pair_p1593840546t_unit: $tType ).
thf(type_def_32,type,
pair_p1765063010t_unit: $tType ).
thf(type_def_33,type,
pair_p125712459t_unit: $tType ).
thf(type_def_34,type,
product_unit: $tType ).
thf(type_def_35,type,
product_prod_nat_nat: $tType ).
thf(type_def_36,type,
produc1572603623od_a_a: $tType ).
thf(func_def_0,type,
set_Pr1295299783od_a_a: $tType ).
thf(func_def_1,type,
pair_p1593840546t_unit: $tType ).
thf(func_def_2,type,
pair_p2041852168t_unit: $tType ).
thf(func_def_3,type,
set_Pr1490359111at_nat: $tType ).
thf(func_def_4,type,
pair_p1765063010t_unit: $tType ).
thf(func_def_5,type,
set_se958357159od_a_a: $tType ).
thf(func_def_6,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_7,type,
pair_p1914262621t_unit: $tType ).
thf(func_def_8,type,
set_Pr1746169692at_nat: $tType ).
thf(func_def_9,type,
produc1572603623od_a_a: $tType ).
thf(func_def_10,type,
pair_p125712459t_unit: $tType ).
thf(func_def_11,type,
set_Pr7688842at_nat: $tType ).
thf(func_def_12,type,
set_Pr894832732_a_nat: $tType ).
thf(func_def_13,type,
set_Pr339609346od_a_a: $tType ).
thf(func_def_14,type,
set_Pr681306928od_a_a: $tType ).
thf(func_def_15,type,
set_Pr1689873822_a_a_a: $tType ).
thf(func_def_16,type,
set_se1612935105at_nat: $tType ).
thf(func_def_17,type,
set_se1596668135od_a_a: $tType ).
thf(func_def_18,type,
set_Pr1986765409at_nat: $tType ).
thf(func_def_19,type,
set_Pr548851891_a_nat: $tType ).
thf(func_def_20,type,
set_Pr967348953_nat_a: $tType ).
thf(func_def_21,type,
set_Product_prod_a_a: $tType ).
thf(func_def_22,type,
product_prod_nat_nat: $tType ).
thf(func_def_23,type,
set_set_nat: $tType ).
thf(func_def_24,type,
product_prod_a_a: $tType ).
thf(func_def_25,type,
set_set_a: $tType ).
thf(func_def_26,type,
set_nat: $tType ).
thf(func_def_27,type,
product_unit: $tType ).
thf(func_def_28,type,
set_a: $tType ).
thf(func_def_29,type,
nat: $tType ).
thf(func_def_30,type,
a: $tType ).
thf(func_def_31,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(func_def_32,type,
finite361944167at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat ).
thf(func_def_33,type,
finite702915405od_a_a: set_Pr1948701895od_a_a > set_se958357159od_a_a ).
thf(func_def_34,type,
finite351630733od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a ).
thf(func_def_35,type,
finite_Fpow_a: set_a > set_set_a ).
thf(func_def_36,type,
finite_card_a: set_a > nat ).
thf(func_def_37,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_38,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(func_def_39,type,
finite277291581at_nat: set_Pr1746169692at_nat > $o ).
thf(func_def_40,type,
finite1297454819od_a_a: set_Pr339609346od_a_a > $o ).
thf(func_def_41,type,
finite1808550458_nat_a: set_Pr967348953_nat_a > $o ).
thf(func_def_42,type,
finite48957584at_nat: set_Pr1490359111at_nat > $o ).
thf(func_def_43,type,
finite256329232od_a_a: set_Pr1295299783od_a_a > $o ).
thf(func_def_44,type,
finite1837575485_a_nat: set_Pr894832732_a_nat > $o ).
thf(func_def_45,type,
finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
thf(func_def_46,type,
finite1919032935_a_a_a: set_Pr1689873822_a_a_a > $o ).
thf(func_def_47,type,
finite1743148308_a_nat: set_Pr548851891_a_nat > $o ).
thf(func_def_48,type,
finite942416723at_nat: set_Pr7688842at_nat > $o ).
thf(func_def_49,type,
finite676513017od_a_a: set_Pr681306928od_a_a > $o ).
thf(func_def_50,type,
finite179568208od_a_a: set_Product_prod_a_a > $o ).
thf(func_def_51,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(func_def_52,type,
finite1457549322at_nat: set_se1612935105at_nat > $o ).
thf(func_def_53,type,
finite323969008od_a_a: set_se958357159od_a_a > $o ).
thf(func_def_54,type,
finite1145471536od_a_a: set_se1596668135od_a_a > $o ).
thf(func_def_55,type,
finite_finite_set_a: set_set_a > $o ).
thf(func_def_56,type,
finite_finite_a: set_a > $o ).
thf(func_def_57,type,
bot_bot_set_nat: set_nat ).
thf(func_def_58,type,
bot_bo2130386637at_nat: set_Pr1986765409at_nat ).
thf(func_def_59,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(func_def_60,type,
bot_bot_set_a: set_a ).
thf(func_def_61,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_62,type,
ord_le1347718902_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > $o ).
thf(func_def_63,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(func_def_64,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_65,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_66,type,
ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_67,type,
ord_le2084554594od_a_a: set_Pr339609346od_a_a > set_Pr339609346od_a_a > $o ).
thf(func_def_68,type,
ord_le344568633_nat_a: set_Pr967348953_nat_a > set_Pr967348953_nat_a > $o ).
thf(func_def_69,type,
ord_le492294332_a_nat: set_Pr894832732_a_nat > set_Pr894832732_a_nat > $o ).
thf(func_def_70,type,
ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_71,type,
ord_le677315902_a_a_a: set_Pr1689873822_a_a_a > set_Pr1689873822_a_a_a > $o ).
thf(func_def_72,type,
ord_le2073555219_a_nat: set_Pr548851891_a_nat > set_Pr548851891_a_nat > $o ).
thf(func_def_73,type,
ord_le1816232656od_a_a: set_Pr681306928od_a_a > set_Pr681306928od_a_a > $o ).
thf(func_def_74,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_75,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_76,type,
pair_p128415500ph_nat: pair_p1914262621t_unit > $o ).
thf(func_def_77,type,
pair_p752841413at_nat: pair_p2041852168t_unit > $o ).
thf(func_def_78,type,
pair_p159207083od_a_a: pair_p1593840546t_unit > $o ).
thf(func_def_79,type,
pair_p374947051od_a_a: pair_p1765063010t_unit > $o ).
thf(func_def_80,type,
pair_p1802376898raph_a: pair_p125712459t_unit > $o ).
thf(func_def_81,type,
pair_p1027063983ms_nat: pair_p1914262621t_unit > $o ).
thf(func_def_82,type,
pair_p56914274at_nat: pair_p2041852168t_unit > $o ).
thf(func_def_83,type,
pair_p1906342088od_a_a: pair_p1593840546t_unit > $o ).
thf(func_def_84,type,
pair_p504738056od_a_a: pair_p1765063010t_unit > $o ).
thf(func_def_85,type,
pair_p1864019935ioms_a: pair_p125712459t_unit > $o ).
thf(func_def_86,type,
pair_p69470259t_unit: pair_p1914262621t_unit > product_unit ).
thf(func_def_87,type,
pair_p1984658862t_unit: pair_p1765063010t_unit > product_unit ).
thf(func_def_88,type,
pair_p1896222615t_unit: pair_p125712459t_unit > product_unit ).
thf(func_def_89,type,
pair_p1167410509t_unit: set_nat > set_Pr1986765409at_nat > product_unit > pair_p1914262621t_unit ).
thf(func_def_90,type,
pair_p398687508t_unit: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_unit > pair_p1765063010t_unit ).
thf(func_def_91,type,
pair_p1621517565t_unit: set_a > set_Product_prod_a_a > product_unit > pair_p125712459t_unit ).
thf(func_def_92,type,
pair_p715279805t_unit: pair_p1914262621t_unit > set_Pr1986765409at_nat ).
thf(func_def_93,type,
pair_p806300874t_unit: pair_p2041852168t_unit > set_Pr1490359111at_nat ).
thf(func_def_94,type,
pair_p1559300324t_unit: pair_p1593840546t_unit > set_Pr1295299783od_a_a ).
thf(func_def_95,type,
pair_p1783210148t_unit: pair_p1765063010t_unit > set_Pr1948701895od_a_a ).
thf(func_def_96,type,
pair_p133601421t_unit: pair_p125712459t_unit > set_Product_prod_a_a ).
thf(func_def_97,type,
pair_p1677060310t_unit: pair_p1914262621t_unit > set_nat ).
thf(func_def_98,type,
pair_p210955889t_unit: pair_p2041852168t_unit > set_Pr1986765409at_nat ).
thf(func_def_99,type,
pair_p1652294923t_unit: pair_p1593840546t_unit > set_Pr1948701895od_a_a ).
thf(func_def_100,type,
pair_p447552203t_unit: pair_p1765063010t_unit > set_Product_prod_a_a ).
thf(func_def_101,type,
pair_p1047056820t_unit: pair_p125712459t_unit > set_a ).
thf(func_def_102,type,
pair_p1515597646ph_nat: pair_p1914262621t_unit > $o ).
thf(func_def_103,type,
pair_p646030121od_a_a: pair_p1765063010t_unit > $o ).
thf(func_def_104,type,
pair_p68905728raph_a: pair_p125712459t_unit > $o ).
thf(func_def_105,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_106,type,
produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
thf(func_def_107,type,
produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
thf(func_def_108,type,
produc1182842125od_a_a: set_nat > ( nat > set_Product_prod_a_a ) > set_Pr339609346od_a_a ).
thf(func_def_109,type,
product_Sigma_nat_a: set_nat > ( nat > set_a ) > set_Pr967348953_nat_a ).
thf(func_def_110,type,
produc931712687_a_nat: set_Product_prod_a_a > ( product_prod_a_a > set_nat ) > set_Pr894832732_a_nat ).
thf(func_def_111,type,
produc304751368od_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_112,type,
produc1282482655_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_a ) > set_Pr1689873822_a_a_a ).
thf(func_def_113,type,
product_Sigma_a_nat: set_a > ( a > set_nat ) > set_Pr548851891_a_nat ).
thf(func_def_114,type,
produc292491723at_nat: set_a > ( a > set_Pr1986765409at_nat ) > set_Pr7688842at_nat ).
thf(func_def_115,type,
produc520147185od_a_a: set_a > ( a > set_Product_prod_a_a ) > set_Pr681306928od_a_a ).
thf(func_def_116,type,
product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
thf(func_def_117,type,
produc1833107820_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
thf(func_def_118,type,
product_product_a_a: set_a > set_a > set_Product_prod_a_a ).
thf(func_def_119,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_120,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(func_def_121,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(func_def_122,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_123,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_124,type,
collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
thf(func_def_125,type,
collec453062450od_a_a: ( set_Pr1948701895od_a_a > $o ) > set_se958357159od_a_a ).
thf(func_def_126,type,
collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
thf(func_def_127,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_128,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_129,type,
member_nat: nat > set_nat > $o ).
thf(func_def_130,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_131,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_132,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_133,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(func_def_134,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_135,type,
member_a: a > set_a > $o ).
thf(func_def_136,type,
g: pair_p125712459t_unit ).
thf(func_def_137,type,
n: nat ).
thf(func_def_153,type,
ph1:
!>[X0: $tType] : X0 ).
thf(f1056,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1055,f1046]) ).
thf(f1046,plain,
( $true
= ( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) ) ),
inference(cnf_transformation,[status(thm)],[f467]) ).
thf(f467,plain,
( $true
= ( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) ) ),
inference(fool_elimination,[status(thm)],[f466]) ).
thf(f466,plain,
( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [X0: a] : ( pair_p1047056820t_unit @ g ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [X0: a] : ( pair_p1047056820t_unit @ g ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',fact_0__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062) ).
thf(f1055,plain,
( $true
!= ( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f1054,f1050]) ).
thf(f1050,plain,
( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1036]) ).
thf(f1036,plain,
( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) )
!= $true ),
inference(flattening,[status(thm)],[f391]) ).
thf(f391,plain,
( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f390]) ).
thf(f390,plain,
~ ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) ),
inference(rectify,[status(thm)],[f356]) ).
thf(f356,negated_conjecture,
~ ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) ),
inference(negated_conjecture,[status(cth)],[f355]) ).
thf(f355,conjecture,
finite179568208od_a_a @ ( pair_p133601421t_unit @ g ),
file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',conj_0) ).
thf(f1054,plain,
( ( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) )
= $true )
| ( $true
!= ( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1053]) ).
thf(f1053,plain,
( ( $true != $true )
| ( $true
!= ( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) ) )
| ( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) )
= $true ) ),
inference(superposition,[status(thm)],[f1051,f1044]) ).
thf(f1044,plain,
( ( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1003]) ).
thf(f1003,plain,
( ( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Y0: a] : ( pair_p1047056820t_unit @ g ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f1002]) ).
thf(f1002,plain,
( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [X0: a] : ( pair_p1047056820t_unit @ g ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [X0: a] : ( pair_p1047056820t_unit @ g ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',fact_1__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062) ).
thf(f1051,plain,
! [X0: set_Product_prod_a_a,X1: set_Product_prod_a_a] :
( ( ( ord_le1824328871od_a_a @ X1 @ X0 )
!= $true )
| ( ( finite179568208od_a_a @ X0 )
!= $true )
| ( $true
= ( finite179568208od_a_a @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1041]) ).
thf(f1041,plain,
! [X0: set_Product_prod_a_a,X1: set_Product_prod_a_a] :
( ( ( finite179568208od_a_a @ X0 )
!= $true )
| ( ( ord_le1824328871od_a_a @ X1 @ X0 )
!= $true )
| ( $true
= ( finite179568208od_a_a @ X1 ) ) ),
inference(flattening,[status(thm)],[f1040]) ).
thf(f1040,plain,
! [X0: set_Product_prod_a_a,X1: set_Product_prod_a_a] :
( ( $true
= ( finite179568208od_a_a @ X1 ) )
| ( ( finite179568208od_a_a @ X0 )
!= $true )
| ( ( ord_le1824328871od_a_a @ X1 @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f581]) ).
thf(f581,plain,
! [X0: set_Product_prod_a_a,X1: set_Product_prod_a_a] :
( ( ( ord_le1824328871od_a_a @ X1 @ X0 )
= $true )
=> ( ( ( finite179568208od_a_a @ X0 )
= $true )
=> ( $true
= ( finite179568208od_a_a @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f580]) ).
thf(f580,plain,
! [X0: set_Product_prod_a_a,X1: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X1 @ X0 )
=> ( ( finite179568208od_a_a @ X0 )
=> ( finite179568208od_a_a @ X1 ) ) ),
inference(rectify,[status(thm)],[f20]) ).
thf(f20,axiom,
! [X4: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X3 @ X4 )
=> ( ( finite179568208od_a_a @ X4 )
=> ( finite179568208od_a_a @ X3 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',fact_19_finite__subset) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP093^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n014.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 08:26:12 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : ITP093^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.46 % Domain : Interactive Theorem Proving
% 0.29/0.46 % Problem : Sledgehammer Kuratowski problem prob_43__5514550_1
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Des21]
% 0.29/0.46 % Names : Kuratowski/prob_43__5514550_1 [Des21]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.44 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% 0.29/0.46 % Syntax : Number of formulae : 493 ( 113 unt; 138 typ; 0 def)
% 0.29/0.46 % Number of atoms : 1236 ( 234 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 12 ( 3 avg)
% 0.29/0.46 % Number of connectives : 3205 ( 91 ~; 31 |; 122 &;2361 @)
% 0.29/0.46 % ( 0 <=>; 600 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 16 ( 7 avg)
% 0.29/0.46 % Number of types : 32 ( 31 usr)
% 0.29/0.46 % Number of type conns : 337 ( 337 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 108 ( 107 usr; 6 con; 0-3 aty)
% 0.29/0.46 % Number of variables : 1153 ( 214 ^; 904 !; 35 ?;1153 :)
% 0.29/0.46 % SPC : TH0_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:44:24.363
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % Could-be-implicit typings (31)
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.29/0.46 set_Pr1295299783od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Ounit_J,type,
% 0.29/0.46 pair_p1593840546t_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Ounit_J,type,
% 0.29/0.46 pair_p2041852168t_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.29/0.46 set_Pr1490359111at_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
% 0.29/0.46 pair_p1765063010t_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.29/0.46 set_se958357159od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 set_Pr1948701895od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
% 0.29/0.46 pair_p1914262621t_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.29/0.46 set_Pr1746169692at_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 produc1572603623od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
% 0.29/0.46 pair_p125712459t_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.29/0.46 set_Pr7688842at_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 set_Pr894832732_a_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 set_Pr339609346od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 set_Pr681306928od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 0.29/0.46 set_Pr1689873822_a_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.29/0.46 set_se1612935105at_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 set_se1596668135od_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 set_Pr1986765409at_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 set_Pr548851891_a_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
% 0.29/0.46 set_Pr967348953_nat_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 set_Product_prod_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 product_prod_nat_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.29/0.46 set_set_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 product_prod_a_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.29/0.46 set_set_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 set_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Product____Type__Ounit,type,
% 0.29/0.46 product_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 set_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.46 nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_tf__a,type,
% 0.29/0.46 a: $tType ).
% 0.29/0.46
% 0.29/0.46 % Explicit typings (107)
% 0.29/0.46 thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
% 0.29/0.46 finite_Fpow_nat: set_nat > set_set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 finite361944167at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 finite702915405od_a_a: set_Pr1948701895od_a_a > set_se958357159od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 finite351630733od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_OFpow_001tf__a,type,
% 0.29/0.46 finite_Fpow_a: set_a > set_set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ocard_001tf__a,type,
% 0.29/0.46 finite_card_a: set_a > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.29/0.46 finite_finite_nat: set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 finite772653738at_nat: set_Pr1986765409at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 finite277291581at_nat: set_Pr1746169692at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 finite1297454819od_a_a: set_Pr339609346od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
% 0.29/0.46 finite1808550458_nat_a: set_Pr967348953_nat_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 finite48957584at_nat: set_Pr1490359111at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 finite256329232od_a_a: set_Pr1295299783od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J,type,
% 0.29/0.46 finite1837575485_a_nat: set_Pr894832732_a_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 0.29/0.46 finite1919032935_a_a_a: set_Pr1689873822_a_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
% 0.29/0.46 finite1743148308_a_nat: set_Pr548851891_a_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 finite942416723at_nat: set_Pr7688842at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 finite676513017od_a_a: set_Pr681306928od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 finite179568208od_a_a: set_Product_prod_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 finite2012248349et_nat: set_set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 finite1457549322at_nat: set_se1612935105at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 finite323969008od_a_a: set_se958357159od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 finite1145471536od_a_a: set_se1596668135od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 finite_finite_set_a: set_set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
% 0.29/0.46 finite_finite_a: set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 bot_bot_set_nat: set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 bot_bo2130386637at_nat: set_Pr1986765409at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 bot_bo2131659635od_a_a: set_Product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 bot_bot_set_a: set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.29/0.46 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
% 0.29/0.46 ord_le1347718902_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.46 ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.29/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 ord_le2084554594od_a_a: set_Pr339609346od_a_a > set_Pr339609346od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
% 0.29/0.46 ord_le344568633_nat_a: set_Pr967348953_nat_a > set_Pr967348953_nat_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 ord_le492294332_a_nat: set_Pr894832732_a_nat > set_Pr894832732_a_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 0.29/0.46 ord_le677315902_a_a_a: set_Pr1689873822_a_a_a > set_Pr1689873822_a_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 ord_le2073555219_a_nat: set_Pr548851891_a_nat > set_Pr548851891_a_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 ord_le1816232656od_a_a: set_Pr681306928od_a_a > set_Pr681306928od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Nat__Onat,type,
% 0.29/0.46 pair_p128415500ph_nat: pair_p1914262621t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 pair_p752841413at_nat: pair_p2041852168t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 pair_p159207083od_a_a: pair_p1593840546t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 pair_p374947051od_a_a: pair_p1765063010t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001tf__a,type,
% 0.29/0.46 pair_p1802376898raph_a: pair_p125712459t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Nat__Onat,type,
% 0.29/0.46 pair_p1027063983ms_nat: pair_p1914262621t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 pair_p56914274at_nat: pair_p2041852168t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 pair_p1906342088od_a_a: pair_p1593840546t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 pair_p504738056od_a_a: pair_p1765063010t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001tf__a,type,
% 0.29/0.46 pair_p1864019935ioms_a: pair_p125712459t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p69470259t_unit: pair_p1914262621t_unit > product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1984658862t_unit: pair_p1765063010t_unit > product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore_001tf__a_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1896222615t_unit: pair_p125712459t_unit > product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1167410509t_unit: set_nat > set_Pr1986765409at_nat > product_unit > pair_p1914262621t_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p398687508t_unit: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_unit > pair_p1765063010t_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext_001tf__a_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1621517565t_unit: set_a > set_Product_prod_a_a > product_unit > pair_p125712459t_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p715279805t_unit: pair_p1914262621t_unit > set_Pr1986765409at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p806300874t_unit: pair_p2041852168t_unit > set_Pr1490359111at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1559300324t_unit: pair_p1593840546t_unit > set_Pr1295299783od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1783210148t_unit: pair_p1765063010t_unit > set_Pr1948701895od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001tf__a_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p133601421t_unit: pair_p125712459t_unit > set_Product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1677060310t_unit: pair_p1914262621t_unit > set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p210955889t_unit: pair_p2041852168t_unit > set_Pr1986765409at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1652294923t_unit: pair_p1593840546t_unit > set_Pr1948701895od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p447552203t_unit: pair_p1765063010t_unit > set_Product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001tf__a_001t__Product____Type__Ounit,type,
% 0.29/0.46 pair_p1047056820t_unit: pair_p125712459t_unit > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__wf__digraph_001t__Nat__Onat,type,
% 0.29/0.46 pair_p1515597646ph_nat: pair_p1914262621t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__wf__digraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 pair_p646030121od_a_a: pair_p1765063010t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__wf__digraph_001tf__a,type,
% 0.29/0.46 pair_p68905728raph_a: pair_p125712459t_unit > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 0.29/0.46 product_Pair_a_a: a > a > product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 produc1182842125od_a_a: set_nat > ( nat > set_Product_prod_a_a ) > set_Pr339609346od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 product_Sigma_nat_a: set_nat > ( nat > set_a ) > set_Pr967348953_nat_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
% 0.29/0.46 produc931712687_a_nat: set_Product_prod_a_a > ( product_prod_a_a > set_nat ) > set_Pr894832732_a_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 produc304751368od_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
% 0.29/0.46 produc1282482655_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_a ) > set_Pr1689873822_a_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001tf__a_001t__Nat__Onat,type,
% 0.29/0.46 product_Sigma_a_nat: set_a > ( a > set_nat ) > set_Pr548851891_a_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 produc292491723at_nat: set_a > ( a > set_Pr1986765409at_nat ) > set_Pr7688842at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 produc520147185od_a_a: set_a > ( a > set_Product_prod_a_a ) > set_Pr681306928od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma_001tf__a_001tf__a,type,
% 0.29/0.46 product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001_Eo,type,
% 0.29/0.46 produc1833107820_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oproduct_001tf__a_001tf__a,type,
% 0.29/0.46 product_product_a_a: set_a > set_a > set_Product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.29/0.46 collect_nat: ( nat > $o ) > set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.46 collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.46 collec453062450od_a_a: ( set_Pr1948701895od_a_a > $o ) > set_se958357159od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 collect_set_a: ( set_a > $o ) > set_set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.29/0.46 collect_a: ( a > $o ) > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Nat__Onat,type,
% 0.29/0.46 member_nat: nat > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.46 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.46 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 member_set_nat: set_nat > set_set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.46 member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 member_set_a: set_a > set_set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001tf__a,type,
% 0.29/0.46 member_a: a > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_G,type,
% 0.29/0.46 g: pair_p125712459t_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_n,type,
% 0.29/0.46 n: nat ).
