%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP154^1 : 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 : n003.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 08:55:59 AM UTC 2026
% Result : Theorem 2.74s 1.60s
% Output : Refutation 2.74s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 2
% Syntax : Number of formulae : 11 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 30 ( 7 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 107 ( 6 ~; 4 |; 1 &; 96 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 130 ( 127 usr; 14 con; 0-3 aty)
% Number of variables : 4 ( 0 ^; 4 !; 0 ?; 4 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Product_prod_a_a: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_7,type,
set_a: $tType ).
thf(type_def_9,type,
set_Pr2045688071_set_a: $tType ).
thf(type_def_10,type,
set_se1596668135od_a_a: $tType ).
thf(type_def_11,type,
set_set_set_a: $tType ).
thf(type_def_12,type,
set_set_a: $tType ).
thf(type_def_13,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_14,type,
set_Pr1295299783od_a_a: $tType ).
thf(type_def_15,type,
product_prod_a_a: $tType ).
thf(type_def_16,type,
produc1691597095_set_a: $tType ).
thf(type_def_17,type,
produc1299534078_set_a: $tType ).
thf(type_def_18,type,
produc1472671646_set_a: $tType ).
thf(type_def_19,type,
produc386552094_set_a: $tType ).
thf(type_def_20,type,
produc275558334_set_a: $tType ).
thf(type_def_21,type,
produc398359056_set_a: $tType ).
thf(type_def_22,type,
produc270245360od_a_a: $tType ).
thf(type_def_23,type,
produc407913767_set_a: $tType ).
thf(type_def_24,type,
produc522143687_set_a: $tType ).
thf(type_def_25,type,
produc304417648_set_a: $tType ).
thf(type_def_26,type,
produc376942416od_a_a: $tType ).
thf(type_def_27,type,
produc1083127431_set_a: $tType ).
thf(type_def_28,type,
produc1572603623od_a_a: $tType ).
thf(type_def_29,type,
produc1990712039od_a_a: $tType ).
thf(func_def_0,type,
set_Pr1295299783od_a_a: $tType ).
thf(func_def_1,type,
produc1990712039od_a_a: $tType ).
thf(func_def_2,type,
produc398359056_set_a: $tType ).
thf(func_def_3,type,
produc1299534078_set_a: $tType ).
thf(func_def_4,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_5,type,
produc304417648_set_a: $tType ).
thf(func_def_6,type,
produc270245360od_a_a: $tType ).
thf(func_def_7,type,
produc386552094_set_a: $tType ).
thf(func_def_8,type,
produc1472671646_set_a: $tType ).
thf(func_def_9,type,
produc1572603623od_a_a: $tType ).
thf(func_def_10,type,
produc376942416od_a_a: $tType ).
thf(func_def_11,type,
produc275558334_set_a: $tType ).
thf(func_def_12,type,
produc407913767_set_a: $tType ).
thf(func_def_13,type,
set_Pr2045688071_set_a: $tType ).
thf(func_def_14,type,
produc1083127431_set_a: $tType ).
thf(func_def_15,type,
produc522143687_set_a: $tType ).
thf(func_def_16,type,
set_se1596668135od_a_a: $tType ).
thf(func_def_17,type,
produc1691597095_set_a: $tType ).
thf(func_def_18,type,
set_Product_prod_a_a: $tType ).
thf(func_def_19,type,
set_set_set_a: $tType ).
thf(func_def_20,type,
product_prod_a_a: $tType ).
thf(func_def_21,type,
set_set_a: $tType ).
thf(func_def_22,type,
set_a: $tType ).
thf(func_def_23,type,
a: $tType ).
thf(func_def_24,type,
bNF_Ca850758276_set_a: set_Product_prod_a_a > ( a > set_a ) > $o ).
thf(func_def_25,type,
finite283787856_set_a: set_Pr2045688071_set_a > $o ).
thf(func_def_26,type,
finite179568208od_a_a: set_Product_prod_a_a > $o ).
thf(func_def_27,type,
finite1145471536od_a_a: set_se1596668135od_a_a > $o ).
thf(func_def_28,type,
finite1606323175_set_a: set_set_set_a > $o ).
thf(func_def_29,type,
finite_finite_set_a: set_set_a > $o ).
thf(func_def_30,type,
finite_finite_a: set_a > $o ).
thf(func_def_31,type,
inf_in920712437od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_32,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(func_def_33,type,
sup_su615935707od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_34,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(func_def_35,type,
lattic1600076923_set_a: set_set_a > set_a ).
thf(func_def_36,type,
lattic501550101_set_a: set_set_a > set_a ).
thf(func_def_37,type,
order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_38,type,
order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_39,type,
order_816746613_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_40,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_41,type,
order_under_a: set_Product_prod_a_a > a > set_a ).
thf(func_def_42,type,
bot_bo1293121322_a_a_o: product_prod_a_a > $o ).
thf(func_def_43,type,
bot_bot_a_o: a > $o ).
thf(func_def_44,type,
bot_bo1519632275_set_a: produc1691597095_set_a ).
thf(func_def_45,type,
bot_bo1116205875od_a_a: set_Pr1948701895od_a_a ).
thf(func_def_46,type,
bot_bo671027059_set_a: set_Pr2045688071_set_a ).
thf(func_def_47,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(func_def_48,type,
bot_bo1424703315od_a_a: set_se1596668135od_a_a ).
thf(func_def_49,type,
bot_bo1264989578_set_a: set_set_set_a ).
thf(func_def_50,type,
bot_bot_set_set_a: set_set_a ).
thf(func_def_51,type,
bot_bot_set_a: set_a ).
thf(func_def_52,type,
ord_le2039441566_set_a: produc1299534078_set_a > produc1299534078_set_a > $o ).
thf(func_def_53,type,
ord_le1729087294_set_a: produc1472671646_set_a > produc1472671646_set_a > $o ).
thf(func_def_54,type,
ord_le642967742_set_a: produc386552094_set_a > produc386552094_set_a > $o ).
thf(func_def_55,type,
ord_le2095302494_set_a: produc275558334_set_a > produc275558334_set_a > $o ).
thf(func_def_56,type,
ord_le1138266544_set_a: produc398359056_set_a > produc398359056_set_a > $o ).
thf(func_def_57,type,
ord_le526661008od_a_a: produc270245360od_a_a > produc270245360od_a_a > $o ).
thf(func_def_58,type,
ord_le723713223_set_a: produc407913767_set_a > produc407913767_set_a > $o ).
thf(func_def_59,type,
ord_le1201700711_set_a: produc522143687_set_a > produc522143687_set_a > $o ).
thf(func_def_60,type,
ord_le560833296_set_a: produc304417648_set_a > produc304417648_set_a > $o ).
thf(func_def_61,type,
ord_le49202928od_a_a: produc376942416od_a_a > produc376942416od_a_a > $o ).
thf(func_def_62,type,
ord_le1762684455_set_a: produc1083127431_set_a > produc1083127431_set_a > $o ).
thf(func_def_63,type,
ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(func_def_64,type,
ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_65,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_66,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_67,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_68,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(func_def_69,type,
prefer1867335523od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_70,type,
prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_71,type,
prefer270414042_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a ).
thf(func_def_72,type,
prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_73,type,
prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_74,type,
prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_75,type,
prefer1203288218_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a ).
thf(func_def_76,type,
prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_77,type,
prefer926864306od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_78,type,
prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_79,type,
prefer1671525929_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a ).
thf(func_def_80,type,
prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_81,type,
prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_82,type,
prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_83,type,
prefer1710114912_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_84,type,
prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_85,type,
prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_86,type,
prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_87,type,
prefer484977842_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_88,type,
prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_89,type,
prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_90,type,
produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a ).
thf(func_def_91,type,
produc759028718_set_a: produc1691597095_set_a > set_set_a > produc1299534078_set_a ).
thf(func_def_92,type,
produc1719072398_set_a: produc1691597095_set_a > set_a > produc1472671646_set_a ).
thf(func_def_93,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_94,type,
produc1599887374_set_a: set_Product_prod_a_a > set_set_a > produc386552094_set_a ).
thf(func_def_95,type,
produc194376366_set_a: set_Product_prod_a_a > set_a > produc275558334_set_a ).
thf(func_def_96,type,
produc760262016_set_a: set_set_a > produc1691597095_set_a > produc398359056_set_a ).
thf(func_def_97,type,
produc1618062688od_a_a: set_set_a > set_Product_prod_a_a > produc270245360od_a_a ).
thf(func_def_98,type,
produc2102896407_set_a: set_set_a > set_set_a > produc407913767_set_a ).
thf(func_def_99,type,
produc32105911_set_a: set_set_a > set_a > produc522143687_set_a ).
thf(func_def_100,type,
produc272677600_set_a: set_a > produc1691597095_set_a > produc304417648_set_a ).
thf(func_def_101,type,
produc1397177024od_a_a: set_a > set_Product_prod_a_a > produc376942416od_a_a ).
thf(func_def_102,type,
produc1560024183_set_a: set_a > set_set_a > produc1083127431_set_a ).
thf(func_def_103,type,
produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
thf(func_def_104,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_105,type,
id_on_1182990020od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(func_def_106,type,
id_on_a: set_a > set_Product_prod_a_a ).
thf(func_def_107,type,
refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_108,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_109,type,
total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_110,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_111,type,
trans_a: set_Product_prod_a_a > $o ).
thf(func_def_112,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(func_def_113,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_114,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_115,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_116,type,
is_empty_a: set_a > $o ).
thf(func_def_117,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_118,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_119,type,
member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_120,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_121,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(func_def_122,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(func_def_123,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_124,type,
member_a: a > set_a > $o ).
thf(func_def_125,type,
carrier: set_a ).
thf(func_def_126,type,
relation: set_Product_prod_a_a ).
thf(func_def_127,type,
x: a ).
thf(func_def_128,type,
y: a ).
thf(func_def_143,type,
sK0: ( product_prod_a_a > $o ) > a ).
thf(func_def_144,type,
sK1: ( product_prod_a_a > $o ) > a ).
thf(func_def_145,type,
sK2: ( set_a > set_a ) > set_a ).
thf(func_def_146,type,
sK3: ( set_a > set_a ) > set_a ).
thf(func_def_147,type,
sK4: set_a > a ).
thf(func_def_148,type,
sK5: ( product_prod_a_a > $o ) > a ).
thf(func_def_149,type,
sK6: ( product_prod_a_a > $o ) > a ).
thf(func_def_150,type,
sK7: set_a > a ).
thf(func_def_151,type,
sK8: ( set_a > set_a ) > set_a ).
thf(func_def_152,type,
sK9: ( set_a > set_a ) > set_a ).
thf(func_def_153,type,
sK10: product_prod_a_a > a ).
thf(func_def_154,type,
sK11: product_prod_a_a > a ).
thf(func_def_155,type,
sK12: product_prod_a_a > a ).
thf(func_def_156,type,
sK13: product_prod_a_a > a ).
thf(func_def_157,type,
sK14: set_a > a ).
thf(func_def_158,type,
sK15: ( set_a > set_a ) > set_a ).
thf(func_def_159,type,
sK16: ( set_a > set_a ) > set_a ).
thf(func_def_160,type,
sK17: set_a > set_a > a ).
thf(func_def_161,type,
sK18: ( set_a > set_a ) > set_a ).
thf(func_def_162,type,
sK19: ( set_a > set_a ) > set_a ).
thf(func_def_163,type,
sK20: set_a > a ).
thf(func_def_165,type,
ph22:
!>[X0: $tType] : X0 ).
thf(f1315,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1273,f1255]) ).
thf(f1255,plain,
! [X0: set_a] :
( $true
= ( ord_less_eq_set_a @ X0 @ X0 ) ),
inference(cnf_transformation,[status(thm)],[f909]) ).
thf(f909,plain,
! [X0: set_a] :
( $true
= ( ord_less_eq_set_a @ X0 @ X0 ) ),
inference(fool_elimination,[status(thm)],[f908]) ).
thf(f908,plain,
! [X0: set_a] : ( ord_less_eq_set_a @ X0 @ X0 ),
inference(rectify,[status(thm)],[f196]) ).
thf(f196,axiom,
! [X13: set_a] : ( ord_less_eq_set_a @ X13 @ X13 ),
file('/export/starexec/sandbox2/tmp/tmp.b3Te47zhCv/DTF2THF_27726.p',fact_195_subset__refl) ).
thf(f1273,plain,
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1066]) ).
thf(f1066,plain,
( ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true )
& ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f767]) ).
thf(f767,plain,
~ ( ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
= $true )
| ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f766]) ).
thf(f766,plain,
~ ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ),
inference(rectify,[status(thm)],[f356]) ).
thf(f356,negated_conjecture,
~ ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ),
inference(negated_conjecture,[status(cth)],[f355]) ).
thf(f355,conjecture,
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.b3Te47zhCv/DTF2THF_27726.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP154^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n003.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:27:55 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.45 ---- Original DTF file ---
% 0.31/0.45 thf(spec,logic,$$dhol).
% 0.31/0.45 %------------------------------------------------------------------------------
% 0.31/0.45 % File : ITP154^1 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.45 % Domain : Interactive Theorem Proving
% 0.31/0.45 % Problem : Sledgehammer Preferences problem prob_205__6250810_1
% 0.31/0.45 % Version : Especial.
% 0.31/0.45 % English :
% 0.31/0.45
% 0.31/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.45 % Source : [Des21]
% 0.31/0.45 % Names : Preferences/prob_205__6250810_1 [Des21]
% 0.31/0.45
% 0.31/0.45 % Status : Theorem
% 0.31/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.31/0.45 % Syntax : Number of formulae : 484 ( 122 unt; 129 typ; 0 def)
% 0.31/0.45 % Number of atoms : 1216 ( 241 equ; 0 cnn)
% 0.31/0.45 % Maximal formula atoms : 10 ( 3 avg)
% 0.31/0.45 % Number of connectives : 3744 ( 144 ~; 24 |; 133 &;2879 @)
% 0.31/0.45 % ( 0 <=>; 564 =>; 0 <=; 0 <~>)
% 0.31/0.45 % Maximal formula depth : 16 ( 7 avg)
% 0.31/0.45 % Number of types : 25 ( 24 usr)
% 0.31/0.45 % Number of type conns : 230 ( 230 >; 0 *; 0 +; 0 <<)
% 0.31/0.45 % Number of symbols : 106 ( 105 usr; 12 con; 0-3 aty)
% 0.31/0.45 % Number of variables : 1036 ( 64 ^; 954 !; 18 ?;1036 :)
% 0.31/0.45 % SPC : TH0_THM_EQU_NAR
% 0.31/0.45
% 0.31/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:44:35.897
% 0.31/0.45 %------------------------------------------------------------------------------
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% 0.31/0.46 % assms(2)
% 0.31/0.46 thf(fact_2_rational__preference__axioms,axiom,
% 0.31/0.46 prefer719835986ence_a @ carrier @ relation ).
% 0.31/0.46
% 0.31/0.46 % rational_preference_axioms
% 0.31/0.46 thf(fact_3_nbt__nest,axiom,
% 0.31/0.46 ! [Y: a,X: a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 0.31/0.46 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nbt_nest
% 0.31/0.46 thf(fact_4_assms_I1_J,axiom,
% 0.31/0.46 finite_finite_a @ carrier ).
% 0.31/0.46
% 0.31/0.46 % assms(1)
% 0.31/0.46 thf(fact_5_preference__axioms,axiom,
% 0.31/0.46 prefer1292084992ence_a @ carrier @ relation ).
% 0.31/0.46
% 0.31/0.46 % preference_axioms
% 0.31/0.46 thf(fact_6_trans__refl,axiom,
% 0.31/0.46 order_preorder_on_a @ carrier @ relation ).
% 0.31/0.46
% 0.31/0.46 % trans_refl
% 0.31/0.46 thf(fact_7_no__better__subset__pref,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % no_better_subset_pref
% 0.31/0.46 thf(fact_8_no__better__thansubset__rel,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member_a @ X @ carrier )
% 0.31/0.46 => ( ( member_a @ Y @ carrier )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % no_better_thansubset_rel
% 0.31/0.46 thf(fact_9_no__better__than__nonepty,axiom,
% 0.31/0.46 ! [X: a] :
% 0.31/0.46 ( ( carrier != bot_bot_set_a )
% 0.31/0.46 => ( ( member_a @ X @ carrier )
% 0.31/0.46 => ( ( prefer1676310729than_a @ X @ carrier @ relation )
% 0.31/0.46 != bot_bot_set_a ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % no_better_than_nonepty
% 0.31/0.46 thf(fact_10_total,axiom,
% 0.31/0.46 total_on_a @ carrier @ relation ).
% 0.31/0.46
% 0.31/0.46 % total
% 0.31/0.46 thf(fact_11_worse__in__no__better,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % worse_in_no_better
% 0.31/0.46 thf(fact_12_reflexivity,axiom,
% 0.31/0.46 refl_on_a @ carrier @ relation ).
% 0.31/0.46
% 0.31/0.46 % reflexivity
% 0.31/0.46 thf(fact_13_compl,axiom,
% 0.31/0.46 ! [X2: a] :
% 0.31/0.46 ( ( member_a @ X2 @ carrier )
% 0.31/0.46 => ! [Xa: a] :
% 0.31/0.46 ( ( member_a @ Xa @ carrier )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
% 0.31/0.46 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % compl
% 0.31/0.46 thf(fact_14_completeD,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member_a @ X @ carrier )
% 0.31/0.46 => ( ( member_a @ Y @ carrier )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % completeD
% 0.31/0.46 thf(fact_15_not__outside,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 => ( member_a @ X @ carrier ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_outside
% 0.31/0.46 thf(fact_16_strict__is__neg__transitive,axiom,
% 0.31/0.46 ! [X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( ( member_a @ X @ carrier )
% 0.31/0.46 & ( member_a @ Y @ carrier )
% 0.31/0.46 & ( member_a @ Z @ carrier ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) )
% 0.31/0.46 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % strict_is_neg_transitive
% 0.31/0.46 thf(fact_17_weak__is__transitive,axiom,
% 0.31/0.46 ! [X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( ( member_a @ X @ carrier )
% 0.31/0.46 & ( member_a @ Y @ carrier )
% 0.31/0.46 & ( member_a @ Z @ carrier ) )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % weak_is_transitive
% 0.31/0.46 thf(fact_18_strict__trans,axiom,
% 0.31/0.46 ! [X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % strict_trans
% 0.31/0.46 thf(fact_19_indiff__trans,axiom,
% 0.31/0.46 ! [X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % indiff_trans
% 0.31/0.46 thf(fact_20_strict__not__refl__weak,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( ( member_a @ X @ carrier )
% 0.31/0.46 & ( member_a @ Y @ carrier ) )
% 0.31/0.46 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.31/0.46 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % strict_not_refl_weak
% 0.31/0.46 thf(fact_21_rational__preference_Oaxioms_I1_J,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.axioms(1)
% 0.31/0.46 thf(fact_22_rational__preference_Oaxioms_I1_J,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.axioms(1)
% 0.31/0.46 thf(fact_23_preference__def,axiom,
% 0.31/0.46 ( prefer1409946857od_a_a
% 0.31/0.46 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 0.31/0.46 ( ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member2057358096od_a_a @ X3 @ Carrier2 ) )
% 0.31/0.46 & ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member2057358096od_a_a @ Y2 @ Carrier2 ) )
% 0.31/0.46 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference_def
% 0.31/0.46 thf(fact_24_preference__def,axiom,
% 0.31/0.46 ( prefer1710114912_set_a
% 0.31/0.46 = ( ^ [Carrier2: set_set_a,Relation2: set_Pr2045688071_set_a] :
% 0.31/0.46 ( ! [X3: set_a,Y2: set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member_set_a @ X3 @ Carrier2 ) )
% 0.31/0.46 & ! [X3: set_a,Y2: set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member_set_a @ Y2 @ Carrier2 ) )
% 0.31/0.46 & ( order_816746613_set_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference_def
% 0.31/0.46 thf(fact_25_preference__def,axiom,
% 0.31/0.46 ( prefer1338892073od_a_a
% 0.31/0.46 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member449909584od_a_a @ X3 @ Carrier2 ) )
% 0.31/0.46 & ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member449909584od_a_a @ Y2 @ Carrier2 ) )
% 0.31/0.46 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference_def
% 0.31/0.46 thf(fact_26_preference__def,axiom,
% 0.31/0.46 ( prefer1292084992ence_a
% 0.31/0.46 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X3: a,Y2: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member_a @ X3 @ Carrier2 ) )
% 0.31/0.46 & ! [X3: a,Y2: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.31/0.46 => ( member_a @ Y2 @ Carrier2 ) )
% 0.31/0.46 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference_def
% 0.31/0.46 thf(fact_27_preference_Ointro,axiom,
% 0.31/0.46 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member2057358096od_a_a @ X4 @ Carrier ) )
% 0.31/0.46 => ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 0.31/0.46 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.intro
% 0.31/0.46 thf(fact_28_preference_Ointro,axiom,
% 0.31/0.46 ! [Relation: set_Pr2045688071_set_a,Carrier: set_set_a] :
% 0.31/0.46 ( ! [X4: set_a,Y3: set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member_set_a @ X4 @ Carrier ) )
% 0.31/0.46 => ( ! [X4: set_a,Y3: set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member_set_a @ Y3 @ Carrier ) )
% 0.31/0.46 => ( ( order_816746613_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer1710114912_set_a @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.intro
% 0.31/0.46 thf(fact_29_preference_Ointro,axiom,
% 0.31/0.46 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member449909584od_a_a @ X4 @ Carrier ) )
% 0.31/0.46 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 0.31/0.46 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.intro
% 0.31/0.46 thf(fact_30_preference_Ointro,axiom,
% 0.31/0.46 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 0.31/0.46 ( ! [X4: a,Y3: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member_a @ X4 @ Carrier ) )
% 0.31/0.46 => ( ! [X4: a,Y3: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 0.31/0.46 => ( member_a @ Y3 @ Carrier ) )
% 0.31/0.46 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.intro
% 0.31/0.46 thf(fact_31_preference_Otrans__refl,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.trans_refl
% 0.31/0.46 thf(fact_32_preference_Otrans__refl,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.trans_refl
% 0.31/0.46 thf(fact_33_preference_Onot__outside,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.not_outside
% 0.31/0.46 thf(fact_34_preference_Onot__outside,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member_set_a @ X @ Carrier ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.not_outside
% 0.31/0.46 thf(fact_35_preference_Onot__outside,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.not_outside
% 0.31/0.46 thf(fact_36_preference_Onot__outside,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member_a @ X @ Carrier ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.not_outside
% 0.31/0.46 thf(fact_37_preference_Oreflexivity,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.reflexivity
% 0.31/0.46 thf(fact_38_preference_Oreflexivity,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.reflexivity
% 0.31/0.46 thf(fact_39_preference_Oindiff__trans,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 0.31/0.46 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 0.31/0.46 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.indiff_trans
% 0.31/0.46 thf(fact_40_preference_Oindiff__trans,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.31/0.46 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.indiff_trans
% 0.31/0.46 thf(fact_41_preference_Oindiff__trans,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.indiff_trans
% 0.31/0.46 thf(fact_42_rational__preference_Ocompl,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ! [X2: set_a] :
% 0.31/0.46 ( ( member_set_a @ X2 @ Carrier )
% 0.31/0.46 => ! [Xa: set_a] :
% 0.31/0.46 ( ( member_set_a @ Xa @ Carrier )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Xa ) @ Relation )
% 0.31/0.46 | ( member593450832_set_a @ ( produc1928581911_set_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.compl
% 0.31/0.46 thf(fact_43_rational__preference_Ocompl,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ! [X2: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.31/0.46 => ! [Xa: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ Xa @ Carrier )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Xa ) @ Relation )
% 0.31/0.46 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.compl
% 0.31/0.46 thf(fact_44_rational__preference_Ocompl,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ! [X2: a] :
% 0.31/0.46 ( ( member_a @ X2 @ Carrier )
% 0.31/0.46 => ! [Xa: a] :
% 0.31/0.46 ( ( member_a @ Xa @ Carrier )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
% 0.31/0.46 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.compl
% 0.31/0.46 thf(fact_45_rational__preference_Ototal,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.total
% 0.31/0.46 thf(fact_46_rational__preference_Ototal,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( total_on_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.total
% 0.31/0.46 thf(fact_47_rational__preference_OcompleteD,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.completeD
% 0.31/0.46 thf(fact_48_rational__preference_OcompleteD,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 => ( ( member_set_a @ Y @ Carrier )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.completeD
% 0.31/0.46 thf(fact_49_rational__preference_OcompleteD,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.completeD
% 0.31/0.46 thf(fact_50_rational__preference_OcompleteD,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_a @ X @ Carrier )
% 0.31/0.46 => ( ( member_a @ Y @ Carrier )
% 0.31/0.46 => ( ( X != Y )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.completeD
% 0.31/0.46 thf(fact_51_rational__preference_Ostrict__trans,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_trans
% 0.31/0.46 thf(fact_52_rational__preference_Ostrict__trans,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_trans
% 0.31/0.46 thf(fact_53_rational__preference_Ostrict__trans,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_trans
% 0.31/0.46 thf(fact_54_rational__preference_Oweak__is__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 & ( member2057358096od_a_a @ Y @ Carrier )
% 0.31/0.46 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.weak_is_transitive
% 0.31/0.46 thf(fact_55_rational__preference_Oweak__is__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 & ( member_set_a @ Y @ Carrier )
% 0.31/0.46 & ( member_set_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 0.31/0.46 => ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.weak_is_transitive
% 0.31/0.46 thf(fact_56_rational__preference_Oweak__is__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 & ( member449909584od_a_a @ Y @ Carrier )
% 0.31/0.46 & ( member449909584od_a_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.weak_is_transitive
% 0.31/0.46 thf(fact_57_rational__preference_Oweak__is__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member_a @ X @ Carrier )
% 0.31/0.46 & ( member_a @ Y @ Carrier )
% 0.31/0.46 & ( member_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.weak_is_transitive
% 0.31/0.46 thf(fact_58_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 & ( member2057358096od_a_a @ Y @ Carrier ) )
% 0.31/0.46 => ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_not_refl_weak
% 0.31/0.46 thf(fact_59_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 & ( member_set_a @ Y @ Carrier ) )
% 0.31/0.46 => ( ( ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_not_refl_weak
% 0.31/0.46 thf(fact_60_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 & ( member449909584od_a_a @ Y @ Carrier ) )
% 0.31/0.46 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_not_refl_weak
% 0.31/0.46 thf(fact_61_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member_a @ X @ Carrier )
% 0.31/0.46 & ( member_a @ Y @ Carrier ) )
% 0.31/0.46 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_not_refl_weak
% 0.31/0.46 thf(fact_62_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 & ( member2057358096od_a_a @ Y @ Carrier )
% 0.31/0.46 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ X ) @ Relation ) )
% 0.31/0.46 | ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_is_neg_transitive
% 0.31/0.46 thf(fact_63_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 & ( member_set_a @ Y @ Carrier )
% 0.31/0.46 & ( member_set_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) )
% 0.31/0.46 | ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_is_neg_transitive
% 0.31/0.46 thf(fact_64_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 & ( member449909584od_a_a @ Y @ Carrier )
% 0.31/0.46 & ( member449909584od_a_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) )
% 0.31/0.46 | ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_is_neg_transitive
% 0.31/0.46 thf(fact_65_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member_a @ X @ Carrier )
% 0.31/0.46 & ( member_a @ Y @ Carrier )
% 0.31/0.46 & ( member_a @ Z @ Carrier ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) )
% 0.31/0.46 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.strict_is_neg_transitive
% 0.31/0.46 thf(fact_66_rational__preference_Oworse__in__no__better,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member2057358096od_a_a @ Y @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.worse_in_no_better
% 0.31/0.46 thf(fact_67_rational__preference_Oworse__in__no__better,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member_set_a @ Y @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.worse_in_no_better
% 0.31/0.46 thf(fact_68_rational__preference_Oworse__in__no__better,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.worse_in_no_better
% 0.31/0.46 thf(fact_69_rational__preference_Oworse__in__no__better,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.worse_in_no_better
% 0.31/0.46 thf(fact_70_rational__preference_Ono__better__than__nonepty,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bo1116205875od_a_a )
% 0.31/0.46 => ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( prefer926864306od_a_a @ X @ Carrier @ Relation )
% 0.31/0.46 != bot_bo1116205875od_a_a ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_than_nonepty
% 0.31/0.46 thf(fact_71_rational__preference_Ono__better__than__nonepty,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bot_set_set_a )
% 0.31/0.46 => ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 => ( ( prefer1671525929_set_a @ X @ Carrier @ Relation )
% 0.31/0.46 != bot_bot_set_set_a ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_than_nonepty
% 0.31/0.46 thf(fact_72_rational__preference_Ono__better__than__nonepty,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bo2131659635od_a_a )
% 0.31/0.46 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( prefer1498087410od_a_a @ X @ Carrier @ Relation )
% 0.31/0.46 != bot_bo2131659635od_a_a ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_than_nonepty
% 0.31/0.46 thf(fact_73_rational__preference_Ono__better__than__nonepty,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bot_set_a )
% 0.31/0.46 => ( ( member_a @ X @ Carrier )
% 0.31/0.46 => ( ( prefer1676310729than_a @ X @ Carrier @ Relation )
% 0.31/0.46 != bot_bot_set_a ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_than_nonepty
% 0.31/0.46 thf(fact_74_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 0.31/0.46 => ( ( ord_le456379495od_a_a @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) @ ( prefer926864306od_a_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_thansubset_rel
% 0.31/0.46 thf(fact_75_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 => ( ( member_set_a @ Y @ Carrier )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 => ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_thansubset_rel
% 0.31/0.46 thf(fact_76_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_thansubset_rel
% 0.31/0.46 thf(fact_77_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_a @ X @ Carrier )
% 0.31/0.46 => ( ( member_a @ Y @ Carrier )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_thansubset_rel
% 0.31/0.46 thf(fact_78_rational__preference_Ono__better__subset__pref,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_subset_pref
% 0.31/0.46 thf(fact_79_rational__preference_Ono__better__subset__pref,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_subset_pref
% 0.31/0.46 thf(fact_80_rational__preference_Ono__better__subset__pref,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.no_better_subset_pref
% 0.31/0.46 thf(fact_81_rational__preference_Onbt__nest,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Y: set_a,X: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 | ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.nbt_nest
% 0.31/0.46 thf(fact_82_rational__preference_Onbt__nest,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Y: product_prod_a_a,X: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 | ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.nbt_nest
% 0.31/0.46 thf(fact_83_rational__preference_Onbt__nest,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 0.31/0.46 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.nbt_nest
% 0.31/0.46 thf(fact_84_at__lst__asgd__not__ge,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( carrier != bot_bot_set_a )
% 0.31/0.46 => ( ( member_a @ X @ carrier )
% 0.31/0.46 => ( ( member_a @ Y @ carrier )
% 0.31/0.46 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
% 0.31/0.46 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_lst_asgd_not_ge
% 0.31/0.46 thf(fact_85_same__at__least__as__equal,axiom,
% 0.31/0.46 ! [Z: a,Y: a] :
% 0.31/0.46 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) )
% 0.31/0.46 => ( ( prefer161218362good_a @ Z @ carrier @ relation )
% 0.31/0.46 = ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % same_at_least_as_equal
% 0.31/0.46 thf(fact_86_pref__in__at__least__as,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 => ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pref_in_at_least_as
% 0.31/0.46 thf(fact_87_empty__subsetI,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A ) ).
% 0.31/0.46
% 0.31/0.46 % empty_subsetI
% 0.31/0.46 thf(fact_88_empty__subsetI,axiom,
% 0.31/0.46 ! [A: set_set_a] : ( ord_le318720350_set_a @ bot_bot_set_set_a @ A ) ).
% 0.31/0.46
% 0.31/0.46 % empty_subsetI
% 0.31/0.46 thf(fact_89_empty__subsetI,axiom,
% 0.31/0.46 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 0.31/0.46
% 0.31/0.46 % empty_subsetI
% 0.31/0.46 thf(fact_90_subset__empty,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A @ bot_bo2131659635od_a_a )
% 0.31/0.46 = ( A = bot_bo2131659635od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_empty
% 0.31/0.46 thf(fact_91_subset__empty,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ bot_bot_set_set_a )
% 0.31/0.46 = ( A = bot_bot_set_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_empty
% 0.31/0.46 thf(fact_92_subset__empty,axiom,
% 0.31/0.46 ! [A: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.31/0.46 = ( A = bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_empty
% 0.31/0.46 thf(fact_93_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A2: produc1572603623od_a_a,P: produc1572603623od_a_a > $o] :
% 0.31/0.46 ( ( member2057358096od_a_a @ A2 @ ( collec1635618130od_a_a @ P ) )
% 0.31/0.46 = ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_94_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A2: set_a,P: set_a > $o] :
% 0.31/0.46 ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
% 0.31/0.46 = ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_95_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A2: a,P: a > $o] :
% 0.31/0.46 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 0.31/0.46 = ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_96_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.31/0.46 ( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
% 0.31/0.46 = ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_97_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( collec1635618130od_a_a
% 0.31/0.46 @ ^ [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A ) )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_98_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( collect_set_a
% 0.31/0.46 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_99_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: set_a] :
% 0.31/0.46 ( ( collect_a
% 0.31/0.46 @ ^ [X3: a] : ( member_a @ X3 @ A ) )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_100_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( collec645855634od_a_a
% 0.31/0.46 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_101_Collect__cong,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.31/0.46 ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ( ( P @ X4 )
% 0.31/0.46 = ( Q @ X4 ) )
% 0.31/0.46 => ( ( collec645855634od_a_a @ P )
% 0.31/0.46 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_cong
% 0.31/0.46 thf(fact_102_Collect__cong,axiom,
% 0.31/0.46 ! [P: a > $o,Q: a > $o] :
% 0.31/0.46 ( ! [X4: a] :
% 0.31/0.46 ( ( P @ X4 )
% 0.31/0.46 = ( Q @ X4 ) )
% 0.31/0.46 => ( ( collect_a @ P )
% 0.31/0.46 = ( collect_a @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_cong
% 0.31/0.46 thf(fact_103_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a,C: set_a,D: set_a] :
% 0.31/0.46 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A2 @ B ) @ ( produc1928581911_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_less_eq_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_104_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_set_a,C: set_a,D: set_set_a] :
% 0.31/0.46 ( ( ord_le1762684455_set_a @ ( produc1560024183_set_a @ A2 @ B ) @ ( produc1560024183_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_less_eq_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_le318720350_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_105_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_set_a,B: set_a,C: set_set_a,D: set_a] :
% 0.31/0.46 ( ( ord_le1201700711_set_a @ ( produc32105911_set_a @ A2 @ B ) @ ( produc32105911_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_le318720350_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_106_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_Product_prod_a_a,C: set_a,D: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le49202928od_a_a @ ( produc1397177024od_a_a @ A2 @ B ) @ ( produc1397177024od_a_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_less_eq_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_le1824328871od_a_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_107_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_Product_prod_a_a,B: set_a,C: set_Product_prod_a_a,D: set_a] :
% 0.31/0.46 ( ( ord_le2095302494_set_a @ ( produc194376366_set_a @ A2 @ B ) @ ( produc194376366_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_le1824328871od_a_a @ A2 @ C )
% 0.31/0.46 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_108_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_set_a,B: set_set_a,C: set_set_a,D: set_set_a] :
% 0.31/0.46 ( ( ord_le723713223_set_a @ ( produc2102896407_set_a @ A2 @ B ) @ ( produc2102896407_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_le318720350_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_le318720350_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_109_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_a,B: produc1691597095_set_a,C: set_a,D: produc1691597095_set_a] :
% 0.31/0.46 ( ( ord_le560833296_set_a @ ( produc272677600_set_a @ A2 @ B ) @ ( produc272677600_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_less_eq_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_le486764743_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_110_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_Product_prod_a_a,B: set_set_a,C: set_Product_prod_a_a,D: set_set_a] :
% 0.31/0.46 ( ( ord_le642967742_set_a @ ( produc1599887374_set_a @ A2 @ B ) @ ( produc1599887374_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_le1824328871od_a_a @ A2 @ C )
% 0.31/0.46 & ( ord_le318720350_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_111_Pair__le,axiom,
% 0.31/0.46 ! [A2: produc1691597095_set_a,B: set_a,C: produc1691597095_set_a,D: set_a] :
% 0.31/0.46 ( ( ord_le1729087294_set_a @ ( produc1719072398_set_a @ A2 @ B ) @ ( produc1719072398_set_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_le486764743_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_112_Pair__le,axiom,
% 0.31/0.46 ! [A2: set_set_a,B: set_Product_prod_a_a,C: set_set_a,D: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le526661008od_a_a @ ( produc1618062688od_a_a @ A2 @ B ) @ ( produc1618062688od_a_a @ C @ D ) )
% 0.31/0.46 = ( ( ord_le318720350_set_a @ A2 @ C )
% 0.31/0.46 & ( ord_le1824328871od_a_a @ B @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_le
% 0.31/0.46 thf(fact_113_as__good__as__sameIff,axiom,
% 0.31/0.46 ! [Z: a,Y: a] :
% 0.31/0.46 ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 0.31/0.46 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % as_good_as_sameIff
% 0.31/0.46 thf(fact_114_as__good__asIff,axiom,
% 0.31/0.46 ! [X: a,Y: a] :
% 0.31/0.46 ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 0.31/0.46 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % as_good_asIff
% 0.31/0.46 thf(fact_115_finite__has__maximal,axiom,
% 0.31/0.46 ! [A: set_se1596668135od_a_a] :
% 0.31/0.46 ( ( finite1145471536od_a_a @ A )
% 0.31/0.46 => ( ( A != bot_bo1424703315od_a_a )
% 0.31/0.46 => ? [X4: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member1838126896od_a_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member1838126896od_a_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ X4 @ Xa )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal
% 0.31/0.46 thf(fact_116_finite__has__maximal,axiom,
% 0.31/0.46 ! [A: set_Pr2045688071_set_a] :
% 0.31/0.46 ( ( finite283787856_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bo671027059_set_a )
% 0.31/0.46 => ? [X4: produc1691597095_set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: produc1691597095_set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_le486764743_set_a @ X4 @ Xa )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal
% 0.31/0.46 thf(fact_117_finite__has__maximal,axiom,
% 0.31/0.46 ! [A: set_set_set_a] :
% 0.31/0.46 ( ( finite1606323175_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bo1264989578_set_a )
% 0.31/0.46 => ? [X4: set_set_a] :
% 0.31/0.46 ( ( member_set_set_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: set_set_a] :
% 0.31/0.46 ( ( member_set_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ X4 @ Xa )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal
% 0.31/0.46 thf(fact_118_finite__has__maximal,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ? [X4: set_a] :
% 0.31/0.46 ( ( member_set_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: set_a] :
% 0.31/0.46 ( ( member_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal
% 0.31/0.46 thf(fact_119_finite__has__minimal,axiom,
% 0.31/0.46 ! [A: set_se1596668135od_a_a] :
% 0.31/0.46 ( ( finite1145471536od_a_a @ A )
% 0.31/0.46 => ( ( A != bot_bo1424703315od_a_a )
% 0.31/0.46 => ? [X4: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member1838126896od_a_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member1838126896od_a_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ Xa @ X4 )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal
% 0.31/0.46 thf(fact_120_finite__has__minimal,axiom,
% 0.31/0.46 ! [A: set_Pr2045688071_set_a] :
% 0.31/0.46 ( ( finite283787856_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bo671027059_set_a )
% 0.31/0.46 => ? [X4: produc1691597095_set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: produc1691597095_set_a] :
% 0.31/0.46 ( ( member593450832_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_le486764743_set_a @ Xa @ X4 )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal
% 0.31/0.46 thf(fact_121_finite__has__minimal,axiom,
% 0.31/0.46 ! [A: set_set_set_a] :
% 0.31/0.46 ( ( finite1606323175_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bo1264989578_set_a )
% 0.31/0.46 => ? [X4: set_set_a] :
% 0.31/0.46 ( ( member_set_set_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: set_set_a] :
% 0.31/0.46 ( ( member_set_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ Xa @ X4 )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal
% 0.31/0.46 thf(fact_122_finite__has__minimal,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ? [X4: set_a] :
% 0.31/0.46 ( ( member_set_a @ X4 @ A )
% 0.31/0.46 & ! [Xa: set_a] :
% 0.31/0.46 ( ( member_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal
% 0.31/0.46 thf(fact_123_empty__Collect__eq,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o] :
% 0.31/0.46 ( ( bot_bo2131659635od_a_a
% 0.31/0.46 = ( collec645855634od_a_a @ P ) )
% 0.31/0.46 = ( ! [X3: product_prod_a_a] :
% 0.31/0.46 ~ ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_Collect_eq
% 0.31/0.46 thf(fact_124_empty__Collect__eq,axiom,
% 0.31/0.46 ! [P: set_a > $o] :
% 0.31/0.46 ( ( bot_bot_set_set_a
% 0.31/0.46 = ( collect_set_a @ P ) )
% 0.31/0.46 = ( ! [X3: set_a] :
% 0.31/0.46 ~ ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_Collect_eq
% 0.31/0.46 thf(fact_125_empty__Collect__eq,axiom,
% 0.31/0.46 ! [P: a > $o] :
% 0.31/0.46 ( ( bot_bot_set_a
% 0.31/0.46 = ( collect_a @ P ) )
% 0.31/0.46 = ( ! [X3: a] :
% 0.31/0.46 ~ ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_Collect_eq
% 0.31/0.46 thf(fact_126_Collect__empty__eq,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o] :
% 0.31/0.46 ( ( ( collec645855634od_a_a @ P )
% 0.31/0.46 = bot_bo2131659635od_a_a )
% 0.31/0.46 = ( ! [X3: product_prod_a_a] :
% 0.31/0.46 ~ ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_empty_eq
% 0.31/0.46 thf(fact_127_Collect__empty__eq,axiom,
% 0.31/0.46 ! [P: set_a > $o] :
% 0.31/0.46 ( ( ( collect_set_a @ P )
% 0.31/0.46 = bot_bot_set_set_a )
% 0.31/0.46 = ( ! [X3: set_a] :
% 0.31/0.46 ~ ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_empty_eq
% 0.31/0.46 thf(fact_128_Collect__empty__eq,axiom,
% 0.31/0.46 ! [P: a > $o] :
% 0.31/0.46 ( ( ( collect_a @ P )
% 0.31/0.46 = bot_bot_set_a )
% 0.31/0.46 = ( ! [X3: a] :
% 0.31/0.46 ~ ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_empty_eq
% 0.31/0.46 thf(fact_129_all__not__in__conv,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( ! [X3: produc1572603623od_a_a] :
% 0.31/0.46 ~ ( member2057358096od_a_a @ X3 @ A ) )
% 0.31/0.46 = ( A = bot_bo1116205875od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_not_in_conv
% 0.31/0.46 thf(fact_130_all__not__in__conv,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( ! [X3: set_a] :
% 0.31/0.46 ~ ( member_set_a @ X3 @ A ) )
% 0.31/0.46 = ( A = bot_bot_set_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_not_in_conv
% 0.31/0.46 thf(fact_131_all__not__in__conv,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ! [X3: product_prod_a_a] :
% 0.31/0.46 ~ ( member449909584od_a_a @ X3 @ A ) )
% 0.31/0.46 = ( A = bot_bo2131659635od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_not_in_conv
% 0.31/0.46 thf(fact_132_all__not__in__conv,axiom,
% 0.31/0.46 ! [A: set_a] :
% 0.31/0.46 ( ( ! [X3: a] :
% 0.31/0.46 ~ ( member_a @ X3 @ A ) )
% 0.31/0.46 = ( A = bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_not_in_conv
% 0.31/0.46 thf(fact_133_empty__iff,axiom,
% 0.31/0.46 ! [C: produc1572603623od_a_a] :
% 0.31/0.46 ~ ( member2057358096od_a_a @ C @ bot_bo1116205875od_a_a ) ).