% 0.29/0.46
% 0.29/0.46 % Relevant facts (354)
% 0.29/0.46 thf(fact_0__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062,axiom,
% 0.29/0.46 ( finite179568208od_a_a
% 0.29/0.46 @ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
% 0.29/0.46 @ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>finite (pverts G \<times> pverts G)\<close>
% 0.29/0.46 thf(fact_1__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062,axiom,
% 0.29/0.46 ( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
% 0.29/0.46 @ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
% 0.29/0.46 @ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>parcs G \<subseteq> pverts G \<times> pverts G\<close>
% 0.29/0.46 thf(fact_2_pair__fin__digraph__axioms__def,axiom,
% 0.29/0.46 ( pair_p56914274at_nat
% 0.29/0.46 = ( ^ [G: pair_p2041852168t_unit] :
% 0.29/0.46 ( ( finite772653738at_nat @ ( pair_p210955889t_unit @ G ) )
% 0.29/0.46 & ( finite48957584at_nat @ ( pair_p806300874t_unit @ G ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms_def
% 0.29/0.46 thf(fact_3_pair__fin__digraph__axioms__def,axiom,
% 0.29/0.46 ( pair_p1906342088od_a_a
% 0.29/0.46 = ( ^ [G: pair_p1593840546t_unit] :
% 0.29/0.46 ( ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G ) )
% 0.29/0.46 & ( finite256329232od_a_a @ ( pair_p1559300324t_unit @ G ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms_def
% 0.29/0.46 thf(fact_4_pair__fin__digraph__axioms__def,axiom,
% 0.29/0.46 ( pair_p504738056od_a_a
% 0.29/0.46 = ( ^ [G: pair_p1765063010t_unit] :
% 0.29/0.46 ( ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G ) )
% 0.29/0.46 & ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms_def
% 0.29/0.46 thf(fact_5_pair__fin__digraph__axioms__def,axiom,
% 0.29/0.46 ( pair_p1027063983ms_nat
% 0.29/0.46 = ( ^ [G: pair_p1914262621t_unit] :
% 0.29/0.46 ( ( finite_finite_nat @ ( pair_p1677060310t_unit @ G ) )
% 0.29/0.46 & ( finite772653738at_nat @ ( pair_p715279805t_unit @ G ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms_def
% 0.29/0.46 thf(fact_6_pair__fin__digraph__axioms__def,axiom,
% 0.29/0.46 ( pair_p1864019935ioms_a
% 0.29/0.46 = ( ^ [G: pair_p125712459t_unit] :
% 0.29/0.46 ( ( finite_finite_a @ ( pair_p1047056820t_unit @ G ) )
% 0.29/0.46 & ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms_def
% 0.29/0.46 thf(fact_7_pair__fin__digraph__axioms_Ointro,axiom,
% 0.29/0.46 ! [G2: pair_p2041852168t_unit] :
% 0.29/0.46 ( ( finite772653738at_nat @ ( pair_p210955889t_unit @ G2 ) )
% 0.29/0.46 => ( ( finite48957584at_nat @ ( pair_p806300874t_unit @ G2 ) )
% 0.29/0.46 => ( pair_p56914274at_nat @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms.intro
% 0.29/0.46 thf(fact_8_pair__fin__digraph__axioms_Ointro,axiom,
% 0.29/0.46 ! [G2: pair_p1593840546t_unit] :
% 0.29/0.46 ( ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G2 ) )
% 0.29/0.46 => ( ( finite256329232od_a_a @ ( pair_p1559300324t_unit @ G2 ) )
% 0.29/0.46 => ( pair_p1906342088od_a_a @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms.intro
% 0.29/0.46 thf(fact_9_pair__fin__digraph__axioms_Ointro,axiom,
% 0.29/0.46 ! [G2: pair_p1765063010t_unit] :
% 0.29/0.46 ( ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G2 ) )
% 0.29/0.46 => ( ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G2 ) )
% 0.29/0.46 => ( pair_p504738056od_a_a @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms.intro
% 0.29/0.46 thf(fact_10_pair__fin__digraph__axioms_Ointro,axiom,
% 0.29/0.46 ! [G2: pair_p1914262621t_unit] :
% 0.29/0.46 ( ( finite_finite_nat @ ( pair_p1677060310t_unit @ G2 ) )
% 0.29/0.46 => ( ( finite772653738at_nat @ ( pair_p715279805t_unit @ G2 ) )
% 0.29/0.46 => ( pair_p1027063983ms_nat @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms.intro
% 0.29/0.46 thf(fact_11_pair__fin__digraph__axioms_Ointro,axiom,
% 0.29/0.46 ! [G2: pair_p125712459t_unit] :
% 0.29/0.46 ( ( finite_finite_a @ ( pair_p1047056820t_unit @ G2 ) )
% 0.29/0.46 => ( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G2 ) )
% 0.29/0.46 => ( pair_p1864019935ioms_a @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms.intro
% 0.29/0.46 thf(fact_12_pair__fin__digraph_Opair__finite__arcs,axiom,
% 0.29/0.46 ! [G2: pair_p1914262621t_unit] :
% 0.29/0.46 ( ( pair_p128415500ph_nat @ G2 )
% 0.29/0.46 => ( finite772653738at_nat @ ( pair_p715279805t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_arcs
% 0.29/0.46 thf(fact_13_pair__fin__digraph_Opair__finite__arcs,axiom,
% 0.29/0.46 ! [G2: pair_p1765063010t_unit] :
% 0.29/0.46 ( ( pair_p374947051od_a_a @ G2 )
% 0.29/0.46 => ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_arcs
% 0.29/0.46 thf(fact_14_pair__fin__digraph_Opair__finite__arcs,axiom,
% 0.29/0.46 ! [G2: pair_p125712459t_unit] :
% 0.29/0.46 ( ( pair_p1802376898raph_a @ G2 )
% 0.29/0.46 => ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_arcs
% 0.29/0.46 thf(fact_15_finite__subset,axiom,
% 0.29/0.46 ! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( ord_le841296385at_nat @ A @ B )
% 0.29/0.46 => ( ( finite772653738at_nat @ B )
% 0.29/0.46 => ( finite772653738at_nat @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_subset
% 0.29/0.46 thf(fact_16_finite__subset,axiom,
% 0.29/0.46 ! [A: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.29/0.46 ( ( ord_le456379495od_a_a @ A @ B )
% 0.29/0.46 => ( ( finite1664988688od_a_a @ B )
% 0.29/0.46 => ( finite1664988688od_a_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_subset
% 0.29/0.46 thf(fact_17_finite__subset,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( finite_finite_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_subset
% 0.29/0.46 thf(fact_18_finite__subset,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( finite_finite_nat @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_subset
% 0.29/0.46 thf(fact_19_finite__subset,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.46 => ( ( finite179568208od_a_a @ B )
% 0.29/0.46 => ( finite179568208od_a_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_subset
% 0.29/0.46 thf(fact_20_infinite__super,axiom,
% 0.29/0.46 ! [S: set_Pr1986765409at_nat,T: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( ord_le841296385at_nat @ S @ T )
% 0.29/0.46 => ( ~ ( finite772653738at_nat @ S )
% 0.29/0.46 => ~ ( finite772653738at_nat @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_super
% 0.29/0.46 thf(fact_21_infinite__super,axiom,
% 0.29/0.46 ! [S: set_Pr1948701895od_a_a,T: set_Pr1948701895od_a_a] :
% 0.29/0.46 ( ( ord_le456379495od_a_a @ S @ T )
% 0.29/0.46 => ( ~ ( finite1664988688od_a_a @ S )
% 0.29/0.46 => ~ ( finite1664988688od_a_a @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_super
% 0.29/0.46 thf(fact_22_infinite__super,axiom,
% 0.29/0.46 ! [S: set_a,T: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ S @ T )
% 0.29/0.46 => ( ~ ( finite_finite_a @ S )
% 0.29/0.46 => ~ ( finite_finite_a @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_super
% 0.29/0.46 thf(fact_23_infinite__super,axiom,
% 0.29/0.46 ! [S: set_nat,T: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ S @ T )
% 0.29/0.46 => ( ~ ( finite_finite_nat @ S )
% 0.29/0.46 => ~ ( finite_finite_nat @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_super
% 0.29/0.46 thf(fact_24_infinite__super,axiom,
% 0.29/0.46 ! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ S @ T )
% 0.29/0.46 => ( ~ ( finite179568208od_a_a @ S )
% 0.29/0.46 => ~ ( finite179568208od_a_a @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_super
% 0.29/0.46 thf(fact_25_rev__finite__subset,axiom,
% 0.29/0.46 ! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( finite772653738at_nat @ B )
% 0.29/0.46 => ( ( ord_le841296385at_nat @ A @ B )
% 0.29/0.46 => ( finite772653738at_nat @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_finite_subset
% 0.29/0.46 thf(fact_26_rev__finite__subset,axiom,
% 0.29/0.46 ! [B: set_Pr1948701895od_a_a,A: set_Pr1948701895od_a_a] :
% 0.29/0.46 ( ( finite1664988688od_a_a @ B )
% 0.29/0.46 => ( ( ord_le456379495od_a_a @ A @ B )
% 0.29/0.46 => ( finite1664988688od_a_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_finite_subset
% 0.29/0.46 thf(fact_27_rev__finite__subset,axiom,
% 0.29/0.46 ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite179568208od_a_a @ B )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.46 => ( finite179568208od_a_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_finite_subset
% 0.29/0.46 thf(fact_28_rev__finite__subset,axiom,
% 0.29/0.46 ! [B: set_nat,A: set_nat] :
% 0.29/0.46 ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.46 => ( finite_finite_nat @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_finite_subset
% 0.29/0.46 thf(fact_29_rev__finite__subset,axiom,
% 0.29/0.46 ! [B: set_a,A: set_a] :
% 0.29/0.46 ( ( finite_finite_a @ B )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.46 => ( finite_finite_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_finite_subset
% 0.29/0.46 thf(fact_30_finite__has__maximal2,axiom,
% 0.29/0.46 ! [A: set_se1596668135od_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite1145471536od_a_a @ A )
% 0.29/0.46 => ( ( member1838126896od_a_a @ A2 @ A )
% 0.29/0.46 => ? [X: set_Product_prod_a_a] :
% 0.29/0.46 ( ( member1838126896od_a_a @ X @ A )
% 0.29/0.46 & ( ord_le1824328871od_a_a @ A2 @ X )
% 0.29/0.46 & ! [Xa: set_Product_prod_a_a] :
% 0.29/0.46 ( ( member1838126896od_a_a @ Xa @ A )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ X @ Xa )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_maximal2
% 0.29/0.46 thf(fact_31_finite__has__maximal2,axiom,
% 0.29/0.46 ! [A: set_nat,A2: nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( member_nat @ A2 @ A )
% 0.29/0.46 => ? [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 & ( ord_less_eq_nat @ A2 @ X )
% 0.29/0.46 & ! [Xa: nat] :
% 0.29/0.46 ( ( member_nat @ Xa @ A )
% 0.29/0.46 => ( ( ord_less_eq_nat @ X @ Xa )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_maximal2
% 0.29/0.46 thf(fact_32_finite__has__maximal2,axiom,
% 0.29/0.46 ! [A: set_set_nat,A2: set_nat] :
% 0.29/0.46 ( ( finite2012248349et_nat @ A )
% 0.29/0.46 => ( ( member_set_nat @ A2 @ A )
% 0.29/0.46 => ? [X: set_nat] :
% 0.29/0.46 ( ( member_set_nat @ X @ A )
% 0.29/0.46 & ( ord_less_eq_set_nat @ A2 @ X )
% 0.29/0.46 & ! [Xa: set_nat] :
% 0.29/0.46 ( ( member_set_nat @ Xa @ A )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ X @ Xa )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_maximal2
% 0.29/0.46 thf(fact_33_finite__has__maximal2,axiom,
% 0.29/0.46 ! [A: set_set_a,A2: set_a] :
% 0.29/0.46 ( ( finite_finite_set_a @ A )
% 0.29/0.46 => ( ( member_set_a @ A2 @ A )
% 0.29/0.46 => ? [X: set_a] :
% 0.29/0.46 ( ( member_set_a @ X @ A )
% 0.29/0.46 & ( ord_less_eq_set_a @ A2 @ X )
% 0.29/0.46 & ! [Xa: set_a] :
% 0.29/0.46 ( ( member_set_a @ Xa @ A )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ X @ Xa )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_maximal2
% 0.29/0.46 thf(fact_34_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.29/0.46 ( ( ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
% 0.29/0.46 | ( finite179568208od_a_a @ ( collec645855634od_a_a @ Q ) ) )
% 0.29/0.46 => ( finite179568208od_a_a
% 0.29/0.46 @ ( collec645855634od_a_a
% 0.29/0.46 @ ^ [X2: product_prod_a_a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 & ( Q @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_conjI
% 0.29/0.46 thf(fact_35_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [P: a > $o,Q: a > $o] :
% 0.29/0.46 ( ( ( finite_finite_a @ ( collect_a @ P ) )
% 0.29/0.46 | ( finite_finite_a @ ( collect_a @ Q ) ) )
% 0.29/0.46 => ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 & ( Q @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_conjI
% 0.29/0.46 thf(fact_36_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.46 ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.29/0.46 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.29/0.46 => ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [X2: nat] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 & ( Q @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_conjI
% 0.29/0.46 thf(fact_37_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.29/0.46 ( ( ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
% 0.29/0.46 | ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) )
% 0.29/0.46 => ( finite772653738at_nat
% 0.29/0.46 @ ( collec7649004at_nat
% 0.29/0.46 @ ^ [X2: product_prod_nat_nat] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 & ( Q @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_conjI
% 0.29/0.46 thf(fact_38_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [P: produc1572603623od_a_a > $o,Q: produc1572603623od_a_a > $o] :
% 0.29/0.46 ( ( ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
% 0.29/0.46 | ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ Q ) ) )
% 0.29/0.46 => ( finite1664988688od_a_a
% 0.29/0.46 @ ( collec1635618130od_a_a
% 0.29/0.46 @ ^ [X2: produc1572603623od_a_a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 & ( Q @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_conjI
% 0.29/0.46 thf(fact_39_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.29/0.46 ( ( finite179568208od_a_a
% 0.29/0.46 @ ( collec645855634od_a_a
% 0.29/0.46 @ ^ [X2: product_prod_a_a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 | ( Q @ X2 ) ) ) )
% 0.29/0.46 = ( ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
% 0.29/0.46 & ( finite179568208od_a_a @ ( collec645855634od_a_a @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_disjI
% 0.29/0.46 thf(fact_40_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P: a > $o,Q: a > $o] :
% 0.29/0.46 ( ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 | ( Q @ X2 ) ) ) )
% 0.29/0.46 = ( ( finite_finite_a @ ( collect_a @ P ) )
% 0.29/0.46 & ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_disjI
% 0.29/0.46 thf(fact_41_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [X2: nat] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 | ( Q @ X2 ) ) ) )
% 0.29/0.46 = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.29/0.46 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_disjI
% 0.29/0.46 thf(fact_42_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.29/0.46 ( ( finite772653738at_nat
% 0.29/0.46 @ ( collec7649004at_nat
% 0.29/0.46 @ ^ [X2: product_prod_nat_nat] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 | ( Q @ X2 ) ) ) )
% 0.29/0.46 = ( ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
% 0.29/0.46 & ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_disjI
% 0.29/0.46 thf(fact_43_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P: produc1572603623od_a_a > $o,Q: produc1572603623od_a_a > $o] :
% 0.29/0.46 ( ( finite1664988688od_a_a
% 0.29/0.46 @ ( collec1635618130od_a_a
% 0.29/0.46 @ ^ [X2: produc1572603623od_a_a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 | ( Q @ X2 ) ) ) )
% 0.29/0.46 = ( ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
% 0.29/0.46 & ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_disjI
% 0.29/0.46 thf(fact_44_finite__Collect__subsets,axiom,
% 0.29/0.46 ! [A: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( finite772653738at_nat @ A )
% 0.29/0.46 => ( finite1457549322at_nat
% 0.29/0.46 @ ( collec1606769740at_nat
% 0.29/0.46 @ ^ [B2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ B2 @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_subsets
% 0.29/0.46 thf(fact_45_finite__Collect__subsets,axiom,
% 0.29/0.46 ! [A: set_Pr1948701895od_a_a] :
% 0.29/0.46 ( ( finite1664988688od_a_a @ A )
% 0.29/0.46 => ( finite323969008od_a_a
% 0.29/0.46 @ ( collec453062450od_a_a
% 0.29/0.46 @ ^ [B2: set_Pr1948701895od_a_a] : ( ord_le456379495od_a_a @ B2 @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_subsets
% 0.29/0.46 thf(fact_46_finite__Collect__subsets,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( finite1145471536od_a_a
% 0.29/0.46 @ ( collec183727474od_a_a
% 0.29/0.46 @ ^ [B2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ B2 @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_subsets
% 0.29/0.46 thf(fact_47_finite__Collect__subsets,axiom,
% 0.29/0.46 ! [A: set_nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( finite2012248349et_nat
% 0.29/0.46 @ ( collect_set_nat
% 0.29/0.46 @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_subsets
% 0.29/0.46 thf(fact_48_finite__Collect__subsets,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( finite_finite_set_a
% 0.29/0.46 @ ( collect_set_a
% 0.29/0.46 @ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_subsets
% 0.29/0.46 thf(fact_49_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_a,B: a > set_nat] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ! [A3: a] :
% 0.29/0.46 ( ( member_a @ A3 @ A )
% 0.29/0.46 => ( finite_finite_nat @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite1743148308_a_nat @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_50_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_nat,B: nat > set_a] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ! [A3: nat] :
% 0.29/0.46 ( ( member_nat @ A3 @ A )
% 0.29/0.46 => ( finite_finite_a @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite1808550458_nat_a @ ( product_Sigma_nat_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_51_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_nat,B: nat > set_nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ! [A3: nat] :
% 0.29/0.46 ( ( member_nat @ A3 @ A )
% 0.29/0.46 => ( finite_finite_nat @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite772653738at_nat @ ( produc45129834at_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_52_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_a,B: a > set_a] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ! [A3: a] :
% 0.29/0.46 ( ( member_a @ A3 @ A )
% 0.29/0.46 => ( finite_finite_a @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite179568208od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_53_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 0.29/0.46 ( ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ! [A3: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ A3 @ A )
% 0.29/0.46 => ( finite_finite_a @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite1919032935_a_a_a @ ( produc1282482655_a_a_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_54_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: product_prod_a_a > set_nat] :
% 0.29/0.46 ( ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ! [A3: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ A3 @ A )
% 0.29/0.46 => ( finite_finite_nat @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite1837575485_a_nat @ ( produc931712687_a_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_55_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_a,B: a > set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ! [A3: a] :
% 0.29/0.46 ( ( member_a @ A3 @ A )
% 0.29/0.46 => ( finite179568208od_a_a @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite676513017od_a_a @ ( produc520147185od_a_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_56_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_a,B: a > set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ! [A3: a] :
% 0.29/0.46 ( ( member_a @ A3 @ A )
% 0.29/0.46 => ( finite772653738at_nat @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite942416723at_nat @ ( produc292491723at_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_57_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_nat,B: nat > set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ! [A3: nat] :
% 0.29/0.46 ( ( member_nat @ A3 @ A )
% 0.29/0.46 => ( finite179568208od_a_a @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite1297454819od_a_a @ ( produc1182842125od_a_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_58_finite__SigmaI,axiom,
% 0.29/0.46 ! [A: set_nat,B: nat > set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ! [A3: nat] :
% 0.29/0.46 ( ( member_nat @ A3 @ A )
% 0.29/0.46 => ( finite772653738at_nat @ ( B @ A3 ) ) )
% 0.29/0.46 => ( finite277291581at_nat @ ( produc894163943at_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_59_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [G2: pair_p125712459t_unit] :
% 0.29/0.46 ( ( pair_p1802376898raph_a @ G2 )
% 0.29/0.46 => ( pair_p1864019935ioms_a @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.axioms(2)
% 0.29/0.46 thf(fact_60_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [G2: pair_p1914262621t_unit] :
% 0.29/0.46 ( ( pair_p128415500ph_nat @ G2 )
% 0.29/0.46 => ( pair_p1027063983ms_nat @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.axioms(2)
% 0.29/0.46 thf(fact_61_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [G2: pair_p1765063010t_unit] :
% 0.29/0.46 ( ( pair_p374947051od_a_a @ G2 )
% 0.29/0.46 => ( pair_p504738056od_a_a @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.axioms(2)
% 0.29/0.46 thf(fact_62_not__finite__existsD,axiom,
% 0.29/0.46 ! [P: product_prod_a_a > $o] :
% 0.29/0.46 ( ~ ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
% 0.29/0.46 => ? [X_1: product_prod_a_a] : ( P @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_finite_existsD
% 0.29/0.46 thf(fact_63_not__finite__existsD,axiom,
% 0.29/0.46 ! [P: a > $o] :
% 0.29/0.46 ( ~ ( finite_finite_a @ ( collect_a @ P ) )
% 0.29/0.46 => ? [X_1: a] : ( P @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_finite_existsD
% 0.29/0.46 thf(fact_64_not__finite__existsD,axiom,
% 0.29/0.46 ! [P: nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.29/0.46 => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_finite_existsD
% 0.29/0.46 thf(fact_65_not__finite__existsD,axiom,
% 0.29/0.46 ! [P: product_prod_nat_nat > $o] :
% 0.29/0.46 ( ~ ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
% 0.29/0.46 => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_finite_existsD
% 0.29/0.46 thf(fact_66_not__finite__existsD,axiom,
% 0.29/0.46 ! [P: produc1572603623od_a_a > $o] :
% 0.29/0.46 ( ~ ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
% 0.29/0.46 => ? [X_1: produc1572603623od_a_a] : ( P @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_finite_existsD
% 0.29/0.46 thf(fact_67_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,R: a > a > $o] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ? [Xa: a] :
% 0.29/0.46 ( ( member_a @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: a] :
% 0.29/0.46 ( ( member_a @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [A4: a] :
% 0.29/0.46 ( ( member_a @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_68_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_a,B: set_nat,R: a > nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ? [Xa: nat] :
% 0.29/0.46 ( ( member_nat @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [A4: a] :
% 0.29/0.46 ( ( member_a @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_69_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_a,R: nat > a > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ? [Xa: a] :
% 0.29/0.46 ( ( member_a @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: a] :
% 0.29/0.46 ( ( member_a @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A4: nat] :
% 0.29/0.46 ( ( member_nat @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_70_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat,R: nat > nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ? [Xa: nat] :
% 0.29/0.46 ( ( member_nat @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A4: nat] :
% 0.29/0.46 ( ( member_nat @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_71_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_a,R: product_prod_a_a > a > $o] :
% 0.29/0.46 ( ~ ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ A )
% 0.29/0.46 => ? [Xa: a] :
% 0.29/0.46 ( ( member_a @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: a] :
% 0.29/0.46 ( ( member_a @ X @ B )
% 0.29/0.46 & ~ ( finite179568208od_a_a
% 0.29/0.46 @ ( collec645855634od_a_a
% 0.29/0.46 @ ^ [A4: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_72_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_nat,R: product_prod_a_a > nat > $o] :
% 0.29/0.46 ( ~ ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ A )
% 0.29/0.46 => ? [Xa: nat] :
% 0.29/0.46 ( ( member_nat @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ B )
% 0.29/0.46 & ~ ( finite179568208od_a_a
% 0.29/0.46 @ ( collec645855634od_a_a
% 0.29/0.46 @ ^ [A4: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_73_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_a,B: set_Product_prod_a_a,R: a > product_prod_a_a > $o] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite179568208od_a_a @ B )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ? [Xa: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [A4: a] :
% 0.29/0.46 ( ( member_a @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_74_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_a,B: set_Pr1986765409at_nat,R: a > product_prod_nat_nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite772653738at_nat @ B )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ? [Xa: product_prod_nat_nat] :
% 0.29/0.46 ( ( member701585322at_nat @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: product_prod_nat_nat] :
% 0.29/0.46 ( ( member701585322at_nat @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [A4: a] :
% 0.29/0.46 ( ( member_a @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_75_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_Product_prod_a_a,R: nat > product_prod_a_a > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite179568208od_a_a @ B )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ? [Xa: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A4: nat] :
% 0.29/0.46 ( ( member_nat @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_76_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_Pr1986765409at_nat,R: nat > product_prod_nat_nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite772653738at_nat @ B )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ? [Xa: product_prod_nat_nat] :
% 0.29/0.46 ( ( member701585322at_nat @ Xa @ B )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: product_prod_nat_nat] :
% 0.29/0.46 ( ( member701585322at_nat @ X @ B )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A4: nat] :
% 0.29/0.46 ( ( member_nat @ A4 @ A )
% 0.29/0.46 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_77_pair__fin__digraph_Opair__fin__digraph,axiom,
% 0.29/0.46 ! [G2: pair_p125712459t_unit] :
% 0.29/0.46 ( ( pair_p1802376898raph_a @ G2 )
% 0.29/0.46 => ( pair_p1802376898raph_a @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_fin_digraph
% 0.29/0.46 thf(fact_78_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_nat] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( finite1743148308_a_nat
% 0.29/0.46 @ ( product_Sigma_a_nat @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_79_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_a] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( finite1808550458_nat_a
% 0.29/0.46 @ ( product_Sigma_nat_a @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_80_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( finite772653738at_nat
% 0.29/0.46 @ ( produc45129834at_nat @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_81_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( finite179568208od_a_a
% 0.29/0.46 @ ( product_Sigma_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_82_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_a] :
% 0.29/0.46 ( ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ( finite_finite_a @ B )
% 0.29/0.46 => ( finite1919032935_a_a_a
% 0.29/0.46 @ ( produc1282482655_a_a_a @ A
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_83_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_nat] :
% 0.29/0.46 ( ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ( finite_finite_nat @ B )
% 0.29/0.46 => ( finite1837575485_a_nat
% 0.29/0.46 @ ( produc931712687_a_nat @ A
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_84_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite179568208od_a_a @ B )
% 0.29/0.46 => ( finite676513017od_a_a
% 0.29/0.46 @ ( produc520147185od_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_85_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( finite_finite_a @ A )
% 0.29/0.46 => ( ( finite772653738at_nat @ B )
% 0.29/0.46 => ( finite942416723at_nat
% 0.29/0.46 @ ( produc292491723at_nat @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_86_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite179568208od_a_a @ B )
% 0.29/0.46 => ( finite1297454819od_a_a
% 0.29/0.46 @ ( produc1182842125od_a_a @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_87_finite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( finite772653738at_nat @ B )
% 0.29/0.46 => ( finite277291581at_nat
% 0.29/0.46 @ ( produc894163943at_nat @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_88_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.29/0.46 ! [G2: pair_p2041852168t_unit] :
% 0.29/0.46 ( ( pair_p752841413at_nat @ G2 )
% 0.29/0.46 => ( finite772653738at_nat @ ( pair_p210955889t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_verts
% 0.29/0.46 thf(fact_89_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.29/0.46 ! [G2: pair_p1593840546t_unit] :
% 0.29/0.46 ( ( pair_p159207083od_a_a @ G2 )
% 0.29/0.46 => ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_verts
% 0.29/0.46 thf(fact_90_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.29/0.46 ! [G2: pair_p125712459t_unit] :
% 0.29/0.46 ( ( pair_p1802376898raph_a @ G2 )
% 0.29/0.46 => ( finite_finite_a @ ( pair_p1047056820t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_verts
% 0.29/0.46 thf(fact_91_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.29/0.46 ! [G2: pair_p1914262621t_unit] :
% 0.29/0.46 ( ( pair_p128415500ph_nat @ G2 )
% 0.29/0.46 => ( finite_finite_nat @ ( pair_p1677060310t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_verts
% 0.29/0.46 thf(fact_92_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.29/0.46 ! [G2: pair_p1765063010t_unit] :
% 0.29/0.46 ( ( pair_p374947051od_a_a @ G2 )
% 0.29/0.46 => ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_verts
% 0.29/0.46 thf(fact_93_finite__has__minimal2,axiom,
% 0.29/0.46 ! [A: set_se1596668135od_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( finite1145471536od_a_a @ A )
% 0.29/0.46 => ( ( member1838126896od_a_a @ A2 @ A )
% 0.29/0.46 => ? [X: set_Product_prod_a_a] :
% 0.29/0.46 ( ( member1838126896od_a_a @ X @ A )
% 0.29/0.46 & ( ord_le1824328871od_a_a @ X @ A2 )
% 0.29/0.46 & ! [Xa: set_Product_prod_a_a] :
% 0.29/0.46 ( ( member1838126896od_a_a @ Xa @ A )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ Xa @ X )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_minimal2
% 0.29/0.46 thf(fact_94_finite__has__minimal2,axiom,
% 0.29/0.46 ! [A: set_nat,A2: nat] :
% 0.29/0.46 ( ( finite_finite_nat @ A )
% 0.29/0.46 => ( ( member_nat @ A2 @ A )
% 0.29/0.46 => ? [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 & ( ord_less_eq_nat @ X @ A2 )
% 0.29/0.46 & ! [Xa: nat] :
% 0.29/0.46 ( ( member_nat @ Xa @ A )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Xa @ X )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_minimal2
% 0.29/0.46 thf(fact_95_finite__has__minimal2,axiom,
% 0.29/0.46 ! [A: set_set_nat,A2: set_nat] :
% 0.29/0.46 ( ( finite2012248349et_nat @ A )
% 0.29/0.46 => ( ( member_set_nat @ A2 @ A )
% 0.29/0.46 => ? [X: set_nat] :
% 0.29/0.46 ( ( member_set_nat @ X @ A )
% 0.29/0.46 & ( ord_less_eq_set_nat @ X @ A2 )
% 0.29/0.46 & ! [Xa: set_nat] :
% 0.29/0.46 ( ( member_set_nat @ Xa @ A )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ Xa @ X )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_minimal2
% 0.29/0.46 thf(fact_96_finite__has__minimal2,axiom,
% 0.29/0.46 ! [A: set_set_a,A2: set_a] :
% 0.29/0.46 ( ( finite_finite_set_a @ A )
% 0.29/0.46 => ( ( member_set_a @ A2 @ A )
% 0.29/0.46 => ? [X: set_a] :
% 0.29/0.46 ( ( member_set_a @ X @ A )
% 0.29/0.46 & ( ord_less_eq_set_a @ X @ A2 )
% 0.29/0.46 & ! [Xa: set_a] :
% 0.29/0.46 ( ( member_set_a @ Xa @ A )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ Xa @ X )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_minimal2
% 0.29/0.46 thf(fact_97_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_nat] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ~ ( finite_finite_nat @ B )
% 0.29/0.46 => ~ ( finite1743148308_a_nat
% 0.29/0.46 @ ( product_Sigma_a_nat @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_98_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_a] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ~ ( finite_finite_a @ B )
% 0.29/0.46 => ~ ( finite1808550458_nat_a
% 0.29/0.46 @ ( product_Sigma_nat_a @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_99_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ~ ( finite_finite_nat @ B )
% 0.29/0.46 => ~ ( finite772653738at_nat
% 0.29/0.46 @ ( produc45129834at_nat @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_100_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ~ ( finite_finite_a @ B )
% 0.29/0.46 => ~ ( finite179568208od_a_a
% 0.29/0.46 @ ( product_Sigma_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_101_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_a] :
% 0.29/0.46 ( ~ ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ~ ( finite_finite_a @ B )
% 0.29/0.46 => ~ ( finite1919032935_a_a_a
% 0.29/0.46 @ ( produc1282482655_a_a_a @ A
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_102_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_nat] :
% 0.29/0.46 ( ~ ( finite179568208od_a_a @ A )
% 0.29/0.46 => ( ~ ( finite_finite_nat @ B )
% 0.29/0.46 => ~ ( finite1837575485_a_nat
% 0.29/0.46 @ ( produc931712687_a_nat @ A
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_103_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ~ ( finite179568208od_a_a @ B )
% 0.29/0.46 => ~ ( finite676513017od_a_a
% 0.29/0.46 @ ( produc520147185od_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_104_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_a,B: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ~ ( finite_finite_a @ A )
% 0.29/0.46 => ( ~ ( finite772653738at_nat @ B )
% 0.29/0.46 => ~ ( finite942416723at_nat
% 0.29/0.46 @ ( produc292491723at_nat @ A
% 0.29/0.46 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_105_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ~ ( finite179568208od_a_a @ B )
% 0.29/0.46 => ~ ( finite1297454819od_a_a
% 0.29/0.46 @ ( produc1182842125od_a_a @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_106_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_Pr1986765409at_nat] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A )
% 0.29/0.46 => ( ~ ( finite772653738at_nat @ B )
% 0.29/0.46 => ~ ( finite277291581at_nat
% 0.29/0.46 @ ( produc894163943at_nat @ A
% 0.29/0.46 @ ^ [Uu: nat] : B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_107_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X3 @ C )
% 0.29/0.46 => ( ( ord_le456379495od_a_a
% 0.29/0.46 @ ( produc304751368od_a_a @ A
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : C )
% 0.29/0.46 @ ( produc304751368od_a_a @ B
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : C ) )
% 0.29/0.46 = ( ord_le1824328871od_a_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_108_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: nat,C: set_nat,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( member_nat @ X3 @ C )
% 0.29/0.46 => ( ( ord_le492294332_a_nat
% 0.29/0.46 @ ( produc931712687_a_nat @ A
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : C )
% 0.29/0.46 @ ( produc931712687_a_nat @ B
% 0.29/0.46 @ ^ [Uu: product_prod_a_a] : C ) )
% 0.29/0.46 = ( ord_le1824328871od_a_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_109_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( member449909584od_a_a @ X3 @ C )
% 0.29/0.46 => ( ( ord_le2084554594od_a_a
% 0.29/0.46 @ ( produc1182842125od_a_a @ A
% 0.29/0.46 @ ^ [Uu: nat] : C )
% 0.29/0.46 @ ( produc1182842125od_a_a @ B
% 0.29/0.46 @ ^ [Uu: nat] : C ) )
% 0.29/0.46 = ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_110_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: nat,C: set_nat,A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( member_nat @ X3 @ C )
% 0.29/0.46 => ( ( ord_le841296385at_nat
% 0.29/0.46 @ ( produc45129834at_nat @ A
% 0.29/0.46 @ ^ [Uu: nat] : C )
% 0.29/0.46 @ ( produc45129834at_nat @ B
% 0.29/0.46 @ ^ [Uu: nat] : C ) )
% 0.29/0.46 = ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_111_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X3 @ C )
% 0.29/0.46 => ( ( ord_le1816232656od_a_a
% 0.29/0.46 @ ( produc520147185od_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : C )
% 0.29/0.46 @ ( produc520147185od_a_a @ B
% 0.29/0.46 @ ^ [Uu: a] : C ) )
% 0.29/0.46 = ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_112_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: nat,C: set_nat,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_nat @ X3 @ C )
% 0.29/0.46 => ( ( ord_le2073555219_a_nat
% 0.29/0.46 @ ( product_Sigma_a_nat @ A
% 0.29/0.46 @ ^ [Uu: a] : C )
% 0.29/0.46 @ ( product_Sigma_a_nat @ B
% 0.29/0.46 @ ^ [Uu: a] : C ) )
% 0.29/0.46 = ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_113_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [X3: a,C: set_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ X3 @ C )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a
% 0.29/0.46 @ ( product_Sigma_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : C )
% 0.29/0.46 @ ( product_Sigma_a_a @ B
% 0.29/0.46 @ ^ [Uu: a] : C ) )
% 0.29/0.46 = ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_114_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a,D: product_prod_a_a > set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ C )
% 0.29/0.46 => ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ A )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le456379495od_a_a @ ( produc304751368od_a_a @ A @ B ) @ ( produc304751368od_a_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_115_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_nat,D: product_prod_a_a > set_nat] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ C )
% 0.29/0.46 => ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ A )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le492294332_a_nat @ ( produc931712687_a_nat @ A @ B ) @ ( produc931712687_a_nat @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_116_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_a,D: product_prod_a_a > set_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ C )