% 0.31/0.46
% 0.31/0.46 % empty_iff
% 0.31/0.46 thf(fact_134_empty__iff,axiom,
% 0.31/0.46 ! [C: set_a] :
% 0.31/0.46 ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% 0.31/0.46
% 0.31/0.46 % empty_iff
% 0.31/0.46 thf(fact_135_empty__iff,axiom,
% 0.31/0.46 ! [C: product_prod_a_a] :
% 0.31/0.46 ~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% 0.31/0.46
% 0.31/0.46 % empty_iff
% 0.31/0.46 thf(fact_136_empty__iff,axiom,
% 0.31/0.46 ! [C: a] :
% 0.31/0.46 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.31/0.46
% 0.31/0.46 % empty_iff
% 0.31/0.46 thf(fact_137_subsetI,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ! [X4: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ X4 @ A )
% 0.31/0.46 => ( member2057358096od_a_a @ X4 @ B2 ) )
% 0.31/0.46 => ( ord_le456379495od_a_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetI
% 0.31/0.46 thf(fact_138_subsetI,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ! [X4: set_a] :
% 0.31/0.46 ( ( member_set_a @ X4 @ A )
% 0.31/0.46 => ( member_set_a @ X4 @ B2 ) )
% 0.31/0.46 => ( ord_le318720350_set_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetI
% 0.31/0.46 thf(fact_139_subsetI,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X4 @ A )
% 0.31/0.46 => ( member449909584od_a_a @ X4 @ B2 ) )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetI
% 0.31/0.46 thf(fact_140_subsetI,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ! [X4: a] :
% 0.31/0.46 ( ( member_a @ X4 @ A )
% 0.31/0.46 => ( member_a @ X4 @ B2 ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetI
% 0.31/0.46 thf(fact_141_subset__antisym,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ B2 @ A )
% 0.31/0.46 => ( A = B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_antisym
% 0.31/0.46 thf(fact_142_subset__antisym,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ B2 @ A )
% 0.31/0.46 => ( A = B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_antisym
% 0.31/0.46 thf(fact_143_subset__antisym,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B2 @ A )
% 0.31/0.46 => ( A = B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_antisym
% 0.31/0.46 thf(fact_144_at__least__as__goodD,axiom,
% 0.31/0.46 ! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B2: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_least_as_goodD
% 0.31/0.46 thf(fact_145_at__least__as__goodD,axiom,
% 0.31/0.46 ! [Z: set_a,Y: set_a,B2: set_set_a,Pr: set_Pr2045688071_set_a] :
% 0.31/0.46 ( ( member_set_a @ Z @ ( prefer1203288218_set_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_least_as_goodD
% 0.31/0.46 thf(fact_146_at__least__as__goodD,axiom,
% 0.31/0.46 ! [Z: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_least_as_goodD
% 0.31/0.46 thf(fact_147_at__least__as__goodD,axiom,
% 0.31/0.46 ! [Z: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_least_as_goodD
% 0.31/0.46 thf(fact_148_at__lst__asgd__ge,axiom,
% 0.31/0.46 ! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B2: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_lst_asgd_ge
% 0.31/0.46 thf(fact_149_at__lst__asgd__ge,axiom,
% 0.31/0.46 ! [X: set_a,Y: set_a,B2: set_set_a,Pr: set_Pr2045688071_set_a] :
% 0.31/0.46 ( ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_lst_asgd_ge
% 0.31/0.46 thf(fact_150_at__lst__asgd__ge,axiom,
% 0.31/0.46 ! [X: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_lst_asgd_ge
% 0.31/0.46 thf(fact_151_at__lst__asgd__ge,axiom,
% 0.31/0.46 ! [X: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 0.31/0.46
% 0.31/0.46 % at_lst_asgd_ge
% 0.31/0.46 thf(fact_152_ex__in__conv,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ( ? [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A ) )
% 0.31/0.46 = ( A != bot_bo1116205875od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_in_conv
% 0.31/0.46 thf(fact_153_ex__in__conv,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
% 0.31/0.46 = ( A != bot_bot_set_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_in_conv
% 0.31/0.46 thf(fact_154_ex__in__conv,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
% 0.31/0.46 = ( A != bot_bo2131659635od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_in_conv
% 0.31/0.46 thf(fact_155_ex__in__conv,axiom,
% 0.31/0.46 ! [A: set_a] :
% 0.31/0.46 ( ( ? [X3: a] : ( member_a @ X3 @ A ) )
% 0.31/0.46 = ( A != bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_in_conv
% 0.31/0.46 thf(fact_156_equals0I,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ! [Y3: produc1572603623od_a_a] :
% 0.31/0.46 ~ ( member2057358096od_a_a @ Y3 @ A )
% 0.31/0.46 => ( A = bot_bo1116205875od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0I
% 0.31/0.46 thf(fact_157_equals0I,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ! [Y3: set_a] :
% 0.31/0.46 ~ ( member_set_a @ Y3 @ A )
% 0.31/0.46 => ( A = bot_bot_set_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0I
% 0.31/0.46 thf(fact_158_equals0I,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [Y3: product_prod_a_a] :
% 0.31/0.46 ~ ( member449909584od_a_a @ Y3 @ A )
% 0.31/0.46 => ( A = bot_bo2131659635od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0I
% 0.31/0.46 thf(fact_159_equals0I,axiom,
% 0.31/0.46 ! [A: set_a] :
% 0.31/0.46 ( ! [Y3: a] :
% 0.31/0.46 ~ ( member_a @ Y3 @ A )
% 0.31/0.46 => ( A = bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0I
% 0.31/0.46 thf(fact_160_equals0D,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a,A2: produc1572603623od_a_a] :
% 0.31/0.46 ( ( A = bot_bo1116205875od_a_a )
% 0.31/0.46 => ~ ( member2057358096od_a_a @ A2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0D
% 0.31/0.46 thf(fact_161_equals0D,axiom,
% 0.31/0.46 ! [A: set_set_a,A2: set_a] :
% 0.31/0.46 ( ( A = bot_bot_set_set_a )
% 0.31/0.46 => ~ ( member_set_a @ A2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0D
% 0.31/0.46 thf(fact_162_equals0D,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
% 0.31/0.46 ( ( A = bot_bo2131659635od_a_a )
% 0.31/0.46 => ~ ( member449909584od_a_a @ A2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0D
% 0.31/0.46 thf(fact_163_equals0D,axiom,
% 0.31/0.46 ! [A: set_a,A2: a] :
% 0.31/0.46 ( ( A = bot_bot_set_a )
% 0.31/0.46 => ~ ( member_a @ A2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0D
% 0.31/0.46 thf(fact_164_emptyE,axiom,
% 0.31/0.46 ! [A2: produc1572603623od_a_a] :
% 0.31/0.46 ~ ( member2057358096od_a_a @ A2 @ bot_bo1116205875od_a_a ) ).
% 0.31/0.46
% 0.31/0.46 % emptyE
% 0.31/0.46 thf(fact_165_emptyE,axiom,
% 0.31/0.46 ! [A2: set_a] :
% 0.31/0.46 ~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% 0.31/0.46
% 0.31/0.46 % emptyE
% 0.31/0.46 thf(fact_166_emptyE,axiom,
% 0.31/0.46 ! [A2: product_prod_a_a] :
% 0.31/0.46 ~ ( member449909584od_a_a @ A2 @ bot_bo2131659635od_a_a ) ).
% 0.31/0.46
% 0.31/0.46 % emptyE
% 0.31/0.46 thf(fact_167_emptyE,axiom,
% 0.31/0.46 ! [A2: a] :
% 0.31/0.46 ~ ( member_a @ A2 @ bot_bot_set_a ) ).
% 0.31/0.46
% 0.31/0.46 % emptyE
% 0.31/0.46 thf(fact_168_in__mono,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a,X: produc1572603623od_a_a] :
% 0.31/0.46 ( ( ord_le456379495od_a_a @ A @ B2 )
% 0.31/0.46 => ( ( member2057358096od_a_a @ X @ A )
% 0.31/0.46 => ( member2057358096od_a_a @ X @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_mono
% 0.31/0.46 thf(fact_169_in__mono,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a,X: set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ( ( member_set_a @ X @ A )
% 0.31/0.46 => ( member_set_a @ X @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_mono
% 0.31/0.46 thf(fact_170_in__mono,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ X @ A )
% 0.31/0.46 => ( member449909584od_a_a @ X @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_mono
% 0.31/0.46 thf(fact_171_in__mono,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a,X: a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ( member_a @ X @ A )
% 0.31/0.46 => ( member_a @ X @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_mono
% 0.31/0.46 thf(fact_172_subsetD,axiom,
% 0.31/0.46 ! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a,C: produc1572603623od_a_a] :
% 0.31/0.46 ( ( ord_le456379495od_a_a @ A @ B2 )
% 0.31/0.46 => ( ( member2057358096od_a_a @ C @ A )
% 0.31/0.46 => ( member2057358096od_a_a @ C @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetD
% 0.31/0.46 thf(fact_173_subsetD,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a,C: set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ( ( member_set_a @ C @ A )
% 0.31/0.46 => ( member_set_a @ C @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetD
% 0.31/0.46 thf(fact_174_subsetD,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ C @ A )
% 0.31/0.46 => ( member449909584od_a_a @ C @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetD
% 0.31/0.46 thf(fact_175_subsetD,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a,C: a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ( member_a @ C @ A )
% 0.31/0.46 => ( member_a @ C @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetD
% 0.31/0.46 thf(fact_176_equalityE,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ~ ( ( ord_le1824328871od_a_a @ A @ B2 )
% 0.31/0.46 => ~ ( ord_le1824328871od_a_a @ B2 @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityE
% 0.31/0.46 thf(fact_177_equalityE,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ~ ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ~ ( ord_le318720350_set_a @ B2 @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityE
% 0.31/0.46 thf(fact_178_equalityE,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ~ ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityE
% 0.31/0.46 thf(fact_179_subset__eq,axiom,
% 0.31/0.46 ( ord_le456379495od_a_a
% 0.31/0.46 = ( ^ [A3: set_Pr1948701895od_a_a,B3: set_Pr1948701895od_a_a] :
% 0.31/0.46 ! [X3: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ X3 @ A3 )
% 0.31/0.46 => ( member2057358096od_a_a @ X3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_eq
% 0.31/0.46 thf(fact_180_subset__eq,axiom,
% 0.31/0.46 ( ord_le318720350_set_a
% 0.31/0.46 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 0.31/0.46 ! [X3: set_a] :
% 0.31/0.46 ( ( member_set_a @ X3 @ A3 )
% 0.31/0.46 => ( member_set_a @ X3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_eq
% 0.31/0.46 thf(fact_181_subset__eq,axiom,
% 0.31/0.46 ( ord_le1824328871od_a_a
% 0.31/0.46 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.31/0.46 ! [X3: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X3 @ A3 )
% 0.31/0.46 => ( member449909584od_a_a @ X3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_eq
% 0.31/0.46 thf(fact_182_subset__eq,axiom,
% 0.31/0.46 ( ord_less_eq_set_a
% 0.31/0.46 = ( ^ [A3: set_a,B3: set_a] :
% 0.31/0.46 ! [X3: a] :
% 0.31/0.46 ( ( member_a @ X3 @ A3 )
% 0.31/0.46 => ( member_a @ X3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_eq
% 0.31/0.46 thf(fact_183_equalityD1,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD1
% 0.31/0.46 thf(fact_184_equalityD1,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ( ord_le318720350_set_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD1
% 0.31/0.46 thf(fact_185_equalityD1,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD1
% 0.31/0.46 thf(fact_186_equalityD2,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ B2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD2
% 0.31/0.46 thf(fact_187_equalityD2,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ( ord_le318720350_set_a @ B2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD2
% 0.31/0.46 thf(fact_188_equalityD2,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( A = B2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ B2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD2
% 0.31/0.46 thf(fact_189_subset__iff,axiom,
% 0.31/0.46 ( ord_le456379495od_a_a
% 0.31/0.46 = ( ^ [A3: set_Pr1948701895od_a_a,B3: set_Pr1948701895od_a_a] :
% 0.31/0.46 ! [T: produc1572603623od_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ T @ A3 )
% 0.31/0.46 => ( member2057358096od_a_a @ T @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff
% 0.31/0.46 thf(fact_190_subset__iff,axiom,
% 0.31/0.46 ( ord_le318720350_set_a
% 0.31/0.46 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 0.31/0.46 ! [T: set_a] :
% 0.31/0.46 ( ( member_set_a @ T @ A3 )
% 0.31/0.46 => ( member_set_a @ T @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff
% 0.31/0.46 thf(fact_191_subset__iff,axiom,
% 0.31/0.46 ( ord_le1824328871od_a_a
% 0.31/0.46 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.31/0.46 ! [T: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ T @ A3 )
% 0.31/0.46 => ( member449909584od_a_a @ T @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff
% 0.31/0.46 thf(fact_192_subset__iff,axiom,
% 0.31/0.46 ( ord_less_eq_set_a
% 0.31/0.46 = ( ^ [A3: set_a,B3: set_a] :
% 0.31/0.46 ! [T: a] :
% 0.31/0.46 ( ( member_a @ T @ A3 )
% 0.31/0.46 => ( member_a @ T @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff
% 0.31/0.46 thf(fact_193_subset__refl,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % subset_refl
% 0.31/0.46 thf(fact_194_subset__refl,axiom,
% 0.31/0.46 ! [A: set_set_a] : ( ord_le318720350_set_a @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % subset_refl
% 0.31/0.46 thf(fact_195_subset__refl,axiom,
% 0.31/0.46 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % subset_refl
% 0.31/0.46 thf(fact_196_Collect__mono,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.31/0.46 ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ( ( P @ X4 )
% 0.31/0.46 => ( Q @ X4 ) )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono
% 0.31/0.46 thf(fact_197_Collect__mono,axiom,
% 0.31/0.46 ! [P: set_a > $o,Q: set_a > $o] :
% 0.31/0.46 ( ! [X4: set_a] :
% 0.31/0.46 ( ( P @ X4 )
% 0.31/0.46 => ( Q @ X4 ) )
% 0.31/0.46 => ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono
% 0.31/0.46 thf(fact_198_Collect__mono,axiom,
% 0.31/0.46 ! [P: a > $o,Q: a > $o] :
% 0.31/0.46 ( ! [X4: a] :
% 0.31/0.46 ( ( P @ X4 )
% 0.31/0.46 => ( Q @ X4 ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono
% 0.31/0.46 thf(fact_199_subset__trans,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ B2 @ C2 )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ A @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_trans
% 0.31/0.46 thf(fact_200_subset__trans,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ B2 @ C2 )
% 0.31/0.46 => ( ord_le318720350_set_a @ A @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_trans
% 0.31/0.46 thf(fact_201_subset__trans,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a,C2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_trans
% 0.31/0.46 thf(fact_202_set__eq__subset,axiom,
% 0.31/0.46 ( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A3 @ B3 )
% 0.31/0.46 & ( ord_le1824328871od_a_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % set_eq_subset
% 0.31/0.46 thf(fact_203_set__eq__subset,axiom,
% 0.31/0.46 ( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A3 @ B3 )
% 0.31/0.46 & ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % set_eq_subset
% 0.31/0.46 thf(fact_204_set__eq__subset,axiom,
% 0.31/0.46 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A3: set_a,B3: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % set_eq_subset
% 0.31/0.46 thf(fact_205_Collect__mono__iff,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) )
% 0.31/0.46 = ( ! [X3: product_prod_a_a] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono_iff
% 0.31/0.46 thf(fact_206_Collect__mono__iff,axiom,
% 0.31/0.46 ! [P: set_a > $o,Q: set_a > $o] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
% 0.31/0.46 = ( ! [X3: set_a] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono_iff
% 0.31/0.46 thf(fact_207_Collect__mono__iff,axiom,
% 0.31/0.46 ! [P: a > $o,Q: a > $o] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.31/0.46 = ( ! [X3: a] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono_iff
% 0.31/0.46 thf(fact_208_rational__preference_Osame__at__least__as__equal,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) )
% 0.31/0.46 => ( ( prefer1203288218_set_a @ Z @ Carrier @ Relation )
% 0.31/0.46 = ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.same_at_least_as_equal
% 0.31/0.46 thf(fact_209_rational__preference_Osame__at__least__as__equal,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) )
% 0.31/0.46 => ( ( prefer477097315od_a_a @ Z @ Carrier @ Relation )
% 0.31/0.46 = ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.same_at_least_as_equal
% 0.31/0.46 thf(fact_210_rational__preference_Osame__at__least__as__equal,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) )
% 0.31/0.46 => ( ( prefer161218362good_a @ Z @ Carrier @ Relation )
% 0.31/0.46 = ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.same_at_least_as_equal
% 0.31/0.46 thf(fact_211_rational__preference_Opref__in__at__least__as,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.pref_in_at_least_as
% 0.31/0.46 thf(fact_212_rational__preference_Opref__in__at__least__as,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.pref_in_at_least_as
% 0.31/0.46 thf(fact_213_rational__preference_Opref__in__at__least__as,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.pref_in_at_least_as
% 0.31/0.46 thf(fact_214_rational__preference_Opref__in__at__least__as,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 => ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.pref_in_at_least_as
% 0.31/0.46 thf(fact_215_rational__preference_Oas__good__as__sameIff,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,Z: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ Z @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_as_sameIff
% 0.31/0.46 thf(fact_216_rational__preference_Oas__good__as__sameIff,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_set_a @ Z @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_as_sameIff
% 0.31/0.46 thf(fact_217_rational__preference_Oas__good__as__sameIff,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_as_sameIff
% 0.31/0.46 thf(fact_218_rational__preference_Oas__good__as__sameIff,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_as_sameIff
% 0.31/0.46 thf(fact_219_rational__preference_Oas__good__asIff,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member2057358096od_a_a @ X @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_asIff
% 0.31/0.46 thf(fact_220_rational__preference_Oas__good__asIff,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_set_a @ X @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_asIff
% 0.31/0.46 thf(fact_221_rational__preference_Oas__good__asIff,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member449909584od_a_a @ X @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_asIff
% 0.31/0.46 thf(fact_222_rational__preference_Oas__good__asIff,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.31/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.as_good_asIff
% 0.31/0.46 thf(fact_223_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.31/0.46 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.31/0.46 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bo1116205875od_a_a )
% 0.31/0.46 => ( ( member2057358096od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 0.31/0.46 => ( ~ ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 => ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.at_lst_asgd_not_ge
% 0.31/0.46 thf(fact_224_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.31/0.46 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 0.31/0.46 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bot_set_set_a )
% 0.31/0.46 => ( ( member_set_a @ X @ Carrier )
% 0.31/0.46 => ( ( member_set_a @ Y @ Carrier )
% 0.31/0.46 => ( ~ ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 => ~ ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.at_lst_asgd_not_ge
% 0.31/0.46 thf(fact_225_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.31/0.46 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bo2131659635od_a_a )
% 0.31/0.46 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.31/0.46 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.31/0.46 => ( ~ ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.at_lst_asgd_not_ge
% 0.31/0.46 thf(fact_226_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( Carrier != bot_bot_set_a )
% 0.31/0.46 => ( ( member_a @ X @ Carrier )
% 0.31/0.46 => ( ( member_a @ Y @ Carrier )
% 0.31/0.46 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
% 0.31/0.46 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.at_lst_asgd_not_ge
% 0.31/0.46 thf(fact_227_Pair__mono,axiom,
% 0.31/0.46 ! [X: produc1691597095_set_a,X5: produc1691597095_set_a,Y: set_set_a,Y5: set_set_a] :
% 0.31/0.46 ( ( ord_le486764743_set_a @ X @ X5 )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ Y @ Y5 )
% 0.31/0.46 => ( ord_le2039441566_set_a @ ( produc759028718_set_a @ X @ Y ) @ ( produc759028718_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_mono
% 0.31/0.46 thf(fact_228_Pair__mono,axiom,
% 0.31/0.46 ! [X: set_set_a,X5: set_set_a,Y: set_a,Y5: set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ X @ X5 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 0.31/0.46 => ( ord_le1201700711_set_a @ ( produc32105911_set_a @ X @ Y ) @ ( produc32105911_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_mono
% 0.31/0.46 thf(fact_229_Pair__mono,axiom,
% 0.31/0.46 ! [X: set_set_a,X5: set_set_a,Y: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ X @ X5 )
% 0.31/0.46 => ( ( ord_le1824328871od_a_a @ Y @ Y5 )
% 0.31/0.46 => ( ord_le526661008od_a_a @ ( produc1618062688od_a_a @ X @ Y ) @ ( produc1618062688od_a_a @ X5 @ Y5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_mono
% 0.31/0.46 thf(fact_230_Pair__mono,axiom,
% 0.31/0.46 ! [X: set_set_a,X5: set_set_a,Y: produc1691597095_set_a,Y5: produc1691597095_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ X @ X5 )
% 0.31/0.46 => ( ( ord_le486764743_set_a @ Y @ Y5 )
% 0.31/0.46 => ( ord_le1138266544_set_a @ ( produc760262016_set_a @ X @ Y ) @ ( produc760262016_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_mono
% 0.31/0.46 thf(fact_231_Pair__mono,axiom,
% 0.31/0.46 ! [X: set_set_a,X5: set_set_a,Y: set_set_a,Y5: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ X @ X5 )
% 0.31/0.46 => ( ( ord_le318720350_set_a @ Y @ Y5 )
% 0.31/0.46 => ( ord_le723713223_set_a @ ( produc2102896407_set_a @ X @ Y ) @ ( produc2102896407_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_mono
% 0.31/0.46 thf(fact_232_Pair__mono,axiom,
% 0.31/0.46 ! [X: set_a,X5: set_a,Y: set_a,Y5: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X @ X5 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 0.31/0.46 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_mono
% 0.31/0.46 thf(fact_233_finite__has__maximal2,axiom,
% 0.31/0.46 ! [A: set_set_a,A2: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( member_set_a @ A2 @ A )
% 0.31/0.46 => ? [X4: set_a] :
% 0.31/0.46 ( ( member_set_a @ X4 @ A )
% 0.31/0.46 & ( ord_less_eq_set_a @ A2 @ X4 )
% 0.31/0.46 & ! [Xa: set_a] :
% 0.31/0.46 ( ( member_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal2
% 0.31/0.46 thf(fact_234_finite__has__minimal2,axiom,
% 0.31/0.46 ! [A: set_set_a,A2: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( member_set_a @ A2 @ A )
% 0.31/0.46 => ? [X4: set_a] :
% 0.31/0.46 ( ( member_set_a @ X4 @ A )
% 0.31/0.46 & ( ord_less_eq_set_a @ X4 @ A2 )
% 0.31/0.46 & ! [Xa: set_a] :
% 0.31/0.46 ( ( member_set_a @ Xa @ A )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 0.31/0.46 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal2
% 0.31/0.46 thf(fact_235_bot__prod__def,axiom,
% 0.31/0.46 ( bot_bo1519632275_set_a
% 0.31/0.46 = ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_prod_def
% 0.31/0.46 thf(fact_236_infinite__imp__nonempty,axiom,
% 0.31/0.46 ! [S: set_a] :
% 0.31/0.46 ( ~ ( finite_finite_a @ S )
% 0.31/0.46 => ( S != bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_imp_nonempty
% 0.31/0.46 thf(fact_237_finite_OemptyI,axiom,
% 0.31/0.46 finite_finite_a @ bot_bot_set_a ).