% 0.29/0.46 => ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ A )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le677315902_a_a_a @ ( produc1282482655_a_a_a @ A @ B ) @ ( produc1282482655_a_a_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_117_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_nat,C: set_nat,B: nat > set_Product_prod_a_a,D: nat > set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A @ C )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le2084554594od_a_a @ ( produc1182842125od_a_a @ A @ B ) @ ( produc1182842125od_a_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_118_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_nat,C: set_nat,B: nat > set_nat,D: nat > set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A @ C )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le841296385at_nat @ ( produc45129834at_nat @ A @ B ) @ ( produc45129834at_nat @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_119_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_nat,C: set_nat,B: nat > set_a,D: nat > set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A @ C )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le344568633_nat_a @ ( product_Sigma_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_120_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_a,C: set_a,B: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A @ C )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le1816232656od_a_a @ ( produc520147185od_a_a @ A @ B ) @ ( produc520147185od_a_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_121_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_a,C: set_a,B: a > set_nat,D: a > set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A @ C )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le2073555219_a_nat @ ( product_Sigma_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_122_Sigma__mono,axiom,
% 0.29/0.46 ! [A: set_a,C: set_a,B: a > set_a,D: a > set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A @ C )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ ( product_Sigma_a_a @ A @ B ) @ ( product_Sigma_a_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_123_subsetI,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X @ A )
% 0.29/0.46 => ( member449909584od_a_a @ X @ B ) )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_124_subsetI,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ! [X: nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ( member_nat @ X @ B ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_125_subsetI,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ( member_a @ X @ B ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_126_subset__antisym,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ B @ A )
% 0.29/0.46 => ( A = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_127_subset__antisym,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B @ A )
% 0.29/0.46 => ( A = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_128_subset__antisym,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.46 => ( A = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_129_order__refl,axiom,
% 0.29/0.46 ! [X3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ X3 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_130_order__refl,axiom,
% 0.29/0.46 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_131_order__refl,axiom,
% 0.29/0.46 ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_132_order__refl,axiom,
% 0.29/0.46 ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_133_pair__pre__digraph_Oequality,axiom,
% 0.29/0.46 ! [R2: pair_p125712459t_unit,R3: pair_p125712459t_unit] :
% 0.29/0.46 ( ( ( pair_p1047056820t_unit @ R2 )
% 0.29/0.46 = ( pair_p1047056820t_unit @ R3 ) )
% 0.29/0.46 => ( ( ( pair_p133601421t_unit @ R2 )
% 0.29/0.46 = ( pair_p133601421t_unit @ R3 ) )
% 0.29/0.46 => ( ( ( pair_p1896222615t_unit @ R2 )
% 0.29/0.46 = ( pair_p1896222615t_unit @ R3 ) )
% 0.29/0.46 => ( R2 = R3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.equality
% 0.29/0.46 thf(fact_134_pair__pre__digraph_Oequality,axiom,
% 0.29/0.46 ! [R2: pair_p1914262621t_unit,R3: pair_p1914262621t_unit] :
% 0.29/0.46 ( ( ( pair_p1677060310t_unit @ R2 )
% 0.29/0.46 = ( pair_p1677060310t_unit @ R3 ) )
% 0.29/0.46 => ( ( ( pair_p715279805t_unit @ R2 )
% 0.29/0.46 = ( pair_p715279805t_unit @ R3 ) )
% 0.29/0.46 => ( ( ( pair_p69470259t_unit @ R2 )
% 0.29/0.46 = ( pair_p69470259t_unit @ R3 ) )
% 0.29/0.46 => ( R2 = R3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.equality
% 0.29/0.46 thf(fact_135_pair__pre__digraph_Oequality,axiom,
% 0.29/0.46 ! [R2: pair_p1765063010t_unit,R3: pair_p1765063010t_unit] :
% 0.29/0.46 ( ( ( pair_p447552203t_unit @ R2 )
% 0.29/0.46 = ( pair_p447552203t_unit @ R3 ) )
% 0.29/0.46 => ( ( ( pair_p1783210148t_unit @ R2 )
% 0.29/0.46 = ( pair_p1783210148t_unit @ R3 ) )
% 0.29/0.46 => ( ( ( pair_p1984658862t_unit @ R2 )
% 0.29/0.46 = ( pair_p1984658862t_unit @ R3 ) )
% 0.29/0.46 => ( R2 = R3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.equality
% 0.29/0.46 thf(fact_136_Sigma__cong,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C: a > set_a,D: a > set_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ! [X: a] :
% 0.29/0.46 ( ( member_a @ X @ B )
% 0.29/0.46 => ( ( C @ X )
% 0.29/0.46 = ( D @ X ) ) )
% 0.29/0.46 => ( ( product_Sigma_a_a @ A @ C )
% 0.29/0.46 = ( product_Sigma_a_a @ B @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_cong
% 0.29/0.46 thf(fact_137_Times__eq__cancel2,axiom,
% 0.29/0.46 ! [X3: a,C: set_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ X3 @ C )
% 0.29/0.46 => ( ( ( product_Sigma_a_a @ A
% 0.29/0.46 @ ^ [Uu: a] : C )
% 0.29/0.46 = ( product_Sigma_a_a @ B
% 0.29/0.46 @ ^ [Uu: a] : C ) )
% 0.29/0.46 = ( A = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_eq_cancel2
% 0.29/0.46 thf(fact_138_Collect__subset,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 0.29/0.46 ( ord_le1824328871od_a_a
% 0.29/0.46 @ ( collec645855634od_a_a
% 0.29/0.46 @ ^ [X2: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X2 @ A )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 @ A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_subset
% 0.29/0.46 thf(fact_139_Collect__subset,axiom,
% 0.29/0.46 ! [A: set_nat,P: nat > $o] :
% 0.29/0.46 ( ord_less_eq_set_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [X2: nat] :
% 0.29/0.46 ( ( member_nat @ X2 @ A )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 @ A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_subset
% 0.29/0.46 thf(fact_140_Collect__subset,axiom,
% 0.29/0.46 ! [A: set_a,P: a > $o] :
% 0.29/0.46 ( ord_less_eq_set_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A )
% 0.29/0.46 & ( P @ X2 ) ) )
% 0.29/0.46 @ A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_subset
% 0.29/0.46 thf(fact_141_less__eq__set__def,axiom,
% 0.29/0.46 ( ord_le1824328871od_a_a
% 0.29/0.46 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.29/0.46 ( ord_le1347718902_a_a_o
% 0.29/0.46 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ A5 )
% 0.29/0.46 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_set_def
% 0.29/0.46 thf(fact_142_less__eq__set__def,axiom,
% 0.29/0.46 ( ord_less_eq_set_nat
% 0.29/0.46 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.29/0.46 ( ord_less_eq_nat_o
% 0.29/0.46 @ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
% 0.29/0.46 @ ^ [X2: nat] : ( member_nat @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_set_def
% 0.29/0.46 thf(fact_143_less__eq__set__def,axiom,
% 0.29/0.46 ( ord_less_eq_set_a
% 0.29/0.46 = ( ^ [A5: set_a,B2: set_a] :
% 0.29/0.46 ( ord_less_eq_a_o
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ A5 )
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_set_def
% 0.29/0.46 thf(fact_144_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_145_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B3: nat,A2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_146_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B3: set_nat,A2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_147_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B3: set_a,A2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_148_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ B4 @ A4 )
% 0.29/0.46 & ( ord_le1824328871od_a_a @ A4 @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_149_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.29/0.46 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_150_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 0.29/0.46 & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_151_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: set_a,B4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ B4 @ A4 )
% 0.29/0.46 & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_152_dual__order_Otrans,axiom,
% 0.29/0.46 ! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ C2 @ B3 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ C2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_153_dual__order_Otrans,axiom,
% 0.29/0.46 ! [B3: nat,A2: nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ C2 @ B3 )
% 0.29/0.46 => ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_154_dual__order_Otrans,axiom,
% 0.29/0.46 ! [B3: set_nat,A2: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ C2 @ B3 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_155_dual__order_Otrans,axiom,
% 0.29/0.46 ! [B3: set_a,A2: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ C2 @ B3 )
% 0.29/0.46 => ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_156_linorder__wlog,axiom,
% 0.29/0.46 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 0.29/0.46 ( ! [A3: nat,B5: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A3 @ B5 )
% 0.29/0.46 => ( P @ A3 @ B5 ) )
% 0.29/0.46 => ( ! [A3: nat,B5: nat] :
% 0.29/0.46 ( ( P @ B5 @ A3 )
% 0.29/0.46 => ( P @ A3 @ B5 ) )
% 0.29/0.46 => ( P @ A2 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_157_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_158_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_159_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_160_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_161_order__trans,axiom,
% 0.29/0.46 ! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ Y2 @ Z2 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_162_order__trans,axiom,
% 0.29/0.46 ! [X3: nat,Y2: nat,Z2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 0.29/0.46 => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_163_order__trans,axiom,
% 0.29/0.46 ! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_164_order__trans,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ Y2 @ Z2 )
% 0.29/0.46 => ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_165_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ B3 @ A2 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_166_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_167_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_168_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_169_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 0.29/0.46 => ( ( B3 = C2 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_170_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( B3 = C2 )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_171_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( B3 = C2 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_172_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( B3 = C2 )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_173_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.29/0.46 ( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
% 0.29/0.46 = ( P @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_174_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A2: nat,P: nat > $o] :
% 0.29/0.46 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.29/0.46 = ( P @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_175_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a] :
% 0.29/0.46 ( ( collec645855634od_a_a
% 0.29/0.46 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ A ) )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_176_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: set_nat] :
% 0.29/0.46 ( ( collect_nat
% 0.29/0.46 @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_177_Collect__cong,axiom,
% 0.29/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.29/0.46 ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 = ( Q @ X ) )
% 0.29/0.46 => ( ( collec645855634od_a_a @ P )
% 0.29/0.46 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_178_Collect__cong,axiom,
% 0.29/0.46 ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.46 ( ! [X: nat] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 = ( Q @ X ) )
% 0.29/0.46 => ( ( collect_nat @ P )
% 0.29/0.46 = ( collect_nat @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_179_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( A2 = B3 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ B3 @ C2 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_180_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( A2 = B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_181_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( A2 = B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_182_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( A2 = B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_183_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A4 @ B4 )
% 0.29/0.46 & ( ord_le1824328871od_a_a @ B4 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_184_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.29/0.46 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_185_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.29/0.46 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_186_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A4: set_a,B4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 0.29/0.46 & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_187_antisym__conv,axiom,
% 0.29/0.46 ! [Y2: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ Y2 @ X3 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
% 0.29/0.46 = ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_188_antisym__conv,axiom,
% 0.29/0.46 ! [Y2: nat,X3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.46 = ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_189_antisym__conv,axiom,
% 0.29/0.46 ! [Y2: set_nat,X3: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ Y2 @ X3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ X3 @ Y2 )
% 0.29/0.46 = ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_190_antisym__conv,axiom,
% 0.29/0.46 ! [Y2: set_a,X3: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ Y2 @ X3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 0.29/0.46 = ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_191_le__cases3,axiom,
% 0.29/0.46 ! [X3: nat,Y2: nat,Z2: nat] :
% 0.29/0.46 ( ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
% 0.29/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.29/0.46 => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
% 0.29/0.46 => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
% 0.29/0.46 => ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
% 0.29/0.46 => ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
% 0.29/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 0.29/0.46 => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
% 0.29/0.46 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_192_order_Otrans,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ B3 @ C2 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_193_order_Otrans,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_194_order_Otrans,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_195_order_Otrans,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_196_le__cases,axiom,
% 0.29/0.46 ! [X3: nat,Y2: nat] :
% 0.29/0.46 ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.46 => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_197_eq__refl,axiom,
% 0.29/0.46 ! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( X3 = Y2 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_198_eq__refl,axiom,
% 0.29/0.46 ! [X3: nat,Y2: nat] :
% 0.29/0.46 ( ( X3 = Y2 )
% 0.29/0.46 => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_199_eq__refl,axiom,
% 0.29/0.46 ! [X3: set_nat,Y2: set_nat] :
% 0.29/0.46 ( ( X3 = Y2 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_200_eq__refl,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( X3 = Y2 )
% 0.29/0.46 => ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_201_linear,axiom,
% 0.29/0.46 ! [X3: nat,Y2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.46 | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_202_antisym,axiom,
% 0.29/0.46 ! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ Y2 @ X3 )
% 0.29/0.46 => ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_203_antisym,axiom,
% 0.29/0.46 ! [X3: nat,Y2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.29/0.46 => ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_204_antisym,axiom,
% 0.29/0.46 ! [X3: set_nat,Y2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ Y2 @ X3 )
% 0.29/0.46 => ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_205_antisym,axiom,
% 0.29/0.46 ! [X3: set_a,Y2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ Y2 @ X3 )
% 0.29/0.46 => ( X3 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_206_eq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ X2 @ Y3 )
% 0.29/0.46 & ( ord_le1824328871od_a_a @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_207_eq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [X2: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.29/0.46 & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_208_eq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [X2: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.29/0.46 & ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_209_eq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [X2: set_a,Y3: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X2 @ Y3 )
% 0.29/0.46 & ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_210_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,F: nat > nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_211_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_212_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,F: nat > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_213_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_214_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,F: set_a > nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_215_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_216_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,F: set_nat > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_217_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,F: set_a > set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_218_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_219_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_220_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: nat,F: nat > nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_221_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_222_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: set_a,F: nat > set_a,B3: nat,C2: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_223_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_224_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: nat,F: set_a > nat,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_225_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_226_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: set_a,F: set_nat > set_a,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_227_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: set_nat,F: set_a > set_nat,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_228_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_229_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: nat,F: set_Product_prod_a_a > nat,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_230_order__subst2,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,F: nat > nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_231_order__subst2,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_232_order__subst2,axiom,
% 0.29/0.46 ! [A2: nat,B3: nat,F: nat > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_233_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_234_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,F: set_a > nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_235_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_236_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_nat,B3: set_nat,F: set_nat > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_237_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,F: set_a > set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_238_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_239_order__subst2,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 0.29/0.46 => ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_240_order__subst1,axiom,
% 0.29/0.46 ! [A2: nat,F: nat > nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_241_order__subst1,axiom,
% 0.29/0.46 ! [A2: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_242_order__subst1,axiom,
% 0.29/0.46 ! [A2: nat,F: set_a > nat,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_243_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_244_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_a,F: nat > set_a,B3: nat,C2: nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_245_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_246_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_nat,F: set_a > set_nat,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_247_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_a,F: set_nat > set_a,B3: set_nat,C2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_nat,Y4: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_248_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: set_a,Y4: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_249_order__subst1,axiom,
% 0.29/0.46 ! [A2: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B3: nat,C2: nat] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 0.29/0.46 => ( ! [X: nat,Y4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_250_Collect__mono__iff,axiom,
% 0.29/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) )
% 0.29/0.46 = ( ! [X2: product_prod_a_a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_251_Collect__mono__iff,axiom,
% 0.29/0.46 ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.29/0.46 = ( ! [X2: nat] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_252_Collect__mono__iff,axiom,
% 0.29/0.46 ! [P: a > $o,Q: a > $o] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_253_set__eq__subset,axiom,
% 0.29/0.46 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A5 @ B2 )
% 0.29/0.46 & ( ord_le1824328871od_a_a @ B2 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_254_set__eq__subset,axiom,
% 0.29/0.46 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 0.29/0.46 & ( ord_less_eq_set_nat @ B2 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_255_set__eq__subset,axiom,
% 0.29/0.46 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 0.29/0.46 = ( ^ [A5: set_a,B2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A5 @ B2 )
% 0.29/0.46 & ( ord_less_eq_set_a @ B2 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_256_subset__trans,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.46 => ( ( ord_le1824328871od_a_a @ B @ C )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_257_subset__trans,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat,C: set_nat] :
% 0.29/0.46 ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_258_subset__trans,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.46 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_259_Collect__mono,axiom,
% 0.29/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.29/0.46 ( ! [X: product_prod_a_a] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( Q @ X ) )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_260_Collect__mono,axiom,
% 0.29/0.46 ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.46 ( ! [X: nat] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( Q @ X ) )
% 0.29/0.46 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_261_Collect__mono,axiom,
% 0.29/0.46 ! [P: a > $o,Q: a > $o] :
% 0.29/0.46 ( ! [X: a] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( Q @ X ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_262_subset__refl,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_263_subset__refl,axiom,
% 0.29/0.46 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_264_subset__refl,axiom,
% 0.29/0.46 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_265_subset__iff,axiom,
% 0.29/0.46 ( ord_le1824328871od_a_a
% 0.29/0.46 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.29/0.46 ! [T2: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ T2 @ A5 )
% 0.29/0.46 => ( member449909584od_a_a @ T2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_266_subset__iff,axiom,
% 0.29/0.46 ( ord_less_eq_set_nat
% 0.29/0.46 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.29/0.46 ! [T2: nat] :
% 0.29/0.46 ( ( member_nat @ T2 @ A5 )
% 0.29/0.46 => ( member_nat @ T2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_267_subset__iff,axiom,
% 0.29/0.46 ( ord_less_eq_set_a
% 0.29/0.46 = ( ^ [A5: set_a,B2: set_a] :
% 0.29/0.46 ! [T2: a] :
% 0.29/0.46 ( ( member_a @ T2 @ A5 )
% 0.29/0.46 => ( member_a @ T2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_268_equalityD2,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_269_equalityD2,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ord_less_eq_set_nat @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_270_equalityD2,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ord_less_eq_set_a @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_271_equalityD1,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ord_le1824328871od_a_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_272_equalityD1,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_273_equalityD1,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_274_subset__eq,axiom,
% 0.29/0.46 ( ord_le1824328871od_a_a
% 0.29/0.46 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.29/0.46 ! [X2: product_prod_a_a] :
% 0.29/0.46 ( ( member449909584od_a_a @ X2 @ A5 )
% 0.29/0.46 => ( member449909584od_a_a @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_275_subset__eq,axiom,
% 0.29/0.46 ( ord_less_eq_set_nat
% 0.29/0.46 = ( ^ [A5: set_nat,B2: set_nat] :
% 0.29/0.46 ! [X2: nat] :
% 0.29/0.46 ( ( member_nat @ X2 @ A5 )
% 0.29/0.46 => ( member_nat @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_276_subset__eq,axiom,
% 0.29/0.46 ( ord_less_eq_set_a
% 0.29/0.46 = ( ^ [A5: set_a,B2: set_a] :
% 0.29/0.46 ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A5 )
% 0.29/0.46 => ( member_a @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_277_equalityE,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ~ ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.46 => ~ ( ord_le1824328871od_a_a @ B @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_278_equalityE,axiom,
% 0.29/0.46 ! [A: set_nat,B: set_nat] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ~ ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.46 => ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_279_equalityE,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ~ ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.46 => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_280_subsetD,axiom,
% 0.29/0.46 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
% 0.29/0.46 ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.46 => ( ( member449909584od_a_a @ C2 @ A )
% 0.29/0.46 => ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetD
% 0.29/0.46 thf(fact_281_subsetD,axiom,
% 0.29/0.47 ! [A: set_nat,B: set_nat,C2: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.47 => ( ( member_nat @ C2 @ A )
% 0.29/0.47 => ( member_nat @ C2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetD
% 0.29/0.47 thf(fact_282_subsetD,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a,C2: a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.47 => ( ( member_a @ C2 @ A )
% 0.29/0.47 => ( member_a @ C2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetD
% 0.29/0.47 thf(fact_283_in__mono,axiom,
% 0.29/0.47 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ A @ B )
% 0.29/0.47 => ( ( member449909584od_a_a @ X3 @ A )
% 0.29/0.47 => ( member449909584od_a_a @ X3 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_mono
% 0.29/0.47 thf(fact_284_in__mono,axiom,
% 0.29/0.47 ! [A: set_nat,B: set_nat,X3: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ A @ B )
% 0.29/0.47 => ( ( member_nat @ X3 @ A )
% 0.29/0.47 => ( member_nat @ X3 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_mono
% 0.29/0.47 thf(fact_285_in__mono,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a,X3: a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.47 => ( ( member_a @ X3 @ A )
% 0.29/0.47 => ( member_a @ X3 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_mono
% 0.29/0.47 thf(fact_286_pred__subset__eq,axiom,
% 0.29/0.47 ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1347718902_a_a_o
% 0.29/0.47 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ R )
% 0.29/0.47 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ S ) )
% 0.29/0.47 = ( ord_le1824328871od_a_a @ R @ S ) ) ).
% 0.29/0.47
% 0.29/0.47 % pred_subset_eq
% 0.29/0.47 thf(fact_287_pred__subset__eq,axiom,
% 0.29/0.47 ! [R: set_nat,S: set_nat] :
% 0.29/0.47 ( ( ord_less_eq_nat_o
% 0.29/0.47 @ ^ [X2: nat] : ( member_nat @ X2 @ R )
% 0.29/0.47 @ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
% 0.29/0.47 = ( ord_less_eq_set_nat @ R @ S ) ) ).
% 0.29/0.47
% 0.29/0.47 % pred_subset_eq
% 0.29/0.47 thf(fact_288_pred__subset__eq,axiom,
% 0.29/0.47 ! [R: set_a,S: set_a] :
% 0.29/0.47 ( ( ord_less_eq_a_o
% 0.29/0.47 @ ^ [X2: a] : ( member_a @ X2 @ R )
% 0.29/0.47 @ ^ [X2: a] : ( member_a @ X2 @ S ) )
% 0.29/0.47 = ( ord_less_eq_set_a @ R @ S ) ) ).
% 0.29/0.47
% 0.29/0.47 % pred_subset_eq
% 0.29/0.47 thf(fact_289_pair__pre__digraph_Osurjective,axiom,
% 0.29/0.47 ! [R2: pair_p125712459t_unit] :
% 0.29/0.47 ( R2
% 0.29/0.47 = ( pair_p1621517565t_unit @ ( pair_p1047056820t_unit @ R2 ) @ ( pair_p133601421t_unit @ R2 ) @ ( pair_p1896222615t_unit @ R2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.surjective
% 0.29/0.47 thf(fact_290_pair__pre__digraph_Osurjective,axiom,
% 0.29/0.47 ! [R2: pair_p1914262621t_unit] :
% 0.29/0.47 ( R2
% 0.29/0.47 = ( pair_p1167410509t_unit @ ( pair_p1677060310t_unit @ R2 ) @ ( pair_p715279805t_unit @ R2 ) @ ( pair_p69470259t_unit @ R2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.surjective
% 0.29/0.47 thf(fact_291_pair__pre__digraph_Osurjective,axiom,
% 0.29/0.47 ! [R2: pair_p1765063010t_unit] :
% 0.29/0.47 ( R2
% 0.29/0.47 = ( pair_p398687508t_unit @ ( pair_p447552203t_unit @ R2 ) @ ( pair_p1783210148t_unit @ R2 ) @ ( pair_p1984658862t_unit @ R2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.surjective
% 0.29/0.47 thf(fact_292_member__product,axiom,
% 0.29/0.47 ! [X3: product_prod_a_a,A: set_a,B: set_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X3 @ ( product_product_a_a @ A @ B ) )
% 0.29/0.47 = ( member449909584od_a_a @ X3
% 0.29/0.47 @ ( product_Sigma_a_a @ A
% 0.29/0.47 @ ^ [Uu: a] : B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % member_product
% 0.29/0.47 thf(fact_293_Product__Type_Oproduct__def,axiom,
% 0.29/0.47 ( product_product_a_a
% 0.29/0.47 = ( ^ [A5: set_a,B2: set_a] :
% 0.29/0.47 ( product_Sigma_a_a @ A5
% 0.29/0.47 @ ^ [Uu: a] : B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Product_Type.product_def
% 0.29/0.47 thf(fact_294_subset__Collect__iff,axiom,
% 0.29/0.47 ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ B @ A )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ B
% 0.29/0.47 @ ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ A )
% 0.29/0.47 & ( P @ X2 ) ) ) )
% 0.29/0.47 = ( ! [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ B )
% 0.29/0.47 => ( P @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_Collect_iff
% 0.29/0.47 thf(fact_295_subset__Collect__iff,axiom,
% 0.29/0.47 ! [B: set_nat,A: set_nat,P: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ B @ A )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ B
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ A )
% 0.29/0.47 & ( P @ X2 ) ) ) )
% 0.29/0.47 = ( ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ B )
% 0.29/0.47 => ( P @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_Collect_iff
% 0.29/0.47 thf(fact_296_subset__Collect__iff,axiom,
% 0.29/0.47 ! [B: set_a,A: set_a,P: a > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ B
% 0.29/0.47 @ ( collect_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ A )
% 0.29/0.47 & ( P @ X2 ) ) ) )
% 0.29/0.47 = ( ! [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ B )
% 0.29/0.47 => ( P @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_Collect_iff
% 0.29/0.47 thf(fact_297_subset__CollectI,axiom,
% 0.29/0.47 ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,Q: product_prod_a_a > $o,P: product_prod_a_a > $o] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ B @ A )
% 0.29/0.47 => ( ! [X: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X @ B )
% 0.29/0.47 => ( ( Q @ X )
% 0.29/0.47 => ( P @ X ) ) )
% 0.29/0.47 => ( ord_le1824328871od_a_a
% 0.29/0.47 @ ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ B )
% 0.29/0.47 & ( Q @ X2 ) ) )
% 0.29/0.47 @ ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ A )
% 0.29/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_CollectI
% 0.29/0.47 thf(fact_298_subset__CollectI,axiom,
% 0.29/0.47 ! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ B @ A )
% 0.29/0.47 => ( ! [X: nat] :
% 0.29/0.47 ( ( member_nat @ X @ B )
% 0.29/0.47 => ( ( Q @ X )
% 0.29/0.47 => ( P @ X ) ) )
% 0.29/0.47 => ( ord_less_eq_set_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ B )
% 0.29/0.47 & ( Q @ X2 ) ) )
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ A )
% 0.29/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_CollectI
% 0.29/0.47 thf(fact_299_subset__CollectI,axiom,
% 0.29/0.47 ! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.47 => ( ! [X: a] :
% 0.29/0.47 ( ( member_a @ X @ B )
% 0.29/0.47 => ( ( Q @ X )
% 0.29/0.47 => ( P @ X ) ) )
% 0.29/0.47 => ( ord_less_eq_set_a
% 0.29/0.47 @ ( collect_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ B )
% 0.29/0.47 & ( Q @ X2 ) ) )
% 0.29/0.47 @ ( collect_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ A )
% 0.29/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_CollectI
% 0.29/0.47 thf(fact_300_Collect__restrict,axiom,
% 0.29/0.47 ! [X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 0.29/0.47 ( ord_le1824328871od_a_a
% 0.29/0.47 @ ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ X4 )
% 0.29/0.47 & ( P @ X2 ) ) )
% 0.29/0.47 @ X4 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_restrict
% 0.29/0.47 thf(fact_301_Collect__restrict,axiom,
% 0.29/0.47 ! [X4: set_nat,P: nat > $o] :
% 0.29/0.47 ( ord_less_eq_set_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ X4 )
% 0.29/0.47 & ( P @ X2 ) ) )
% 0.29/0.47 @ X4 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_restrict
% 0.29/0.47 thf(fact_302_Collect__restrict,axiom,
% 0.29/0.47 ! [X4: set_a,P: a > $o] :
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 @ ( collect_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ X4 )
% 0.29/0.47 & ( P @ X2 ) ) )
% 0.29/0.47 @ X4 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_restrict
% 0.29/0.47 thf(fact_303_prop__restrict,axiom,
% 0.29/0.47 ! [X3: product_prod_a_a,Z3: set_Product_prod_a_a,X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 0.29/0.47 ( ( member449909584od_a_a @ X3 @ Z3 )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ Z3
% 0.29/0.47 @ ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ X4 )
% 0.29/0.47 & ( P @ X2 ) ) ) )
% 0.29/0.47 => ( P @ X3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prop_restrict
% 0.29/0.47 thf(fact_304_prop__restrict,axiom,
% 0.29/0.47 ! [X3: nat,Z3: set_nat,X4: set_nat,P: nat > $o] :
% 0.29/0.47 ( ( member_nat @ X3 @ Z3 )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ Z3
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ X4 )
% 0.29/0.47 & ( P @ X2 ) ) ) )
% 0.29/0.47 => ( P @ X3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prop_restrict
% 0.29/0.47 thf(fact_305_prop__restrict,axiom,
% 0.29/0.47 ! [X3: a,Z3: set_a,X4: set_a,P: a > $o] :
% 0.29/0.47 ( ( member_a @ X3 @ Z3 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ Z3
% 0.29/0.47 @ ( collect_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ X4 )
% 0.29/0.47 & ( P @ X2 ) ) ) )
% 0.29/0.47 => ( P @ X3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prop_restrict
% 0.29/0.47 thf(fact_306_conj__subset__def,axiom,
% 0.29/0.47 ! [A: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ A
% 0.29/0.47 @ ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X2: product_prod_a_a] :
% 0.29/0.47 ( ( P @ X2 )
% 0.29/0.47 & ( Q @ X2 ) ) ) )
% 0.29/0.47 = ( ( ord_le1824328871od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.29/0.47 & ( ord_le1824328871od_a_a @ A @ ( collec645855634od_a_a @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % conj_subset_def
% 0.29/0.47 thf(fact_307_conj__subset__def,axiom,
% 0.29/0.47 ! [A: set_nat,P: nat > $o,Q: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ A
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [X2: nat] :
% 0.29/0.47 ( ( P @ X2 )
% 0.29/0.47 & ( Q @ X2 ) ) ) )
% 0.29/0.47 = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
% 0.29/0.47 & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % conj_subset_def
% 0.29/0.47 thf(fact_308_conj__subset__def,axiom,
% 0.29/0.47 ! [A: set_a,P: a > $o,Q: a > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A
% 0.29/0.47 @ ( collect_a
% 0.29/0.47 @ ^ [X2: a] :
% 0.29/0.47 ( ( P @ X2 )
% 0.29/0.47 & ( Q @ X2 ) ) ) )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
% 0.29/0.47 & ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % conj_subset_def
% 0.29/0.47 thf(fact_309_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 0.29/0.47 ! [Pverts: set_a,Parcs: set_Product_prod_a_a,More: product_unit] :
% 0.29/0.47 ( ( pair_p133601421t_unit @ ( pair_p1621517565t_unit @ Pverts @ Parcs @ More ) )
% 0.29/0.47 = Parcs ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.select_convs(2)
% 0.29/0.47 thf(fact_310_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 0.29/0.47 ! [Pverts: set_nat,Parcs: set_Pr1986765409at_nat,More: product_unit] :
% 0.29/0.47 ( ( pair_p715279805t_unit @ ( pair_p1167410509t_unit @ Pverts @ Parcs @ More ) )
% 0.29/0.47 = Parcs ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.select_convs(2)
% 0.29/0.47 thf(fact_311_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 0.29/0.47 ! [Pverts: set_Product_prod_a_a,Parcs: set_Pr1948701895od_a_a,More: product_unit] :
% 0.29/0.47 ( ( pair_p1783210148t_unit @ ( pair_p398687508t_unit @ Pverts @ Parcs @ More ) )
% 0.29/0.47 = Parcs ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.select_convs(2)
% 0.29/0.47 thf(fact_312_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 0.29/0.47 ! [Pverts: set_a,Parcs: set_Product_prod_a_a,More: product_unit] :
% 0.29/0.47 ( ( pair_p1047056820t_unit @ ( pair_p1621517565t_unit @ Pverts @ Parcs @ More ) )
% 0.29/0.47 = Pverts ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.select_convs(1)
% 0.29/0.47 thf(fact_313_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 0.29/0.47 ! [Pverts: set_nat,Parcs: set_Pr1986765409at_nat,More: product_unit] :
% 0.29/0.47 ( ( pair_p1677060310t_unit @ ( pair_p1167410509t_unit @ Pverts @ Parcs @ More ) )
% 0.29/0.47 = Pverts ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.select_convs(1)
% 0.29/0.47 thf(fact_314_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 0.29/0.47 ! [Pverts: set_Product_prod_a_a,Parcs: set_Pr1948701895od_a_a,More: product_unit] :
% 0.29/0.47 ( ( pair_p447552203t_unit @ ( pair_p398687508t_unit @ Pverts @ Parcs @ More ) )
% 0.29/0.47 = Pverts ) ).