% 0.31/0.46
% 0.31/0.46 % finite.emptyI
% 0.31/0.46 thf(fact_238_finite__subset,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ( finite_finite_a @ B2 )
% 0.31/0.46 => ( finite_finite_a @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset
% 0.31/0.46 thf(fact_239_infinite__super,axiom,
% 0.31/0.46 ! [S: set_a,T2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ S @ T2 )
% 0.31/0.46 => ( ~ ( finite_finite_a @ S )
% 0.31/0.46 => ~ ( finite_finite_a @ T2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_super
% 0.31/0.46 thf(fact_240_rev__finite__subset,axiom,
% 0.31/0.46 ! [B2: set_a,A: set_a] :
% 0.31/0.46 ( ( finite_finite_a @ B2 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( finite_finite_a @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rev_finite_subset
% 0.31/0.46 thf(fact_241_old_Oprod_Oinject,axiom,
% 0.31/0.46 ! [A2: a,B: a,A4: a,B4: a] :
% 0.31/0.46 ( ( ( product_Pair_a_a @ A2 @ B )
% 0.31/0.46 = ( product_Pair_a_a @ A4 @ B4 ) )
% 0.31/0.46 = ( ( A2 = A4 )
% 0.31/0.46 & ( B = B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.prod.inject
% 0.31/0.46 thf(fact_242_prod_Oinject,axiom,
% 0.31/0.46 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 0.31/0.46 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 0.31/0.46 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 0.31/0.46 = ( ( X1 = Y1 )
% 0.31/0.46 & ( X22 = Y22 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod.inject
% 0.31/0.46 thf(fact_243_order__refl,axiom,
% 0.31/0.46 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 0.31/0.46
% 0.31/0.46 % order_refl
% 0.31/0.46 thf(fact_244_preorder__on__empty,axiom,
% 0.31/0.46 order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.31/0.46
% 0.31/0.46 % preorder_on_empty
% 0.31/0.46 thf(fact_245_total__on__def,axiom,
% 0.31/0.46 ( total_on_a
% 0.31/0.46 = ( ^ [A3: set_a,R: set_Product_prod_a_a] :
% 0.31/0.46 ! [X3: a] :
% 0.31/0.46 ( ( member_a @ X3 @ A3 )
% 0.31/0.46 => ! [Y2: a] :
% 0.31/0.46 ( ( member_a @ Y2 @ A3 )
% 0.31/0.46 => ( ( X3 != Y2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
% 0.31/0.46 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % total_on_def
% 0.31/0.46 thf(fact_246_total__onI,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 0.31/0.46 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X4 @ A )
% 0.31/0.46 => ( ( member449909584od_a_a @ Y3 @ A )
% 0.31/0.46 => ( ( X4 != Y3 )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R2 )
% 0.31/0.46 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 0.31/0.46 => ( total_1490170027od_a_a @ A @ R2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % total_onI
% 0.31/0.46 thf(fact_247_total__onI,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X4: a,Y3: a] :
% 0.31/0.46 ( ( member_a @ X4 @ A )
% 0.31/0.46 => ( ( member_a @ Y3 @ A )
% 0.31/0.46 => ( ( X4 != Y3 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
% 0.31/0.46 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 0.31/0.46 => ( total_on_a @ A @ R2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % total_onI
% 0.31/0.46 thf(fact_248_total__on__empty,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R2 ) ).
% 0.31/0.46
% 0.31/0.46 % total_on_empty
% 0.31/0.46 thf(fact_249_bot__empty__eq,axiom,
% 0.31/0.46 ( bot_bo1293121322_a_a_o
% 0.31/0.46 = ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_empty_eq
% 0.31/0.46 thf(fact_250_bot__empty__eq,axiom,
% 0.31/0.46 ( bot_bot_a_o
% 0.31/0.46 = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_empty_eq
% 0.31/0.46 thf(fact_251_bot__set__def,axiom,
% 0.31/0.46 ( bot_bot_set_a
% 0.31/0.46 = ( collect_a @ bot_bot_a_o ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_set_def
% 0.31/0.46 thf(fact_252_dual__order_Oantisym,axiom,
% 0.31/0.46 ! [B: set_a,A2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ B @ A2 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ A2 @ B )
% 0.31/0.46 => ( A2 = B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.antisym
% 0.31/0.46 thf(fact_253_dual__order_Oeq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A5: set_a,B5: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ B5 @ A5 )
% 0.31/0.46 & ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.eq_iff
% 0.31/0.46 thf(fact_254_dual__order_Otrans,axiom,
% 0.31/0.46 ! [B: set_a,A2: set_a,C: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ B @ A2 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ C @ B )
% 0.31/0.46 => ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.trans
% 0.31/0.46 thf(fact_255_dual__order_Orefl,axiom,
% 0.31/0.46 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.refl
% 0.31/0.46 thf(fact_256_order__trans,axiom,
% 0.31/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X @ Y )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ Y @ Z )
% 0.31/0.46 => ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_trans
% 0.31/0.46 thf(fact_257_order__class_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ B )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B @ A2 )
% 0.31/0.46 => ( A2 = B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.antisym
% 0.31/0.46 thf(fact_258_ord__le__eq__trans,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a,C: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_trans
% 0.31/0.46 thf(fact_259_ord__eq__le__trans,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a,C: set_a] :
% 0.31/0.46 ( ( A2 = B )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B @ C )
% 0.31/0.46 => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_trans
% 0.31/0.46 thf(fact_260_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [A5: set_a,B5: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.31/0.46 & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.eq_iff
% 0.31/0.46 thf(fact_261_antisym__conv,axiom,
% 0.31/0.46 ! [Y: set_a,X: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ Y @ X )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ X @ Y )
% 0.31/0.46 = ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv
% 0.31/0.46 thf(fact_262_order_Otrans,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a,C: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ B )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B @ C )
% 0.31/0.46 => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.trans
% 0.31/0.46 thf(fact_263_eq__refl,axiom,
% 0.31/0.46 ! [X: set_a,Y: set_a] :
% 0.31/0.46 ( ( X = Y )
% 0.31/0.46 => ( ord_less_eq_set_a @ X @ Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_refl
% 0.31/0.46 thf(fact_264_antisym,axiom,
% 0.31/0.46 ! [X: set_a,Y: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X @ Y )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ Y @ X )
% 0.31/0.46 => ( X = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym
% 0.31/0.46 thf(fact_265_eq__iff,axiom,
% 0.31/0.46 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.31/0.46 = ( ^ [X3: set_a,Y2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 0.31/0.46 & ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff
% 0.31/0.46 thf(fact_266_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ B )
% 0.31/0.46 => ( ( ( F @ B )
% 0.31/0.46 = C )
% 0.31/0.46 => ( ! [X4: set_a,Y3: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_267_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B @ C )
% 0.31/0.46 => ( ! [X4: set_a,Y3: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_268_order__subst2,axiom,
% 0.31/0.46 ! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ B )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X4: set_a,Y3: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_269_order__subst1,axiom,
% 0.31/0.46 ! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B @ C )
% 0.31/0.46 => ( ! [X4: set_a,Y3: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_270_surj__pair,axiom,
% 0.31/0.46 ! [P2: product_prod_a_a] :
% 0.31/0.46 ? [X4: a,Y3: a] :
% 0.31/0.46 ( P2
% 0.31/0.46 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % surj_pair
% 0.31/0.46 thf(fact_271_prod__cases,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.31/0.46 ( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
% 0.31/0.46 => ( P @ P2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases
% 0.31/0.46 thf(fact_272_Pair__inject,axiom,
% 0.31/0.46 ! [A2: a,B: a,A4: a,B4: a] :
% 0.31/0.46 ( ( ( product_Pair_a_a @ A2 @ B )
% 0.31/0.46 = ( product_Pair_a_a @ A4 @ B4 ) )
% 0.31/0.46 => ~ ( ( A2 = A4 )
% 0.31/0.46 => ( B != B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_inject
% 0.31/0.46 thf(fact_273_old_Oprod_Oexhaust,axiom,
% 0.31/0.46 ! [Y: product_prod_a_a] :
% 0.31/0.46 ~ ! [A6: a,B6: a] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair_a_a @ A6 @ B6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.prod.exhaust
% 0.31/0.46 thf(fact_274_old_Oprod_Oinducts,axiom,
% 0.31/0.46 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.31/0.46 ( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
% 0.31/0.46 => ( P @ Prod ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.prod.inducts
% 0.31/0.46 thf(fact_275_subrelI,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X4: a,Y3: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S2 ) )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ R2 @ S2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subrelI
% 0.31/0.46 thf(fact_276_bot_Oextremum__uniqueI,axiom,
% 0.31/0.46 ! [A2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.31/0.46 => ( A2 = bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_uniqueI
% 0.31/0.46 thf(fact_277_bot_Oextremum__unique,axiom,
% 0.31/0.46 ! [A2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.31/0.46 = ( A2 = bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_unique
% 0.31/0.46 thf(fact_278_bot_Oextremum,axiom,
% 0.31/0.46 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum
% 0.31/0.46 thf(fact_279_refl__on__empty,axiom,
% 0.31/0.46 refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.31/0.46
% 0.31/0.46 % refl_on_empty
% 0.31/0.46 thf(fact_280_refl__onD,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A2: product_prod_a_a] :
% 0.31/0.46 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ A2 @ A )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_onD
% 0.31/0.46 thf(fact_281_refl__onD,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a,A2: a] :
% 0.31/0.46 ( ( refl_on_a @ A @ R2 )
% 0.31/0.46 => ( ( member_a @ A2 @ A )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_onD
% 0.31/0.46 thf(fact_282_refl__onD1,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 0.31/0.46 => ( member449909584od_a_a @ X @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_onD1
% 0.31/0.46 thf(fact_283_refl__onD1,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( refl_on_a @ A @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.31/0.46 => ( member_a @ X @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_onD1
% 0.31/0.46 thf(fact_284_refl__onD2,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.31/0.46 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 0.31/0.46 => ( member449909584od_a_a @ Y @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_onD2
% 0.31/0.46 thf(fact_285_refl__onD2,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 0.31/0.46 ( ( refl_on_a @ A @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.31/0.46 => ( member_a @ Y @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_onD2
% 0.31/0.46 thf(fact_286_refl__on__domain,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A2: product_prod_a_a,B: product_prod_a_a] :
% 0.31/0.46 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 0.31/0.46 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ B ) @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ A2 @ A )
% 0.31/0.46 & ( member449909584od_a_a @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_on_domain
% 0.31/0.46 thf(fact_287_refl__on__domain,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a,A2: a,B: a] :
% 0.31/0.46 ( ( refl_on_a @ A @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
% 0.31/0.46 => ( ( member_a @ A2 @ A )
% 0.31/0.46 & ( member_a @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_on_domain
% 0.31/0.46 thf(fact_288_rational__preference__def,axiom,
% 0.31/0.46 ( prefer719835986ence_a
% 0.31/0.46 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 0.31/0.46 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference_def
% 0.31/0.46 thf(fact_289_rational__preference_Ointro,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.intro
% 0.31/0.46 thf(fact_290_finite__transitivity__chain,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,R3: product_prod_a_a > product_prod_a_a > $o] :
% 0.31/0.46 ( ( finite179568208od_a_a @ A )
% 0.31/0.46 => ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ~ ( R3 @ X4 @ X4 )
% 0.31/0.46 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
% 0.31/0.46 ( ( R3 @ X4 @ Y3 )
% 0.31/0.46 => ( ( R3 @ Y3 @ Z3 )
% 0.31/0.46 => ( R3 @ X4 @ Z3 ) ) )
% 0.31/0.46 => ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X4 @ A )
% 0.31/0.46 => ? [Y6: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ Y6 @ A )
% 0.31/0.46 & ( R3 @ X4 @ Y6 ) ) )
% 0.31/0.46 => ( A = bot_bo2131659635od_a_a ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_transitivity_chain
% 0.31/0.46 thf(fact_291_finite__transitivity__chain,axiom,
% 0.31/0.46 ! [A: set_a,R3: a > a > $o] :
% 0.31/0.46 ( ( finite_finite_a @ A )
% 0.31/0.46 => ( ! [X4: a] :
% 0.31/0.46 ~ ( R3 @ X4 @ X4 )
% 0.31/0.46 => ( ! [X4: a,Y3: a,Z3: a] :
% 0.31/0.46 ( ( R3 @ X4 @ Y3 )
% 0.31/0.46 => ( ( R3 @ Y3 @ Z3 )
% 0.31/0.46 => ( R3 @ X4 @ Z3 ) ) )
% 0.31/0.46 => ( ! [X4: a] :
% 0.31/0.46 ( ( member_a @ X4 @ A )
% 0.31/0.46 => ? [Y6: a] :
% 0.31/0.46 ( ( member_a @ Y6 @ A )
% 0.31/0.46 & ( R3 @ X4 @ Y6 ) ) )
% 0.31/0.46 => ( A = bot_bot_set_a ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_transitivity_chain
% 0.31/0.46 thf(fact_292_subset__emptyI,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ~ ( member449909584od_a_a @ X4 @ A )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_emptyI
% 0.31/0.46 thf(fact_293_subset__emptyI,axiom,
% 0.31/0.46 ! [A: set_a] :
% 0.31/0.46 ( ! [X4: a] :
% 0.31/0.46 ~ ( member_a @ X4 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_emptyI
% 0.31/0.46 thf(fact_294_ssubst__Pair__rhs,axiom,
% 0.31/0.46 ! [R2: a,S2: a,R3: set_Product_prod_a_a,S3: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ R3 )
% 0.31/0.46 => ( ( S3 = S2 )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ssubst_Pair_rhs
% 0.31/0.46 thf(fact_295_rational__preference__axioms_Ointro,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( total_on_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference_axioms.intro
% 0.31/0.46 thf(fact_296_rational__preference__axioms__def,axiom,
% 0.31/0.46 prefer580340847ioms_a = total_on_a ).
% 0.31/0.46
% 0.31/0.46 % rational_preference_axioms_def
% 0.31/0.46 thf(fact_297_rational__preference_Oaxioms_I2_J,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % rational_preference.axioms(2)
% 0.31/0.46 thf(fact_298_Collect__empty__eq__bot,axiom,
% 0.31/0.46 ! [P: a > $o] :
% 0.31/0.46 ( ( ( collect_a @ P )
% 0.31/0.46 = bot_bot_set_a )
% 0.31/0.46 = ( P = bot_bot_a_o ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_empty_eq_bot
% 0.31/0.46 thf(fact_299_transitivity,axiom,
% 0.31/0.46 trans_a @ relation ).
% 0.31/0.46
% 0.31/0.46 % transitivity
% 0.31/0.46 thf(fact_300_Set_Ois__empty__def,axiom,
% 0.31/0.46 ( is_empty_a
% 0.31/0.46 = ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Set.is_empty_def
% 0.31/0.46 thf(fact_301_preference_Otransitivity,axiom,
% 0.31/0.46 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.31/0.46 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.31/0.46 => ( trans_a @ Relation ) ) ).
% 0.31/0.46
% 0.31/0.46 % preference.transitivity
% 0.31/0.46 thf(fact_302_trans__empty,axiom,
% 0.31/0.46 trans_a @ bot_bo2131659635od_a_a ).
% 0.31/0.46
% 0.31/0.46 % trans_empty
% 0.31/0.46 thf(fact_303_transD,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( trans_a @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % transD
% 0.31/0.46 thf(fact_304_transE,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.31/0.46 ( ( trans_a @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % transE
% 0.31/0.46 thf(fact_305_transI,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a] :
% 0.31/0.46 ( ! [X4: a,Y3: a,Z3: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R2 )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R2 ) ) )
% 0.31/0.46 => ( trans_a @ R2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % transI
% 0.31/0.46 thf(fact_306_trans__def,axiom,
% 0.31/0.46 ( trans_a
% 0.31/0.46 = ( ^ [R: set_Product_prod_a_a] :
% 0.31/0.46 ! [X3: a,Y2: a,Z4: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trans_def
% 0.31/0.46 thf(fact_307_preorder__on__def,axiom,
% 0.31/0.46 ( order_preorder_on_a
% 0.31/0.46 = ( ^ [A3: set_a,R: set_Product_prod_a_a] :
% 0.31/0.46 ( ( refl_on_a @ A3 @ R )
% 0.31/0.46 & ( trans_a @ R ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % preorder_on_def
% 0.31/0.46 thf(fact_308_relChain__def,axiom,
% 0.31/0.46 ( bNF_Ca850758276_set_a
% 0.31/0.46 = ( ^ [R: set_Product_prod_a_a,As: a > set_a] :
% 0.31/0.46 ! [I: a,J: a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ I @ J ) @ R )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % relChain_def
% 0.31/0.46 thf(fact_309_Id__on__empty,axiom,
% 0.31/0.46 ( ( id_on_a @ bot_bot_set_a )
% 0.31/0.46 = bot_bo2131659635od_a_a ) ).
% 0.31/0.46
% 0.31/0.46 % Id_on_empty
% 0.31/0.46 thf(fact_310_under__incr,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a,A2: a,B: a] :
% 0.31/0.46 ( ( trans_a @ R2 )
% 0.31/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( order_under_a @ R2 @ A2 ) @ ( order_under_a @ R2 @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % under_incr
% 0.31/0.46 thf(fact_311_Id__onI,axiom,
% 0.31/0.46 ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ A2 @ A )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ A2 ) @ ( id_on_1182990020od_a_a @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_onI
% 0.31/0.46 thf(fact_312_Id__onI,axiom,
% 0.31/0.46 ! [A2: a,A: set_a] :
% 0.31/0.46 ( ( member_a @ A2 @ A )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( id_on_a @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_onI
% 0.31/0.46 thf(fact_313_trans__Id__on,axiom,
% 0.31/0.46 ! [A: set_a] : ( trans_a @ ( id_on_a @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % trans_Id_on
% 0.31/0.46 thf(fact_314_Id__on__iff,axiom,
% 0.31/0.46 ! [X: product_prod_a_a,Y: product_prod_a_a,A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( id_on_1182990020od_a_a @ A ) )
% 0.31/0.46 = ( ( X = Y )
% 0.31/0.46 & ( member449909584od_a_a @ X @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_on_iff
% 0.31/0.46 thf(fact_315_Id__on__iff,axiom,
% 0.31/0.46 ! [X: a,Y: a,A: set_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( id_on_a @ A ) )
% 0.31/0.46 = ( ( X = Y )
% 0.31/0.46 & ( member_a @ X @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_on_iff
% 0.31/0.46 thf(fact_316_Id__on__eqI,axiom,
% 0.31/0.46 ! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( A2 = B )
% 0.31/0.46 => ( ( member449909584od_a_a @ A2 @ A )
% 0.31/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ B ) @ ( id_on_1182990020od_a_a @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_on_eqI
% 0.31/0.46 thf(fact_317_Id__on__eqI,axiom,
% 0.31/0.46 ! [A2: a,B: a,A: set_a] :
% 0.31/0.46 ( ( A2 = B )
% 0.31/0.46 => ( ( member_a @ A2 @ A )
% 0.31/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( id_on_a @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_on_eqI
% 0.31/0.46 thf(fact_318_Id__onE,axiom,
% 0.31/0.46 ! [C: produc1572603623od_a_a,A: set_Product_prod_a_a] :
% 0.31/0.46 ( ( member2057358096od_a_a @ C @ ( id_on_1182990020od_a_a @ A ) )
% 0.31/0.46 => ~ ! [X4: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X4 @ A )
% 0.31/0.46 => ( C
% 0.31/0.46 != ( produc1474507607od_a_a @ X4 @ X4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_onE
% 0.31/0.46 thf(fact_319_Id__onE,axiom,
% 0.31/0.46 ! [C: product_prod_a_a,A: set_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ C @ ( id_on_a @ A ) )
% 0.31/0.46 => ~ ! [X4: a] :
% 0.31/0.46 ( ( member_a @ X4 @ A )
% 0.31/0.46 => ( C
% 0.31/0.46 != ( product_Pair_a_a @ X4 @ X4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_onE
% 0.31/0.46 thf(fact_320_refl__on__Id__on,axiom,
% 0.31/0.46 ! [A: set_a] : ( refl_on_a @ A @ ( id_on_a @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_on_Id_on
% 0.31/0.46 thf(fact_321_Greatest__equality,axiom,
% 0.31/0.46 ! [P: set_a > $o,X: set_a] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 => ( ! [Y3: set_a] :
% 0.31/0.46 ( ( P @ Y3 )
% 0.31/0.46 => ( ord_less_eq_set_a @ Y3 @ X ) )
% 0.31/0.46 => ( ( order_Greatest_set_a @ P )
% 0.31/0.46 = X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Greatest_equality
% 0.31/0.46 thf(fact_322_GreatestI2__order,axiom,
% 0.31/0.46 ! [P: set_a > $o,X: set_a,Q: set_a > $o] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 => ( ! [Y3: set_a] :
% 0.31/0.46 ( ( P @ Y3 )
% 0.31/0.46 => ( ord_less_eq_set_a @ Y3 @ X ) )
% 0.31/0.46 => ( ! [X4: set_a] :
% 0.31/0.46 ( ( P @ X4 )
% 0.31/0.46 => ( ! [Y6: set_a] :
% 0.31/0.46 ( ( P @ Y6 )
% 0.31/0.46 => ( ord_less_eq_set_a @ Y6 @ X4 ) )
% 0.31/0.46 => ( Q @ X4 ) ) )
% 0.31/0.46 => ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % GreatestI2_order
% 0.31/0.46 thf(fact_323_Inf__fin_Osubset__imp,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ( finite_finite_set_a @ B2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ B2 ) @ ( lattic1600076923_set_a @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Inf_fin.subset_imp
% 0.31/0.46 thf(fact_324_Sup__fin_Osubset__imp,axiom,
% 0.31/0.46 ! [A: set_set_a,B2: set_set_a] :
% 0.31/0.46 ( ( ord_le318720350_set_a @ A @ B2 )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ( finite_finite_set_a @ B2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ ( lattic501550101_set_a @ B2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Sup_fin.subset_imp
% 0.31/0.46 thf(fact_325_Inf__fin__le__Sup__fin,axiom,
% 0.31/0.46 ! [A: set_set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ A ) @ ( lattic501550101_set_a @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Inf_fin_le_Sup_fin
% 0.31/0.46 thf(fact_326_Sup__fin_OcoboundedI,axiom,
% 0.31/0.46 ! [A: set_set_a,A2: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( member_set_a @ A2 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ A2 @ ( lattic501550101_set_a @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Sup_fin.coboundedI
% 0.31/0.46 thf(fact_327_Inf__fin_OcoboundedI,axiom,
% 0.31/0.46 ! [A: set_set_a,A2: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( member_set_a @ A2 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ A ) @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Inf_fin.coboundedI
% 0.31/0.46 thf(fact_328_Inf__fin_OboundedE,axiom,
% 0.31/0.46 ! [A: set_set_a,X: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) )
% 0.31/0.46 => ! [A7: set_a] :
% 0.31/0.46 ( ( member_set_a @ A7 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ X @ A7 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Inf_fin.boundedE
% 0.31/0.46 thf(fact_329_Inf__fin_OboundedI,axiom,
% 0.31/0.46 ! [A: set_set_a,X: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ! [A6: set_a] :
% 0.31/0.46 ( ( member_set_a @ A6 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ X @ A6 ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Inf_fin.boundedI
% 0.31/0.46 thf(fact_330_Sup__fin_OboundedE,axiom,
% 0.31/0.46 ! [A: set_set_a,X: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X )
% 0.31/0.46 => ! [A7: set_a] :
% 0.31/0.46 ( ( member_set_a @ A7 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ A7 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Sup_fin.boundedE
% 0.31/0.46 thf(fact_331_Sup__fin_OboundedI,axiom,
% 0.31/0.46 ! [A: set_set_a,X: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ! [A6: set_a] :
% 0.31/0.46 ( ( member_set_a @ A6 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ A6 @ X ) )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Sup_fin.boundedI
% 0.31/0.46 thf(fact_332_Inf__fin_Obounded__iff,axiom,
% 0.31/0.46 ! [A: set_set_a,X: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) )
% 0.31/0.46 = ( ! [X3: set_a] :
% 0.31/0.46 ( ( member_set_a @ X3 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ X @ X3 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Inf_fin.bounded_iff
% 0.31/0.46 thf(fact_333_Sup__fin_Obounded__iff,axiom,
% 0.31/0.46 ! [A: set_set_a,X: set_a] :
% 0.31/0.46 ( ( finite_finite_set_a @ A )
% 0.31/0.46 => ( ( A != bot_bot_set_set_a )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X )
% 0.31/0.46 = ( ! [X3: set_a] :
% 0.31/0.46 ( ( member_set_a @ X3 @ A )
% 0.31/0.46 => ( ord_less_eq_set_a @ X3 @ X ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Sup_fin.bounded_iff
% 0.31/0.46 thf(fact_334_Un__empty,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( ( sup_sup_set_a @ A @ B2 )
% 0.31/0.46 = bot_bot_set_a )
% 0.31/0.46 = ( ( A = bot_bot_set_a )
% 0.31/0.46 & ( B2 = bot_bot_set_a ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_empty
% 0.31/0.46 thf(fact_335_Int__subset__iff,axiom,
% 0.31/0.46 ! [C2: set_a,A: set_a,B2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
% 0.31/0.46 = ( ( ord_less_eq_set_a @ C2 @ A )
% 0.31/0.46 & ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_subset_iff
% 0.31/0.46 thf(fact_336_Un__subset__iff,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a,C2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
% 0.31/0.46 = ( ( ord_less_eq_set_a @ A @ C2 )
% 0.31/0.46 & ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_subset_iff
% 0.31/0.46 thf(fact_337_finite__Int,axiom,
% 0.31/0.46 ! [F2: set_a,G: set_a] :
% 0.31/0.46 ( ( ( finite_finite_a @ F2 )
% 0.31/0.46 | ( finite_finite_a @ G ) )
% 0.31/0.46 => ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_Int
% 0.31/0.46 thf(fact_338_finite__Un,axiom,
% 0.31/0.46 ! [F2: set_a,G: set_a] :
% 0.31/0.46 ( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
% 0.31/0.46 = ( ( finite_finite_a @ F2 )
% 0.31/0.46 & ( finite_finite_a @ G ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_Un
% 0.31/0.46 thf(fact_339_trans__Int,axiom,
% 0.31/0.46 ! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( trans_a @ R2 )
% 0.31/0.46 => ( ( trans_a @ S2 )
% 0.31/0.46 => ( trans_a @ ( inf_in920712437od_a_a @ R2 @ S2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trans_Int
% 0.31/0.46 thf(fact_340_refl__on__Un,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a,B2: set_a,S2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( refl_on_a @ A @ R2 )
% 0.31/0.46 => ( ( refl_on_a @ B2 @ S2 )
% 0.31/0.46 => ( refl_on_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_su615935707od_a_a @ R2 @ S2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_on_Un
% 0.31/0.46 thf(fact_341_refl__on__Int,axiom,
% 0.31/0.46 ! [A: set_a,R2: set_Product_prod_a_a,B2: set_a,S2: set_Product_prod_a_a] :
% 0.31/0.46 ( ( refl_on_a @ A @ R2 )
% 0.31/0.46 => ( ( refl_on_a @ B2 @ S2 )
% 0.31/0.46 => ( refl_on_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_in920712437od_a_a @ R2 @ S2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % refl_on_Int
% 0.31/0.46 thf(fact_342_finite__UnI,axiom,
% 0.31/0.46 ! [F2: set_a,G: set_a] :
% 0.31/0.46 ( ( finite_finite_a @ F2 )
% 0.31/0.46 => ( ( finite_finite_a @ G )
% 0.31/0.46 => ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_UnI
% 0.31/0.46 thf(fact_343_Un__infinite,axiom,
% 0.31/0.46 ! [S: set_a,T2: set_a] :
% 0.31/0.46 ( ~ ( finite_finite_a @ S )
% 0.31/0.46 => ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_infinite
% 0.31/0.46 thf(fact_344_infinite__Un,axiom,
% 0.31/0.46 ! [S: set_a,T2: set_a] :
% 0.31/0.46 ( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
% 0.31/0.46 = ( ~ ( finite_finite_a @ S )
% 0.31/0.46 | ~ ( finite_finite_a @ T2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_Un
% 0.31/0.46 thf(fact_345_Un__Int__assoc__eq,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a,C2: set_a] :
% 0.31/0.46 ( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
% 0.31/0.46 = ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) )
% 0.31/0.46 = ( ord_less_eq_set_a @ C2 @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_Int_assoc_eq
% 0.31/0.46 thf(fact_346_Int__mono,axiom,
% 0.31/0.46 ! [A: set_a,C2: set_a,B2: set_a,D2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ C2 )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ B2 @ D2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_mono
% 0.31/0.46 thf(fact_347_Int__lower1,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% 0.31/0.46
% 0.31/0.46 % Int_lower1
% 0.31/0.46 thf(fact_348_Int__lower2,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% 0.31/0.46
% 0.31/0.46 % Int_lower2
% 0.31/0.46 thf(fact_349_Int__absorb1,axiom,
% 0.31/0.46 ! [B2: set_a,A: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ B2 @ A )
% 0.31/0.46 => ( ( inf_inf_set_a @ A @ B2 )
% 0.31/0.46 = B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_absorb1
% 0.31/0.46 thf(fact_350_Int__absorb2,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ( inf_inf_set_a @ A @ B2 )
% 0.31/0.46 = A ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_absorb2
% 0.31/0.46 thf(fact_351_Int__greatest,axiom,
% 0.31/0.46 ! [C2: set_a,A: set_a,B2: set_a] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ C2 @ A )
% 0.31/0.46 => ( ( ord_less_eq_set_a @ C2 @ B2 )
% 0.31/0.46 => ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_greatest
% 0.31/0.46 thf(fact_352_Int__Collect__mono,axiom,
% 0.31/0.46 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.31/0.46 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 0.31/0.46 => ( ! [X4: product_prod_a_a] :
% 0.31/0.46 ( ( member449909584od_a_a @ X4 @ A )
% 0.31/0.46 => ( ( P @ X4 )
% 0.31/0.46 => ( Q @ X4 ) ) )
% 0.31/0.46 => ( ord_le1824328871od_a_a @ ( inf_in920712437od_a_a @ A @ ( collec645855634od_a_a @ P ) ) @ ( inf_in920712437od_a_a @ B2 @ ( collec645855634od_a_a @ Q ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_Collect_mono
% 0.31/0.46 thf(fact_353_Int__Collect__mono,axiom,
% 0.31/0.46 ! [A: set_a,B2: set_a,P: a > $o,Q: a > $o] :
% 0.31/0.46 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.31/0.46 => ( ! [X4: a] :
% 0.31/0.46 ( ( member_a @ X4 @ A )
% 0.31/0.46 => ( ( P @ X4 )
% 0.31/0.46 => ( Q @ X4 ) ) )
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% 2.57/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.b3Te47zhCv/DTF2DTF27726.p
% See solution above
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% 2.69/1.50
% 2.69/1.50 thf(fact_12_reflexivity,axiom,
% 2.69/1.50 refl_on_a @ carrier @ relation ).
% 2.69/1.50
% 2.69/1.50 thf(fact_13_compl,axiom,
% 2.69/1.50 ! [X2: a] :
% 2.69/1.50 ( ( member_a @ X2 @ carrier )
% 2.69/1.50 => ! [Xa: a] :
% 2.69/1.50 ( ( member_a @ Xa @ carrier )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
% 2.69/1.50 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_14_completeD,axiom,
% 2.69/1.50 ! [X: a,Y: a] :
% 2.69/1.50 ( ( member_a @ X @ carrier )
% 2.69/1.50 => ( ( member_a @ Y @ carrier )
% 2.69/1.50 => ( ( X != Y )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_15_not__outside,axiom,
% 2.69/1.50 ! [X: a,Y: a] :
% 2.69/1.50 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 => ( member_a @ X @ carrier ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_16_strict__is__neg__transitive,axiom,
% 2.69/1.50 ! [X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( ( member_a @ X @ carrier )
% 2.69/1.50 & ( member_a @ Y @ carrier )
% 2.69/1.50 & ( member_a @ Z @ carrier ) )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) )
% 2.69/1.50 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_17_weak__is__transitive,axiom,
% 2.69/1.50 ! [X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( ( member_a @ X @ carrier )
% 2.69/1.50 & ( member_a @ Y @ carrier )
% 2.69/1.50 & ( member_a @ Z @ carrier ) )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.69/1.50 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_18_strict__trans,axiom,
% 2.69/1.50 ! [X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_19_indiff__trans,axiom,
% 2.69/1.50 ! [X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.69/1.50 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.69/1.50 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_20_strict__not__refl__weak,axiom,
% 2.69/1.50 ! [X: a,Y: a] :
% 2.69/1.50 ( ( ( member_a @ X @ carrier )
% 2.69/1.50 & ( member_a @ Y @ carrier ) )
% 2.69/1.50 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.69/1.50 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_21_rational__preference_Oaxioms_I1_J,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_22_rational__preference_Oaxioms_I1_J,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.50 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_23_preference__def,axiom,
% 2.69/1.50 ( prefer1409946857od_a_a
% 2.69/1.50 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 2.69/1.50 ( ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 2.69/1.50 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member2057358096od_a_a @ X3 @ Carrier2 ) )
% 2.69/1.50 & ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 2.69/1.50 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member2057358096od_a_a @ Y2 @ Carrier2 ) )
% 2.69/1.50 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_24_preference__def,axiom,
% 2.69/1.50 ( prefer1710114912_set_a
% 2.69/1.50 = ( ^ [Carrier2: set_set_a,Relation2: set_Pr2045688071_set_a] :
% 2.69/1.50 ( ! [X3: set_a,Y2: set_a] :
% 2.69/1.50 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member_set_a @ X3 @ Carrier2 ) )
% 2.69/1.50 & ! [X3: set_a,Y2: set_a] :
% 2.69/1.50 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member_set_a @ Y2 @ Carrier2 ) )
% 2.69/1.50 & ( order_816746613_set_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_25_preference__def,axiom,
% 2.69/1.50 ( prefer1338892073od_a_a
% 2.69/1.50 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 2.69/1.50 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member449909584od_a_a @ X3 @ Carrier2 ) )
% 2.69/1.50 & ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 2.69/1.50 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member449909584od_a_a @ Y2 @ Carrier2 ) )
% 2.69/1.50 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_26_preference__def,axiom,
% 2.69/1.50 ( prefer1292084992ence_a
% 2.69/1.50 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 2.69/1.50 ( ! [X3: a,Y2: a] :
% 2.69/1.50 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member_a @ X3 @ Carrier2 ) )
% 2.69/1.50 & ! [X3: a,Y2: a] :
% 2.69/1.50 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.69/1.50 => ( member_a @ Y2 @ Carrier2 ) )
% 2.69/1.50 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_27_preference_Ointro,axiom,
% 2.69/1.50 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.69/1.50 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member2057358096od_a_a @ X4 @ Carrier ) )
% 2.69/1.50 => ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.69/1.50 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 2.69/1.50 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_28_preference_Ointro,axiom,
% 2.69/1.50 ! [Relation: set_Pr2045688071_set_a,Carrier: set_set_a] :
% 2.69/1.50 ( ! [X4: set_a,Y3: set_a] :
% 2.69/1.50 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member_set_a @ X4 @ Carrier ) )
% 2.69/1.50 => ( ! [X4: set_a,Y3: set_a] :
% 2.69/1.50 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member_set_a @ Y3 @ Carrier ) )
% 2.69/1.50 => ( ( order_816746613_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( prefer1710114912_set_a @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_29_preference_Ointro,axiom,
% 2.69/1.50 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 2.69/1.50 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.69/1.50 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member449909584od_a_a @ X4 @ Carrier ) )
% 2.69/1.50 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.69/1.50 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 2.69/1.50 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_30_preference_Ointro,axiom,
% 2.69/1.50 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 2.69/1.50 ( ! [X4: a,Y3: a] :
% 2.69/1.50 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member_a @ X4 @ Carrier ) )
% 2.69/1.50 => ( ! [X4: a,Y3: a] :
% 2.69/1.50 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 2.69/1.50 => ( member_a @ Y3 @ Carrier ) )
% 2.69/1.50 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 2.69/1.50 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_31_preference_Otrans__refl,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_32_preference_Otrans__refl,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.50 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_33_preference_Onot__outside,axiom,
% 2.69/1.50 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.50 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_34_preference_Onot__outside,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.50 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( member_set_a @ X @ Carrier ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_35_preference_Onot__outside,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.50 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_36_preference_Onot__outside,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.50 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( member_a @ X @ Carrier ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_37_preference_Oreflexivity,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_38_preference_Oreflexivity,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.50 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_39_preference_Oindiff__trans,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 2.69/1.50 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 2.69/1.50 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
% 2.69/1.50 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_40_preference_Oindiff__trans,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.69/1.50 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 2.69/1.50 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_41_preference_Oindiff__trans,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 2.69/1.50 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_42_rational__preference_Ocompl,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a] :
% 2.69/1.50 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.50 => ! [X2: set_a] :
% 2.69/1.50 ( ( member_set_a @ X2 @ Carrier )
% 2.69/1.50 => ! [Xa: set_a] :
% 2.69/1.50 ( ( member_set_a @ Xa @ Carrier )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Xa ) @ Relation )
% 2.69/1.50 | ( member593450832_set_a @ ( produc1928581911_set_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_43_rational__preference_Ocompl,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ! [X2: product_prod_a_a] :
% 2.69/1.50 ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.69/1.50 => ! [Xa: product_prod_a_a] :
% 2.69/1.50 ( ( member449909584od_a_a @ Xa @ Carrier )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Xa ) @ Relation )
% 2.69/1.50 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_44_rational__preference_Ocompl,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.50 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.50 => ! [X2: a] :
% 2.69/1.50 ( ( member_a @ X2 @ Carrier )
% 2.69/1.50 => ! [Xa: a] :
% 2.69/1.50 ( ( member_a @ Xa @ Carrier )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
% 2.69/1.50 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_45_rational__preference_Ototal,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.69/1.50 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_46_rational__preference_Ototal,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.50 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( total_on_a @ Carrier @ Relation ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_47_rational__preference_OcompleteD,axiom,
% 2.69/1.50 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.50 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.50 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 2.69/1.50 => ( ( X != Y )
% 2.69/1.50 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_48_rational__preference_OcompleteD,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.50 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member_set_a @ X @ Carrier )
% 2.69/1.50 => ( ( member_set_a @ Y @ Carrier )
% 2.69/1.50 => ( ( X != Y )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.50 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_49_rational__preference_OcompleteD,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.50 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.50 => ( ( member449909584od_a_a @ Y @ Carrier )
% 2.69/1.50 => ( ( X != Y )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_50_rational__preference_OcompleteD,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.50 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( member_a @ X @ Carrier )
% 2.69/1.50 => ( ( member_a @ Y @ Carrier )
% 2.69/1.50 => ( ( X != Y )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_51_rational__preference_Ostrict__trans,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 2.69/1.50 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 2.69/1.50 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
% 2.69/1.50 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_52_rational__preference_Ostrict__trans,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.69/1.50 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 2.69/1.50 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_53_rational__preference_Ostrict__trans,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 2.69/1.50 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_54_rational__preference_Oweak__is__transitive,axiom,
% 2.69/1.50 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 2.69/1.50 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.50 & ( member2057358096od_a_a @ Y @ Carrier )
% 2.69/1.50 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 2.69/1.50 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_55_rational__preference_Oweak__is__transitive,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 2.69/1.50 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member_set_a @ X @ Carrier )
% 2.69/1.50 & ( member_set_a @ Y @ Carrier )
% 2.69/1.50 & ( member_set_a @ Z @ Carrier ) )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 2.69/1.50 => ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_56_rational__preference_Oweak__is__transitive,axiom,
% 2.69/1.50 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.69/1.50 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.50 & ( member449909584od_a_a @ Y @ Carrier )
% 2.69/1.50 & ( member449909584od_a_a @ Z @ Carrier ) )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_57_rational__preference_Oweak__is__transitive,axiom,
% 2.69/1.50 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.69/1.50 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member_a @ X @ Carrier )
% 2.69/1.50 & ( member_a @ Y @ Carrier )
% 2.69/1.50 & ( member_a @ Z @ Carrier ) )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.69/1.50 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_58_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.69/1.50 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.50 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.50 & ( member2057358096od_a_a @ Y @ Carrier ) )
% 2.69/1.50 => ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.50 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.50 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.50
% 2.69/1.50 thf(fact_59_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.69/1.50 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.50 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.50 => ( ( ( member_set_a @ X @ Carrier )
% 2.69/1.50 & ( member_set_a @ Y @ Carrier ) )
% 2.69/1.51 => ( ( ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_60_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.51 & ( member449909584od_a_a @ Y @ Carrier ) )
% 2.69/1.51 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_61_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member_a @ X @ Carrier )
% 2.69/1.51 & ( member_a @ Y @ Carrier ) )
% 2.69/1.51 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_62_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.51 & ( member2057358096od_a_a @ Y @ Carrier )