% 0.29/0.47
% 0.29/0.47 % pair_pre_digraph.select_convs(1)
% 0.29/0.47 thf(fact_315_Fpow__def,axiom,
% 0.29/0.47 ( finite361944167at_nat
% 0.29/0.47 = ( ^ [A5: set_Pr1986765409at_nat] :
% 0.29/0.47 ( collec1606769740at_nat
% 0.29/0.47 @ ^ [X5: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( ord_le841296385at_nat @ X5 @ A5 )
% 0.29/0.47 & ( finite772653738at_nat @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Fpow_def
% 0.29/0.47 thf(fact_316_Fpow__def,axiom,
% 0.29/0.47 ( finite702915405od_a_a
% 0.29/0.47 = ( ^ [A5: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( collec453062450od_a_a
% 0.29/0.47 @ ^ [X5: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( ( ord_le456379495od_a_a @ X5 @ A5 )
% 0.29/0.47 & ( finite1664988688od_a_a @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Fpow_def
% 0.29/0.47 thf(fact_317_Fpow__def,axiom,
% 0.29/0.47 ( finite351630733od_a_a
% 0.29/0.47 = ( ^ [A5: set_Product_prod_a_a] :
% 0.29/0.47 ( collec183727474od_a_a
% 0.29/0.47 @ ^ [X5: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ X5 @ A5 )
% 0.29/0.47 & ( finite179568208od_a_a @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Fpow_def
% 0.29/0.47 thf(fact_318_Fpow__def,axiom,
% 0.29/0.47 ( finite_Fpow_nat
% 0.29/0.47 = ( ^ [A5: set_nat] :
% 0.29/0.47 ( collect_set_nat
% 0.29/0.47 @ ^ [X5: set_nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ X5 @ A5 )
% 0.29/0.47 & ( finite_finite_nat @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Fpow_def
% 0.29/0.47 thf(fact_319_Fpow__def,axiom,
% 0.29/0.47 ( finite_Fpow_a
% 0.29/0.47 = ( ^ [A5: set_a] :
% 0.29/0.47 ( collect_set_a
% 0.29/0.47 @ ^ [X5: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X5 @ A5 )
% 0.29/0.47 & ( finite_finite_a @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Fpow_def
% 0.29/0.47 thf(fact_320_pair__fin__digraph__def,axiom,
% 0.29/0.47 ( pair_p1802376898raph_a
% 0.29/0.47 = ( ^ [G: pair_p125712459t_unit] :
% 0.29/0.47 ( ( pair_p68905728raph_a @ G )
% 0.29/0.47 & ( pair_p1864019935ioms_a @ G ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_fin_digraph_def
% 0.29/0.47 thf(fact_321_pair__fin__digraph__def,axiom,
% 0.29/0.47 ( pair_p128415500ph_nat
% 0.29/0.47 = ( ^ [G: pair_p1914262621t_unit] :
% 0.29/0.47 ( ( pair_p1515597646ph_nat @ G )
% 0.29/0.47 & ( pair_p1027063983ms_nat @ G ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_fin_digraph_def
% 0.29/0.47 thf(fact_322_pair__fin__digraph__def,axiom,
% 0.29/0.47 ( pair_p374947051od_a_a
% 0.29/0.47 = ( ^ [G: pair_p1765063010t_unit] :
% 0.29/0.47 ( ( pair_p646030121od_a_a @ G )
% 0.29/0.47 & ( pair_p504738056od_a_a @ G ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_fin_digraph_def
% 0.29/0.47 thf(fact_323_pair__fin__digraph_Ointro,axiom,
% 0.29/0.47 ! [G2: pair_p125712459t_unit] :
% 0.29/0.47 ( ( pair_p68905728raph_a @ G2 )
% 0.29/0.47 => ( ( pair_p1864019935ioms_a @ G2 )
% 0.29/0.47 => ( pair_p1802376898raph_a @ G2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_fin_digraph.intro
% 0.29/0.47 thf(fact_324_pair__fin__digraph_Ointro,axiom,
% 0.29/0.47 ! [G2: pair_p1914262621t_unit] :
% 0.29/0.47 ( ( pair_p1515597646ph_nat @ G2 )
% 0.29/0.47 => ( ( pair_p1027063983ms_nat @ G2 )
% 0.29/0.47 => ( pair_p128415500ph_nat @ G2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_fin_digraph.intro
% 0.29/0.47 thf(fact_325_pair__fin__digraph_Ointro,axiom,
% 0.29/0.47 ! [G2: pair_p1765063010t_unit] :
% 0.29/0.47 ( ( pair_p646030121od_a_a @ G2 )
% 0.29/0.47 => ( ( pair_p504738056od_a_a @ G2 )
% 0.29/0.47 => ( pair_p374947051od_a_a @ G2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_fin_digraph.intro
% 0.29/0.47 thf(fact_326_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_a,B: set_nat] :
% 0.29/0.47 ( ( finite1743148308_a_nat
% 0.29/0.47 @ ( product_Sigma_a_nat @ A
% 0.29/0.47 @ ^ [Uu: a] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_a )
% 0.29/0.47 | ( B = bot_bot_set_nat )
% 0.29/0.47 | ( ( finite_finite_a @ A )
% 0.29/0.47 & ( finite_finite_nat @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_327_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_nat,B: set_a] :
% 0.29/0.47 ( ( finite1808550458_nat_a
% 0.29/0.47 @ ( product_Sigma_nat_a @ A
% 0.29/0.47 @ ^ [Uu: nat] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_nat )
% 0.29/0.47 | ( B = bot_bot_set_a )
% 0.29/0.47 | ( ( finite_finite_nat @ A )
% 0.29/0.47 & ( finite_finite_a @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_328_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_nat,B: set_nat] :
% 0.29/0.47 ( ( finite772653738at_nat
% 0.29/0.47 @ ( produc45129834at_nat @ A
% 0.29/0.47 @ ^ [Uu: nat] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_nat )
% 0.29/0.47 | ( B = bot_bot_set_nat )
% 0.29/0.47 | ( ( finite_finite_nat @ A )
% 0.29/0.47 & ( finite_finite_nat @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_329_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a] :
% 0.29/0.47 ( ( finite179568208od_a_a
% 0.29/0.47 @ ( product_Sigma_a_a @ A
% 0.29/0.47 @ ^ [Uu: a] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_a )
% 0.29/0.47 | ( B = bot_bot_set_a )
% 0.29/0.47 | ( ( finite_finite_a @ A )
% 0.29/0.47 & ( finite_finite_a @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_330_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_Product_prod_a_a,B: set_a] :
% 0.29/0.47 ( ( finite1919032935_a_a_a
% 0.29/0.47 @ ( produc1282482655_a_a_a @ A
% 0.29/0.47 @ ^ [Uu: product_prod_a_a] : B ) )
% 0.29/0.47 = ( ( A = bot_bo2131659635od_a_a )
% 0.29/0.47 | ( B = bot_bot_set_a )
% 0.29/0.47 | ( ( finite179568208od_a_a @ A )
% 0.29/0.47 & ( finite_finite_a @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_331_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_Product_prod_a_a,B: set_nat] :
% 0.29/0.47 ( ( finite1837575485_a_nat
% 0.29/0.47 @ ( produc931712687_a_nat @ A
% 0.29/0.47 @ ^ [Uu: product_prod_a_a] : B ) )
% 0.29/0.47 = ( ( A = bot_bo2131659635od_a_a )
% 0.29/0.47 | ( B = bot_bot_set_nat )
% 0.29/0.47 | ( ( finite179568208od_a_a @ A )
% 0.29/0.47 & ( finite_finite_nat @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_332_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_a,B: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite676513017od_a_a
% 0.29/0.47 @ ( produc520147185od_a_a @ A
% 0.29/0.47 @ ^ [Uu: a] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_a )
% 0.29/0.47 | ( B = bot_bo2131659635od_a_a )
% 0.29/0.47 | ( ( finite_finite_a @ A )
% 0.29/0.47 & ( finite179568208od_a_a @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_333_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_a,B: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( finite942416723at_nat
% 0.29/0.47 @ ( produc292491723at_nat @ A
% 0.29/0.47 @ ^ [Uu: a] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_a )
% 0.29/0.47 | ( B = bot_bo2130386637at_nat )
% 0.29/0.47 | ( ( finite_finite_a @ A )
% 0.29/0.47 & ( finite772653738at_nat @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_334_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_nat,B: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite1297454819od_a_a
% 0.29/0.47 @ ( produc1182842125od_a_a @ A
% 0.29/0.47 @ ^ [Uu: nat] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_nat )
% 0.29/0.47 | ( B = bot_bo2131659635od_a_a )
% 0.29/0.47 | ( ( finite_finite_nat @ A )
% 0.29/0.47 & ( finite179568208od_a_a @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_335_finite__cartesian__product__iff,axiom,
% 0.29/0.47 ! [A: set_nat,B: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( finite277291581at_nat
% 0.29/0.47 @ ( produc894163943at_nat @ A
% 0.29/0.47 @ ^ [Uu: nat] : B ) )
% 0.29/0.47 = ( ( A = bot_bot_set_nat )
% 0.29/0.47 | ( B = bot_bo2130386637at_nat )
% 0.29/0.47 | ( ( finite_finite_nat @ A )
% 0.29/0.47 & ( finite772653738at_nat @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_cartesian_product_iff
% 0.29/0.47 thf(fact_336_empty__iff,axiom,
% 0.29/0.47 ! [C2: product_prod_a_a] :
% 0.29/0.47 ~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% 0.29/0.47
% 0.29/0.47 % empty_iff
% 0.29/0.47 thf(fact_337_empty__iff,axiom,
% 0.29/0.47 ! [C2: nat] :
% 0.29/0.47 ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% 0.29/0.47
% 0.29/0.47 % empty_iff
% 0.29/0.47 thf(fact_338_all__not__in__conv,axiom,
% 0.29/0.47 ! [A: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ! [X2: product_prod_a_a] :
% 0.29/0.47 ~ ( member449909584od_a_a @ X2 @ A ) )
% 0.29/0.47 = ( A = bot_bo2131659635od_a_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % all_not_in_conv
% 0.29/0.47 thf(fact_339_all__not__in__conv,axiom,
% 0.29/0.47 ! [A: set_nat] :
% 0.29/0.47 ( ( ! [X2: nat] :
% 0.29/0.47 ~ ( member_nat @ X2 @ A ) )
% 0.29/0.47 = ( A = bot_bot_set_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % all_not_in_conv
% 0.29/0.47 thf(fact_340_Collect__empty__eq,axiom,
% 0.29/0.47 ! [P: product_prod_a_a > $o] :
% 0.29/0.47 ( ( ( collec645855634od_a_a @ P )
% 0.29/0.47 = bot_bo2131659635od_a_a )
% 0.29/0.47 = ( ! [X2: product_prod_a_a] :
% 0.29/0.47 ~ ( P @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_empty_eq
% 0.29/0.47 thf(fact_341_Collect__empty__eq,axiom,
% 0.29/0.47 ! [P: nat > $o] :
% 0.29/0.47 ( ( ( collect_nat @ P )
% 0.29/0.47 = bot_bot_set_nat )
% 0.29/0.47 = ( ! [X2: nat] :
% 0.29/0.47 ~ ( P @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_empty_eq
% 0.29/0.47 thf(fact_342_empty__Collect__eq,axiom,
% 0.29/0.47 ! [P: product_prod_a_a > $o] :
% 0.29/0.47 ( ( bot_bo2131659635od_a_a
% 0.29/0.47 = ( collec645855634od_a_a @ P ) )
% 0.29/0.47 = ( ! [X2: product_prod_a_a] :
% 0.29/0.47 ~ ( P @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % empty_Collect_eq
% 0.29/0.47 thf(fact_343_empty__Collect__eq,axiom,
% 0.29/0.47 ! [P: nat > $o] :
% 0.29/0.47 ( ( bot_bot_set_nat
% 0.29/0.47 = ( collect_nat @ P ) )
% 0.29/0.47 = ( ! [X2: nat] :
% 0.29/0.47 ~ ( P @ X2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % empty_Collect_eq
% 0.29/0.47 thf(fact_344_empty__subsetI,axiom,
% 0.29/0.47 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_345_empty__subsetI,axiom,
% 0.29/0.47 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_346_empty__subsetI,axiom,
% 0.29/0.47 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_347_subset__empty,axiom,
% 0.29/0.47 ! [A: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.29/0.47 = ( A = bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_empty
% 0.29/0.47 thf(fact_348_calculation,axiom,
% 0.29/0.47 ( ( finite_finite_a @ ( pair_p1047056820t_unit @ g ) )
% 0.29/0.47 & ( ( finite_card_a @ ( pair_p1047056820t_unit @ g ) )
% 0.29/0.47 = n )
% 0.29/0.47 & ( ( pair_p133601421t_unit @ g )
% 0.29/0.47 = ( collec645855634od_a_a
% 0.29/0.47 @ ( produc1833107820_a_a_o
% 0.29/0.47 @ ^ [U: a,V: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ U @ V )
% 0.29/0.47 @ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
% 0.29/0.47 @ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) )
% 0.29/0.47 & ( U != V ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % calculation
% 0.29/0.47 thf(fact_349_finite__Collect__le__nat,axiom,
% 0.29/0.47 ! [K: nat] :
% 0.29/0.47 ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_Collect_le_nat
% 0.29/0.47 thf(fact_350_finite__nat__set__iff__bounded__le,axiom,
% 0.29/0.47 ( finite_finite_nat
% 0.29/0.47 = ( ^ [N2: set_nat] :
% 0.29/0.47 ? [M: nat] :
% 0.29/0.47 ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ N2 )
% 0.29/0.47 => ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_nat_set_iff_bounded_le
% 0.29/0.47 thf(fact_351_infinite__nat__iff__unbounded__le,axiom,
% 0.29/0.47 ! [S: set_nat] :
% 0.29/0.47 ( ( ~ ( finite_finite_nat @ S ) )
% 0.29/0.47 = ( ! [M: nat] :
% 0.29/0.47 ? [N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 & ( member_nat @ N @ S ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_nat_iff_unbounded_le
% 0.29/0.47 thf(fact_352_bounded__Max__nat,axiom,
% 0.29/0.47 ! [P: nat > $o,X3: nat,M2: nat] :
% 0.29/0.47 ( ( P @ X3 )
% 0.29/0.47 => ( ! [X: nat] :
% 0.29/0.47 ( ( P @ X )
% 0.29/0.47 => ( ord_less_eq_nat @ X @ M2 ) )
% 0.29/0.47 => ~ ! [M3: nat] :
% 0.29/0.47 ( ( P @ M3 )
% 0.29/0.47 => ~ ! [X6: nat] :
% 0.29/0.47 ( ( P @ X6 )
% 0.29/0.47 => ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bounded_Max_nat
% 0.29/0.47 thf(fact_353_finite__less__ub,axiom,
% 0.29/0.47 ! [F: nat > nat,U2: nat] :
% 0.29/0.47 ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 0.29/0.47 => ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_less_ub
% 0.29/0.47
% 0.29/0.47 % Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.98/1.56 ---- Embedded in THF ---
% 2.98/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.98/1.56 %%% Generated on Tue Feb 24 08:26:13 EST 2026
% 2.98/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.98/1.56 %%% Logic specification used:
% 2.98/1.56 %%% thf(spec, logic, $$dhol).
% 2.98/1.56
% 2.98/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2DTF5057.p
% See solution above
% 2.98/1.57 ------------------------
% 3.12/1.62 ---- Cleaned THF ---
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 3.12/1.62 set_Pr1295299783od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Ounit_J,type,
% 3.12/1.62 pair_p1593840546t_unit: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Ounit_J,type,
% 3.12/1.62 pair_p2041852168t_unit: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.12/1.62 set_Pr1490359111at_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
% 3.12/1.62 pair_p1765063010t_unit: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 3.12/1.62 set_se958357159od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 set_Pr1948701895od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
% 3.12/1.62 pair_p1914262621t_unit: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.12/1.62 set_Pr1746169692at_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 produc1572603623od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Pair____Digraph__Opair____pre____digraph__Opair____pre____digraph____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
% 3.12/1.62 pair_p125712459t_unit: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.12/1.62 set_Pr7688842at_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 set_Pr894832732_a_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 set_Pr339609346od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 set_Pr681306928od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 3.12/1.62 set_Pr1689873822_a_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 3.12/1.62 set_se1612935105at_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 set_se1596668135od_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 set_Pr1986765409at_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 set_Pr548851891_a_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
% 3.12/1.62 set_Pr967348953_nat_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 set_Product_prod_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 product_prod_nat_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 3.12/1.62 set_set_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 product_prod_a_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 3.12/1.62 set_set_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 3.12/1.62 set_nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Product____Type__Ounit,type,
% 3.12/1.62 product_unit: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 3.12/1.62 set_a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_t__Nat__Onat,type,
% 3.12/1.62 nat: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(ty_n_tf__a,type,
% 3.12/1.62 a: $tType ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
% 3.12/1.62 finite_Fpow_nat: set_nat > set_set_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 finite361944167at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 finite702915405od_a_a: set_Pr1948701895od_a_a > set_se958357159od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 finite351630733od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_OFpow_001tf__a,type,
% 3.12/1.62 finite_Fpow_a: set_a > set_set_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ocard_001tf__a,type,
% 3.12/1.62 finite_card_a: set_a > nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 3.12/1.62 finite_finite_nat: set_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 finite772653738at_nat: set_Pr1986765409at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 finite277291581at_nat: set_Pr1746169692at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 finite1297454819od_a_a: set_Pr339609346od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
% 3.12/1.62 finite1808550458_nat_a: set_Pr967348953_nat_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 finite48957584at_nat: set_Pr1490359111at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 finite256329232od_a_a: set_Pr1295299783od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J,type,
% 3.12/1.62 finite1837575485_a_nat: set_Pr894832732_a_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 3.12/1.62 finite1919032935_a_a_a: set_Pr1689873822_a_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
% 3.12/1.62 finite1743148308_a_nat: set_Pr548851891_a_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 finite942416723at_nat: set_Pr7688842at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 finite676513017od_a_a: set_Pr681306928od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 finite179568208od_a_a: set_Product_prod_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.12/1.62 finite2012248349et_nat: set_set_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 finite1457549322at_nat: set_se1612935105at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 finite323969008od_a_a: set_se958357159od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 finite1145471536od_a_a: set_se1596668135od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
% 3.12/1.62 finite_finite_set_a: set_set_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
% 3.12/1.62 finite_finite_a: set_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.12/1.62 bot_bot_set_nat: set_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 bot_bo2130386637at_nat: set_Pr1986765409at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 bot_bo2131659635od_a_a: set_Product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 3.12/1.62 bot_bot_set_a: set_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 3.12/1.62 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
% 3.12/1.62 ord_le1347718902_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
% 3.12/1.62 ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.12/1.62 ord_less_eq_nat: nat > nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.12/1.62 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 ord_le2084554594od_a_a: set_Pr339609346od_a_a > set_Pr339609346od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
% 3.12/1.62 ord_le344568633_nat_a: set_Pr967348953_nat_a > set_Pr967348953_nat_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 ord_le492294332_a_nat: set_Pr894832732_a_nat > set_Pr894832732_a_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 3.12/1.62 ord_le677315902_a_a_a: set_Pr1689873822_a_a_a > set_Pr1689873822_a_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 ord_le2073555219_a_nat: set_Pr548851891_a_nat > set_Pr548851891_a_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 ord_le1816232656od_a_a: set_Pr681306928od_a_a > set_Pr681306928od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 3.12/1.62 ord_less_eq_set_a: set_a > set_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Nat__Onat,type,
% 3.12/1.62 pair_p128415500ph_nat: pair_p1914262621t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 pair_p752841413at_nat: pair_p2041852168t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 pair_p159207083od_a_a: pair_p1593840546t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 pair_p374947051od_a_a: pair_p1765063010t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph_001tf__a,type,
% 3.12/1.62 pair_p1802376898raph_a: pair_p125712459t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Nat__Onat,type,
% 3.12/1.62 pair_p1027063983ms_nat: pair_p1914262621t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 pair_p56914274at_nat: pair_p2041852168t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 pair_p1906342088od_a_a: pair_p1593840546t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 pair_p504738056od_a_a: pair_p1765063010t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms_001tf__a,type,
% 3.12/1.62 pair_p1864019935ioms_a: pair_p125712459t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p69470259t_unit: pair_p1914262621t_unit > product_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1984658862t_unit: pair_p1765063010t_unit > product_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore_001tf__a_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1896222615t_unit: pair_p125712459t_unit > product_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1167410509t_unit: set_nat > set_Pr1986765409at_nat > product_unit > pair_p1914262621t_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p398687508t_unit: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_unit > pair_p1765063010t_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext_001tf__a_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1621517565t_unit: set_a > set_Product_prod_a_a > product_unit > pair_p125712459t_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p715279805t_unit: pair_p1914262621t_unit > set_Pr1986765409at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p806300874t_unit: pair_p2041852168t_unit > set_Pr1490359111at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1559300324t_unit: pair_p1593840546t_unit > set_Pr1295299783od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1783210148t_unit: pair_p1765063010t_unit > set_Pr1948701895od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001tf__a_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p133601421t_unit: pair_p125712459t_unit > set_Product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1677060310t_unit: pair_p1914262621t_unit > set_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p210955889t_unit: pair_p2041852168t_unit > set_Pr1986765409at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1652294923t_unit: pair_p1593840546t_unit > set_Pr1948701895od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p447552203t_unit: pair_p1765063010t_unit > set_Product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001tf__a_001t__Product____Type__Ounit,type,
% 3.12/1.62 pair_p1047056820t_unit: pair_p125712459t_unit > set_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__wf__digraph_001t__Nat__Onat,type,
% 3.12/1.62 pair_p1515597646ph_nat: pair_p1914262621t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__wf__digraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 pair_p646030121od_a_a: pair_p1765063010t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Pair__Digraph_Opair__wf__digraph_001tf__a,type,
% 3.12/1.62 pair_p68905728raph_a: pair_p125712459t_unit > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 3.12/1.62 product_Pair_a_a: a > a > product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 3.12/1.62 produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 produc1182842125od_a_a: set_nat > ( nat > set_Product_prod_a_a ) > set_Pr339609346od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001tf__a,type,
% 3.12/1.62 product_Sigma_nat_a: set_nat > ( nat > set_a ) > set_Pr967348953_nat_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
% 3.12/1.62 produc931712687_a_nat: set_Product_prod_a_a > ( product_prod_a_a > set_nat ) > set_Pr894832732_a_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 produc304751368od_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
% 3.12/1.62 produc1282482655_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_a ) > set_Pr1689873822_a_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001tf__a_001t__Nat__Onat,type,
% 3.12/1.62 product_Sigma_a_nat: set_a > ( a > set_nat ) > set_Pr548851891_a_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 produc292491723at_nat: set_a > ( a > set_Pr1986765409at_nat ) > set_Pr7688842at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 produc520147185od_a_a: set_a > ( a > set_Product_prod_a_a ) > set_Pr681306928od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_OSigma_001tf__a_001tf__a,type,
% 3.12/1.62 product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001_Eo,type,
% 3.12/1.62 produc1833107820_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Product__Type_Oproduct_001tf__a_001tf__a,type,
% 3.12/1.62 product_product_a_a: set_a > set_a > set_Product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 3.12/1.62 collect_nat: ( nat > $o ) > set_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.12/1.62 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.12/1.62 collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.12/1.62 collec453062450od_a_a: ( set_Pr1948701895od_a_a > $o ) > set_se958357159od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 3.12/1.62 collect_set_a: ( set_a > $o ) > set_set_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.12/1.62 collect_a: ( a > $o ) > set_a ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001t__Nat__Onat,type,
% 3.12/1.62 member_nat: nat > set_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.12/1.62 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.12/1.62 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.12/1.62 member_set_nat: set_nat > set_set_nat > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.12/1.62 member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 3.12/1.62 member_set_a: set_a > set_set_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_c_member_001tf__a,type,
% 3.12/1.62 member_a: a > set_a > $o ).
% 3.12/1.62
% 3.12/1.62 thf(sy_v_G,type,
% 3.12/1.62 g: pair_p125712459t_unit ).
% 3.12/1.62
% 3.12/1.62 thf(sy_v_n,type,
% 3.12/1.62 n: nat ).
% 3.12/1.62
% 3.12/1.62 thf(fact_0__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062,axiom,
% 3.12/1.62 ( finite179568208od_a_a
% 3.12/1.62 @ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
% 3.12/1.62 @ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_1__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062,axiom,
% 3.12/1.62 ( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
% 3.12/1.62 @ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
% 3.12/1.62 @ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_2_pair__fin__digraph__axioms__def,axiom,
% 3.12/1.62 ( pair_p56914274at_nat
% 3.12/1.62 = ( ^ [G: pair_p2041852168t_unit] :
% 3.12/1.62 ( ( finite772653738at_nat @ ( pair_p210955889t_unit @ G ) )
% 3.12/1.62 & ( finite48957584at_nat @ ( pair_p806300874t_unit @ G ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_3_pair__fin__digraph__axioms__def,axiom,
% 3.12/1.62 ( pair_p1906342088od_a_a
% 3.12/1.62 = ( ^ [G: pair_p1593840546t_unit] :
% 3.12/1.62 ( ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G ) )
% 3.12/1.62 & ( finite256329232od_a_a @ ( pair_p1559300324t_unit @ G ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_4_pair__fin__digraph__axioms__def,axiom,
% 3.12/1.62 ( pair_p504738056od_a_a
% 3.12/1.62 = ( ^ [G: pair_p1765063010t_unit] :
% 3.12/1.62 ( ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G ) )
% 3.12/1.62 & ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_5_pair__fin__digraph__axioms__def,axiom,
% 3.12/1.62 ( pair_p1027063983ms_nat
% 3.12/1.62 = ( ^ [G: pair_p1914262621t_unit] :
% 3.12/1.62 ( ( finite_finite_nat @ ( pair_p1677060310t_unit @ G ) )
% 3.12/1.62 & ( finite772653738at_nat @ ( pair_p715279805t_unit @ G ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_6_pair__fin__digraph__axioms__def,axiom,
% 3.12/1.62 ( pair_p1864019935ioms_a
% 3.12/1.62 = ( ^ [G: pair_p125712459t_unit] :
% 3.12/1.62 ( ( finite_finite_a @ ( pair_p1047056820t_unit @ G ) )
% 3.12/1.62 & ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_7_pair__fin__digraph__axioms_Ointro,axiom,
% 3.12/1.62 ! [G2: pair_p2041852168t_unit] :
% 3.12/1.62 ( ( finite772653738at_nat @ ( pair_p210955889t_unit @ G2 ) )
% 3.12/1.62 => ( ( finite48957584at_nat @ ( pair_p806300874t_unit @ G2 ) )
% 3.12/1.62 => ( pair_p56914274at_nat @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_8_pair__fin__digraph__axioms_Ointro,axiom,
% 3.12/1.62 ! [G2: pair_p1593840546t_unit] :
% 3.12/1.62 ( ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G2 ) )
% 3.12/1.62 => ( ( finite256329232od_a_a @ ( pair_p1559300324t_unit @ G2 ) )
% 3.12/1.62 => ( pair_p1906342088od_a_a @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_9_pair__fin__digraph__axioms_Ointro,axiom,
% 3.12/1.62 ! [G2: pair_p1765063010t_unit] :
% 3.12/1.62 ( ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G2 ) )
% 3.12/1.62 => ( ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G2 ) )
% 3.12/1.62 => ( pair_p504738056od_a_a @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_10_pair__fin__digraph__axioms_Ointro,axiom,
% 3.12/1.62 ! [G2: pair_p1914262621t_unit] :
% 3.12/1.62 ( ( finite_finite_nat @ ( pair_p1677060310t_unit @ G2 ) )
% 3.12/1.62 => ( ( finite772653738at_nat @ ( pair_p715279805t_unit @ G2 ) )
% 3.12/1.62 => ( pair_p1027063983ms_nat @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_11_pair__fin__digraph__axioms_Ointro,axiom,
% 3.12/1.62 ! [G2: pair_p125712459t_unit] :
% 3.12/1.62 ( ( finite_finite_a @ ( pair_p1047056820t_unit @ G2 ) )
% 3.12/1.62 => ( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G2 ) )
% 3.12/1.62 => ( pair_p1864019935ioms_a @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_12_pair__fin__digraph_Opair__finite__arcs,axiom,
% 3.12/1.62 ! [G2: pair_p1914262621t_unit] :
% 3.12/1.62 ( ( pair_p128415500ph_nat @ G2 )
% 3.12/1.62 => ( finite772653738at_nat @ ( pair_p715279805t_unit @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_13_pair__fin__digraph_Opair__finite__arcs,axiom,
% 3.12/1.62 ! [G2: pair_p1765063010t_unit] :
% 3.12/1.62 ( ( pair_p374947051od_a_a @ G2 )
% 3.12/1.62 => ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_14_pair__fin__digraph_Opair__finite__arcs,axiom,
% 3.12/1.62 ! [G2: pair_p125712459t_unit] :
% 3.12/1.62 ( ( pair_p1802376898raph_a @ G2 )
% 3.12/1.62 => ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G2 ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_15_finite__subset,axiom,
% 3.12/1.62 ! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
% 3.12/1.62 ( ( ord_le841296385at_nat @ A @ B )
% 3.12/1.62 => ( ( finite772653738at_nat @ B )
% 3.12/1.62 => ( finite772653738at_nat @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_16_finite__subset,axiom,
% 3.12/1.62 ! [A: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.12/1.62 ( ( ord_le456379495od_a_a @ A @ B )
% 3.12/1.62 => ( ( finite1664988688od_a_a @ B )
% 3.12/1.62 => ( finite1664988688od_a_a @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_17_finite__subset,axiom,
% 3.12/1.62 ! [A: set_a,B: set_a] :
% 3.12/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.62 => ( ( finite_finite_a @ B )
% 3.12/1.62 => ( finite_finite_a @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_18_finite__subset,axiom,
% 3.12/1.62 ! [A: set_nat,B: set_nat] :
% 3.12/1.62 ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.62 => ( ( finite_finite_nat @ B )
% 3.12/1.62 => ( finite_finite_nat @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_19_finite__subset,axiom,
% 3.12/1.62 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.62 ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.62 => ( ( finite179568208od_a_a @ B )
% 3.12/1.62 => ( finite179568208od_a_a @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_20_infinite__super,axiom,
% 3.12/1.62 ! [S: set_Pr1986765409at_nat,T: set_Pr1986765409at_nat] :
% 3.12/1.62 ( ( ord_le841296385at_nat @ S @ T )
% 3.12/1.62 => ( ~ ( finite772653738at_nat @ S )
% 3.12/1.62 => ~ ( finite772653738at_nat @ T ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_21_infinite__super,axiom,
% 3.12/1.62 ! [S: set_Pr1948701895od_a_a,T: set_Pr1948701895od_a_a] :
% 3.12/1.62 ( ( ord_le456379495od_a_a @ S @ T )
% 3.12/1.62 => ( ~ ( finite1664988688od_a_a @ S )
% 3.12/1.62 => ~ ( finite1664988688od_a_a @ T ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_22_infinite__super,axiom,
% 3.12/1.62 ! [S: set_a,T: set_a] :
% 3.12/1.62 ( ( ord_less_eq_set_a @ S @ T )
% 3.12/1.62 => ( ~ ( finite_finite_a @ S )
% 3.12/1.62 => ~ ( finite_finite_a @ T ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_23_infinite__super,axiom,
% 3.12/1.62 ! [S: set_nat,T: set_nat] :
% 3.12/1.62 ( ( ord_less_eq_set_nat @ S @ T )
% 3.12/1.62 => ( ~ ( finite_finite_nat @ S )
% 3.12/1.62 => ~ ( finite_finite_nat @ T ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_24_infinite__super,axiom,
% 3.12/1.62 ! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
% 3.12/1.62 ( ( ord_le1824328871od_a_a @ S @ T )
% 3.12/1.62 => ( ~ ( finite179568208od_a_a @ S )
% 3.12/1.62 => ~ ( finite179568208od_a_a @ T ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_25_rev__finite__subset,axiom,
% 3.12/1.62 ! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 3.12/1.62 ( ( finite772653738at_nat @ B )
% 3.12/1.62 => ( ( ord_le841296385at_nat @ A @ B )
% 3.12/1.62 => ( finite772653738at_nat @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_26_rev__finite__subset,axiom,
% 3.12/1.62 ! [B: set_Pr1948701895od_a_a,A: set_Pr1948701895od_a_a] :
% 3.12/1.62 ( ( finite1664988688od_a_a @ B )
% 3.12/1.62 => ( ( ord_le456379495od_a_a @ A @ B )
% 3.12/1.62 => ( finite1664988688od_a_a @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_27_rev__finite__subset,axiom,
% 3.12/1.62 ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 3.12/1.62 ( ( finite179568208od_a_a @ B )
% 3.12/1.62 => ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.62 => ( finite179568208od_a_a @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_28_rev__finite__subset,axiom,
% 3.12/1.62 ! [B: set_nat,A: set_nat] :
% 3.12/1.62 ( ( finite_finite_nat @ B )
% 3.12/1.62 => ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.62 => ( finite_finite_nat @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_29_rev__finite__subset,axiom,
% 3.12/1.62 ! [B: set_a,A: set_a] :
% 3.12/1.62 ( ( finite_finite_a @ B )
% 3.12/1.62 => ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.62 => ( finite_finite_a @ A ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_30_finite__has__maximal2,axiom,
% 3.12/1.62 ! [A: set_se1596668135od_a_a,A2: set_Product_prod_a_a] :
% 3.12/1.62 ( ( finite1145471536od_a_a @ A )
% 3.12/1.62 => ( ( member1838126896od_a_a @ A2 @ A )
% 3.12/1.62 => ? [X: set_Product_prod_a_a] :
% 3.12/1.62 ( ( member1838126896od_a_a @ X @ A )
% 3.12/1.62 & ( ord_le1824328871od_a_a @ A2 @ X )
% 3.12/1.62 & ! [Xa: set_Product_prod_a_a] :
% 3.12/1.62 ( ( member1838126896od_a_a @ Xa @ A )
% 3.12/1.62 => ( ( ord_le1824328871od_a_a @ X @ Xa )
% 3.12/1.62 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_31_finite__has__maximal2,axiom,
% 3.12/1.62 ! [A: set_nat,A2: nat] :
% 3.12/1.62 ( ( finite_finite_nat @ A )
% 3.12/1.62 => ( ( member_nat @ A2 @ A )
% 3.12/1.62 => ? [X: nat] :
% 3.12/1.62 ( ( member_nat @ X @ A )
% 3.12/1.62 & ( ord_less_eq_nat @ A2 @ X )
% 3.12/1.62 & ! [Xa: nat] :
% 3.12/1.62 ( ( member_nat @ Xa @ A )
% 3.12/1.62 => ( ( ord_less_eq_nat @ X @ Xa )
% 3.12/1.62 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_32_finite__has__maximal2,axiom,
% 3.12/1.62 ! [A: set_set_nat,A2: set_nat] :
% 3.12/1.62 ( ( finite2012248349et_nat @ A )
% 3.12/1.62 => ( ( member_set_nat @ A2 @ A )
% 3.12/1.62 => ? [X: set_nat] :
% 3.12/1.62 ( ( member_set_nat @ X @ A )
% 3.12/1.62 & ( ord_less_eq_set_nat @ A2 @ X )
% 3.12/1.62 & ! [Xa: set_nat] :
% 3.12/1.62 ( ( member_set_nat @ Xa @ A )
% 3.12/1.62 => ( ( ord_less_eq_set_nat @ X @ Xa )
% 3.12/1.62 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_33_finite__has__maximal2,axiom,
% 3.12/1.62 ! [A: set_set_a,A2: set_a] :
% 3.12/1.62 ( ( finite_finite_set_a @ A )
% 3.12/1.62 => ( ( member_set_a @ A2 @ A )
% 3.12/1.62 => ? [X: set_a] :
% 3.12/1.62 ( ( member_set_a @ X @ A )
% 3.12/1.62 & ( ord_less_eq_set_a @ A2 @ X )
% 3.12/1.62 & ! [Xa: set_a] :
% 3.12/1.62 ( ( member_set_a @ Xa @ A )
% 3.12/1.62 => ( ( ord_less_eq_set_a @ X @ Xa )
% 3.12/1.62 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_34_finite__Collect__conjI,axiom,
% 3.12/1.62 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.12/1.62 ( ( ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
% 3.12/1.62 | ( finite179568208od_a_a @ ( collec645855634od_a_a @ Q ) ) )
% 3.12/1.62 => ( finite179568208od_a_a
% 3.12/1.62 @ ( collec645855634od_a_a
% 3.12/1.62 @ ^ [X2: product_prod_a_a] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 & ( Q @ X2 ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_35_finite__Collect__conjI,axiom,
% 3.12/1.62 ! [P: a > $o,Q: a > $o] :
% 3.12/1.62 ( ( ( finite_finite_a @ ( collect_a @ P ) )
% 3.12/1.62 | ( finite_finite_a @ ( collect_a @ Q ) ) )
% 3.12/1.62 => ( finite_finite_a
% 3.12/1.62 @ ( collect_a
% 3.12/1.62 @ ^ [X2: a] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 & ( Q @ X2 ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_36_finite__Collect__conjI,axiom,
% 3.12/1.62 ! [P: nat > $o,Q: nat > $o] :
% 3.12/1.62 ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 3.12/1.62 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 3.12/1.62 => ( finite_finite_nat
% 3.12/1.62 @ ( collect_nat
% 3.12/1.62 @ ^ [X2: nat] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 & ( Q @ X2 ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_37_finite__Collect__conjI,axiom,
% 3.12/1.62 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 3.12/1.62 ( ( ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
% 3.12/1.62 | ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) )
% 3.12/1.62 => ( finite772653738at_nat
% 3.12/1.62 @ ( collec7649004at_nat
% 3.12/1.62 @ ^ [X2: product_prod_nat_nat] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 & ( Q @ X2 ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_38_finite__Collect__conjI,axiom,
% 3.12/1.62 ! [P: produc1572603623od_a_a > $o,Q: produc1572603623od_a_a > $o] :
% 3.12/1.62 ( ( ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
% 3.12/1.62 | ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ Q ) ) )
% 3.12/1.62 => ( finite1664988688od_a_a
% 3.12/1.62 @ ( collec1635618130od_a_a
% 3.12/1.62 @ ^ [X2: produc1572603623od_a_a] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 & ( Q @ X2 ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_39_finite__Collect__disjI,axiom,
% 3.12/1.62 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.12/1.62 ( ( finite179568208od_a_a
% 3.12/1.62 @ ( collec645855634od_a_a
% 3.12/1.62 @ ^ [X2: product_prod_a_a] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 | ( Q @ X2 ) ) ) )
% 3.12/1.62 = ( ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
% 3.12/1.62 & ( finite179568208od_a_a @ ( collec645855634od_a_a @ Q ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_40_finite__Collect__disjI,axiom,
% 3.12/1.62 ! [P: a > $o,Q: a > $o] :
% 3.12/1.62 ( ( finite_finite_a
% 3.12/1.62 @ ( collect_a
% 3.12/1.62 @ ^ [X2: a] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 | ( Q @ X2 ) ) ) )
% 3.12/1.62 = ( ( finite_finite_a @ ( collect_a @ P ) )
% 3.12/1.62 & ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_41_finite__Collect__disjI,axiom,
% 3.12/1.62 ! [P: nat > $o,Q: nat > $o] :
% 3.12/1.62 ( ( finite_finite_nat
% 3.12/1.62 @ ( collect_nat
% 3.12/1.62 @ ^ [X2: nat] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 | ( Q @ X2 ) ) ) )
% 3.12/1.62 = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 3.12/1.62 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_42_finite__Collect__disjI,axiom,
% 3.12/1.62 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 3.12/1.62 ( ( finite772653738at_nat
% 3.12/1.62 @ ( collec7649004at_nat
% 3.12/1.62 @ ^ [X2: product_prod_nat_nat] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 | ( Q @ X2 ) ) ) )
% 3.12/1.62 = ( ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
% 3.12/1.62 & ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_43_finite__Collect__disjI,axiom,
% 3.12/1.62 ! [P: produc1572603623od_a_a > $o,Q: produc1572603623od_a_a > $o] :
% 3.12/1.62 ( ( finite1664988688od_a_a
% 3.12/1.62 @ ( collec1635618130od_a_a
% 3.12/1.62 @ ^ [X2: produc1572603623od_a_a] :
% 3.12/1.62 ( ( P @ X2 )
% 3.12/1.62 | ( Q @ X2 ) ) ) )
% 3.12/1.62 = ( ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
% 3.12/1.62 & ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ Q ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_44_finite__Collect__subsets,axiom,
% 3.12/1.62 ! [A: set_Pr1986765409at_nat] :
% 3.12/1.62 ( ( finite772653738at_nat @ A )
% 3.12/1.62 => ( finite1457549322at_nat
% 3.12/1.62 @ ( collec1606769740at_nat
% 3.12/1.62 @ ^ [B2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ B2 @ A ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_45_finite__Collect__subsets,axiom,
% 3.12/1.62 ! [A: set_Pr1948701895od_a_a] :
% 3.12/1.62 ( ( finite1664988688od_a_a @ A )
% 3.12/1.62 => ( finite323969008od_a_a
% 3.12/1.62 @ ( collec453062450od_a_a
% 3.12/1.62 @ ^ [B2: set_Pr1948701895od_a_a] : ( ord_le456379495od_a_a @ B2 @ A ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_46_finite__Collect__subsets,axiom,
% 3.12/1.62 ! [A: set_Product_prod_a_a] :
% 3.12/1.62 ( ( finite179568208od_a_a @ A )
% 3.12/1.62 => ( finite1145471536od_a_a
% 3.12/1.62 @ ( collec183727474od_a_a
% 3.12/1.62 @ ^ [B2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ B2 @ A ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_47_finite__Collect__subsets,axiom,
% 3.12/1.62 ! [A: set_nat] :
% 3.12/1.62 ( ( finite_finite_nat @ A )
% 3.12/1.62 => ( finite2012248349et_nat
% 3.12/1.62 @ ( collect_set_nat
% 3.12/1.62 @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_48_finite__Collect__subsets,axiom,
% 3.12/1.62 ! [A: set_a] :
% 3.12/1.62 ( ( finite_finite_a @ A )
% 3.12/1.62 => ( finite_finite_set_a
% 3.12/1.62 @ ( collect_set_a
% 3.12/1.62 @ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_49_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_a,B: a > set_nat] :
% 3.12/1.62 ( ( finite_finite_a @ A )
% 3.12/1.62 => ( ! [A3: a] :
% 3.12/1.62 ( ( member_a @ A3 @ A )
% 3.12/1.62 => ( finite_finite_nat @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite1743148308_a_nat @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_50_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_nat,B: nat > set_a] :
% 3.12/1.62 ( ( finite_finite_nat @ A )
% 3.12/1.62 => ( ! [A3: nat] :
% 3.12/1.62 ( ( member_nat @ A3 @ A )
% 3.12/1.62 => ( finite_finite_a @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite1808550458_nat_a @ ( product_Sigma_nat_a @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_51_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_nat,B: nat > set_nat] :
% 3.12/1.62 ( ( finite_finite_nat @ A )
% 3.12/1.62 => ( ! [A3: nat] :
% 3.12/1.62 ( ( member_nat @ A3 @ A )
% 3.12/1.62 => ( finite_finite_nat @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite772653738at_nat @ ( produc45129834at_nat @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_52_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_a,B: a > set_a] :
% 3.12/1.62 ( ( finite_finite_a @ A )
% 3.12/1.62 => ( ! [A3: a] :
% 3.12/1.62 ( ( member_a @ A3 @ A )
% 3.12/1.62 => ( finite_finite_a @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite179568208od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_53_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 3.12/1.62 ( ( finite179568208od_a_a @ A )
% 3.12/1.62 => ( ! [A3: product_prod_a_a] :
% 3.12/1.62 ( ( member449909584od_a_a @ A3 @ A )
% 3.12/1.62 => ( finite_finite_a @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite1919032935_a_a_a @ ( produc1282482655_a_a_a @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_54_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_Product_prod_a_a,B: product_prod_a_a > set_nat] :
% 3.12/1.62 ( ( finite179568208od_a_a @ A )
% 3.12/1.62 => ( ! [A3: product_prod_a_a] :
% 3.12/1.62 ( ( member449909584od_a_a @ A3 @ A )
% 3.12/1.62 => ( finite_finite_nat @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite1837575485_a_nat @ ( produc931712687_a_nat @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_55_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_a,B: a > set_Product_prod_a_a] :
% 3.12/1.62 ( ( finite_finite_a @ A )
% 3.12/1.62 => ( ! [A3: a] :
% 3.12/1.62 ( ( member_a @ A3 @ A )
% 3.12/1.62 => ( finite179568208od_a_a @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite676513017od_a_a @ ( produc520147185od_a_a @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_56_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_a,B: a > set_Pr1986765409at_nat] :
% 3.12/1.62 ( ( finite_finite_a @ A )
% 3.12/1.62 => ( ! [A3: a] :
% 3.12/1.62 ( ( member_a @ A3 @ A )
% 3.12/1.62 => ( finite772653738at_nat @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite942416723at_nat @ ( produc292491723at_nat @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_57_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_nat,B: nat > set_Product_prod_a_a] :
% 3.12/1.62 ( ( finite_finite_nat @ A )
% 3.12/1.62 => ( ! [A3: nat] :
% 3.12/1.62 ( ( member_nat @ A3 @ A )
% 3.12/1.62 => ( finite179568208od_a_a @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite1297454819od_a_a @ ( produc1182842125od_a_a @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_58_finite__SigmaI,axiom,
% 3.12/1.62 ! [A: set_nat,B: nat > set_Pr1986765409at_nat] :
% 3.12/1.62 ( ( finite_finite_nat @ A )
% 3.12/1.62 => ( ! [A3: nat] :
% 3.12/1.62 ( ( member_nat @ A3 @ A )
% 3.12/1.62 => ( finite772653738at_nat @ ( B @ A3 ) ) )
% 3.12/1.62 => ( finite277291581at_nat @ ( produc894163943at_nat @ A @ B ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_59_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 3.12/1.62 ! [G2: pair_p125712459t_unit] :
% 3.12/1.62 ( ( pair_p1802376898raph_a @ G2 )
% 3.12/1.62 => ( pair_p1864019935ioms_a @ G2 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_60_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 3.12/1.62 ! [G2: pair_p1914262621t_unit] :
% 3.12/1.62 ( ( pair_p128415500ph_nat @ G2 )
% 3.12/1.62 => ( pair_p1027063983ms_nat @ G2 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_61_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 3.12/1.62 ! [G2: pair_p1765063010t_unit] :
% 3.12/1.62 ( ( pair_p374947051od_a_a @ G2 )
% 3.12/1.62 => ( pair_p504738056od_a_a @ G2 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_62_not__finite__existsD,axiom,
% 3.12/1.62 ! [P: product_prod_a_a > $o] :
% 3.12/1.62 ( ~ ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
% 3.12/1.62 => ? [X_1: product_prod_a_a] : ( P @ X_1 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_63_not__finite__existsD,axiom,
% 3.12/1.62 ! [P: a > $o] :
% 3.12/1.62 ( ~ ( finite_finite_a @ ( collect_a @ P ) )
% 3.12/1.62 => ? [X_1: a] : ( P @ X_1 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_64_not__finite__existsD,axiom,
% 3.12/1.62 ! [P: nat > $o] :
% 3.12/1.62 ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 3.12/1.62 => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_65_not__finite__existsD,axiom,
% 3.12/1.62 ! [P: product_prod_nat_nat > $o] :
% 3.12/1.62 ( ~ ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
% 3.12/1.62 => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_66_not__finite__existsD,axiom,
% 3.12/1.62 ! [P: produc1572603623od_a_a > $o] :
% 3.12/1.62 ( ~ ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
% 3.12/1.62 => ? [X_1: produc1572603623od_a_a] : ( P @ X_1 ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_67_pigeonhole__infinite__rel,axiom,
% 3.12/1.62 ! [A: set_a,B: set_a,R: a > a > $o] :
% 3.12/1.62 ( ~ ( finite_finite_a @ A )
% 3.12/1.62 => ( ( finite_finite_a @ B )
% 3.12/1.62 => ( ! [X: a] :
% 3.12/1.62 ( ( member_a @ X @ A )
% 3.12/1.62 => ? [Xa: a] :
% 3.12/1.62 ( ( member_a @ Xa @ B )
% 3.12/1.62 & ( R @ X @ Xa ) ) )
% 3.12/1.62 => ? [X: a] :
% 3.12/1.62 ( ( member_a @ X @ B )
% 3.12/1.62 & ~ ( finite_finite_a
% 3.12/1.62 @ ( collect_a
% 3.12/1.62 @ ^ [A4: a] :
% 3.12/1.62 ( ( member_a @ A4 @ A )
% 3.12/1.62 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_68_pigeonhole__infinite__rel,axiom,
% 3.12/1.62 ! [A: set_a,B: set_nat,R: a > nat > $o] :
% 3.12/1.62 ( ~ ( finite_finite_a @ A )
% 3.12/1.62 => ( ( finite_finite_nat @ B )
% 3.12/1.62 => ( ! [X: a] :
% 3.12/1.62 ( ( member_a @ X @ A )
% 3.12/1.62 => ? [Xa: nat] :
% 3.12/1.62 ( ( member_nat @ Xa @ B )
% 3.12/1.62 & ( R @ X @ Xa ) ) )
% 3.12/1.62 => ? [X: nat] :
% 3.12/1.62 ( ( member_nat @ X @ B )
% 3.12/1.62 & ~ ( finite_finite_a
% 3.12/1.62 @ ( collect_a
% 3.12/1.62 @ ^ [A4: a] :
% 3.12/1.62 ( ( member_a @ A4 @ A )
% 3.12/1.62 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.62
% 3.12/1.62 thf(fact_69_pigeonhole__infinite__rel,axiom,
% 3.12/1.62 ! [A: set_nat,B: set_a,R: nat > a > $o] :
% 3.12/1.62 ( ~ ( finite_finite_nat @ A )
% 3.12/1.62 => ( ( finite_finite_a @ B )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ? [Xa: a] :
% 3.12/1.63 ( ( member_a @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: a] :
% 3.12/1.63 ( ( member_a @ X @ B )
% 3.12/1.63 & ~ ( finite_finite_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [A4: nat] :
% 3.12/1.63 ( ( member_nat @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_70_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat,R: nat > nat > $o] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite_finite_nat @ B )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ? [Xa: nat] :
% 3.12/1.63 ( ( member_nat @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ B )
% 3.12/1.63 & ~ ( finite_finite_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [A4: nat] :
% 3.12/1.63 ( ( member_nat @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_71_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_a,R: product_prod_a_a > a > $o] :
% 3.12/1.63 ( ~ ( finite179568208od_a_a @ A )
% 3.12/1.63 => ( ( finite_finite_a @ B )
% 3.12/1.63 => ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ A )
% 3.12/1.63 => ? [Xa: a] :
% 3.12/1.63 ( ( member_a @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: a] :
% 3.12/1.63 ( ( member_a @ X @ B )
% 3.12/1.63 & ~ ( finite179568208od_a_a
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [A4: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_72_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_nat,R: product_prod_a_a > nat > $o] :
% 3.12/1.63 ( ~ ( finite179568208od_a_a @ A )
% 3.12/1.63 => ( ( finite_finite_nat @ B )
% 3.12/1.63 => ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ A )
% 3.12/1.63 => ? [Xa: nat] :
% 3.12/1.63 ( ( member_nat @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ B )
% 3.12/1.63 & ~ ( finite179568208od_a_a
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [A4: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_73_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Product_prod_a_a,R: a > product_prod_a_a > $o] :
% 3.12/1.63 ( ~ ( finite_finite_a @ A )
% 3.12/1.63 => ( ( finite179568208od_a_a @ B )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ A )
% 3.12/1.63 => ? [Xa: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ B )
% 3.12/1.63 & ~ ( finite_finite_a
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [A4: a] :
% 3.12/1.63 ( ( member_a @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_74_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Pr1986765409at_nat,R: a > product_prod_nat_nat > $o] :
% 3.12/1.63 ( ~ ( finite_finite_a @ A )
% 3.12/1.63 => ( ( finite772653738at_nat @ B )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ A )
% 3.12/1.63 => ? [Xa: product_prod_nat_nat] :
% 3.12/1.63 ( ( member701585322at_nat @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: product_prod_nat_nat] :
% 3.12/1.63 ( ( member701585322at_nat @ X @ B )
% 3.12/1.63 & ~ ( finite_finite_a
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [A4: a] :
% 3.12/1.63 ( ( member_a @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_75_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Product_prod_a_a,R: nat > product_prod_a_a > $o] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite179568208od_a_a @ B )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ? [Xa: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ B )
% 3.12/1.63 & ~ ( finite_finite_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [A4: nat] :
% 3.12/1.63 ( ( member_nat @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_76_pigeonhole__infinite__rel,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Pr1986765409at_nat,R: nat > product_prod_nat_nat > $o] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite772653738at_nat @ B )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ? [Xa: product_prod_nat_nat] :
% 3.12/1.63 ( ( member701585322at_nat @ Xa @ B )
% 3.12/1.63 & ( R @ X @ Xa ) ) )
% 3.12/1.63 => ? [X: product_prod_nat_nat] :
% 3.12/1.63 ( ( member701585322at_nat @ X @ B )
% 3.12/1.63 & ~ ( finite_finite_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [A4: nat] :
% 3.12/1.63 ( ( member_nat @ A4 @ A )
% 3.12/1.63 & ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_77_pair__fin__digraph_Opair__fin__digraph,axiom,
% 3.12/1.63 ! [G2: pair_p125712459t_unit] :
% 3.12/1.63 ( ( pair_p1802376898raph_a @ G2 )
% 3.12/1.63 => ( pair_p1802376898raph_a @ G2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_78_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_nat] :
% 3.12/1.63 ( ( finite_finite_a @ A )
% 3.12/1.63 => ( ( finite_finite_nat @ B )
% 3.12/1.63 => ( finite1743148308_a_nat
% 3.12/1.63 @ ( product_Sigma_a_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_79_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_a] :
% 3.12/1.63 ( ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite_finite_a @ B )
% 3.12/1.63 => ( finite1808550458_nat_a
% 3.12/1.63 @ ( product_Sigma_nat_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_80_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite_finite_nat @ B )
% 3.12/1.63 => ( finite772653738at_nat
% 3.12/1.63 @ ( produc45129834at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_81_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ( finite_finite_a @ A )
% 3.12/1.63 => ( ( finite_finite_a @ B )
% 3.12/1.63 => ( finite179568208od_a_a
% 3.12/1.63 @ ( product_Sigma_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_82_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_a] :
% 3.12/1.63 ( ( finite179568208od_a_a @ A )
% 3.12/1.63 => ( ( finite_finite_a @ B )
% 3.12/1.63 => ( finite1919032935_a_a_a
% 3.12/1.63 @ ( produc1282482655_a_a_a @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_83_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_nat] :
% 3.12/1.63 ( ( finite179568208od_a_a @ A )
% 3.12/1.63 => ( ( finite_finite_nat @ B )
% 3.12/1.63 => ( finite1837575485_a_nat
% 3.12/1.63 @ ( produc931712687_a_nat @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_84_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( finite_finite_a @ A )
% 3.12/1.63 => ( ( finite179568208od_a_a @ B )
% 3.12/1.63 => ( finite676513017od_a_a
% 3.12/1.63 @ ( produc520147185od_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_85_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ( finite_finite_a @ A )
% 3.12/1.63 => ( ( finite772653738at_nat @ B )
% 3.12/1.63 => ( finite942416723at_nat
% 3.12/1.63 @ ( produc292491723at_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_86_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite179568208od_a_a @ B )
% 3.12/1.63 => ( finite1297454819od_a_a
% 3.12/1.63 @ ( produc1182842125od_a_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_87_finite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( finite772653738at_nat @ B )
% 3.12/1.63 => ( finite277291581at_nat
% 3.12/1.63 @ ( produc894163943at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_88_pair__fin__digraph_Opair__finite__verts,axiom,
% 3.12/1.63 ! [G2: pair_p2041852168t_unit] :
% 3.12/1.63 ( ( pair_p752841413at_nat @ G2 )
% 3.12/1.63 => ( finite772653738at_nat @ ( pair_p210955889t_unit @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_89_pair__fin__digraph_Opair__finite__verts,axiom,
% 3.12/1.63 ! [G2: pair_p1593840546t_unit] :
% 3.12/1.63 ( ( pair_p159207083od_a_a @ G2 )
% 3.12/1.63 => ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_90_pair__fin__digraph_Opair__finite__verts,axiom,
% 3.12/1.63 ! [G2: pair_p125712459t_unit] :
% 3.12/1.63 ( ( pair_p1802376898raph_a @ G2 )
% 3.12/1.63 => ( finite_finite_a @ ( pair_p1047056820t_unit @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_91_pair__fin__digraph_Opair__finite__verts,axiom,
% 3.12/1.63 ! [G2: pair_p1914262621t_unit] :
% 3.12/1.63 ( ( pair_p128415500ph_nat @ G2 )
% 3.12/1.63 => ( finite_finite_nat @ ( pair_p1677060310t_unit @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_92_pair__fin__digraph_Opair__finite__verts,axiom,
% 3.12/1.63 ! [G2: pair_p1765063010t_unit] :
% 3.12/1.63 ( ( pair_p374947051od_a_a @ G2 )
% 3.12/1.63 => ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_93_finite__has__minimal2,axiom,
% 3.12/1.63 ! [A: set_se1596668135od_a_a,A2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( finite1145471536od_a_a @ A )
% 3.12/1.63 => ( ( member1838126896od_a_a @ A2 @ A )
% 3.12/1.63 => ? [X: set_Product_prod_a_a] :
% 3.12/1.63 ( ( member1838126896od_a_a @ X @ A )
% 3.12/1.63 & ( ord_le1824328871od_a_a @ X @ A2 )
% 3.12/1.63 & ! [Xa: set_Product_prod_a_a] :
% 3.12/1.63 ( ( member1838126896od_a_a @ Xa @ A )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ Xa @ X )
% 3.12/1.63 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_94_finite__has__minimal2,axiom,
% 3.12/1.63 ! [A: set_nat,A2: nat] :
% 3.12/1.63 ( ( finite_finite_nat @ A )
% 3.12/1.63 => ( ( member_nat @ A2 @ A )
% 3.12/1.63 => ? [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 & ( ord_less_eq_nat @ X @ A2 )
% 3.12/1.63 & ! [Xa: nat] :
% 3.12/1.63 ( ( member_nat @ Xa @ A )
% 3.12/1.63 => ( ( ord_less_eq_nat @ Xa @ X )
% 3.12/1.63 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_95_finite__has__minimal2,axiom,
% 3.12/1.63 ! [A: set_set_nat,A2: set_nat] :
% 3.12/1.63 ( ( finite2012248349et_nat @ A )
% 3.12/1.63 => ( ( member_set_nat @ A2 @ A )
% 3.12/1.63 => ? [X: set_nat] :
% 3.12/1.63 ( ( member_set_nat @ X @ A )
% 3.12/1.63 & ( ord_less_eq_set_nat @ X @ A2 )
% 3.12/1.63 & ! [Xa: set_nat] :
% 3.12/1.63 ( ( member_set_nat @ Xa @ A )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ Xa @ X )
% 3.12/1.63 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_96_finite__has__minimal2,axiom,
% 3.12/1.63 ! [A: set_set_a,A2: set_a] :
% 3.12/1.63 ( ( finite_finite_set_a @ A )
% 3.12/1.63 => ( ( member_set_a @ A2 @ A )
% 3.12/1.63 => ? [X: set_a] :
% 3.12/1.63 ( ( member_set_a @ X @ A )
% 3.12/1.63 & ( ord_less_eq_set_a @ X @ A2 )
% 3.12/1.63 & ! [Xa: set_a] :
% 3.12/1.63 ( ( member_set_a @ Xa @ A )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ Xa @ X )
% 3.12/1.63 => ( X = Xa ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_97_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_nat] :
% 3.12/1.63 ( ~ ( finite_finite_a @ A )
% 3.12/1.63 => ( ~ ( finite_finite_nat @ B )
% 3.12/1.63 => ~ ( finite1743148308_a_nat
% 3.12/1.63 @ ( product_Sigma_a_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_98_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_a] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ~ ( finite_finite_a @ B )
% 3.12/1.63 => ~ ( finite1808550458_nat_a
% 3.12/1.63 @ ( product_Sigma_nat_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_99_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ~ ( finite_finite_nat @ B )
% 3.12/1.63 => ~ ( finite772653738at_nat
% 3.12/1.63 @ ( produc45129834at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_100_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ~ ( finite_finite_a @ A )
% 3.12/1.63 => ( ~ ( finite_finite_a @ B )
% 3.12/1.63 => ~ ( finite179568208od_a_a
% 3.12/1.63 @ ( product_Sigma_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_101_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_a] :
% 3.12/1.63 ( ~ ( finite179568208od_a_a @ A )