% 2.69/1.51 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 2.69/1.51 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Z ) @ Relation )
% 2.69/1.51 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ X ) @ Relation ) )
% 2.69/1.51 | ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_63_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member_set_a @ X @ Carrier )
% 2.69/1.51 & ( member_set_a @ Y @ Carrier )
% 2.69/1.51 & ( member_set_a @ Z @ Carrier ) )
% 2.69/1.51 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
% 2.69/1.51 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) )
% 2.69/1.51 | ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_64_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.51 & ( member449909584od_a_a @ Y @ Carrier )
% 2.69/1.51 & ( member449909584od_a_a @ Z @ Carrier ) )
% 2.69/1.51 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 2.69/1.51 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) )
% 2.69/1.51 | ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_65_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member_a @ X @ Carrier )
% 2.69/1.51 & ( member_a @ Y @ Carrier )
% 2.69/1.51 & ( member_a @ Z @ Carrier ) )
% 2.69/1.51 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.69/1.51 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 2.69/1.51 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) )
% 2.69/1.51 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_66_rational__preference_Oworse__in__no__better,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member2057358096od_a_a @ Y @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_67_rational__preference_Oworse__in__no__better,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member_set_a @ Y @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_68_rational__preference_Oworse__in__no__better,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_69_rational__preference_Oworse__in__no__better,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_70_rational__preference_Ono__better__than__nonepty,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bo1116205875od_a_a )
% 2.69/1.51 => ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.51 => ( ( prefer926864306od_a_a @ X @ Carrier @ Relation )
% 2.69/1.51 != bot_bo1116205875od_a_a ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_71_rational__preference_Ono__better__than__nonepty,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bot_set_set_a )
% 2.69/1.51 => ( ( member_set_a @ X @ Carrier )
% 2.69/1.51 => ( ( prefer1671525929_set_a @ X @ Carrier @ Relation )
% 2.69/1.51 != bot_bot_set_set_a ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_72_rational__preference_Ono__better__than__nonepty,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bo2131659635od_a_a )
% 2.69/1.51 => ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.51 => ( ( prefer1498087410od_a_a @ X @ Carrier @ Relation )
% 2.69/1.51 != bot_bo2131659635od_a_a ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_73_rational__preference_Ono__better__than__nonepty,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bot_set_a )
% 2.69/1.51 => ( ( member_a @ X @ Carrier )
% 2.69/1.51 => ( ( prefer1676310729than_a @ X @ Carrier @ Relation )
% 2.69/1.51 != bot_bot_set_a ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_74_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.51 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 2.69/1.51 => ( ( ord_le456379495od_a_a @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) @ ( prefer926864306od_a_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_75_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member_set_a @ X @ Carrier )
% 2.69/1.51 => ( ( member_set_a @ Y @ Carrier )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 => ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_76_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.51 => ( ( member449909584od_a_a @ Y @ Carrier )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_77_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member_a @ X @ Carrier )
% 2.69/1.51 => ( ( member_a @ Y @ Carrier )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_78_rational__preference_Ono__better__subset__pref,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_79_rational__preference_Ono__better__subset__pref,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_80_rational__preference_Ono__better__subset__pref,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_81_rational__preference_Onbt__nest,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Y: set_a,X: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 | ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_82_rational__preference_Onbt__nest,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Y: product_prod_a_a,X: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 | ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_83_rational__preference_Onbt__nest,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 2.69/1.51 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_84_at__lst__asgd__not__ge,axiom,
% 2.69/1.51 ! [X: a,Y: a] :
% 2.69/1.51 ( ( carrier != bot_bot_set_a )
% 2.69/1.51 => ( ( member_a @ X @ carrier )
% 2.69/1.51 => ( ( member_a @ Y @ carrier )
% 2.69/1.51 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
% 2.69/1.51 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_85_same__at__least__as__equal,axiom,
% 2.69/1.51 ! [Z: a,Y: a] :
% 2.69/1.51 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 2.69/1.51 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) )
% 2.69/1.51 => ( ( prefer161218362good_a @ Z @ carrier @ relation )
% 2.69/1.51 = ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_86_pref__in__at__least__as,axiom,
% 2.69/1.51 ! [X: a,Y: a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.51 => ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_87_empty__subsetI,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_88_empty__subsetI,axiom,
% 2.69/1.51 ! [A: set_set_a] : ( ord_le318720350_set_a @ bot_bot_set_set_a @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_89_empty__subsetI,axiom,
% 2.69/1.51 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_90_subset__empty,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A @ bot_bo2131659635od_a_a )
% 2.69/1.51 = ( A = bot_bo2131659635od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_91_subset__empty,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ bot_bot_set_set_a )
% 2.69/1.51 = ( A = bot_bot_set_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_92_subset__empty,axiom,
% 2.69/1.51 ! [A: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 2.69/1.51 = ( A = bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_93_mem__Collect__eq,axiom,
% 2.69/1.51 ! [A2: produc1572603623od_a_a,P: produc1572603623od_a_a > $o] :
% 2.69/1.51 ( ( member2057358096od_a_a @ A2 @ ( collec1635618130od_a_a @ P ) )
% 2.69/1.51 = ( P @ A2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_94_mem__Collect__eq,axiom,
% 2.69/1.51 ! [A2: set_a,P: set_a > $o] :
% 2.69/1.51 ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
% 2.69/1.51 = ( P @ A2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_95_mem__Collect__eq,axiom,
% 2.69/1.51 ! [A2: a,P: a > $o] :
% 2.69/1.51 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 2.69/1.51 = ( P @ A2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_96_mem__Collect__eq,axiom,
% 2.69/1.51 ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
% 2.69/1.51 ( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
% 2.69/1.51 = ( P @ A2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_97_Collect__mem__eq,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ( collec1635618130od_a_a
% 2.69/1.51 @ ^ [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A ) )
% 2.69/1.51 = A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_98_Collect__mem__eq,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( collect_set_a
% 2.69/1.51 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
% 2.69/1.51 = A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_99_Collect__mem__eq,axiom,
% 2.69/1.51 ! [A: set_a] :
% 2.69/1.51 ( ( collect_a
% 2.69/1.51 @ ^ [X3: a] : ( member_a @ X3 @ A ) )
% 2.69/1.51 = A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_100_Collect__mem__eq,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( collec645855634od_a_a
% 2.69/1.51 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
% 2.69/1.51 = A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_101_Collect__cong,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.69/1.51 ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 = ( Q @ X4 ) )
% 2.69/1.51 => ( ( collec645855634od_a_a @ P )
% 2.69/1.51 = ( collec645855634od_a_a @ Q ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_102_Collect__cong,axiom,
% 2.69/1.51 ! [P: a > $o,Q: a > $o] :
% 2.69/1.51 ( ! [X4: a] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 = ( Q @ X4 ) )
% 2.69/1.51 => ( ( collect_a @ P )
% 2.69/1.51 = ( collect_a @ Q ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_103_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a,C: set_a,D: set_a] :
% 2.69/1.51 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A2 @ B ) @ ( produc1928581911_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_less_eq_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_104_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_set_a,C: set_a,D: set_set_a] :
% 2.69/1.51 ( ( ord_le1762684455_set_a @ ( produc1560024183_set_a @ A2 @ B ) @ ( produc1560024183_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_less_eq_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_le318720350_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_105_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_set_a,B: set_a,C: set_set_a,D: set_a] :
% 2.69/1.51 ( ( ord_le1201700711_set_a @ ( produc32105911_set_a @ A2 @ B ) @ ( produc32105911_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_le318720350_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_106_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_Product_prod_a_a,C: set_a,D: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le49202928od_a_a @ ( produc1397177024od_a_a @ A2 @ B ) @ ( produc1397177024od_a_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_less_eq_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_le1824328871od_a_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_107_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_Product_prod_a_a,B: set_a,C: set_Product_prod_a_a,D: set_a] :
% 2.69/1.51 ( ( ord_le2095302494_set_a @ ( produc194376366_set_a @ A2 @ B ) @ ( produc194376366_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_le1824328871od_a_a @ A2 @ C )
% 2.69/1.51 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_108_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_set_a,B: set_set_a,C: set_set_a,D: set_set_a] :
% 2.69/1.51 ( ( ord_le723713223_set_a @ ( produc2102896407_set_a @ A2 @ B ) @ ( produc2102896407_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_le318720350_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_le318720350_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_109_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_a,B: produc1691597095_set_a,C: set_a,D: produc1691597095_set_a] :
% 2.69/1.51 ( ( ord_le560833296_set_a @ ( produc272677600_set_a @ A2 @ B ) @ ( produc272677600_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_less_eq_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_le486764743_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_110_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_Product_prod_a_a,B: set_set_a,C: set_Product_prod_a_a,D: set_set_a] :
% 2.69/1.51 ( ( ord_le642967742_set_a @ ( produc1599887374_set_a @ A2 @ B ) @ ( produc1599887374_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_le1824328871od_a_a @ A2 @ C )
% 2.69/1.51 & ( ord_le318720350_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_111_Pair__le,axiom,
% 2.69/1.51 ! [A2: produc1691597095_set_a,B: set_a,C: produc1691597095_set_a,D: set_a] :
% 2.69/1.51 ( ( ord_le1729087294_set_a @ ( produc1719072398_set_a @ A2 @ B ) @ ( produc1719072398_set_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_le486764743_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_112_Pair__le,axiom,
% 2.69/1.51 ! [A2: set_set_a,B: set_Product_prod_a_a,C: set_set_a,D: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le526661008od_a_a @ ( produc1618062688od_a_a @ A2 @ B ) @ ( produc1618062688od_a_a @ C @ D ) )
% 2.69/1.51 = ( ( ord_le318720350_set_a @ A2 @ C )
% 2.69/1.51 & ( ord_le1824328871od_a_a @ B @ D ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_113_as__good__as__sameIff,axiom,
% 2.69/1.51 ! [Z: a,Y: a] :
% 2.69/1.51 ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 2.69/1.51 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 2.69/1.51 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_114_as__good__asIff,axiom,
% 2.69/1.51 ! [X: a,Y: a] :
% 2.69/1.51 ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 2.69/1.51 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.69/1.51 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_115_finite__has__maximal,axiom,
% 2.69/1.51 ! [A: set_se1596668135od_a_a] :
% 2.69/1.51 ( ( finite1145471536od_a_a @ A )
% 2.69/1.51 => ( ( A != bot_bo1424703315od_a_a )
% 2.69/1.51 => ? [X4: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member1838126896od_a_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member1838126896od_a_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ X4 @ Xa )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_116_finite__has__maximal,axiom,
% 2.69/1.51 ! [A: set_Pr2045688071_set_a] :
% 2.69/1.51 ( ( finite283787856_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bo671027059_set_a )
% 2.69/1.51 => ? [X4: produc1691597095_set_a] :
% 2.69/1.51 ( ( member593450832_set_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: produc1691597095_set_a] :
% 2.69/1.51 ( ( member593450832_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_le486764743_set_a @ X4 @ Xa )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_117_finite__has__maximal,axiom,
% 2.69/1.51 ! [A: set_set_set_a] :
% 2.69/1.51 ( ( finite1606323175_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bo1264989578_set_a )
% 2.69/1.51 => ? [X4: set_set_a] :
% 2.69/1.51 ( ( member_set_set_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: set_set_a] :
% 2.69/1.51 ( ( member_set_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ X4 @ Xa )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_118_finite__has__maximal,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ? [X4: set_a] :
% 2.69/1.51 ( ( member_set_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: set_a] :
% 2.69/1.51 ( ( member_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_119_finite__has__minimal,axiom,
% 2.69/1.51 ! [A: set_se1596668135od_a_a] :
% 2.69/1.51 ( ( finite1145471536od_a_a @ A )
% 2.69/1.51 => ( ( A != bot_bo1424703315od_a_a )
% 2.69/1.51 => ? [X4: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member1838126896od_a_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member1838126896od_a_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ Xa @ X4 )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_120_finite__has__minimal,axiom,
% 2.69/1.51 ! [A: set_Pr2045688071_set_a] :
% 2.69/1.51 ( ( finite283787856_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bo671027059_set_a )
% 2.69/1.51 => ? [X4: produc1691597095_set_a] :
% 2.69/1.51 ( ( member593450832_set_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: produc1691597095_set_a] :
% 2.69/1.51 ( ( member593450832_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_le486764743_set_a @ Xa @ X4 )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_121_finite__has__minimal,axiom,
% 2.69/1.51 ! [A: set_set_set_a] :
% 2.69/1.51 ( ( finite1606323175_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bo1264989578_set_a )
% 2.69/1.51 => ? [X4: set_set_a] :
% 2.69/1.51 ( ( member_set_set_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: set_set_a] :
% 2.69/1.51 ( ( member_set_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ Xa @ X4 )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_122_finite__has__minimal,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ? [X4: set_a] :
% 2.69/1.51 ( ( member_set_a @ X4 @ A )
% 2.69/1.51 & ! [Xa: set_a] :
% 2.69/1.51 ( ( member_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_123_empty__Collect__eq,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o] :
% 2.69/1.51 ( ( bot_bo2131659635od_a_a
% 2.69/1.51 = ( collec645855634od_a_a @ P ) )
% 2.69/1.51 = ( ! [X3: product_prod_a_a] :
% 2.69/1.51 ~ ( P @ X3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_124_empty__Collect__eq,axiom,
% 2.69/1.51 ! [P: set_a > $o] :
% 2.69/1.51 ( ( bot_bot_set_set_a
% 2.69/1.51 = ( collect_set_a @ P ) )
% 2.69/1.51 = ( ! [X3: set_a] :
% 2.69/1.51 ~ ( P @ X3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_125_empty__Collect__eq,axiom,
% 2.69/1.51 ! [P: a > $o] :
% 2.69/1.51 ( ( bot_bot_set_a
% 2.69/1.51 = ( collect_a @ P ) )
% 2.69/1.51 = ( ! [X3: a] :
% 2.69/1.51 ~ ( P @ X3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_126_Collect__empty__eq,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o] :
% 2.69/1.51 ( ( ( collec645855634od_a_a @ P )
% 2.69/1.51 = bot_bo2131659635od_a_a )
% 2.69/1.51 = ( ! [X3: product_prod_a_a] :
% 2.69/1.51 ~ ( P @ X3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_127_Collect__empty__eq,axiom,
% 2.69/1.51 ! [P: set_a > $o] :
% 2.69/1.51 ( ( ( collect_set_a @ P )
% 2.69/1.51 = bot_bot_set_set_a )
% 2.69/1.51 = ( ! [X3: set_a] :
% 2.69/1.51 ~ ( P @ X3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_128_Collect__empty__eq,axiom,
% 2.69/1.51 ! [P: a > $o] :
% 2.69/1.51 ( ( ( collect_a @ P )
% 2.69/1.51 = bot_bot_set_a )
% 2.69/1.51 = ( ! [X3: a] :
% 2.69/1.51 ~ ( P @ X3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_129_all__not__in__conv,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ( ! [X3: produc1572603623od_a_a] :
% 2.69/1.51 ~ ( member2057358096od_a_a @ X3 @ A ) )
% 2.69/1.51 = ( A = bot_bo1116205875od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_130_all__not__in__conv,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( ! [X3: set_a] :
% 2.69/1.51 ~ ( member_set_a @ X3 @ A ) )
% 2.69/1.51 = ( A = bot_bot_set_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_131_all__not__in__conv,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ! [X3: product_prod_a_a] :
% 2.69/1.51 ~ ( member449909584od_a_a @ X3 @ A ) )
% 2.69/1.51 = ( A = bot_bo2131659635od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_132_all__not__in__conv,axiom,
% 2.69/1.51 ! [A: set_a] :
% 2.69/1.51 ( ( ! [X3: a] :
% 2.69/1.51 ~ ( member_a @ X3 @ A ) )
% 2.69/1.51 = ( A = bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_133_empty__iff,axiom,
% 2.69/1.51 ! [C: produc1572603623od_a_a] :
% 2.69/1.51 ~ ( member2057358096od_a_a @ C @ bot_bo1116205875od_a_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_134_empty__iff,axiom,
% 2.69/1.51 ! [C: set_a] :
% 2.69/1.51 ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_135_empty__iff,axiom,
% 2.69/1.51 ! [C: product_prod_a_a] :
% 2.69/1.51 ~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_136_empty__iff,axiom,
% 2.69/1.51 ! [C: a] :
% 2.69/1.51 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_137_subsetI,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ! [X4: produc1572603623od_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ X4 @ A )
% 2.69/1.51 => ( member2057358096od_a_a @ X4 @ B2 ) )
% 2.69/1.51 => ( ord_le456379495od_a_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_138_subsetI,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ! [X4: set_a] :
% 2.69/1.51 ( ( member_set_a @ X4 @ A )
% 2.69/1.51 => ( member_set_a @ X4 @ B2 ) )
% 2.69/1.51 => ( ord_le318720350_set_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_139_subsetI,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 2.69/1.51 ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X4 @ A )
% 2.69/1.51 => ( member449909584od_a_a @ X4 @ B2 ) )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_140_subsetI,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ! [X4: a] :
% 2.69/1.51 ( ( member_a @ X4 @ A )
% 2.69/1.51 => ( member_a @ X4 @ B2 ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_141_subset__antisym,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ B2 @ A )
% 2.69/1.51 => ( A = B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_142_subset__antisym,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ B2 @ A )
% 2.69/1.51 => ( A = B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_143_subset__antisym,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B2 @ A )
% 2.69/1.51 => ( A = B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_144_at__least__as__goodD,axiom,
% 2.69/1.51 ! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B2: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_145_at__least__as__goodD,axiom,
% 2.69/1.51 ! [Z: set_a,Y: set_a,B2: set_set_a,Pr: set_Pr2045688071_set_a] :
% 2.69/1.51 ( ( member_set_a @ Z @ ( prefer1203288218_set_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_146_at__least__as__goodD,axiom,
% 2.69/1.51 ! [Z: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_147_at__least__as__goodD,axiom,
% 2.69/1.51 ! [Z: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_148_at__lst__asgd__ge,axiom,
% 2.69/1.51 ! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B2: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_149_at__lst__asgd__ge,axiom,
% 2.69/1.51 ! [X: set_a,Y: set_a,B2: set_set_a,Pr: set_Pr2045688071_set_a] :
% 2.69/1.51 ( ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_150_at__lst__asgd__ge,axiom,
% 2.69/1.51 ! [X: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_151_at__lst__asgd__ge,axiom,
% 2.69/1.51 ! [X: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_152_ex__in__conv,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ( ? [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A ) )
% 2.69/1.51 = ( A != bot_bo1116205875od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_153_ex__in__conv,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
% 2.69/1.51 = ( A != bot_bot_set_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_154_ex__in__conv,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
% 2.69/1.51 = ( A != bot_bo2131659635od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_155_ex__in__conv,axiom,
% 2.69/1.51 ! [A: set_a] :
% 2.69/1.51 ( ( ? [X3: a] : ( member_a @ X3 @ A ) )
% 2.69/1.51 = ( A != bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_156_equals0I,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ! [Y3: produc1572603623od_a_a] :
% 2.69/1.51 ~ ( member2057358096od_a_a @ Y3 @ A )
% 2.69/1.51 => ( A = bot_bo1116205875od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_157_equals0I,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ! [Y3: set_a] :
% 2.69/1.51 ~ ( member_set_a @ Y3 @ A )
% 2.69/1.51 => ( A = bot_bot_set_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_158_equals0I,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] :
% 2.69/1.51 ( ! [Y3: product_prod_a_a] :
% 2.69/1.51 ~ ( member449909584od_a_a @ Y3 @ A )
% 2.69/1.51 => ( A = bot_bo2131659635od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_159_equals0I,axiom,
% 2.69/1.51 ! [A: set_a] :
% 2.69/1.51 ( ! [Y3: a] :
% 2.69/1.51 ~ ( member_a @ Y3 @ A )
% 2.69/1.51 => ( A = bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_160_equals0D,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a,A2: produc1572603623od_a_a] :
% 2.69/1.51 ( ( A = bot_bo1116205875od_a_a )
% 2.69/1.51 => ~ ( member2057358096od_a_a @ A2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_161_equals0D,axiom,
% 2.69/1.51 ! [A: set_set_a,A2: set_a] :
% 2.69/1.51 ( ( A = bot_bot_set_set_a )
% 2.69/1.51 => ~ ( member_set_a @ A2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_162_equals0D,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
% 2.69/1.51 ( ( A = bot_bo2131659635od_a_a )
% 2.69/1.51 => ~ ( member449909584od_a_a @ A2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_163_equals0D,axiom,
% 2.69/1.51 ! [A: set_a,A2: a] :
% 2.69/1.51 ( ( A = bot_bot_set_a )
% 2.69/1.51 => ~ ( member_a @ A2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_164_emptyE,axiom,
% 2.69/1.51 ! [A2: produc1572603623od_a_a] :
% 2.69/1.51 ~ ( member2057358096od_a_a @ A2 @ bot_bo1116205875od_a_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_165_emptyE,axiom,
% 2.69/1.51 ! [A2: set_a] :
% 2.69/1.51 ~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_166_emptyE,axiom,
% 2.69/1.51 ! [A2: product_prod_a_a] :
% 2.69/1.51 ~ ( member449909584od_a_a @ A2 @ bot_bo2131659635od_a_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_167_emptyE,axiom,
% 2.69/1.51 ! [A2: a] :
% 2.69/1.51 ~ ( member_a @ A2 @ bot_bot_set_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_168_in__mono,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a,X: produc1572603623od_a_a] :
% 2.69/1.51 ( ( ord_le456379495od_a_a @ A @ B2 )
% 2.69/1.51 => ( ( member2057358096od_a_a @ X @ A )
% 2.69/1.51 => ( member2057358096od_a_a @ X @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_169_in__mono,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a,X: set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ( ( member_set_a @ X @ A )
% 2.69/1.51 => ( member_set_a @ X @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_170_in__mono,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ X @ A )
% 2.69/1.51 => ( member449909584od_a_a @ X @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_171_in__mono,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a,X: a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ( member_a @ X @ A )
% 2.69/1.51 => ( member_a @ X @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_172_subsetD,axiom,
% 2.69/1.51 ! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a,C: produc1572603623od_a_a] :
% 2.69/1.51 ( ( ord_le456379495od_a_a @ A @ B2 )
% 2.69/1.51 => ( ( member2057358096od_a_a @ C @ A )
% 2.69/1.51 => ( member2057358096od_a_a @ C @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_173_subsetD,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a,C: set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ( ( member_set_a @ C @ A )
% 2.69/1.51 => ( member_set_a @ C @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_174_subsetD,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ C @ A )
% 2.69/1.51 => ( member449909584od_a_a @ C @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_175_subsetD,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a,C: a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ( member_a @ C @ A )
% 2.69/1.51 => ( member_a @ C @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_176_equalityE,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ~ ( ( ord_le1824328871od_a_a @ A @ B2 )
% 2.69/1.51 => ~ ( ord_le1824328871od_a_a @ B2 @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_177_equalityE,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ~ ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ~ ( ord_le318720350_set_a @ B2 @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_178_equalityE,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ~ ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_179_subset__eq,axiom,