% 3.12/1.63 => ( ~ ( finite_finite_a @ B )
% 3.12/1.63 => ~ ( finite1919032935_a_a_a
% 3.12/1.63 @ ( produc1282482655_a_a_a @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_102_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_nat] :
% 3.12/1.63 ( ~ ( finite179568208od_a_a @ A )
% 3.12/1.63 => ( ~ ( finite_finite_nat @ B )
% 3.12/1.63 => ~ ( finite1837575485_a_nat
% 3.12/1.63 @ ( produc931712687_a_nat @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_103_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ~ ( finite_finite_a @ A )
% 3.12/1.63 => ( ~ ( finite179568208od_a_a @ B )
% 3.12/1.63 => ~ ( finite676513017od_a_a
% 3.12/1.63 @ ( produc520147185od_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_104_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ~ ( finite_finite_a @ A )
% 3.12/1.63 => ( ~ ( finite772653738at_nat @ B )
% 3.12/1.63 => ~ ( finite942416723at_nat
% 3.12/1.63 @ ( produc292491723at_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_105_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ~ ( finite179568208od_a_a @ B )
% 3.12/1.63 => ~ ( finite1297454819od_a_a
% 3.12/1.63 @ ( produc1182842125od_a_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_106_infinite__cartesian__product,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ~ ( finite_finite_nat @ A )
% 3.12/1.63 => ( ~ ( finite772653738at_nat @ B )
% 3.12/1.63 => ~ ( finite277291581at_nat
% 3.12/1.63 @ ( produc894163943at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_107_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X3 @ C )
% 3.12/1.63 => ( ( ord_le456379495od_a_a
% 3.12/1.63 @ ( produc304751368od_a_a @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : C )
% 3.12/1.63 @ ( produc304751368od_a_a @ B
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : C ) )
% 3.12/1.63 = ( ord_le1824328871od_a_a @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_108_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: nat,C: set_nat,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( member_nat @ X3 @ C )
% 3.12/1.63 => ( ( ord_le492294332_a_nat
% 3.12/1.63 @ ( produc931712687_a_nat @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : C )
% 3.12/1.63 @ ( produc931712687_a_nat @ B
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : C ) )
% 3.12/1.63 = ( ord_le1824328871od_a_a @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_109_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( member449909584od_a_a @ X3 @ C )
% 3.12/1.63 => ( ( ord_le2084554594od_a_a
% 3.12/1.63 @ ( produc1182842125od_a_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : C )
% 3.12/1.63 @ ( produc1182842125od_a_a @ B
% 3.12/1.63 @ ^ [Uu: nat] : C ) )
% 3.12/1.63 = ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_110_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: nat,C: set_nat,A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( member_nat @ X3 @ C )
% 3.12/1.63 => ( ( ord_le841296385at_nat
% 3.12/1.63 @ ( produc45129834at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : C )
% 3.12/1.63 @ ( produc45129834at_nat @ B
% 3.12/1.63 @ ^ [Uu: nat] : C ) )
% 3.12/1.63 = ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_111_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_a,B: set_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X3 @ C )
% 3.12/1.63 => ( ( ord_le1816232656od_a_a
% 3.12/1.63 @ ( produc520147185od_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : C )
% 3.12/1.63 @ ( produc520147185od_a_a @ B
% 3.12/1.63 @ ^ [Uu: a] : C ) )
% 3.12/1.63 = ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_112_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: nat,C: set_nat,A: set_a,B: set_a] :
% 3.12/1.63 ( ( member_nat @ X3 @ C )
% 3.12/1.63 => ( ( ord_le2073555219_a_nat
% 3.12/1.63 @ ( product_Sigma_a_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : C )
% 3.12/1.63 @ ( product_Sigma_a_nat @ B
% 3.12/1.63 @ ^ [Uu: a] : C ) )
% 3.12/1.63 = ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_113_Times__subset__cancel2,axiom,
% 3.12/1.63 ! [X3: a,C: set_a,A: set_a,B: set_a] :
% 3.12/1.63 ( ( member_a @ X3 @ C )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a
% 3.12/1.63 @ ( product_Sigma_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : C )
% 3.12/1.63 @ ( product_Sigma_a_a @ B
% 3.12/1.63 @ ^ [Uu: a] : C ) )
% 3.12/1.63 = ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_114_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a,D: product_prod_a_a > set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ C )
% 3.12/1.63 => ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ A )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le456379495od_a_a @ ( produc304751368od_a_a @ A @ B ) @ ( produc304751368od_a_a @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_115_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_nat,D: product_prod_a_a > set_nat] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ C )
% 3.12/1.63 => ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ A )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le492294332_a_nat @ ( produc931712687_a_nat @ A @ B ) @ ( produc931712687_a_nat @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_116_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_a,D: product_prod_a_a > set_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ C )
% 3.12/1.63 => ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ A )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le677315902_a_a_a @ ( produc1282482655_a_a_a @ A @ B ) @ ( produc1282482655_a_a_a @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_117_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_nat,C: set_nat,B: nat > set_Product_prod_a_a,D: nat > set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ C )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le2084554594od_a_a @ ( produc1182842125od_a_a @ A @ B ) @ ( produc1182842125od_a_a @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_118_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_nat,C: set_nat,B: nat > set_nat,D: nat > set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ C )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le841296385at_nat @ ( produc45129834at_nat @ A @ B ) @ ( produc45129834at_nat @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_119_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_nat,C: set_nat,B: nat > set_a,D: nat > set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ C )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le344568633_nat_a @ ( product_Sigma_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_120_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_a,C: set_a,B: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ C )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ A )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le1816232656od_a_a @ ( produc520147185od_a_a @ A @ B ) @ ( produc520147185od_a_a @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_121_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_a,C: set_a,B: a > set_nat,D: a > set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ C )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ A )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le2073555219_a_nat @ ( product_Sigma_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_122_Sigma__mono,axiom,
% 3.12/1.63 ! [A: set_a,C: set_a,B: a > set_a,D: a > set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ C )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ A )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ ( product_Sigma_a_a @ A @ B ) @ ( product_Sigma_a_a @ C @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_123_subsetI,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ A )
% 3.12/1.63 => ( member449909584od_a_a @ X @ B ) )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A @ B ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_124_subsetI,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ A )
% 3.12/1.63 => ( member_nat @ X @ B ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_125_subsetI,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ A )
% 3.12/1.63 => ( member_a @ X @ B ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_126_subset__antisym,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B @ A )
% 3.12/1.63 => ( A = B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_127_subset__antisym,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B @ A )
% 3.12/1.63 => ( A = B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_128_subset__antisym,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B @ A )
% 3.12/1.63 => ( A = B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_129_order__refl,axiom,
% 3.12/1.63 ! [X3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ X3 @ X3 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_130_order__refl,axiom,
% 3.12/1.63 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_131_order__refl,axiom,
% 3.12/1.63 ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_132_order__refl,axiom,
% 3.12/1.63 ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_133_pair__pre__digraph_Oequality,axiom,
% 3.12/1.63 ! [R2: pair_p125712459t_unit,R3: pair_p125712459t_unit] :
% 3.12/1.63 ( ( ( pair_p1047056820t_unit @ R2 )
% 3.12/1.63 = ( pair_p1047056820t_unit @ R3 ) )
% 3.12/1.63 => ( ( ( pair_p133601421t_unit @ R2 )
% 3.12/1.63 = ( pair_p133601421t_unit @ R3 ) )
% 3.12/1.63 => ( ( ( pair_p1896222615t_unit @ R2 )
% 3.12/1.63 = ( pair_p1896222615t_unit @ R3 ) )
% 3.12/1.63 => ( R2 = R3 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_134_pair__pre__digraph_Oequality,axiom,
% 3.12/1.63 ! [R2: pair_p1914262621t_unit,R3: pair_p1914262621t_unit] :
% 3.12/1.63 ( ( ( pair_p1677060310t_unit @ R2 )
% 3.12/1.63 = ( pair_p1677060310t_unit @ R3 ) )
% 3.12/1.63 => ( ( ( pair_p715279805t_unit @ R2 )
% 3.12/1.63 = ( pair_p715279805t_unit @ R3 ) )
% 3.12/1.63 => ( ( ( pair_p69470259t_unit @ R2 )
% 3.12/1.63 = ( pair_p69470259t_unit @ R3 ) )
% 3.12/1.63 => ( R2 = R3 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_135_pair__pre__digraph_Oequality,axiom,
% 3.12/1.63 ! [R2: pair_p1765063010t_unit,R3: pair_p1765063010t_unit] :
% 3.12/1.63 ( ( ( pair_p447552203t_unit @ R2 )
% 3.12/1.63 = ( pair_p447552203t_unit @ R3 ) )
% 3.12/1.63 => ( ( ( pair_p1783210148t_unit @ R2 )
% 3.12/1.63 = ( pair_p1783210148t_unit @ R3 ) )
% 3.12/1.63 => ( ( ( pair_p1984658862t_unit @ R2 )
% 3.12/1.63 = ( pair_p1984658862t_unit @ R3 ) )
% 3.12/1.63 => ( R2 = R3 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_136_Sigma__cong,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a,C: a > set_a,D: a > set_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ B )
% 3.12/1.63 => ( ( C @ X )
% 3.12/1.63 = ( D @ X ) ) )
% 3.12/1.63 => ( ( product_Sigma_a_a @ A @ C )
% 3.12/1.63 = ( product_Sigma_a_a @ B @ D ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_137_Times__eq__cancel2,axiom,
% 3.12/1.63 ! [X3: a,C: set_a,A: set_a,B: set_a] :
% 3.12/1.63 ( ( member_a @ X3 @ C )
% 3.12/1.63 => ( ( ( product_Sigma_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : C )
% 3.12/1.63 = ( product_Sigma_a_a @ B
% 3.12/1.63 @ ^ [Uu: a] : C ) )
% 3.12/1.63 = ( A = B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_138_Collect__subset,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 3.12/1.63 ( ord_le1824328871od_a_a
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) )
% 3.12/1.63 @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_139_Collect__subset,axiom,
% 3.12/1.63 ! [A: set_nat,P: nat > $o] :
% 3.12/1.63 ( ord_less_eq_set_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) )
% 3.12/1.63 @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_140_Collect__subset,axiom,
% 3.12/1.63 ! [A: set_a,P: a > $o] :
% 3.12/1.63 ( ord_less_eq_set_a
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) )
% 3.12/1.63 @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_141_less__eq__set__def,axiom,
% 3.12/1.63 ( ord_le1824328871od_a_a
% 3.12/1.63 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 3.12/1.63 ( ord_le1347718902_a_a_o
% 3.12/1.63 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ A5 )
% 3.12/1.63 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_142_less__eq__set__def,axiom,
% 3.12/1.63 ( ord_less_eq_set_nat
% 3.12/1.63 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.12/1.63 ( ord_less_eq_nat_o
% 3.12/1.63 @ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
% 3.12/1.63 @ ^ [X2: nat] : ( member_nat @ X2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_143_less__eq__set__def,axiom,
% 3.12/1.63 ( ord_less_eq_set_a
% 3.12/1.63 = ( ^ [A5: set_a,B2: set_a] :
% 3.12/1.63 ( ord_less_eq_a_o
% 3.12/1.63 @ ^ [X2: a] : ( member_a @ X2 @ A5 )
% 3.12/1.63 @ ^ [X2: a] : ( member_a @ X2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_144_dual__order_Oantisym,axiom,
% 3.12/1.63 ! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_145_dual__order_Oantisym,axiom,
% 3.12/1.63 ! [B3: nat,A2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_146_dual__order_Oantisym,axiom,
% 3.12/1.63 ! [B3: set_nat,A2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_147_dual__order_Oantisym,axiom,
% 3.12/1.63 ! [B3: set_a,A2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_148_dual__order_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ B4 @ A4 )
% 3.12/1.63 & ( ord_le1824328871od_a_a @ A4 @ B4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_149_dual__order_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: nat,B4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ B4 @ A4 )
% 3.12/1.63 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_150_dual__order_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: set_nat,B4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 3.12/1.63 & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_151_dual__order_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: set_a,B4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ B4 @ A4 )
% 3.12/1.63 & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_152_dual__order_Otrans,axiom,
% 3.12/1.63 ! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ C2 @ B3 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ C2 @ A2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_153_dual__order_Otrans,axiom,
% 3.12/1.63 ! [B3: nat,A2: nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ C2 @ B3 )
% 3.12/1.63 => ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_154_dual__order_Otrans,axiom,
% 3.12/1.63 ! [B3: set_nat,A2: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ C2 @ B3 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_155_dual__order_Otrans,axiom,
% 3.12/1.63 ! [B3: set_a,A2: set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ B3 @ A2 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ C2 @ B3 )
% 3.12/1.63 => ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_156_linorder__wlog,axiom,
% 3.12/1.63 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 3.12/1.63 ( ! [A3: nat,B5: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A3 @ B5 )
% 3.12/1.63 => ( P @ A3 @ B5 ) )
% 3.12/1.63 => ( ! [A3: nat,B5: nat] :
% 3.12/1.63 ( ( P @ B5 @ A3 )
% 3.12/1.63 => ( P @ A3 @ B5 ) )
% 3.12/1.63 => ( P @ A2 @ B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_157_dual__order_Orefl,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A2 @ A2 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_158_dual__order_Orefl,axiom,
% 3.12/1.63 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_159_dual__order_Orefl,axiom,
% 3.12/1.63 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_160_dual__order_Orefl,axiom,
% 3.12/1.63 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_161_order__trans,axiom,
% 3.12/1.63 ! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ Y2 @ Z2 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ X3 @ Z2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_162_order__trans,axiom,
% 3.12/1.63 ! [X3: nat,Y2: nat,Z2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 3.12/1.63 => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_163_order__trans,axiom,
% 3.12/1.63 ! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_164_order__trans,axiom,
% 3.12/1.63 ! [X3: set_a,Y2: set_a,Z2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ Y2 @ Z2 )
% 3.12/1.63 => ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_165_order__class_Oorder_Oantisym,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B3 @ A2 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_166_order__class_Oorder_Oantisym,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_167_order__class_Oorder_Oantisym,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_168_order__class_Oorder_Oantisym,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 3.12/1.63 => ( A2 = B3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_169_ord__le__eq__trans,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 3.12/1.63 => ( ( B3 = C2 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_170_ord__le__eq__trans,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( B3 = C2 )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_171_ord__le__eq__trans,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( B3 = C2 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_172_ord__le__eq__trans,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( B3 = C2 )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_173_mem__Collect__eq,axiom,
% 3.12/1.63 ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
% 3.12/1.63 ( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
% 3.12/1.63 = ( P @ A2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_174_mem__Collect__eq,axiom,
% 3.12/1.63 ! [A2: nat,P: nat > $o] :
% 3.12/1.63 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 3.12/1.63 = ( P @ A2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_175_Collect__mem__eq,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a] :
% 3.12/1.63 ( ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ A ) )
% 3.12/1.63 = A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_176_Collect__mem__eq,axiom,
% 3.12/1.63 ! [A: set_nat] :
% 3.12/1.63 ( ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
% 3.12/1.63 = A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_177_Collect__cong,axiom,
% 3.12/1.63 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.12/1.63 ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( P @ X )
% 3.12/1.63 = ( Q @ X ) )
% 3.12/1.63 => ( ( collec645855634od_a_a @ P )
% 3.12/1.63 = ( collec645855634od_a_a @ Q ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_178_Collect__cong,axiom,
% 3.12/1.63 ! [P: nat > $o,Q: nat > $o] :
% 3.12/1.63 ( ! [X: nat] :
% 3.12/1.63 ( ( P @ X )
% 3.12/1.63 = ( Q @ X ) )
% 3.12/1.63 => ( ( collect_nat @ P )
% 3.12/1.63 = ( collect_nat @ Q ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_179_ord__eq__le__trans,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( A2 = B3 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B3 @ C2 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_180_ord__eq__le__trans,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( A2 = B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_181_ord__eq__le__trans,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( A2 = B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_182_ord__eq__le__trans,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( A2 = B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_183_order__class_Oorder_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A4 @ B4 )
% 3.12/1.63 & ( ord_le1824328871od_a_a @ B4 @ A4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_184_order__class_Oorder_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: nat,B4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A4 @ B4 )
% 3.12/1.63 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_185_order__class_Oorder_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: set_nat,B4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 3.12/1.63 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_186_order__class_Oorder_Oeq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A4: set_a,B4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 3.12/1.63 & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_187_antisym__conv,axiom,
% 3.12/1.63 ! [Y2: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ Y2 @ X3 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
% 3.12/1.63 = ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_188_antisym__conv,axiom,
% 3.12/1.63 ! [Y2: nat,X3: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ Y2 @ X3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.12/1.63 = ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_189_antisym__conv,axiom,
% 3.12/1.63 ! [Y2: set_nat,X3: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ Y2 @ X3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ X3 @ Y2 )
% 3.12/1.63 = ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_190_antisym__conv,axiom,
% 3.12/1.63 ! [Y2: set_a,X3: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ Y2 @ X3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 3.12/1.63 = ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_191_le__cases3,axiom,
% 3.12/1.63 ! [X3: nat,Y2: nat,Z2: nat] :
% 3.12/1.63 ( ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.12/1.63 => ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
% 3.12/1.63 => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
% 3.12/1.63 => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
% 3.12/1.63 => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
% 3.12/1.63 => ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
% 3.12/1.63 => ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
% 3.12/1.63 => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
% 3.12/1.63 => ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 3.12/1.63 => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
% 3.12/1.63 => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
% 3.12/1.63 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_192_order_Otrans,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B3 @ C2 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_193_order_Otrans,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_194_order_Otrans,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_195_order_Otrans,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_196_le__cases,axiom,
% 3.12/1.63 ! [X3: nat,Y2: nat] :
% 3.12/1.63 ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
% 3.12/1.63 => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_197_eq__refl,axiom,
% 3.12/1.63 ! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( X3 = Y2 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ X3 @ Y2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_198_eq__refl,axiom,
% 3.12/1.63 ! [X3: nat,Y2: nat] :
% 3.12/1.63 ( ( X3 = Y2 )
% 3.12/1.63 => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_199_eq__refl,axiom,
% 3.12/1.63 ! [X3: set_nat,Y2: set_nat] :
% 3.12/1.63 ( ( X3 = Y2 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_200_eq__refl,axiom,
% 3.12/1.63 ! [X3: set_a,Y2: set_a] :
% 3.12/1.63 ( ( X3 = Y2 )
% 3.12/1.63 => ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_201_linear,axiom,
% 3.12/1.63 ! [X3: nat,Y2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.12/1.63 | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_202_antisym,axiom,
% 3.12/1.63 ! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ Y2 @ X3 )
% 3.12/1.63 => ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_203_antisym,axiom,
% 3.12/1.63 ! [X3: nat,Y2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ Y2 @ X3 )
% 3.12/1.63 => ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_204_antisym,axiom,
% 3.12/1.63 ! [X3: set_nat,Y2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ Y2 @ X3 )
% 3.12/1.63 => ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_205_antisym,axiom,
% 3.12/1.63 ! [X3: set_a,Y2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ Y2 @ X3 )
% 3.12/1.63 => ( X3 = Y2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_206_eq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X2 @ Y3 )
% 3.12/1.63 & ( ord_le1824328871od_a_a @ Y3 @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_207_eq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [X2: nat,Y3: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 3.12/1.63 & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_208_eq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [X2: set_nat,Y3: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 3.12/1.63 & ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_209_eq__iff,axiom,
% 3.12/1.63 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [X2: set_a,Y3: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X2 @ Y3 )
% 3.12/1.63 & ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_210_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,F: nat > nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_211_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_212_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,F: nat > set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_213_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_214_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,F: set_a > nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_215_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_216_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,F: set_nat > set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_217_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,F: set_a > set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_218_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_219_ord__le__eq__subst,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ( F @ B3 )
% 3.12/1.63 = C2 )
% 3.12/1.63 => ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_220_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: nat,F: nat > nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_221_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_222_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: set_a,F: nat > set_a,B3: nat,C2: nat] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_223_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_224_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: nat,F: set_a > nat,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_225_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_226_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: set_a,F: set_nat > set_a,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_227_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: set_nat,F: set_a > set_nat,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_228_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_229_ord__eq__le__subst,axiom,
% 3.12/1.63 ! [A2: nat,F: set_Product_prod_a_a > nat,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( A2
% 3.12/1.63 = ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_230_order__subst2,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,F: nat > nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_231_order__subst2,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_232_order__subst2,axiom,
% 3.12/1.63 ! [A2: nat,B3: nat,F: nat > set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_233_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_234_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,F: set_a > nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_235_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_236_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_nat,B3: set_nat,F: set_nat > set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_237_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,F: set_a > set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_238_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_239_order__subst2,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A2 @ B3 )
% 3.12/1.63 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
% 3.12/1.63 => ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_240_order__subst1,axiom,
% 3.12/1.63 ! [A2: nat,F: nat > nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_241_order__subst1,axiom,
% 3.12/1.63 ! [A2: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_242_order__subst1,axiom,
% 3.12/1.63 ! [A2: nat,F: set_a > nat,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_243_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_244_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_a,F: nat > set_a,B3: nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_245_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_246_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_nat,F: set_a > set_nat,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_247_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_a,F: set_nat > set_a,B3: set_nat,C2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_nat,Y4: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_248_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: set_a,Y4: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X @ Y4 )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_249_order__subst1,axiom,
% 3.12/1.63 ! [A2: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B3: nat,C2: nat] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A2 @ ( F @ B3 ) )
% 3.12/1.63 => ( ( ord_less_eq_nat @ B3 @ C2 )
% 3.12/1.63 => ( ! [X: nat,Y4: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ X @ Y4 )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ ( F @ X ) @ ( F @ Y4 ) ) )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_250_Collect__mono__iff,axiom,
% 3.12/1.63 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) )
% 3.12/1.63 = ( ! [X2: product_prod_a_a] :
% 3.12/1.63 ( ( P @ X2 )
% 3.12/1.63 => ( Q @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_251_Collect__mono__iff,axiom,
% 3.12/1.63 ! [P: nat > $o,Q: nat > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 3.12/1.63 = ( ! [X2: nat] :
% 3.12/1.63 ( ( P @ X2 )
% 3.12/1.63 => ( Q @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_252_Collect__mono__iff,axiom,
% 3.12/1.63 ! [P: a > $o,Q: a > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 3.12/1.63 = ( ! [X2: a] :
% 3.12/1.63 ( ( P @ X2 )
% 3.12/1.63 => ( Q @ X2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_253_set__eq__subset,axiom,
% 3.12/1.63 ( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A5 @ B2 )
% 3.12/1.63 & ( ord_le1824328871od_a_a @ B2 @ A5 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_254_set__eq__subset,axiom,
% 3.12/1.63 ( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 3.12/1.63 & ( ord_less_eq_set_nat @ B2 @ A5 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_255_set__eq__subset,axiom,
% 3.12/1.63 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 3.12/1.63 = ( ^ [A5: set_a,B2: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A5 @ B2 )
% 3.12/1.63 & ( ord_less_eq_set_a @ B2 @ A5 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_256_subset__trans,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B @ C )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A @ C ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_257_subset__trans,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat,C: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B @ C )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_258_subset__trans,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a,C: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B @ C )
% 3.12/1.63 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_259_Collect__mono,axiom,
% 3.12/1.63 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.12/1.63 ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( P @ X )
% 3.12/1.63 => ( Q @ X ) )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_260_Collect__mono,axiom,
% 3.12/1.63 ! [P: nat > $o,Q: nat > $o] :
% 3.12/1.63 ( ! [X: nat] :
% 3.12/1.63 ( ( P @ X )
% 3.12/1.63 => ( Q @ X ) )
% 3.12/1.63 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_261_Collect__mono,axiom,
% 3.12/1.63 ! [P: a > $o,Q: a > $o] :
% 3.12/1.63 ( ! [X: a] :
% 3.12/1.63 ( ( P @ X )
% 3.12/1.63 => ( Q @ X ) )
% 3.12/1.63 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_262_subset__refl,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_263_subset__refl,axiom,
% 3.12/1.63 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_264_subset__refl,axiom,
% 3.12/1.63 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_265_subset__iff,axiom,
% 3.12/1.63 ( ord_le1824328871od_a_a
% 3.12/1.63 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 3.12/1.63 ! [T2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ T2 @ A5 )
% 3.12/1.63 => ( member449909584od_a_a @ T2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_266_subset__iff,axiom,
% 3.12/1.63 ( ord_less_eq_set_nat
% 3.12/1.63 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.12/1.63 ! [T2: nat] :
% 3.12/1.63 ( ( member_nat @ T2 @ A5 )
% 3.12/1.63 => ( member_nat @ T2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_267_subset__iff,axiom,
% 3.12/1.63 ( ord_less_eq_set_a
% 3.12/1.63 = ( ^ [A5: set_a,B2: set_a] :
% 3.12/1.63 ! [T2: a] :
% 3.12/1.63 ( ( member_a @ T2 @ A5 )
% 3.12/1.63 => ( member_a @ T2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_268_equalityD2,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ B @ A ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_269_equalityD2,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ord_less_eq_set_nat @ B @ A ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_270_equalityD2,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ord_less_eq_set_a @ B @ A ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_271_equalityD1,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ord_le1824328871od_a_a @ A @ B ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_272_equalityD1,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_273_equalityD1,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_274_subset__eq,axiom,
% 3.12/1.63 ( ord_le1824328871od_a_a
% 3.12/1.63 = ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 3.12/1.63 ! [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ A5 )