% 2.69/1.51 ( ord_le456379495od_a_a
% 2.69/1.51 = ( ^ [A3: set_Pr1948701895od_a_a,B3: set_Pr1948701895od_a_a] :
% 2.69/1.51 ! [X3: produc1572603623od_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ X3 @ A3 )
% 2.69/1.51 => ( member2057358096od_a_a @ X3 @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_180_subset__eq,axiom,
% 2.69/1.51 ( ord_le318720350_set_a
% 2.69/1.51 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 2.69/1.51 ! [X3: set_a] :
% 2.69/1.51 ( ( member_set_a @ X3 @ A3 )
% 2.69/1.51 => ( member_set_a @ X3 @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_181_subset__eq,axiom,
% 2.69/1.51 ( ord_le1824328871od_a_a
% 2.69/1.51 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.69/1.51 ! [X3: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X3 @ A3 )
% 2.69/1.51 => ( member449909584od_a_a @ X3 @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_182_subset__eq,axiom,
% 2.69/1.51 ( ord_less_eq_set_a
% 2.69/1.51 = ( ^ [A3: set_a,B3: set_a] :
% 2.69/1.51 ! [X3: a] :
% 2.69/1.51 ( ( member_a @ X3 @ A3 )
% 2.69/1.51 => ( member_a @ X3 @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_183_equalityD1,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_184_equalityD1,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ( ord_le318720350_set_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_185_equalityD1,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ A @ B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_186_equalityD2,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ B2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_187_equalityD2,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ( ord_le318720350_set_a @ B2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_188_equalityD2,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( A = B2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ B2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_189_subset__iff,axiom,
% 2.69/1.51 ( ord_le456379495od_a_a
% 2.69/1.51 = ( ^ [A3: set_Pr1948701895od_a_a,B3: set_Pr1948701895od_a_a] :
% 2.69/1.51 ! [T: produc1572603623od_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ T @ A3 )
% 2.69/1.51 => ( member2057358096od_a_a @ T @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_190_subset__iff,axiom,
% 2.69/1.51 ( ord_le318720350_set_a
% 2.69/1.51 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 2.69/1.51 ! [T: set_a] :
% 2.69/1.51 ( ( member_set_a @ T @ A3 )
% 2.69/1.51 => ( member_set_a @ T @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_191_subset__iff,axiom,
% 2.69/1.51 ( ord_le1824328871od_a_a
% 2.69/1.51 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.69/1.51 ! [T: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ T @ A3 )
% 2.69/1.51 => ( member449909584od_a_a @ T @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_192_subset__iff,axiom,
% 2.69/1.51 ( ord_less_eq_set_a
% 2.69/1.51 = ( ^ [A3: set_a,B3: set_a] :
% 2.69/1.51 ! [T: a] :
% 2.69/1.51 ( ( member_a @ T @ A3 )
% 2.69/1.51 => ( member_a @ T @ B3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_193_subset__refl,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_194_subset__refl,axiom,
% 2.69/1.51 ! [A: set_set_a] : ( ord_le318720350_set_a @ A @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_195_subset__refl,axiom,
% 2.69/1.51 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_196_Collect__mono,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.69/1.51 ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 => ( Q @ X4 ) )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_197_Collect__mono,axiom,
% 2.69/1.51 ! [P: set_a > $o,Q: set_a > $o] :
% 2.69/1.51 ( ! [X4: set_a] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 => ( Q @ X4 ) )
% 2.69/1.51 => ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_198_Collect__mono,axiom,
% 2.69/1.51 ! [P: a > $o,Q: a > $o] :
% 2.69/1.51 ( ! [X4: a] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 => ( Q @ X4 ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_199_subset__trans,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ B2 @ C2 )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ A @ C2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_200_subset__trans,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ B2 @ C2 )
% 2.69/1.51 => ( ord_le318720350_set_a @ A @ C2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_201_subset__trans,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a,C2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_202_set__eq__subset,axiom,
% 2.69/1.51 ( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y4 = Z2 ) )
% 2.69/1.51 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A3 @ B3 )
% 2.69/1.51 & ( ord_le1824328871od_a_a @ B3 @ A3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_203_set__eq__subset,axiom,
% 2.69/1.51 ( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
% 2.69/1.51 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A3 @ B3 )
% 2.69/1.51 & ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_204_set__eq__subset,axiom,
% 2.69/1.51 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.69/1.51 = ( ^ [A3: set_a,B3: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.69/1.51 & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_205_Collect__mono__iff,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) )
% 2.69/1.51 = ( ! [X3: product_prod_a_a] :
% 2.69/1.51 ( ( P @ X3 )
% 2.69/1.51 => ( Q @ X3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_206_Collect__mono__iff,axiom,
% 2.69/1.51 ! [P: set_a > $o,Q: set_a > $o] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
% 2.69/1.51 = ( ! [X3: set_a] :
% 2.69/1.51 ( ( P @ X3 )
% 2.69/1.51 => ( Q @ X3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_207_Collect__mono__iff,axiom,
% 2.69/1.51 ! [P: a > $o,Q: a > $o] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 2.69/1.51 = ( ! [X3: a] :
% 2.69/1.51 ( ( P @ X3 )
% 2.69/1.51 => ( Q @ X3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_208_rational__preference_Osame__at__least__as__equal,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) )
% 2.69/1.51 => ( ( prefer1203288218_set_a @ Z @ Carrier @ Relation )
% 2.69/1.51 = ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_209_rational__preference_Osame__at__least__as__equal,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) )
% 2.69/1.51 => ( ( prefer477097315od_a_a @ Z @ Carrier @ Relation )
% 2.69/1.51 = ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_210_rational__preference_Osame__at__least__as__equal,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) )
% 2.69/1.51 => ( ( prefer161218362good_a @ Z @ Carrier @ Relation )
% 2.69/1.51 = ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_211_rational__preference_Opref__in__at__least__as,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_212_rational__preference_Opref__in__at__least__as,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_213_rational__preference_Opref__in__at__least__as,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_214_rational__preference_Opref__in__at__least__as,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 => ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_215_rational__preference_Oas__good__as__sameIff,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,Z: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member2057358096od_a_a @ Z @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_216_rational__preference_Oas__good__as__sameIff,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member_set_a @ Z @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_217_rational__preference_Oas__good__as__sameIff,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_218_rational__preference_Oas__good__as__sameIff,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 2.69/1.51 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_219_rational__preference_Oas__good__asIff,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member2057358096od_a_a @ X @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_220_rational__preference_Oas__good__asIff,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member_set_a @ X @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_221_rational__preference_Oas__good__asIff,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member449909584od_a_a @ X @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_222_rational__preference_Oas__good__asIff,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.69/1.51 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_223_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.69/1.51 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.69/1.51 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bo1116205875od_a_a )
% 2.69/1.51 => ( ( member2057358096od_a_a @ X @ Carrier )
% 2.69/1.51 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 2.69/1.51 => ( ~ ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 => ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_224_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.69/1.51 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
% 2.69/1.51 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bot_set_set_a )
% 2.69/1.51 => ( ( member_set_a @ X @ Carrier )
% 2.69/1.51 => ( ( member_set_a @ Y @ Carrier )
% 2.69/1.51 => ( ~ ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 => ~ ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_225_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.69/1.51 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bo2131659635od_a_a )
% 2.69/1.51 => ( ( member449909584od_a_a @ X @ Carrier )
% 2.69/1.51 => ( ( member449909584od_a_a @ Y @ Carrier )
% 2.69/1.51 => ( ~ ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_226_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( Carrier != bot_bot_set_a )
% 2.69/1.51 => ( ( member_a @ X @ Carrier )
% 2.69/1.51 => ( ( member_a @ Y @ Carrier )
% 2.69/1.51 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
% 2.69/1.51 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_227_Pair__mono,axiom,
% 2.69/1.51 ! [X: produc1691597095_set_a,X5: produc1691597095_set_a,Y: set_set_a,Y5: set_set_a] :
% 2.69/1.51 ( ( ord_le486764743_set_a @ X @ X5 )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ Y @ Y5 )
% 2.69/1.51 => ( ord_le2039441566_set_a @ ( produc759028718_set_a @ X @ Y ) @ ( produc759028718_set_a @ X5 @ Y5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_228_Pair__mono,axiom,
% 2.69/1.51 ! [X: set_set_a,X5: set_set_a,Y: set_a,Y5: set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ X @ X5 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 2.69/1.51 => ( ord_le1201700711_set_a @ ( produc32105911_set_a @ X @ Y ) @ ( produc32105911_set_a @ X5 @ Y5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_229_Pair__mono,axiom,
% 2.69/1.51 ! [X: set_set_a,X5: set_set_a,Y: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ X @ X5 )
% 2.69/1.51 => ( ( ord_le1824328871od_a_a @ Y @ Y5 )
% 2.69/1.51 => ( ord_le526661008od_a_a @ ( produc1618062688od_a_a @ X @ Y ) @ ( produc1618062688od_a_a @ X5 @ Y5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_230_Pair__mono,axiom,
% 2.69/1.51 ! [X: set_set_a,X5: set_set_a,Y: produc1691597095_set_a,Y5: produc1691597095_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ X @ X5 )
% 2.69/1.51 => ( ( ord_le486764743_set_a @ Y @ Y5 )
% 2.69/1.51 => ( ord_le1138266544_set_a @ ( produc760262016_set_a @ X @ Y ) @ ( produc760262016_set_a @ X5 @ Y5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_231_Pair__mono,axiom,
% 2.69/1.51 ! [X: set_set_a,X5: set_set_a,Y: set_set_a,Y5: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ X @ X5 )
% 2.69/1.51 => ( ( ord_le318720350_set_a @ Y @ Y5 )
% 2.69/1.51 => ( ord_le723713223_set_a @ ( produc2102896407_set_a @ X @ Y ) @ ( produc2102896407_set_a @ X5 @ Y5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_232_Pair__mono,axiom,
% 2.69/1.51 ! [X: set_a,X5: set_a,Y: set_a,Y5: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X @ X5 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 2.69/1.51 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_233_finite__has__maximal2,axiom,
% 2.69/1.51 ! [A: set_set_a,A2: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( member_set_a @ A2 @ A )
% 2.69/1.51 => ? [X4: set_a] :
% 2.69/1.51 ( ( member_set_a @ X4 @ A )
% 2.69/1.51 & ( ord_less_eq_set_a @ A2 @ X4 )
% 2.69/1.51 & ! [Xa: set_a] :
% 2.69/1.51 ( ( member_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_234_finite__has__minimal2,axiom,
% 2.69/1.51 ! [A: set_set_a,A2: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( member_set_a @ A2 @ A )
% 2.69/1.51 => ? [X4: set_a] :
% 2.69/1.51 ( ( member_set_a @ X4 @ A )
% 2.69/1.51 & ( ord_less_eq_set_a @ X4 @ A2 )
% 2.69/1.51 & ! [Xa: set_a] :
% 2.69/1.51 ( ( member_set_a @ Xa @ A )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 2.69/1.51 => ( X4 = Xa ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_235_bot__prod__def,axiom,
% 2.69/1.51 ( bot_bo1519632275_set_a
% 2.69/1.51 = ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_236_infinite__imp__nonempty,axiom,
% 2.69/1.51 ! [S: set_a] :
% 2.69/1.51 ( ~ ( finite_finite_a @ S )
% 2.69/1.51 => ( S != bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_237_finite_OemptyI,axiom,
% 2.69/1.51 finite_finite_a @ bot_bot_set_a ).
% 2.69/1.51
% 2.69/1.51 thf(fact_238_finite__subset,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ( finite_finite_a @ B2 )
% 2.69/1.51 => ( finite_finite_a @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_239_infinite__super,axiom,
% 2.69/1.51 ! [S: set_a,T2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ S @ T2 )
% 2.69/1.51 => ( ~ ( finite_finite_a @ S )
% 2.69/1.51 => ~ ( finite_finite_a @ T2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_240_rev__finite__subset,axiom,
% 2.69/1.51 ! [B2: set_a,A: set_a] :
% 2.69/1.51 ( ( finite_finite_a @ B2 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( finite_finite_a @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_241_old_Oprod_Oinject,axiom,
% 2.69/1.51 ! [A2: a,B: a,A4: a,B4: a] :
% 2.69/1.51 ( ( ( product_Pair_a_a @ A2 @ B )
% 2.69/1.51 = ( product_Pair_a_a @ A4 @ B4 ) )
% 2.69/1.51 = ( ( A2 = A4 )
% 2.69/1.51 & ( B = B4 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_242_prod_Oinject,axiom,
% 2.69/1.51 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 2.69/1.51 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 2.69/1.51 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 2.69/1.51 = ( ( X1 = Y1 )
% 2.69/1.51 & ( X22 = Y22 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_243_order__refl,axiom,
% 2.69/1.51 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_244_preorder__on__empty,axiom,
% 2.69/1.51 order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 2.69/1.51
% 2.69/1.51 thf(fact_245_total__on__def,axiom,
% 2.69/1.51 ( total_on_a
% 2.69/1.51 = ( ^ [A3: set_a,R: set_Product_prod_a_a] :
% 2.69/1.51 ! [X3: a] :
% 2.69/1.51 ( ( member_a @ X3 @ A3 )
% 2.69/1.51 => ! [Y2: a] :
% 2.69/1.51 ( ( member_a @ Y2 @ A3 )
% 2.69/1.51 => ( ( X3 != Y2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
% 2.69/1.51 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R ) ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_246_total__onI,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 2.69/1.51 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X4 @ A )
% 2.69/1.51 => ( ( member449909584od_a_a @ Y3 @ A )
% 2.69/1.51 => ( ( X4 != Y3 )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R2 )
% 2.69/1.51 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 2.69/1.51 => ( total_1490170027od_a_a @ A @ R2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_247_total__onI,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a] :
% 2.69/1.51 ( ! [X4: a,Y3: a] :
% 2.69/1.51 ( ( member_a @ X4 @ A )
% 2.69/1.51 => ( ( member_a @ Y3 @ A )
% 2.69/1.51 => ( ( X4 != Y3 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
% 2.69/1.51 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 2.69/1.51 => ( total_on_a @ A @ R2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_248_total__on__empty,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R2 ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_249_bot__empty__eq,axiom,
% 2.69/1.51 ( bot_bo1293121322_a_a_o
% 2.69/1.51 = ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_250_bot__empty__eq,axiom,
% 2.69/1.51 ( bot_bot_a_o
% 2.69/1.51 = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_251_bot__set__def,axiom,
% 2.69/1.51 ( bot_bot_set_a
% 2.69/1.51 = ( collect_a @ bot_bot_a_o ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_252_dual__order_Oantisym,axiom,
% 2.69/1.51 ! [B: set_a,A2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ B @ A2 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ A2 @ B )
% 2.69/1.51 => ( A2 = B ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_253_dual__order_Oeq__iff,axiom,
% 2.69/1.51 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.69/1.51 = ( ^ [A5: set_a,B5: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ B5 @ A5 )
% 2.69/1.51 & ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_254_dual__order_Otrans,axiom,
% 2.69/1.51 ! [B: set_a,A2: set_a,C: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ B @ A2 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ C @ B )
% 2.69/1.51 => ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_255_dual__order_Orefl,axiom,
% 2.69/1.51 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_256_order__trans,axiom,
% 2.69/1.51 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X @ Y )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ Y @ Z )
% 2.69/1.51 => ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_257_order__class_Oorder_Oantisym,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B @ A2 )
% 2.69/1.51 => ( A2 = B ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_258_ord__le__eq__trans,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a,C: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ B )
% 2.69/1.51 => ( ( B = C )
% 2.69/1.51 => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_259_ord__eq__le__trans,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a,C: set_a] :
% 2.69/1.51 ( ( A2 = B )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B @ C )
% 2.69/1.51 => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_260_order__class_Oorder_Oeq__iff,axiom,
% 2.69/1.51 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.69/1.51 = ( ^ [A5: set_a,B5: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 2.69/1.51 & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_261_antisym__conv,axiom,
% 2.69/1.51 ! [Y: set_a,X: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ Y @ X )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ X @ Y )
% 2.69/1.51 = ( X = Y ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_262_order_Otrans,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a,C: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B @ C )
% 2.69/1.51 => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_263_eq__refl,axiom,
% 2.69/1.51 ! [X: set_a,Y: set_a] :
% 2.69/1.51 ( ( X = Y )
% 2.69/1.51 => ( ord_less_eq_set_a @ X @ Y ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_264_antisym,axiom,
% 2.69/1.51 ! [X: set_a,Y: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X @ Y )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ Y @ X )
% 2.69/1.51 => ( X = Y ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_265_eq__iff,axiom,
% 2.69/1.51 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.69/1.51 = ( ^ [X3: set_a,Y2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 2.69/1.51 & ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_266_ord__le__eq__subst,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ B )
% 2.69/1.51 => ( ( ( F @ B )
% 2.69/1.51 = C )
% 2.69/1.51 => ( ! [X4: set_a,Y3: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_267_ord__eq__le__subst,axiom,
% 2.69/1.51 ! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
% 2.69/1.51 ( ( A2
% 2.69/1.51 = ( F @ B ) )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B @ C )
% 2.69/1.51 => ( ! [X4: set_a,Y3: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_268_order__subst2,axiom,
% 2.69/1.51 ! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ B )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
% 2.69/1.51 => ( ! [X4: set_a,Y3: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_269_order__subst1,axiom,
% 2.69/1.51 ! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B @ C )
% 2.69/1.51 => ( ! [X4: set_a,Y3: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_270_surj__pair,axiom,
% 2.69/1.51 ! [P2: product_prod_a_a] :
% 2.69/1.51 ? [X4: a,Y3: a] :
% 2.69/1.51 ( P2
% 2.69/1.51 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_271_prod__cases,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 2.69/1.51 ( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
% 2.69/1.51 => ( P @ P2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_272_Pair__inject,axiom,
% 2.69/1.51 ! [A2: a,B: a,A4: a,B4: a] :
% 2.69/1.51 ( ( ( product_Pair_a_a @ A2 @ B )
% 2.69/1.51 = ( product_Pair_a_a @ A4 @ B4 ) )
% 2.69/1.51 => ~ ( ( A2 = A4 )
% 2.69/1.51 => ( B != B4 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_273_old_Oprod_Oexhaust,axiom,
% 2.69/1.51 ! [Y: product_prod_a_a] :
% 2.69/1.51 ~ ! [A6: a,B6: a] :
% 2.69/1.51 ( Y
% 2.69/1.51 != ( product_Pair_a_a @ A6 @ B6 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_274_old_Oprod_Oinducts,axiom,
% 2.69/1.51 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 2.69/1.51 ( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
% 2.69/1.51 => ( P @ Prod ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_275_subrelI,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 2.69/1.51 ( ! [X4: a,Y3: a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S2 ) )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ R2 @ S2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_276_bot_Oextremum__uniqueI,axiom,
% 2.69/1.51 ! [A2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 2.69/1.51 => ( A2 = bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_277_bot_Oextremum__unique,axiom,
% 2.69/1.51 ! [A2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 2.69/1.51 = ( A2 = bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_278_bot_Oextremum,axiom,
% 2.69/1.51 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_279_refl__on__empty,axiom,
% 2.69/1.51 refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 2.69/1.51
% 2.69/1.51 thf(fact_280_refl__onD,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A2: product_prod_a_a] :
% 2.69/1.51 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ A2 @ A )
% 2.69/1.51 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ A2 ) @ R2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_281_refl__onD,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a,A2: a] :
% 2.69/1.51 ( ( refl_on_a @ A @ R2 )
% 2.69/1.51 => ( ( member_a @ A2 @ A )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_282_refl__onD1,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 2.69/1.51 => ( member449909584od_a_a @ X @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_283_refl__onD1,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( refl_on_a @ A @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.69/1.51 => ( member_a @ X @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_284_refl__onD2,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.69/1.51 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 2.69/1.51 => ( member449909584od_a_a @ Y @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_285_refl__onD2,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 2.69/1.51 ( ( refl_on_a @ A @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.69/1.51 => ( member_a @ Y @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_286_refl__on__domain,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A2: product_prod_a_a,B: product_prod_a_a] :
% 2.69/1.51 ( ( refl_o1298442278od_a_a @ A @ R2 )
% 2.69/1.51 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ B ) @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ A2 @ A )
% 2.69/1.51 & ( member449909584od_a_a @ B @ A ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_287_refl__on__domain,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a,A2: a,B: a] :
% 2.69/1.51 ( ( refl_on_a @ A @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
% 2.69/1.51 => ( ( member_a @ A2 @ A )
% 2.69/1.51 & ( member_a @ B @ A ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_288_rational__preference__def,axiom,
% 2.69/1.51 ( prefer719835986ence_a
% 2.69/1.51 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 2.69/1.51 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_289_rational__preference_Ointro,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.51 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 2.69/1.51 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_290_finite__transitivity__chain,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,R3: product_prod_a_a > product_prod_a_a > $o] :
% 2.69/1.51 ( ( finite179568208od_a_a @ A )
% 2.69/1.51 => ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ~ ( R3 @ X4 @ X4 )
% 2.69/1.51 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
% 2.69/1.51 ( ( R3 @ X4 @ Y3 )
% 2.69/1.51 => ( ( R3 @ Y3 @ Z3 )
% 2.69/1.51 => ( R3 @ X4 @ Z3 ) ) )
% 2.69/1.51 => ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X4 @ A )
% 2.69/1.51 => ? [Y6: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ Y6 @ A )
% 2.69/1.51 & ( R3 @ X4 @ Y6 ) ) )
% 2.69/1.51 => ( A = bot_bo2131659635od_a_a ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_291_finite__transitivity__chain,axiom,
% 2.69/1.51 ! [A: set_a,R3: a > a > $o] :
% 2.69/1.51 ( ( finite_finite_a @ A )
% 2.69/1.51 => ( ! [X4: a] :
% 2.69/1.51 ~ ( R3 @ X4 @ X4 )
% 2.69/1.51 => ( ! [X4: a,Y3: a,Z3: a] :
% 2.69/1.51 ( ( R3 @ X4 @ Y3 )
% 2.69/1.51 => ( ( R3 @ Y3 @ Z3 )
% 2.69/1.51 => ( R3 @ X4 @ Z3 ) ) )
% 2.69/1.51 => ( ! [X4: a] :
% 2.69/1.51 ( ( member_a @ X4 @ A )
% 2.69/1.51 => ? [Y6: a] :
% 2.69/1.51 ( ( member_a @ Y6 @ A )
% 2.69/1.51 & ( R3 @ X4 @ Y6 ) ) )
% 2.69/1.51 => ( A = bot_bot_set_a ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_292_subset__emptyI,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a] :
% 2.69/1.51 ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ~ ( member449909584od_a_a @ X4 @ A )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_293_subset__emptyI,axiom,
% 2.69/1.51 ! [A: set_a] :
% 2.69/1.51 ( ! [X4: a] :
% 2.69/1.51 ~ ( member_a @ X4 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_294_ssubst__Pair__rhs,axiom,
% 2.69/1.51 ! [R2: a,S2: a,R3: set_Product_prod_a_a,S3: a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ R3 )
% 2.69/1.51 => ( ( S3 = S2 )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S3 ) @ R3 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_295_rational__preference__axioms_Ointro,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.51 ( ( total_on_a @ Carrier @ Relation )
% 2.69/1.51 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_296_rational__preference__axioms__def,axiom,
% 2.69/1.51 prefer580340847ioms_a = total_on_a ).