% 3.12/1.63 => ( member449909584od_a_a @ X2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_275_subset__eq,axiom,
% 3.12/1.63 ( ord_less_eq_set_nat
% 3.12/1.63 = ( ^ [A5: set_nat,B2: set_nat] :
% 3.12/1.63 ! [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ A5 )
% 3.12/1.63 => ( member_nat @ X2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_276_subset__eq,axiom,
% 3.12/1.63 ( ord_less_eq_set_a
% 3.12/1.63 = ( ^ [A5: set_a,B2: set_a] :
% 3.12/1.63 ! [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ A5 )
% 3.12/1.63 => ( member_a @ X2 @ B2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_277_equalityE,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ~ ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.63 => ~ ( ord_le1824328871od_a_a @ B @ A ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_278_equalityE,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ~ ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.63 => ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_279_equalityE,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ( A = B )
% 3.12/1.63 => ~ ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.63 => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_280_subsetD,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.63 => ( ( member449909584od_a_a @ C2 @ A )
% 3.12/1.63 => ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_281_subsetD,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat,C2: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.63 => ( ( member_nat @ C2 @ A )
% 3.12/1.63 => ( member_nat @ C2 @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_282_subsetD,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a,C2: a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.63 => ( ( member_a @ C2 @ A )
% 3.12/1.63 => ( member_a @ C2 @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_283_in__mono,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A @ B )
% 3.12/1.63 => ( ( member449909584od_a_a @ X3 @ A )
% 3.12/1.63 => ( member449909584od_a_a @ X3 @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_284_in__mono,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat,X3: nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A @ B )
% 3.12/1.63 => ( ( member_nat @ X3 @ A )
% 3.12/1.63 => ( member_nat @ X3 @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_285_in__mono,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a,X3: a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 3.12/1.63 => ( ( member_a @ X3 @ A )
% 3.12/1.63 => ( member_a @ X3 @ B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_286_pred__subset__eq,axiom,
% 3.12/1.63 ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1347718902_a_a_o
% 3.12/1.63 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ R )
% 3.12/1.63 @ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ S ) )
% 3.12/1.63 = ( ord_le1824328871od_a_a @ R @ S ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_287_pred__subset__eq,axiom,
% 3.12/1.63 ! [R: set_nat,S: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_nat_o
% 3.12/1.63 @ ^ [X2: nat] : ( member_nat @ X2 @ R )
% 3.12/1.63 @ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
% 3.12/1.63 = ( ord_less_eq_set_nat @ R @ S ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_288_pred__subset__eq,axiom,
% 3.12/1.63 ! [R: set_a,S: set_a] :
% 3.12/1.63 ( ( ord_less_eq_a_o
% 3.12/1.63 @ ^ [X2: a] : ( member_a @ X2 @ R )
% 3.12/1.63 @ ^ [X2: a] : ( member_a @ X2 @ S ) )
% 3.12/1.63 = ( ord_less_eq_set_a @ R @ S ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_289_pair__pre__digraph_Osurjective,axiom,
% 3.12/1.63 ! [R2: pair_p125712459t_unit] :
% 3.12/1.63 ( R2
% 3.12/1.63 = ( pair_p1621517565t_unit @ ( pair_p1047056820t_unit @ R2 ) @ ( pair_p133601421t_unit @ R2 ) @ ( pair_p1896222615t_unit @ R2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_290_pair__pre__digraph_Osurjective,axiom,
% 3.12/1.63 ! [R2: pair_p1914262621t_unit] :
% 3.12/1.63 ( R2
% 3.12/1.63 = ( pair_p1167410509t_unit @ ( pair_p1677060310t_unit @ R2 ) @ ( pair_p715279805t_unit @ R2 ) @ ( pair_p69470259t_unit @ R2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_291_pair__pre__digraph_Osurjective,axiom,
% 3.12/1.63 ! [R2: pair_p1765063010t_unit] :
% 3.12/1.63 ( R2
% 3.12/1.63 = ( pair_p398687508t_unit @ ( pair_p447552203t_unit @ R2 ) @ ( pair_p1783210148t_unit @ R2 ) @ ( pair_p1984658862t_unit @ R2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_292_member__product,axiom,
% 3.12/1.63 ! [X3: product_prod_a_a,A: set_a,B: set_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X3 @ ( product_product_a_a @ A @ B ) )
% 3.12/1.63 = ( member449909584od_a_a @ X3
% 3.12/1.63 @ ( product_Sigma_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_293_Product__Type_Oproduct__def,axiom,
% 3.12/1.63 ( product_product_a_a
% 3.12/1.63 = ( ^ [A5: set_a,B2: set_a] :
% 3.12/1.63 ( product_Sigma_a_a @ A5
% 3.12/1.63 @ ^ [Uu: a] : B2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_294_subset__Collect__iff,axiom,
% 3.12/1.63 ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ B @ A )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ B
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) ) )
% 3.12/1.63 = ( ! [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ B )
% 3.12/1.63 => ( P @ X2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_295_subset__Collect__iff,axiom,
% 3.12/1.63 ! [B: set_nat,A: set_nat,P: nat > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ B @ A )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ B
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) ) )
% 3.12/1.63 = ( ! [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ B )
% 3.12/1.63 => ( P @ X2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_296_subset__Collect__iff,axiom,
% 3.12/1.63 ! [B: set_a,A: set_a,P: a > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ B @ A )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ B
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) ) )
% 3.12/1.63 = ( ! [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ B )
% 3.12/1.63 => ( P @ X2 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_297_subset__CollectI,axiom,
% 3.12/1.63 ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,Q: product_prod_a_a > $o,P: product_prod_a_a > $o] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ B @ A )
% 3.12/1.63 => ( ! [X: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X @ B )
% 3.12/1.63 => ( ( Q @ X )
% 3.12/1.63 => ( P @ X ) ) )
% 3.12/1.63 => ( ord_le1824328871od_a_a
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ B )
% 3.12/1.63 & ( Q @ X2 ) ) )
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_298_subset__CollectI,axiom,
% 3.12/1.63 ! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ B @ A )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( member_nat @ X @ B )
% 3.12/1.63 => ( ( Q @ X )
% 3.12/1.63 => ( P @ X ) ) )
% 3.12/1.63 => ( ord_less_eq_set_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ B )
% 3.12/1.63 & ( Q @ X2 ) ) )
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_299_subset__CollectI,axiom,
% 3.12/1.63 ! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ B @ A )
% 3.12/1.63 => ( ! [X: a] :
% 3.12/1.63 ( ( member_a @ X @ B )
% 3.12/1.63 => ( ( Q @ X )
% 3.12/1.63 => ( P @ X ) ) )
% 3.12/1.63 => ( ord_less_eq_set_a
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ B )
% 3.12/1.63 & ( Q @ X2 ) ) )
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ A )
% 3.12/1.63 & ( P @ X2 ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_300_Collect__restrict,axiom,
% 3.12/1.63 ! [X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 3.12/1.63 ( ord_le1824328871od_a_a
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ X4 )
% 3.12/1.63 & ( P @ X2 ) ) )
% 3.12/1.63 @ X4 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_301_Collect__restrict,axiom,
% 3.12/1.63 ! [X4: set_nat,P: nat > $o] :
% 3.12/1.63 ( ord_less_eq_set_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ X4 )
% 3.12/1.63 & ( P @ X2 ) ) )
% 3.12/1.63 @ X4 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_302_Collect__restrict,axiom,
% 3.12/1.63 ! [X4: set_a,P: a > $o] :
% 3.12/1.63 ( ord_less_eq_set_a
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ X4 )
% 3.12/1.63 & ( P @ X2 ) ) )
% 3.12/1.63 @ X4 ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_303_prop__restrict,axiom,
% 3.12/1.63 ! [X3: product_prod_a_a,Z3: set_Product_prod_a_a,X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
% 3.12/1.63 ( ( member449909584od_a_a @ X3 @ Z3 )
% 3.12/1.63 => ( ( ord_le1824328871od_a_a @ Z3
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( member449909584od_a_a @ X2 @ X4 )
% 3.12/1.63 & ( P @ X2 ) ) ) )
% 3.12/1.63 => ( P @ X3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_304_prop__restrict,axiom,
% 3.12/1.63 ! [X3: nat,Z3: set_nat,X4: set_nat,P: nat > $o] :
% 3.12/1.63 ( ( member_nat @ X3 @ Z3 )
% 3.12/1.63 => ( ( ord_less_eq_set_nat @ Z3
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ X4 )
% 3.12/1.63 & ( P @ X2 ) ) ) )
% 3.12/1.63 => ( P @ X3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_305_prop__restrict,axiom,
% 3.12/1.63 ! [X3: a,Z3: set_a,X4: set_a,P: a > $o] :
% 3.12/1.63 ( ( member_a @ X3 @ Z3 )
% 3.12/1.63 => ( ( ord_less_eq_set_a @ Z3
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( member_a @ X2 @ X4 )
% 3.12/1.63 & ( P @ X2 ) ) ) )
% 3.12/1.63 => ( P @ X3 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_306_conj__subset__def,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ A
% 3.12/1.63 @ ( collec645855634od_a_a
% 3.12/1.63 @ ^ [X2: product_prod_a_a] :
% 3.12/1.63 ( ( P @ X2 )
% 3.12/1.63 & ( Q @ X2 ) ) ) )
% 3.12/1.63 = ( ( ord_le1824328871od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 3.12/1.63 & ( ord_le1824328871od_a_a @ A @ ( collec645855634od_a_a @ Q ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_307_conj__subset__def,axiom,
% 3.12/1.63 ! [A: set_nat,P: nat > $o,Q: nat > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ A
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [X2: nat] :
% 3.12/1.63 ( ( P @ X2 )
% 3.12/1.63 & ( Q @ X2 ) ) ) )
% 3.12/1.63 = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
% 3.12/1.63 & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_308_conj__subset__def,axiom,
% 3.12/1.63 ! [A: set_a,P: a > $o,Q: a > $o] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A
% 3.12/1.63 @ ( collect_a
% 3.12/1.63 @ ^ [X2: a] :
% 3.12/1.63 ( ( P @ X2 )
% 3.12/1.63 & ( Q @ X2 ) ) ) )
% 3.12/1.63 = ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
% 3.12/1.63 & ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_309_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 3.12/1.63 ! [Pverts: set_a,Parcs: set_Product_prod_a_a,More: product_unit] :
% 3.12/1.63 ( ( pair_p133601421t_unit @ ( pair_p1621517565t_unit @ Pverts @ Parcs @ More ) )
% 3.12/1.63 = Parcs ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_310_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 3.12/1.63 ! [Pverts: set_nat,Parcs: set_Pr1986765409at_nat,More: product_unit] :
% 3.12/1.63 ( ( pair_p715279805t_unit @ ( pair_p1167410509t_unit @ Pverts @ Parcs @ More ) )
% 3.12/1.63 = Parcs ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_311_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 3.12/1.63 ! [Pverts: set_Product_prod_a_a,Parcs: set_Pr1948701895od_a_a,More: product_unit] :
% 3.12/1.63 ( ( pair_p1783210148t_unit @ ( pair_p398687508t_unit @ Pverts @ Parcs @ More ) )
% 3.12/1.63 = Parcs ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_312_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 3.12/1.63 ! [Pverts: set_a,Parcs: set_Product_prod_a_a,More: product_unit] :
% 3.12/1.63 ( ( pair_p1047056820t_unit @ ( pair_p1621517565t_unit @ Pverts @ Parcs @ More ) )
% 3.12/1.63 = Pverts ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_313_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 3.12/1.63 ! [Pverts: set_nat,Parcs: set_Pr1986765409at_nat,More: product_unit] :
% 3.12/1.63 ( ( pair_p1677060310t_unit @ ( pair_p1167410509t_unit @ Pverts @ Parcs @ More ) )
% 3.12/1.63 = Pverts ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_314_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 3.12/1.63 ! [Pverts: set_Product_prod_a_a,Parcs: set_Pr1948701895od_a_a,More: product_unit] :
% 3.12/1.63 ( ( pair_p447552203t_unit @ ( pair_p398687508t_unit @ Pverts @ Parcs @ More ) )
% 3.12/1.63 = Pverts ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_315_Fpow__def,axiom,
% 3.12/1.63 ( finite361944167at_nat
% 3.12/1.63 = ( ^ [A5: set_Pr1986765409at_nat] :
% 3.12/1.63 ( collec1606769740at_nat
% 3.12/1.63 @ ^ [X5: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ( ord_le841296385at_nat @ X5 @ A5 )
% 3.12/1.63 & ( finite772653738at_nat @ X5 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_316_Fpow__def,axiom,
% 3.12/1.63 ( finite702915405od_a_a
% 3.12/1.63 = ( ^ [A5: set_Pr1948701895od_a_a] :
% 3.12/1.63 ( collec453062450od_a_a
% 3.12/1.63 @ ^ [X5: set_Pr1948701895od_a_a] :
% 3.12/1.63 ( ( ord_le456379495od_a_a @ X5 @ A5 )
% 3.12/1.63 & ( finite1664988688od_a_a @ X5 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_317_Fpow__def,axiom,
% 3.12/1.63 ( finite351630733od_a_a
% 3.12/1.63 = ( ^ [A5: set_Product_prod_a_a] :
% 3.12/1.63 ( collec183727474od_a_a
% 3.12/1.63 @ ^ [X5: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ord_le1824328871od_a_a @ X5 @ A5 )
% 3.12/1.63 & ( finite179568208od_a_a @ X5 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_318_Fpow__def,axiom,
% 3.12/1.63 ( finite_Fpow_nat
% 3.12/1.63 = ( ^ [A5: set_nat] :
% 3.12/1.63 ( collect_set_nat
% 3.12/1.63 @ ^ [X5: set_nat] :
% 3.12/1.63 ( ( ord_less_eq_set_nat @ X5 @ A5 )
% 3.12/1.63 & ( finite_finite_nat @ X5 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_319_Fpow__def,axiom,
% 3.12/1.63 ( finite_Fpow_a
% 3.12/1.63 = ( ^ [A5: set_a] :
% 3.12/1.63 ( collect_set_a
% 3.12/1.63 @ ^ [X5: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ X5 @ A5 )
% 3.12/1.63 & ( finite_finite_a @ X5 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_320_pair__fin__digraph__def,axiom,
% 3.12/1.63 ( pair_p1802376898raph_a
% 3.12/1.63 = ( ^ [G: pair_p125712459t_unit] :
% 3.12/1.63 ( ( pair_p68905728raph_a @ G )
% 3.12/1.63 & ( pair_p1864019935ioms_a @ G ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_321_pair__fin__digraph__def,axiom,
% 3.12/1.63 ( pair_p128415500ph_nat
% 3.12/1.63 = ( ^ [G: pair_p1914262621t_unit] :
% 3.12/1.63 ( ( pair_p1515597646ph_nat @ G )
% 3.12/1.63 & ( pair_p1027063983ms_nat @ G ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_322_pair__fin__digraph__def,axiom,
% 3.12/1.63 ( pair_p374947051od_a_a
% 3.12/1.63 = ( ^ [G: pair_p1765063010t_unit] :
% 3.12/1.63 ( ( pair_p646030121od_a_a @ G )
% 3.12/1.63 & ( pair_p504738056od_a_a @ G ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_323_pair__fin__digraph_Ointro,axiom,
% 3.12/1.63 ! [G2: pair_p125712459t_unit] :
% 3.12/1.63 ( ( pair_p68905728raph_a @ G2 )
% 3.12/1.63 => ( ( pair_p1864019935ioms_a @ G2 )
% 3.12/1.63 => ( pair_p1802376898raph_a @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_324_pair__fin__digraph_Ointro,axiom,
% 3.12/1.63 ! [G2: pair_p1914262621t_unit] :
% 3.12/1.63 ( ( pair_p1515597646ph_nat @ G2 )
% 3.12/1.63 => ( ( pair_p1027063983ms_nat @ G2 )
% 3.12/1.63 => ( pair_p128415500ph_nat @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_325_pair__fin__digraph_Ointro,axiom,
% 3.12/1.63 ! [G2: pair_p1765063010t_unit] :
% 3.12/1.63 ( ( pair_p646030121od_a_a @ G2 )
% 3.12/1.63 => ( ( pair_p504738056od_a_a @ G2 )
% 3.12/1.63 => ( pair_p374947051od_a_a @ G2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_326_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_a,B: set_nat] :
% 3.12/1.63 ( ( finite1743148308_a_nat
% 3.12/1.63 @ ( product_Sigma_a_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_a )
% 3.12/1.63 | ( B = bot_bot_set_nat )
% 3.12/1.63 | ( ( finite_finite_a @ A )
% 3.12/1.63 & ( finite_finite_nat @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_327_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_a] :
% 3.12/1.63 ( ( finite1808550458_nat_a
% 3.12/1.63 @ ( product_Sigma_nat_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_nat )
% 3.12/1.63 | ( B = bot_bot_set_a )
% 3.12/1.63 | ( ( finite_finite_nat @ A )
% 3.12/1.63 & ( finite_finite_a @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_328_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_nat] :
% 3.12/1.63 ( ( finite772653738at_nat
% 3.12/1.63 @ ( produc45129834at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_nat )
% 3.12/1.63 | ( B = bot_bot_set_nat )
% 3.12/1.63 | ( ( finite_finite_nat @ A )
% 3.12/1.63 & ( finite_finite_nat @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_329_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_a,B: set_a] :
% 3.12/1.63 ( ( finite179568208od_a_a
% 3.12/1.63 @ ( product_Sigma_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_a )
% 3.12/1.63 | ( B = bot_bot_set_a )
% 3.12/1.63 | ( ( finite_finite_a @ A )
% 3.12/1.63 & ( finite_finite_a @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_330_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_a] :
% 3.12/1.63 ( ( finite1919032935_a_a_a
% 3.12/1.63 @ ( produc1282482655_a_a_a @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : B ) )
% 3.12/1.63 = ( ( A = bot_bo2131659635od_a_a )
% 3.12/1.63 | ( B = bot_bot_set_a )
% 3.12/1.63 | ( ( finite179568208od_a_a @ A )
% 3.12/1.63 & ( finite_finite_a @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_331_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a,B: set_nat] :
% 3.12/1.63 ( ( finite1837575485_a_nat
% 3.12/1.63 @ ( produc931712687_a_nat @ A
% 3.12/1.63 @ ^ [Uu: product_prod_a_a] : B ) )
% 3.12/1.63 = ( ( A = bot_bo2131659635od_a_a )
% 3.12/1.63 | ( B = bot_bot_set_nat )
% 3.12/1.63 | ( ( finite179568208od_a_a @ A )
% 3.12/1.63 & ( finite_finite_nat @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_332_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( finite676513017od_a_a
% 3.12/1.63 @ ( produc520147185od_a_a @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_a )
% 3.12/1.63 | ( B = bot_bo2131659635od_a_a )
% 3.12/1.63 | ( ( finite_finite_a @ A )
% 3.12/1.63 & ( finite179568208od_a_a @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_333_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_a,B: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ( finite942416723at_nat
% 3.12/1.63 @ ( produc292491723at_nat @ A
% 3.12/1.63 @ ^ [Uu: a] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_a )
% 3.12/1.63 | ( B = bot_bo2130386637at_nat )
% 3.12/1.63 | ( ( finite_finite_a @ A )
% 3.12/1.63 & ( finite772653738at_nat @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_334_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Product_prod_a_a] :
% 3.12/1.63 ( ( finite1297454819od_a_a
% 3.12/1.63 @ ( produc1182842125od_a_a @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_nat )
% 3.12/1.63 | ( B = bot_bo2131659635od_a_a )
% 3.12/1.63 | ( ( finite_finite_nat @ A )
% 3.12/1.63 & ( finite179568208od_a_a @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_335_finite__cartesian__product__iff,axiom,
% 3.12/1.63 ! [A: set_nat,B: set_Pr1986765409at_nat] :
% 3.12/1.63 ( ( finite277291581at_nat
% 3.12/1.63 @ ( produc894163943at_nat @ A
% 3.12/1.63 @ ^ [Uu: nat] : B ) )
% 3.12/1.63 = ( ( A = bot_bot_set_nat )
% 3.12/1.63 | ( B = bot_bo2130386637at_nat )
% 3.12/1.63 | ( ( finite_finite_nat @ A )
% 3.12/1.63 & ( finite772653738at_nat @ B ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_336_empty__iff,axiom,
% 3.12/1.63 ! [C2: product_prod_a_a] :
% 3.12/1.63 ~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_337_empty__iff,axiom,
% 3.12/1.63 ! [C2: nat] :
% 3.12/1.63 ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_338_all__not__in__conv,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a] :
% 3.12/1.63 ( ( ! [X2: product_prod_a_a] :
% 3.12/1.63 ~ ( member449909584od_a_a @ X2 @ A ) )
% 3.12/1.63 = ( A = bot_bo2131659635od_a_a ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_339_all__not__in__conv,axiom,
% 3.12/1.63 ! [A: set_nat] :
% 3.12/1.63 ( ( ! [X2: nat] :
% 3.12/1.63 ~ ( member_nat @ X2 @ A ) )
% 3.12/1.63 = ( A = bot_bot_set_nat ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_340_Collect__empty__eq,axiom,
% 3.12/1.63 ! [P: product_prod_a_a > $o] :
% 3.12/1.63 ( ( ( collec645855634od_a_a @ P )
% 3.12/1.63 = bot_bo2131659635od_a_a )
% 3.12/1.63 = ( ! [X2: product_prod_a_a] :
% 3.12/1.63 ~ ( P @ X2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_341_Collect__empty__eq,axiom,
% 3.12/1.63 ! [P: nat > $o] :
% 3.12/1.63 ( ( ( collect_nat @ P )
% 3.12/1.63 = bot_bot_set_nat )
% 3.12/1.63 = ( ! [X2: nat] :
% 3.12/1.63 ~ ( P @ X2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_342_empty__Collect__eq,axiom,
% 3.12/1.63 ! [P: product_prod_a_a > $o] :
% 3.12/1.63 ( ( bot_bo2131659635od_a_a
% 3.12/1.63 = ( collec645855634od_a_a @ P ) )
% 3.12/1.63 = ( ! [X2: product_prod_a_a] :
% 3.12/1.63 ~ ( P @ X2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_343_empty__Collect__eq,axiom,
% 3.12/1.63 ! [P: nat > $o] :
% 3.12/1.63 ( ( bot_bot_set_nat
% 3.12/1.63 = ( collect_nat @ P ) )
% 3.12/1.63 = ( ! [X2: nat] :
% 3.12/1.63 ~ ( P @ X2 ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_344_empty__subsetI,axiom,
% 3.12/1.63 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_345_empty__subsetI,axiom,
% 3.12/1.63 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_346_empty__subsetI,axiom,
% 3.12/1.63 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_347_subset__empty,axiom,
% 3.12/1.63 ! [A: set_a] :
% 3.12/1.63 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 3.12/1.63 = ( A = bot_bot_set_a ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_348_calculation,axiom,
% 3.12/1.63 ( ( finite_finite_a @ ( pair_p1047056820t_unit @ g ) )
% 3.12/1.63 & ( ( finite_card_a @ ( pair_p1047056820t_unit @ g ) )
% 3.12/1.63 = n )
% 3.12/1.63 & ( ( pair_p133601421t_unit @ g )
% 3.12/1.63 = ( collec645855634od_a_a
% 3.12/1.63 @ ( produc1833107820_a_a_o
% 3.12/1.63 @ ^ [U: a,V: a] :
% 3.12/1.63 ( ( member449909584od_a_a @ ( product_Pair_a_a @ U @ V )
% 3.12/1.63 @ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
% 3.12/1.63 @ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) )
% 3.12/1.63 & ( U != V ) ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_349_finite__Collect__le__nat,axiom,
% 3.12/1.63 ! [K: nat] :
% 3.12/1.63 ( finite_finite_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_350_finite__nat__set__iff__bounded__le,axiom,
% 3.12/1.63 ( finite_finite_nat
% 3.12/1.63 = ( ^ [N2: set_nat] :
% 3.12/1.63 ? [M: nat] :
% 3.12/1.63 ! [X2: nat] :
% 3.12/1.63 ( ( member_nat @ X2 @ N2 )
% 3.12/1.63 => ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_351_infinite__nat__iff__unbounded__le,axiom,
% 3.12/1.63 ! [S: set_nat] :
% 3.12/1.63 ( ( ~ ( finite_finite_nat @ S ) )
% 3.12/1.63 = ( ! [M: nat] :
% 3.12/1.63 ? [N: nat] :
% 3.12/1.63 ( ( ord_less_eq_nat @ M @ N )
% 3.12/1.63 & ( member_nat @ N @ S ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_352_bounded__Max__nat,axiom,
% 3.12/1.63 ! [P: nat > $o,X3: nat,M2: nat] :
% 3.12/1.63 ( ( P @ X3 )
% 3.12/1.63 => ( ! [X: nat] :
% 3.12/1.63 ( ( P @ X )
% 3.12/1.63 => ( ord_less_eq_nat @ X @ M2 ) )
% 3.12/1.63 => ~ ! [M3: nat] :
% 3.12/1.63 ( ( P @ M3 )
% 3.12/1.63 => ~ ! [X6: nat] :
% 3.12/1.63 ( ( P @ X6 )
% 3.12/1.63 => ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(fact_353_finite__less__ub,axiom,
% 3.12/1.63 ! [F: nat > nat,U2: nat] :
% 3.12/1.63 ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 3.12/1.63 => ( finite_finite_nat
% 3.12/1.63 @ ( collect_nat
% 3.12/1.63 @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U2 ) ) ) ) ).
% 3.12/1.63
% 3.12/1.63 thf(conj_0,conjecture,
% 3.12/1.63 finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) ).
% 3.12/1.63 ------------------------
% 3.12/1.67 % (5248)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/2Mi)
% 3.12/1.67 % (5247)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/27Mi)
% 3.12/1.67 % (5246)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/4Mi)
% 3.12/1.67 % (5250)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/275Mi)
% 3.12/1.67 % (5251)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/18Mi)
% 3.12/1.67 % (5248)Instruction limit reached!
% 3.12/1.67 % (5248)------------------------------
% 3.12/1.67 % (5248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.67 % (5248)Termination reason: Unknown
% 3.12/1.67 % (5248)Termination phase: shuffling
% 3.12/1.67
% 3.12/1.67 % (5248)Memory used [KB]: 1535
% 3.12/1.67 % (5248)Time elapsed: 0.003 s
% 3.12/1.67 % (5248)Instructions burned: 2 (million)
% 3.12/1.67 % (5248)------------------------------
% 3.12/1.67 % (5248)------------------------------
% 3.12/1.67 % (5249)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/2Mi)
% 3.12/1.67 % (5246)Instruction limit reached!
% 3.12/1.67 % (5246)------------------------------
% 3.12/1.67 % (5246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.67 % (5246)Termination reason: Unknown
% 3.12/1.67 % (5246)Termination phase: shuffling
% 3.12/1.67
% 3.12/1.67 % (5246)Memory used [KB]: 1535
% 3.12/1.67 % (5246)Time elapsed: 0.004 s
% 3.12/1.67 % (5246)Instructions burned: 4 (million)
% 3.12/1.67 % (5246)------------------------------
% 3.12/1.67 % (5246)------------------------------
% 3.12/1.67 % (5249)Instruction limit reached!
% 3.12/1.67 % (5249)------------------------------
% 3.12/1.67 % (5249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.67 % (5249)Termination reason: Unknown
% 3.12/1.67 % (5249)Termination phase: shuffling
% 3.12/1.67
% 3.12/1.67 % (5249)Memory used [KB]: 1535
% 3.12/1.67 % (5249)Time elapsed: 0.003 s
% 3.12/1.67 % (5245)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/183Mi)
% 3.12/1.67 % (5249)Instructions burned: 2 (million)
% 3.12/1.67 % (5249)------------------------------
% 3.12/1.67 % (5249)------------------------------
% 3.12/1.68 % (5251)Instruction limit reached!
% 3.12/1.68 % (5251)------------------------------
% 3.12/1.68 % (5251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.68 % (5251)Termination reason: Unknown
% 3.12/1.68 % (5251)Termination phase: shuffling
% 3.12/1.68
% 3.12/1.68 % (5251)Memory used [KB]: 1918
% 3.12/1.68 % (5251)Time elapsed: 0.011 s
% 3.12/1.68 % (5251)Instructions burned: 18 (million)
% 3.12/1.68 % (5251)------------------------------
% 3.12/1.68 % (5251)------------------------------
% 3.12/1.69 % (5247)Instruction limit reached!
% 3.12/1.69 % (5247)------------------------------
% 3.12/1.69 % (5247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.69 % (5247)Termination reason: Unknown
% 3.12/1.69 % (5247)Termination phase: shuffling
% 3.12/1.69
% 3.12/1.69 % (5247)Memory used [KB]: 2046
% 3.12/1.69 % (5247)Time elapsed: 0.016 s
% 3.12/1.69 % (5247)Instructions burned: 28 (million)
% 3.12/1.69 % (5247)------------------------------
% 3.12/1.69 % (5247)------------------------------
% 3.12/1.69 % (5253)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/37Mi)
% 3.12/1.69 % (5252)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/3Mi)
% 3.12/1.69 % (5254)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/15Mi)
% 3.12/1.69 % (5252)Instruction limit reached!
% 3.12/1.69 % (5252)------------------------------
% 3.12/1.69 % (5252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.69 % (5252)Termination reason: Unknown
% 3.12/1.69 % (5252)Termination phase: shuffling
% 3.12/1.69
% 3.12/1.69 % (5252)Memory used [KB]: 1535
% 3.12/1.69 % (5252)Time elapsed: 0.004 s
% 3.12/1.69 % (5252)Instructions burned: 3 (million)
% 3.12/1.69 % (5252)------------------------------
% 3.12/1.69 % (5252)------------------------------
% 3.17/1.70 % (5255)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/3Mi)
% 3.17/1.70 % (5254)Instruction limit reached!
% 3.17/1.70 % (5254)------------------------------
% 3.17/1.70 % (5254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.70 % (5254)Termination reason: Unknown
% 3.17/1.70 % (5254)Termination phase: shuffling
% 3.17/1.70
% 3.17/1.70 % (5254)Memory used [KB]: 1918
% 3.17/1.70 % (5254)Time elapsed: 0.010 s
% 3.17/1.70 % (5254)Instructions burned: 16 (million)
% 3.17/1.70 % (5254)------------------------------
% 3.17/1.70 % (5254)------------------------------
% 3.17/1.70 % (5250)Refutation not found, incomplete strategy
% 3.17/1.70 % (5250)------------------------------
% 3.17/1.70 % (5250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.70 % (5250)Termination reason: Refutation not found, incomplete strategy
% 3.17/1.70
% 3.17/1.70
% 3.17/1.70 % (5250)Memory used [KB]: 6652
% 3.17/1.70 % (5250)Time elapsed: 0.029 s
% 3.17/1.70 % (5250)Instructions burned: 56 (million)
% 3.17/1.70 % (5250)------------------------------
% 3.17/1.70 % (5250)------------------------------
% 3.17/1.70 % (5255)Instruction limit reached!
% 3.17/1.70 % (5255)------------------------------
% 3.17/1.70 % (5255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.70 % (5255)Termination reason: Unknown
% 3.17/1.70 % (5255)Termination phase: shuffling
% 3.17/1.70
% 3.17/1.70 % (5255)Memory used [KB]: 1535
% 3.17/1.70 % (5255)Time elapsed: 0.004 s
% 3.17/1.70 % (5255)Instructions burned: 4 (million)
% 3.17/1.70 % (5255)------------------------------
% 3.17/1.70 % (5255)------------------------------
% 3.17/1.70 % (5257)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/1041Mi)
% 3.17/1.71 % (5253)Instruction limit reached!
% 3.17/1.71 % (5253)------------------------------
% 3.17/1.71 % (5253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.71 % (5253)Termination reason: Unknown
% 3.17/1.71 % (5253)Termination phase: shuffling
% 3.17/1.71
% 3.17/1.71 % (5253)Memory used [KB]: 2302
% 3.17/1.71 % (5253)Time elapsed: 0.018 s
% 3.17/1.71 % (5253)Instructions burned: 37 (million)
% 3.17/1.71 % (5253)------------------------------
% 3.17/1.71 % (5253)------------------------------
% 3.17/1.71 % (5258)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/7Mi)
% 3.17/1.71 % (5258)Instruction limit reached!
% 3.17/1.71 % (5258)------------------------------
% 3.17/1.71 % (5258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.71 % (5258)Termination reason: Unknown
% 3.17/1.71 % (5258)Termination phase: shuffling
% 3.17/1.71
% 3.17/1.71 % (5258)Memory used [KB]: 1535
% 3.17/1.71 % (5258)Time elapsed: 0.006 s
% 3.17/1.71 % (5258)Instructions burned: 7 (million)
% 3.17/1.71 % (5258)------------------------------
% 3.17/1.71 % (5258)------------------------------
% 3.17/1.72 % (5261)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/3Mi)
% 3.17/1.72 % (5262)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/3Mi)
% 3.17/1.72 % (5260)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/16Mi)
% 3.17/1.72 % (5261)Instruction limit reached!
% 3.17/1.72 % (5261)------------------------------
% 3.17/1.72 % (5261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.72 % (5261)Termination reason: Unknown
% 3.17/1.72 % (5261)Termination phase: shuffling
% 3.17/1.72
% 3.17/1.72 % (5261)Memory used [KB]: 1535
% 3.17/1.72 % (5261)Time elapsed: 0.003 s
% 3.17/1.72 % (5261)Instructions burned: 3 (million)
% 3.17/1.72 % (5261)------------------------------
% 3.17/1.72 % (5261)------------------------------
% 3.17/1.72 % (5262)Instruction limit reached!
% 3.17/1.72 % (5262)------------------------------
% 3.17/1.72 % (5262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.72 % (5262)Termination reason: Unknown
% 3.17/1.72 % (5262)Termination phase: shuffling
% 3.17/1.72
% 3.17/1.72 % (5262)Memory used [KB]: 1535
% 3.17/1.72 % (5262)Time elapsed: 0.003 s
% 3.17/1.72 % (5262)Instructions burned: 3 (million)
% 3.17/1.72 % (5262)------------------------------
% 3.17/1.72 % (5262)------------------------------
% 3.17/1.72 % (5263)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/7Mi)
% 3.17/1.73 % (5260)Instruction limit reached!
% 3.17/1.73 % (5260)------------------------------
% 3.17/1.73 % (5260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.73 % (5260)Termination reason: Unknown
% 3.17/1.73 % (5260)Termination phase: shuffling
% 3.17/1.73
% 3.17/1.73 % (5260)Memory used [KB]: 1918
% 3.17/1.73 % (5260)Time elapsed: 0.011 s
% 3.17/1.73 % (5260)Instructions burned: 17 (million)
% 3.17/1.73 % (5260)------------------------------
% 3.17/1.73 % (5260)------------------------------
% 3.17/1.73 % (5263)Instruction limit reached!
% 3.17/1.73 % (5263)------------------------------
% 3.17/1.73 % (5263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.73 % (5263)Termination reason: Unknown
% 3.17/1.73 % (5263)Termination phase: shuffling
% 3.17/1.73
% 3.17/1.73 % (5263)Memory used [KB]: 1663
% 3.17/1.73 % (5263)Time elapsed: 0.005 s
% 3.17/1.73 % (5263)Instructions burned: 8 (million)
% 3.17/1.73 % (5263)------------------------------
% 3.17/1.73 % (5263)------------------------------
% 3.17/1.73 % (5267)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/18Mi)
% 3.17/1.73 % (5265)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/3Mi)
% 3.17/1.73 % (5266)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/4Mi)
% 3.17/1.73 % (5265)Instruction limit reached!
% 3.17/1.73 % (5265)------------------------------
% 3.17/1.73 % (5265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.73 % (5267)Instruction limit reached!
% 3.17/1.73 % (5267)------------------------------
% 3.17/1.73 % (5267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.73 % (5267)Termination reason: Unknown
% 3.17/1.73 % (5265)Termination reason: Unknown
% 3.17/1.73 % (5267)Termination phase: shuffling
% 3.17/1.73
% 3.17/1.73 % (5267)Memory used [KB]: 1918
% 3.17/1.73 % (5265)Termination phase: shuffling
% 3.17/1.73 % (5267)Time elapsed: 0.007 s
% 3.17/1.73
% 3.17/1.73 % (5267)Instructions burned: 18 (million)
% 3.17/1.73 % (5267)------------------------------
% 3.17/1.73 % (5267)------------------------------
% 3.17/1.73 % (5265)Memory used [KB]: 1535
% 3.17/1.73 % (5265)Time elapsed: 0.004 s
% 3.17/1.73 % (5265)Instructions burned: 4 (million)
% 3.17/1.73 % (5265)------------------------------
% 3.17/1.73 % (5265)------------------------------
% 3.17/1.74 % (5266)Instruction limit reached!
% 3.17/1.74 % (5266)------------------------------
% 3.17/1.74 % (5266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.74 % (5266)Termination reason: Unknown
% 3.17/1.74 % (5266)Termination phase: shuffling
% 3.17/1.74
% 3.17/1.74 % (5266)Memory used [KB]: 1535
% 3.17/1.74 % (5266)Time elapsed: 0.004 s
% 3.17/1.74 % (5266)Instructions burned: 5 (million)
% 3.17/1.74 % (5266)------------------------------
% 3.17/1.74 % (5266)------------------------------
% 3.17/1.74 % (5269)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/710Mi)
% 3.17/1.74 % (5270)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/6Mi)
% 3.17/1.74 % (5271)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/902Mi)
% 3.17/1.75 % (5270)Instruction limit reached!
% 3.17/1.75 % (5270)------------------------------
% 3.17/1.75 % (5270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.75 % (5270)Termination reason: Unknown
% 3.17/1.75 % (5270)Termination phase: shuffling
% 3.17/1.75
% 3.17/1.75 % (5270)Memory used [KB]: 1535
% 3.17/1.75 % (5270)Time elapsed: 0.005 s
% 3.17/1.75 % (5270)Instructions burned: 7 (million)
% 3.17/1.75 % (5270)------------------------------
% 3.17/1.75 % (5270)------------------------------
% 3.17/1.75 % (5274)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/5Mi)
% 3.17/1.75 % (5245)Instruction limit reached!
% 3.17/1.75 % (5245)------------------------------
% 3.17/1.75 % (5245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.75 % (5245)Termination reason: Unknown
% 3.17/1.75 % (5245)Termination phase: Saturation
% 3.17/1.75
% 3.17/1.75 % (5245)Memory used [KB]: 7547
% 3.17/1.75 % (5245)Time elapsed: 0.081 s
% 3.17/1.75 % (5245)Instructions burned: 184 (million)
% 3.17/1.75 % (5245)------------------------------
% 3.17/1.75 % (5245)------------------------------
% 3.17/1.75 % (5274)Instruction limit reached!
% 3.17/1.75 % (5274)------------------------------
% 3.17/1.75 % (5274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.75 % (5274)Termination reason: Unknown
% 3.17/1.75 % (5274)Termination phase: shuffling
% 3.17/1.75
% 3.17/1.75 % (5274)Memory used [KB]: 1535
% 3.17/1.75 % (5274)Time elapsed: 0.004 s
% 3.17/1.75 % (5274)Instructions burned: 5 (million)
% 3.17/1.75 % (5274)------------------------------
% 3.17/1.75 % (5274)------------------------------
% 3.17/1.76 % (5272)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_5057 for (2999ds/21Mi)
% 3.17/1.76 % (5275)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_5057 for (2998ds/6Mi)
% 3.17/1.76 % (5272)Instruction limit reached!
% 3.17/1.76 % (5272)------------------------------
% 3.17/1.76 % (5272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.76 % (5272)Termination reason: Unknown
% 3.17/1.76 % (5272)Termination phase: shuffling
% 3.17/1.76
% 3.17/1.76 % (5272)Memory used [KB]: 1918
% 3.17/1.76 % (5272)Time elapsed: 0.008 s
% 3.17/1.76 % (5272)Instructions burned: 22 (million)
% 3.17/1.76 % (5272)------------------------------
% 3.17/1.76 % (5272)------------------------------
% 3.17/1.77 % (5275)Instruction limit reached!
% 3.17/1.77 % (5275)------------------------------
% 3.17/1.77 % (5275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.77 % (5275)Termination reason: Unknown
% 3.17/1.77 % (5275)Termination phase: shuffling
% 3.17/1.77
% 3.17/1.77 % (5275)Memory used [KB]: 1663
% 3.17/1.77 % (5275)Time elapsed: 0.005 s
% 3.17/1.77 % (5275)Instructions burned: 7 (million)
% 3.17/1.77 % (5275)------------------------------
% 3.17/1.77 % (5275)------------------------------
% 3.17/1.77 % (5269)First to succeed.
% 3.17/1.77 % (5277)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_5057 for (2998ds/779Mi)
% 3.17/1.77 % (5276)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_5057 for (2998ds/377Mi)
% 3.17/1.77 % (5269)Refutation found. Thanks to Tanya!
% 3.17/1.77 % SZS status Theorem for DTF2THF_5057
% 3.17/1.77 % SZS output start Proof for DTF2THF_5057
% 3.17/1.77 thf(type_def_5, type, set_nat: $tType).
% 3.17/1.77 thf(type_def_6, type, set_set_nat: $tType).
% 3.17/1.77 thf(type_def_8, type, set_Pr1986765409at_nat: $tType).
% 3.17/1.77 thf(type_def_9, type, set_se1612935105at_nat: $tType).
% 3.17/1.77 thf(type_def_10, type, set_Pr1948701895od_a_a: $tType).
% 3.17/1.77 thf(type_def_11, type, set_se958357159od_a_a: $tType).
% 3.17/1.77 thf(type_def_12, type, set_Product_prod_a_a: $tType).
% 3.17/1.77 thf(type_def_13, type, set_se1596668135od_a_a: $tType).
% 3.17/1.77 thf(type_def_14, type, set_a: $tType).
% 3.17/1.77 thf(type_def_15, type, set_set_a: $tType).
% 3.17/1.77 thf(type_def_16, type, nat: $tType).
% 3.17/1.77 thf(type_def_17, type, set_Pr1746169692at_nat: $tType).
% 3.17/1.77 thf(type_def_18, type, set_Pr339609346od_a_a: $tType).
% 3.17/1.77 thf(type_def_19, type, set_Pr967348953_nat_a: $tType).
% 3.17/1.77 thf(type_def_20, type, set_Pr1490359111at_nat: $tType).
% 3.17/1.77 thf(type_def_21, type, set_Pr1295299783od_a_a: $tType).
% 3.17/1.77 thf(type_def_22, type, set_Pr894832732_a_nat: $tType).
% 3.17/1.77 thf(type_def_23, type, set_Pr1689873822_a_a_a: $tType).
% 3.17/1.77 thf(type_def_24, type, set_Pr548851891_a_nat: $tType).
% 3.17/1.77 thf(type_def_25, type, set_Pr7688842at_nat: $tType).
% 3.17/1.77 thf(type_def_26, type, set_Pr681306928od_a_a: $tType).
% 3.17/1.77 thf(type_def_27, type, product_prod_a_a: $tType).
% 3.17/1.77 thf(type_def_28, type, a: $tType).
% 3.17/1.77 thf(type_def_29, type, pair_p1914262621t_unit: $tType).
% 3.17/1.77 thf(type_def_30, type, pair_p2041852168t_unit: $tType).
% 3.17/1.77 thf(type_def_31, type, pair_p1593840546t_unit: $tType).
% 3.17/1.77 thf(type_def_32, type, pair_p1765063010t_unit: $tType).
% 3.17/1.77 thf(type_def_33, type, pair_p125712459t_unit: $tType).
% 3.17/1.77 thf(type_def_34, type, product_unit: $tType).
% 3.17/1.77 thf(type_def_35, type, product_prod_nat_nat: $tType).
% 3.17/1.77 thf(type_def_36, type, produc1572603623od_a_a: $tType).
% 3.17/1.77 thf(func_def_0, type, set_Pr1295299783od_a_a: $tType).
% 3.17/1.77 thf(func_def_1, type, pair_p1593840546t_unit: $tType).
% 3.17/1.77 thf(func_def_2, type, pair_p2041852168t_unit: $tType).
% 3.17/1.77 thf(func_def_3, type, set_Pr1490359111at_nat: $tType).
% 3.17/1.77 thf(func_def_4, type, pair_p1765063010t_unit: $tType).
% 3.17/1.77 thf(func_def_5, type, set_se958357159od_a_a: $tType).
% 3.17/1.77 thf(func_def_6, type, set_Pr1948701895od_a_a: $tType).
% 3.17/1.77 thf(func_def_7, type, pair_p1914262621t_unit: $tType).
% 3.17/1.77 thf(func_def_8, type, set_Pr1746169692at_nat: $tType).
% 3.17/1.77 thf(func_def_9, type, produc1572603623od_a_a: $tType).
% 3.17/1.77 thf(func_def_10, type, pair_p125712459t_unit: $tType).
% 3.17/1.77 thf(func_def_11, type, set_Pr7688842at_nat: $tType).
% 3.17/1.77 thf(func_def_12, type, set_Pr894832732_a_nat: $tType).
% 3.17/1.77 thf(func_def_13, type, set_Pr339609346od_a_a: $tType).
% 3.17/1.77 thf(func_def_14, type, set_Pr681306928od_a_a: $tType).
% 3.17/1.77 thf(func_def_15, type, set_Pr1689873822_a_a_a: $tType).
% 3.17/1.77 thf(func_def_16, type, set_se1612935105at_nat: $tType).
% 3.17/1.77 thf(func_def_17, type, set_se1596668135od_a_a: $tType).
% 3.17/1.77 thf(func_def_18, type, set_Pr1986765409at_nat: $tType).
% 3.17/1.77 thf(func_def_19, type, set_Pr548851891_a_nat: $tType).
% 3.17/1.77 thf(func_def_20, type, set_Pr967348953_nat_a: $tType).
% 3.17/1.77 thf(func_def_21, type, set_Product_prod_a_a: $tType).
% 3.17/1.77 thf(func_def_22, type, product_prod_nat_nat: $tType).
% 3.17/1.77 thf(func_def_23, type, set_set_nat: $tType).
% 3.17/1.77 thf(func_def_24, type, product_prod_a_a: $tType).
% 3.17/1.77 thf(func_def_25, type, set_set_a: $tType).
% 3.17/1.77 thf(func_def_26, type, set_nat: $tType).
% 3.17/1.77 thf(func_def_27, type, product_unit: $tType).
% 3.17/1.77 thf(func_def_28, type, set_a: $tType).
% 3.17/1.77 thf(func_def_29, type, nat: $tType).
% 3.17/1.77 thf(func_def_30, type, a: $tType).
% 3.17/1.77 thf(func_def_31, type, finite_Fpow_nat: set_nat > set_set_nat).
% 3.17/1.77 thf(func_def_32, type, finite361944167at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat).
% 3.17/1.77 thf(func_def_33, type, finite702915405od_a_a: set_Pr1948701895od_a_a > set_se958357159od_a_a).
% 3.17/1.77 thf(func_def_34, type, finite351630733od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a).
% 3.17/1.77 thf(func_def_35, type, finite_Fpow_a: set_a > set_set_a).
% 3.17/1.77 thf(func_def_36, type, finite_card_a: set_a > nat).
% 3.17/1.77 thf(func_def_37, type, finite_finite_nat: set_nat > $o).
% 3.17/1.77 thf(func_def_38, type, finite772653738at_nat: set_Pr1986765409at_nat > $o).
% 3.17/1.77 thf(func_def_39, type, finite277291581at_nat: set_Pr1746169692at_nat > $o).
% 3.17/1.77 thf(func_def_40, type, finite1297454819od_a_a: set_Pr339609346od_a_a > $o).
% 3.17/1.77 thf(func_def_41, type, finite1808550458_nat_a: set_Pr967348953_nat_a > $o).
% 3.17/1.77 thf(func_def_42, type, finite48957584at_nat: set_Pr1490359111at_nat > $o).
% 3.17/1.77 thf(func_def_43, type, finite256329232od_a_a: set_Pr1295299783od_a_a > $o).
% 3.17/1.77 thf(func_def_44, type, finite1837575485_a_nat: set_Pr894832732_a_nat > $o).
% 3.17/1.77 thf(func_def_45, type, finite1664988688od_a_a: set_Pr1948701895od_a_a > $o).
% 3.17/1.77 thf(func_def_46, type, finite1919032935_a_a_a: set_Pr1689873822_a_a_a > $o).
% 3.17/1.77 thf(func_def_47, type, finite1743148308_a_nat: set_Pr548851891_a_nat > $o).
% 3.17/1.77 thf(func_def_48, type, finite942416723at_nat: set_Pr7688842at_nat > $o).
% 3.17/1.77 thf(func_def_49, type, finite676513017od_a_a: set_Pr681306928od_a_a > $o).
% 3.17/1.77 thf(func_def_50, type, finite179568208od_a_a: set_Product_prod_a_a > $o).
% 3.17/1.77 thf(func_def_51, type, finite2012248349et_nat: set_set_nat > $o).
% 3.17/1.77 thf(func_def_52, type, finite1457549322at_nat: set_se1612935105at_nat > $o).
% 3.17/1.77 thf(func_def_53, type, finite323969008od_a_a: set_se958357159od_a_a > $o).
% 3.17/1.77 thf(func_def_54, type, finite1145471536od_a_a: set_se1596668135od_a_a > $o).
% 3.17/1.77 thf(func_def_55, type, finite_finite_set_a: set_set_a > $o).
% 3.17/1.77 thf(func_def_56, type, finite_finite_a: set_a > $o).
% 3.17/1.77 thf(func_def_57, type, bot_bot_set_nat: set_nat).
% 3.17/1.77 thf(func_def_58, type, bot_bo2130386637at_nat: set_Pr1986765409at_nat).
% 3.17/1.77 thf(func_def_59, type, bot_bo2131659635od_a_a: set_Product_prod_a_a).
% 3.17/1.77 thf(func_def_60, type, bot_bot_set_a: set_a).
% 3.17/1.77 thf(func_def_61, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 3.17/1.77 thf(func_def_62, type, ord_le1347718902_a_a_o: (product_prod_a_a > $o) > (product_prod_a_a > $o) > $o).
% 3.17/1.77 thf(func_def_63, type, ord_less_eq_a_o: (a > $o) > (a > $o) > $o).
% 3.17/1.77 thf(func_def_64, type, ord_less_eq_nat: nat > nat > $o).
% 3.17/1.77 thf(func_def_65, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 3.17/1.77 thf(func_def_66, type, ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o).
% 3.17/1.77 thf(func_def_67, type, ord_le2084554594od_a_a: set_Pr339609346od_a_a > set_Pr339609346od_a_a > $o).
% 3.17/1.77 thf(func_def_68, type, ord_le344568633_nat_a: set_Pr967348953_nat_a > set_Pr967348953_nat_a > $o).
% 3.17/1.77 thf(func_def_69, type, ord_le492294332_a_nat: set_Pr894832732_a_nat > set_Pr894832732_a_nat > $o).
% 3.17/1.77 thf(func_def_70, type, ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o).
% 3.17/1.77 thf(func_def_71, type, ord_le677315902_a_a_a: set_Pr1689873822_a_a_a > set_Pr1689873822_a_a_a > $o).
% 3.17/1.77 thf(func_def_72, type, ord_le2073555219_a_nat: set_Pr548851891_a_nat > set_Pr548851891_a_nat > $o).
% 3.17/1.77 thf(func_def_73, type, ord_le1816232656od_a_a: set_Pr681306928od_a_a > set_Pr681306928od_a_a > $o).
% 3.17/1.77 thf(func_def_74, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 3.17/1.77 thf(func_def_75, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.17/1.77 thf(func_def_76, type, pair_p128415500ph_nat: pair_p1914262621t_unit > $o).
% 3.17/1.77 thf(func_def_77, type, pair_p752841413at_nat: pair_p2041852168t_unit > $o).
% 3.17/1.77 thf(func_def_78, type, pair_p159207083od_a_a: pair_p1593840546t_unit > $o).
% 3.17/1.77 thf(func_def_79, type, pair_p374947051od_a_a: pair_p1765063010t_unit > $o).
% 3.17/1.77 thf(func_def_80, type, pair_p1802376898raph_a: pair_p125712459t_unit > $o).
% 3.17/1.77 thf(func_def_81, type, pair_p1027063983ms_nat: pair_p1914262621t_unit > $o).
% 3.17/1.77 thf(func_def_82, type, pair_p56914274at_nat: pair_p2041852168t_unit > $o).
% 3.17/1.77 thf(func_def_83, type, pair_p1906342088od_a_a: pair_p1593840546t_unit > $o).
% 3.17/1.77 thf(func_def_84, type, pair_p504738056od_a_a: pair_p1765063010t_unit > $o).
% 3.17/1.77 thf(func_def_85, type, pair_p1864019935ioms_a: pair_p125712459t_unit > $o).
% 3.17/1.77 thf(func_def_86, type, pair_p69470259t_unit: pair_p1914262621t_unit > product_unit).
% 3.17/1.77 thf(func_def_87, type, pair_p1984658862t_unit: pair_p1765063010t_unit > product_unit).
% 3.17/1.77 thf(func_def_88, type, pair_p1896222615t_unit: pair_p125712459t_unit > product_unit).
% 3.17/1.77 thf(func_def_89, type, pair_p1167410509t_unit: set_nat > set_Pr1986765409at_nat > product_unit > pair_p1914262621t_unit).
% 3.17/1.77 thf(func_def_90, type, pair_p398687508t_unit: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_unit > pair_p1765063010t_unit).
% 3.17/1.77 thf(func_def_91, type, pair_p1621517565t_unit: set_a > set_Product_prod_a_a > product_unit > pair_p125712459t_unit).
% 3.17/1.77 thf(func_def_92, type, pair_p715279805t_unit: pair_p1914262621t_unit > set_Pr1986765409at_nat).
% 3.17/1.77 thf(func_def_93, type, pair_p806300874t_unit: pair_p2041852168t_unit > set_Pr1490359111at_nat).
% 3.17/1.77 thf(func_def_94, type, pair_p1559300324t_unit: pair_p1593840546t_unit > set_Pr1295299783od_a_a).
% 3.17/1.77 thf(func_def_95, type, pair_p1783210148t_unit: pair_p1765063010t_unit > set_Pr1948701895od_a_a).
% 3.17/1.77 thf(func_def_96, type, pair_p133601421t_unit: pair_p125712459t_unit > set_Product_prod_a_a).
% 3.17/1.77 thf(func_def_97, type, pair_p1677060310t_unit: pair_p1914262621t_unit > set_nat).
% 3.17/1.77 thf(func_def_98, type, pair_p210955889t_unit: pair_p2041852168t_unit > set_Pr1986765409at_nat).
% 3.17/1.77 thf(func_def_99, type, pair_p1652294923t_unit: pair_p1593840546t_unit > set_Pr1948701895od_a_a).
% 3.17/1.77 thf(func_def_100, type, pair_p447552203t_unit: pair_p1765063010t_unit > set_Product_prod_a_a).
% 3.17/1.77 thf(func_def_101, type, pair_p1047056820t_unit: pair_p125712459t_unit > set_a).
% 3.17/1.77 thf(func_def_102, type, pair_p1515597646ph_nat: pair_p1914262621t_unit > $o).
% 3.17/1.77 thf(func_def_103, type, pair_p646030121od_a_a: pair_p1765063010t_unit > $o).
% 3.17/1.77 thf(func_def_104, type, pair_p68905728raph_a: pair_p125712459t_unit > $o).
% 3.17/1.77 thf(func_def_105, type, product_Pair_a_a: a > a > product_prod_a_a).
% 3.17/1.77 thf(func_def_106, type, produc45129834at_nat: set_nat > (nat > set_nat) > set_Pr1986765409at_nat).
% 3.17/1.77 thf(func_def_107, type, produc894163943at_nat: set_nat > (nat > set_Pr1986765409at_nat) > set_Pr1746169692at_nat).
% 3.17/1.77 thf(func_def_108, type, produc1182842125od_a_a: set_nat > (nat > set_Product_prod_a_a) > set_Pr339609346od_a_a).
% 3.17/1.77 thf(func_def_109, type, product_Sigma_nat_a: set_nat > (nat > set_a) > set_Pr967348953_nat_a).
% 3.17/1.77 thf(func_def_110, type, produc931712687_a_nat: set_Product_prod_a_a > (product_prod_a_a > set_nat) > set_Pr894832732_a_nat).
% 3.17/1.77 thf(func_def_111, type, produc304751368od_a_a: set_Product_prod_a_a > (product_prod_a_a > set_Product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.17/1.77 thf(func_def_112, type, produc1282482655_a_a_a: set_Product_prod_a_a > (product_prod_a_a > set_a) > set_Pr1689873822_a_a_a).
% 3.17/1.77 thf(func_def_113, type, product_Sigma_a_nat: set_a > (a > set_nat) > set_Pr548851891_a_nat).
% 3.17/1.77 thf(func_def_114, type, produc292491723at_nat: set_a > (a > set_Pr1986765409at_nat) > set_Pr7688842at_nat).
% 3.17/1.77 thf(func_def_115, type, produc520147185od_a_a: set_a > (a > set_Product_prod_a_a) > set_Pr681306928od_a_a).
% 3.17/1.77 thf(func_def_116, type, product_Sigma_a_a: set_a > (a > set_a) > set_Product_prod_a_a).
% 3.17/1.77 thf(func_def_117, type, produc1833107820_a_a_o: (a > a > $o) > product_prod_a_a > $o).
% 3.17/1.77 thf(func_def_118, type, product_product_a_a: set_a > set_a > set_Product_prod_a_a).
% 3.17/1.77 thf(func_def_119, type, collect_nat: (nat > $o) > set_nat).
% 3.17/1.77 thf(func_def_120, type, collec7649004at_nat: (product_prod_nat_nat > $o) > set_Pr1986765409at_nat).
% 3.17/1.77 thf(func_def_121, type, collec1635618130od_a_a: (produc1572603623od_a_a > $o) > set_Pr1948701895od_a_a).
% 3.17/1.77 thf(func_def_122, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 3.17/1.77 thf(func_def_123, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 3.17/1.77 thf(func_def_124, type, collec1606769740at_nat: (set_Pr1986765409at_nat > $o) > set_se1612935105at_nat).
% 3.17/1.77 thf(func_def_125, type, collec453062450od_a_a: (set_Pr1948701895od_a_a > $o) > set_se958357159od_a_a).
% 3.17/1.77 thf(func_def_126, type, collec183727474od_a_a: (set_Product_prod_a_a > $o) > set_se1596668135od_a_a).
% 3.17/1.77 thf(func_def_127, type, collect_set_a: (set_a > $o) > set_set_a).
% 3.17/1.77 thf(func_def_128, type, collect_a: (a > $o) > set_a).
% 3.17/1.77 thf(func_def_129, type, member_nat: nat > set_nat > $o).
% 3.17/1.77 thf(func_def_130, type, member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o).
% 3.17/1.77 thf(func_def_131, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 3.17/1.77 thf(func_def_132, type, member_set_nat: set_nat > set_set_nat > $o).
% 3.17/1.77 thf(func_def_133, type, member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o).
% 3.17/1.77 thf(func_def_134, type, member_set_a: set_a > set_set_a > $o).
% 3.17/1.77 thf(func_def_135, type, member_a: a > set_a > $o).
% 3.17/1.77 thf(func_def_136, type, g: pair_p125712459t_unit).
% 3.17/1.77 thf(func_def_137, type, n: nat).
% 3.17/1.77 thf(func_def_153, type, ph1: !>[X0: $tType]:(X0)).
% 3.17/1.77 thf(f1056,plain,(
% 3.17/1.77 $false),
% 3.17/1.77 inference(subsumption_resolution,[status(thm)],[f1055,f1046])).
% 3.17/1.77 thf(f1046,plain,(
% 3.17/1.77 ($true = (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g)))))),
% 3.17/1.77 inference(cnf_transformation,[status(thm)],[f467])).
% 3.17/1.77 thf(f467,plain,(
% 3.17/1.77 ($true = (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g)))))),
% 3.17/1.77 inference(fool_elimination,[status(thm)],[f466])).
% 3.17/1.77 thf(f466,plain,(
% 3.17/1.77 (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[X0 : a] : (pair_p1047056820t_unit @ g))))),
% 3.17/1.77 inference(rectify,[status(thm)],[f1])).
% 3.17/1.77 thf(f1,axiom,(
% 3.17/1.77 (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[X0 : a] : (pair_p1047056820t_unit @ g))))),
% 3.17/1.77 file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',fact_0__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062)).
% 3.17/1.77 thf(f1055,plain,(
% 3.17/1.77 ($true != (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g)))))),
% 3.17/1.77 inference(subsumption_resolution,[status(thm)],[f1054,f1050])).
% 3.17/1.77 thf(f1050,plain,(
% 3.17/1.77 ((finite179568208od_a_a @ (pair_p133601421t_unit @ g)) != $true)),
% 3.17/1.77 inference(cnf_transformation,[status(thm)],[f1036])).
% 3.17/1.77 thf(f1036,plain,(
% 3.17/1.77 ((finite179568208od_a_a @ (pair_p133601421t_unit @ g)) != $true)),
% 3.17/1.77 inference(flattening,[status(thm)],[f391])).
% 3.17/1.77 thf(f391,plain,(
% 3.17/1.77 ~((finite179568208od_a_a @ (pair_p133601421t_unit @ g)) = $true)),
% 3.17/1.77 inference(fool_elimination,[status(thm)],[f390])).
% 3.17/1.77 thf(f390,plain,(
% 3.17/1.77 ~(finite179568208od_a_a @ (pair_p133601421t_unit @ g))),
% 3.17/1.77 inference(rectify,[status(thm)],[f356])).
% 3.17/1.77 thf(f356,negated_conjecture,(
% 3.17/1.77 ~(finite179568208od_a_a @ (pair_p133601421t_unit @ g))),
% 3.17/1.77 inference(negated_conjecture,[status(cth)],[f355])).
% 3.17/1.77 thf(f355,conjecture,(
% 3.17/1.77 (finite179568208od_a_a @ (pair_p133601421t_unit @ g))),
% 3.17/1.77 file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',conj_0)).
% 3.17/1.77 thf(f1054,plain,(
% 3.17/1.77 ((finite179568208od_a_a @ (pair_p133601421t_unit @ g)) = $true) | ($true != (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g)))))),
% 3.17/1.77 inference(trivial_inequality_removal,[status(thm)],[f1053])).
% 3.17/1.77 thf(f1053,plain,(
% 3.17/1.77 ($true != $true) | ($true != (finite179568208od_a_a @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g))))) | ((finite179568208od_a_a @ (pair_p133601421t_unit @ g)) = $true)),
% 3.17/1.77 inference(superposition,[status(thm)],[f1051,f1044])).
% 3.17/1.77 thf(f1044,plain,(
% 3.17/1.77 ((ord_le1824328871od_a_a @ (pair_p133601421t_unit @ g) @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g)))) = $true)),
% 3.17/1.77 inference(cnf_transformation,[status(thm)],[f1003])).
% 3.17/1.77 thf(f1003,plain,(
% 3.17/1.77 ((ord_le1824328871od_a_a @ (pair_p133601421t_unit @ g) @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[Y0 : a]: (pair_p1047056820t_unit @ g)))) = $true)),
% 3.17/1.77 inference(fool_elimination,[status(thm)],[f1002])).
% 3.17/1.77 thf(f1002,plain,(
% 3.17/1.77 (ord_le1824328871od_a_a @ (pair_p133601421t_unit @ g) @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[X0 : a] : (pair_p1047056820t_unit @ g))))),
% 3.17/1.77 inference(rectify,[status(thm)],[f2])).
% 3.17/1.77 thf(f2,axiom,(
% 3.17/1.77 (ord_le1824328871od_a_a @ (pair_p133601421t_unit @ g) @ (product_Sigma_a_a @ (pair_p1047056820t_unit @ g) @ (^[X0 : a] : (pair_p1047056820t_unit @ g))))),
% 3.17/1.77 file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',fact_1__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062)).
% 3.17/1.77 thf(f1051,plain,(
% 3.17/1.77 ( ! [X0 : set_Product_prod_a_a,X1 : set_Product_prod_a_a] : (((ord_le1824328871od_a_a @ X1 @ X0) != $true) | ((finite179568208od_a_a @ X0) != $true) | ($true = (finite179568208od_a_a @ X1))) )),
% 3.17/1.77 inference(cnf_transformation,[status(thm)],[f1041])).
% 3.17/1.77 thf(f1041,plain,(
% 3.17/1.77 ! [X0 : set_Product_prod_a_a,X1 : set_Product_prod_a_a] : (((finite179568208od_a_a @ X0) != $true) | ((ord_le1824328871od_a_a @ X1 @ X0) != $true) | ($true = (finite179568208od_a_a @ X1)))),
% 3.17/1.77 inference(flattening,[status(thm)],[f1040])).
% 3.17/1.77 thf(f1040,plain,(
% 3.17/1.77 ! [X0 : set_Product_prod_a_a,X1 : set_Product_prod_a_a] : ((($true = (finite179568208od_a_a @ X1)) | ((finite179568208od_a_a @ X0) != $true)) | ((ord_le1824328871od_a_a @ X1 @ X0) != $true))),
% 3.17/1.77 inference(ennf_transformation,[status(thm)],[f581])).
% 3.17/1.77 thf(f581,plain,(
% 3.17/1.77 ! [X0 : set_Product_prod_a_a,X1 : set_Product_prod_a_a] : (((ord_le1824328871od_a_a @ X1 @ X0) = $true) => (((finite179568208od_a_a @ X0) = $true) => ($true = (finite179568208od_a_a @ X1))))),
% 3.17/1.77 inference(fool_elimination,[status(thm)],[f580])).
% 3.17/1.77 thf(f580,plain,(
% 3.17/1.77 ! [X0 : set_Product_prod_a_a,X1 : set_Product_prod_a_a] : ((ord_le1824328871od_a_a @ X1 @ X0) => ((finite179568208od_a_a @ X0) => (finite179568208od_a_a @ X1)))),
% 3.17/1.77 inference(rectify,[status(thm)],[f20])).
% 3.17/1.77 thf(f20,axiom,(
% 3.17/1.77 ! [X4 : set_Product_prod_a_a,X3 : set_Product_prod_a_a] : ((ord_le1824328871od_a_a @ X3 @ X4) => ((finite179568208od_a_a @ X4) => (finite179568208od_a_a @ X3)))),
% 3.17/1.77 file('/export/starexec/sandbox2/tmp/tmp.WsvTUfDMER/DTF2THF_5057.p',fact_19_finite__subset)).
% 3.17/1.77 % SZS output end Proof for DTF2THF_5057
% 3.17/1.77 % (5269)------------------------------
% 3.17/1.77 % (5269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.77 % (5269)Termination reason: Refutation
% 3.17/1.77
% 3.17/1.77 % (5269)Memory used [KB]: 6652
% 3.17/1.77 % (5269)Time elapsed: 0.029 s
% 3.17/1.77 % (5269)Instructions burned: 53 (million)
% 3.17/1.77 % (5269)------------------------------
% 3.17/1.77 % (5269)------------------------------
% 3.17/1.77 % (5243)Success in time 0.13 s
%------------------------------------------------------------------------------