% 2.69/1.51
% 2.69/1.51 thf(fact_297_rational__preference_Oaxioms_I2_J,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.51 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_298_Collect__empty__eq__bot,axiom,
% 2.69/1.51 ! [P: a > $o] :
% 2.69/1.51 ( ( ( collect_a @ P )
% 2.69/1.51 = bot_bot_set_a )
% 2.69/1.51 = ( P = bot_bot_a_o ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_299_transitivity,axiom,
% 2.69/1.51 trans_a @ relation ).
% 2.69/1.51
% 2.69/1.51 thf(fact_300_Set_Ois__empty__def,axiom,
% 2.69/1.51 ( is_empty_a
% 2.69/1.51 = ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_301_preference_Otransitivity,axiom,
% 2.69/1.51 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.69/1.51 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.69/1.51 => ( trans_a @ Relation ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_302_trans__empty,axiom,
% 2.69/1.51 trans_a @ bot_bo2131659635od_a_a ).
% 2.69/1.51
% 2.69/1.51 thf(fact_303_transD,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.69/1.51 ( ( trans_a @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_304_transE,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.69/1.51 ( ( trans_a @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_305_transI,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a] :
% 2.69/1.51 ( ! [X4: a,Y3: a,Z3: a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R2 )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R2 ) ) )
% 2.69/1.51 => ( trans_a @ R2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_306_trans__def,axiom,
% 2.69/1.51 ( trans_a
% 2.69/1.51 = ( ^ [R: set_Product_prod_a_a] :
% 2.69/1.51 ! [X3: a,Y2: a,Z4: a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_307_preorder__on__def,axiom,
% 2.69/1.51 ( order_preorder_on_a
% 2.69/1.51 = ( ^ [A3: set_a,R: set_Product_prod_a_a] :
% 2.69/1.51 ( ( refl_on_a @ A3 @ R )
% 2.69/1.51 & ( trans_a @ R ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_308_relChain__def,axiom,
% 2.69/1.51 ( bNF_Ca850758276_set_a
% 2.69/1.51 = ( ^ [R: set_Product_prod_a_a,As: a > set_a] :
% 2.69/1.51 ! [I: a,J: a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ I @ J ) @ R )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_309_Id__on__empty,axiom,
% 2.69/1.51 ( ( id_on_a @ bot_bot_set_a )
% 2.69/1.51 = bot_bo2131659635od_a_a ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_310_under__incr,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a,A2: a,B: a] :
% 2.69/1.51 ( ( trans_a @ R2 )
% 2.69/1.51 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( order_under_a @ R2 @ A2 ) @ ( order_under_a @ R2 @ B ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_311_Id__onI,axiom,
% 2.69/1.51 ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ A2 @ A )
% 2.69/1.51 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ A2 ) @ ( id_on_1182990020od_a_a @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_312_Id__onI,axiom,
% 2.69/1.51 ! [A2: a,A: set_a] :
% 2.69/1.51 ( ( member_a @ A2 @ A )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( id_on_a @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_313_trans__Id__on,axiom,
% 2.69/1.51 ! [A: set_a] : ( trans_a @ ( id_on_a @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_314_Id__on__iff,axiom,
% 2.69/1.51 ! [X: product_prod_a_a,Y: product_prod_a_a,A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( id_on_1182990020od_a_a @ A ) )
% 2.69/1.51 = ( ( X = Y )
% 2.69/1.51 & ( member449909584od_a_a @ X @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_315_Id__on__iff,axiom,
% 2.69/1.51 ! [X: a,Y: a,A: set_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( id_on_a @ A ) )
% 2.69/1.51 = ( ( X = Y )
% 2.69/1.51 & ( member_a @ X @ A ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_316_Id__on__eqI,axiom,
% 2.69/1.51 ! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( A2 = B )
% 2.69/1.51 => ( ( member449909584od_a_a @ A2 @ A )
% 2.69/1.51 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ B ) @ ( id_on_1182990020od_a_a @ A ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_317_Id__on__eqI,axiom,
% 2.69/1.51 ! [A2: a,B: a,A: set_a] :
% 2.69/1.51 ( ( A2 = B )
% 2.69/1.51 => ( ( member_a @ A2 @ A )
% 2.69/1.51 => ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( id_on_a @ A ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_318_Id__onE,axiom,
% 2.69/1.51 ! [C: produc1572603623od_a_a,A: set_Product_prod_a_a] :
% 2.69/1.51 ( ( member2057358096od_a_a @ C @ ( id_on_1182990020od_a_a @ A ) )
% 2.69/1.51 => ~ ! [X4: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X4 @ A )
% 2.69/1.51 => ( C
% 2.69/1.51 != ( produc1474507607od_a_a @ X4 @ X4 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_319_Id__onE,axiom,
% 2.69/1.51 ! [C: product_prod_a_a,A: set_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ C @ ( id_on_a @ A ) )
% 2.69/1.51 => ~ ! [X4: a] :
% 2.69/1.51 ( ( member_a @ X4 @ A )
% 2.69/1.51 => ( C
% 2.69/1.51 != ( product_Pair_a_a @ X4 @ X4 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_320_refl__on__Id__on,axiom,
% 2.69/1.51 ! [A: set_a] : ( refl_on_a @ A @ ( id_on_a @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_321_Greatest__equality,axiom,
% 2.69/1.51 ! [P: set_a > $o,X: set_a] :
% 2.69/1.51 ( ( P @ X )
% 2.69/1.51 => ( ! [Y3: set_a] :
% 2.69/1.51 ( ( P @ Y3 )
% 2.69/1.51 => ( ord_less_eq_set_a @ Y3 @ X ) )
% 2.69/1.51 => ( ( order_Greatest_set_a @ P )
% 2.69/1.51 = X ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_322_GreatestI2__order,axiom,
% 2.69/1.51 ! [P: set_a > $o,X: set_a,Q: set_a > $o] :
% 2.69/1.51 ( ( P @ X )
% 2.69/1.51 => ( ! [Y3: set_a] :
% 2.69/1.51 ( ( P @ Y3 )
% 2.69/1.51 => ( ord_less_eq_set_a @ Y3 @ X ) )
% 2.69/1.51 => ( ! [X4: set_a] :
% 2.69/1.51 ( ( P @ X4 )
% 2.69/1.51 => ( ! [Y6: set_a] :
% 2.69/1.51 ( ( P @ Y6 )
% 2.69/1.51 => ( ord_less_eq_set_a @ Y6 @ X4 ) )
% 2.69/1.51 => ( Q @ X4 ) ) )
% 2.69/1.51 => ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_323_Inf__fin_Osubset__imp,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ( finite_finite_set_a @ B2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ B2 ) @ ( lattic1600076923_set_a @ A ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_324_Sup__fin_Osubset__imp,axiom,
% 2.69/1.51 ! [A: set_set_a,B2: set_set_a] :
% 2.69/1.51 ( ( ord_le318720350_set_a @ A @ B2 )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ( finite_finite_set_a @ B2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ ( lattic501550101_set_a @ B2 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_325_Inf__fin__le__Sup__fin,axiom,
% 2.69/1.51 ! [A: set_set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ A ) @ ( lattic501550101_set_a @ A ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_326_Sup__fin_OcoboundedI,axiom,
% 2.69/1.51 ! [A: set_set_a,A2: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( member_set_a @ A2 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ A2 @ ( lattic501550101_set_a @ A ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_327_Inf__fin_OcoboundedI,axiom,
% 2.69/1.51 ! [A: set_set_a,A2: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( member_set_a @ A2 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ A ) @ A2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_328_Inf__fin_OboundedE,axiom,
% 2.69/1.51 ! [A: set_set_a,X: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) )
% 2.69/1.51 => ! [A7: set_a] :
% 2.69/1.51 ( ( member_set_a @ A7 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ X @ A7 ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_329_Inf__fin_OboundedI,axiom,
% 2.69/1.51 ! [A: set_set_a,X: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ! [A6: set_a] :
% 2.69/1.51 ( ( member_set_a @ A6 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ X @ A6 ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_330_Sup__fin_OboundedE,axiom,
% 2.69/1.51 ! [A: set_set_a,X: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X )
% 2.69/1.51 => ! [A7: set_a] :
% 2.69/1.51 ( ( member_set_a @ A7 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ A7 @ X ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_331_Sup__fin_OboundedI,axiom,
% 2.69/1.51 ! [A: set_set_a,X: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ! [A6: set_a] :
% 2.69/1.51 ( ( member_set_a @ A6 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ A6 @ X ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_332_Inf__fin_Obounded__iff,axiom,
% 2.69/1.51 ! [A: set_set_a,X: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) )
% 2.69/1.51 = ( ! [X3: set_a] :
% 2.69/1.51 ( ( member_set_a @ X3 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ X @ X3 ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_333_Sup__fin_Obounded__iff,axiom,
% 2.69/1.51 ! [A: set_set_a,X: set_a] :
% 2.69/1.51 ( ( finite_finite_set_a @ A )
% 2.69/1.51 => ( ( A != bot_bot_set_set_a )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X )
% 2.69/1.51 = ( ! [X3: set_a] :
% 2.69/1.51 ( ( member_set_a @ X3 @ A )
% 2.69/1.51 => ( ord_less_eq_set_a @ X3 @ X ) ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_334_Un__empty,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( ( sup_sup_set_a @ A @ B2 )
% 2.69/1.51 = bot_bot_set_a )
% 2.69/1.51 = ( ( A = bot_bot_set_a )
% 2.69/1.51 & ( B2 = bot_bot_set_a ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_335_Int__subset__iff,axiom,
% 2.69/1.51 ! [C2: set_a,A: set_a,B2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
% 2.69/1.51 = ( ( ord_less_eq_set_a @ C2 @ A )
% 2.69/1.51 & ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_336_Un__subset__iff,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a,C2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
% 2.69/1.51 = ( ( ord_less_eq_set_a @ A @ C2 )
% 2.69/1.51 & ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_337_finite__Int,axiom,
% 2.69/1.51 ! [F2: set_a,G: set_a] :
% 2.69/1.51 ( ( ( finite_finite_a @ F2 )
% 2.69/1.51 | ( finite_finite_a @ G ) )
% 2.69/1.51 => ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_338_finite__Un,axiom,
% 2.69/1.51 ! [F2: set_a,G: set_a] :
% 2.69/1.51 ( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
% 2.69/1.51 = ( ( finite_finite_a @ F2 )
% 2.69/1.51 & ( finite_finite_a @ G ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_339_trans__Int,axiom,
% 2.69/1.51 ! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( trans_a @ R2 )
% 2.69/1.51 => ( ( trans_a @ S2 )
% 2.69/1.51 => ( trans_a @ ( inf_in920712437od_a_a @ R2 @ S2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_340_refl__on__Un,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a,B2: set_a,S2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( refl_on_a @ A @ R2 )
% 2.69/1.51 => ( ( refl_on_a @ B2 @ S2 )
% 2.69/1.51 => ( refl_on_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_su615935707od_a_a @ R2 @ S2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_341_refl__on__Int,axiom,
% 2.69/1.51 ! [A: set_a,R2: set_Product_prod_a_a,B2: set_a,S2: set_Product_prod_a_a] :
% 2.69/1.51 ( ( refl_on_a @ A @ R2 )
% 2.69/1.51 => ( ( refl_on_a @ B2 @ S2 )
% 2.69/1.51 => ( refl_on_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_in920712437od_a_a @ R2 @ S2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_342_finite__UnI,axiom,
% 2.69/1.51 ! [F2: set_a,G: set_a] :
% 2.69/1.51 ( ( finite_finite_a @ F2 )
% 2.69/1.51 => ( ( finite_finite_a @ G )
% 2.69/1.51 => ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_343_Un__infinite,axiom,
% 2.69/1.51 ! [S: set_a,T2: set_a] :
% 2.69/1.51 ( ~ ( finite_finite_a @ S )
% 2.69/1.51 => ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_344_infinite__Un,axiom,
% 2.69/1.51 ! [S: set_a,T2: set_a] :
% 2.69/1.51 ( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
% 2.69/1.51 = ( ~ ( finite_finite_a @ S )
% 2.69/1.51 | ~ ( finite_finite_a @ T2 ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_345_Un__Int__assoc__eq,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a,C2: set_a] :
% 2.69/1.51 ( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
% 2.69/1.51 = ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) )
% 2.69/1.51 = ( ord_less_eq_set_a @ C2 @ A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_346_Int__mono,axiom,
% 2.69/1.51 ! [A: set_a,C2: set_a,B2: set_a,D2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ C2 )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ B2 @ D2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_347_Int__lower1,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_348_Int__lower2,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_349_Int__absorb1,axiom,
% 2.69/1.51 ! [B2: set_a,A: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ B2 @ A )
% 2.69/1.51 => ( ( inf_inf_set_a @ A @ B2 )
% 2.69/1.51 = B2 ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_350_Int__absorb2,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ( inf_inf_set_a @ A @ B2 )
% 2.69/1.51 = A ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_351_Int__greatest,axiom,
% 2.69/1.51 ! [C2: set_a,A: set_a,B2: set_a] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ C2 @ A )
% 2.69/1.51 => ( ( ord_less_eq_set_a @ C2 @ B2 )
% 2.69/1.51 => ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_352_Int__Collect__mono,axiom,
% 2.69/1.51 ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.69/1.51 ( ( ord_le1824328871od_a_a @ A @ B2 )
% 2.69/1.51 => ( ! [X4: product_prod_a_a] :
% 2.69/1.51 ( ( member449909584od_a_a @ X4 @ A )
% 2.69/1.51 => ( ( P @ X4 )
% 2.69/1.51 => ( Q @ X4 ) ) )
% 2.69/1.51 => ( ord_le1824328871od_a_a @ ( inf_in920712437od_a_a @ A @ ( collec645855634od_a_a @ P ) ) @ ( inf_in920712437od_a_a @ B2 @ ( collec645855634od_a_a @ Q ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(fact_353_Int__Collect__mono,axiom,
% 2.69/1.51 ! [A: set_a,B2: set_a,P: a > $o,Q: a > $o] :
% 2.69/1.51 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.69/1.51 => ( ! [X4: a] :
% 2.69/1.51 ( ( member_a @ X4 @ A )
% 2.69/1.51 => ( ( P @ X4 )
% 2.69/1.51 => ( Q @ X4 ) ) )
% 2.69/1.51 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% 2.69/1.51
% 2.69/1.51 thf(conj_0,conjecture,
% 2.69/1.51 ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
% 2.69/1.51 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ) ).
% 2.69/1.51 ------------------------
% 2.69/1.55 % (27950)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/2Mi)
% 2.69/1.55 % (27953)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/18Mi)
% 2.69/1.55 % (27949)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_27726 for (2999ds/27Mi)
% 2.69/1.55 % (27950)Instruction limit reached!
% 2.69/1.55 % (27950)------------------------------
% 2.69/1.55 % (27950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.55 % (27950)Termination reason: Unknown
% 2.69/1.55 % (27950)Termination phase: shuffling
% 2.69/1.55
% 2.69/1.55 % (27950)Memory used [KB]: 1535
% 2.69/1.55 % (27950)Time elapsed: 0.003 s
% 2.69/1.55 % (27950)Instructions burned: 4 (million)
% 2.69/1.55 % (27950)------------------------------
% 2.69/1.55 % (27950)------------------------------
% 2.69/1.55 % (27953)Instruction limit reached!
% 2.69/1.55 % (27953)------------------------------
% 2.69/1.55 % (27953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.55 % (27953)Termination reason: Unknown
% 2.69/1.55 % (27953)Termination phase: shuffling
% 2.69/1.55
% 2.69/1.55 % (27953)Memory used [KB]: 1918
% 2.69/1.55 % (27953)Time elapsed: 0.008 s
% 2.69/1.55 % (27953)Instructions burned: 20 (million)
% 2.69/1.55 % (27953)------------------------------
% 2.69/1.55 % (27953)------------------------------
% 2.69/1.55 % (27947)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/183Mi)
% 2.69/1.55 % (27948)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_27726 for (2999ds/4Mi)
% 2.69/1.56 % (27951)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_27726 for (2999ds/2Mi)
% 2.69/1.56 % (27952)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_27726 for (2999ds/275Mi)
% 2.69/1.56 % (27949)Instruction limit reached!
% 2.69/1.56 % (27949)------------------------------
% 2.69/1.56 % (27949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (27949)Termination reason: Unknown
% 2.69/1.56 % (27949)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (27949)Memory used [KB]: 2046
% 2.69/1.56 % (27949)Time elapsed: 0.010 s
% 2.69/1.56 % (27949)Instructions burned: 28 (million)
% 2.69/1.56 % (27949)------------------------------
% 2.69/1.56 % (27949)------------------------------
% 2.69/1.56 % (27951)Instruction limit reached!
% 2.69/1.56 % (27951)------------------------------
% 2.69/1.56 % (27951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (27951)Termination reason: Unknown
% 2.69/1.56 % (27951)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (27951)Memory used [KB]: 1535
% 2.69/1.56 % (27951)Time elapsed: 0.004 s
% 2.69/1.56 % (27951)Instructions burned: 3 (million)
% 2.69/1.56 % (27951)------------------------------
% 2.69/1.56 % (27951)------------------------------
% 2.69/1.56 % (27948)Instruction limit reached!
% 2.69/1.56 % (27948)------------------------------
% 2.69/1.56 % (27948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (27948)Termination reason: Unknown
% 2.69/1.56 % (27948)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (27948)Memory used [KB]: 1535
% 2.69/1.56 % (27948)Time elapsed: 0.005 s
% 2.69/1.56 % (27948)Instructions burned: 5 (million)
% 2.69/1.56 % (27948)------------------------------
% 2.69/1.56 % (27948)------------------------------
% 2.69/1.56 % (27954)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/3Mi)
% 2.69/1.56 % (27954)Instruction limit reached!
% 2.69/1.56 % (27954)------------------------------
% 2.69/1.56 % (27954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (27954)Termination reason: Unknown
% 2.69/1.56 % (27954)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (27954)Memory used [KB]: 1535
% 2.69/1.56 % (27954)Time elapsed: 0.003 s
% 2.69/1.56 % (27954)Instructions burned: 6 (million)
% 2.69/1.56 % (27954)------------------------------
% 2.69/1.56 % (27954)------------------------------
% 2.69/1.56 % (27955)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_27726 for (2999ds/37Mi)
% 2.69/1.57 % (27956)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_27726 for (2999ds/15Mi)
% 2.69/1.57 % (27959)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/7Mi)
% 2.69/1.57 % (27956)Instruction limit reached!
% 2.69/1.57 % (27956)------------------------------
% 2.69/1.57 % (27956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (27956)Termination reason: Unknown
% 2.69/1.57 % (27956)Termination phase: shuffling
% 2.69/1.57
% 2.69/1.57 % (27956)Memory used [KB]: 1791
% 2.69/1.57 % (27956)Time elapsed: 0.007 s
% 2.69/1.57 % (27956)Instructions burned: 17 (million)
% 2.69/1.57 % (27956)------------------------------
% 2.69/1.57 % (27956)------------------------------
% 2.69/1.57 % (27959)Instruction limit reached!
% 2.69/1.57 % (27959)------------------------------
% 2.69/1.57 % (27959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (27959)Termination reason: Unknown
% 2.69/1.57 % (27959)Termination phase: shuffling
% 2.69/1.57
% 2.69/1.57 % (27959)Memory used [KB]: 1535
% 2.69/1.57 % (27959)Time elapsed: 0.004 s
% 2.69/1.57 % (27959)Instructions burned: 8 (million)
% 2.69/1.57 % (27959)------------------------------
% 2.69/1.57 % (27959)------------------------------
% 2.69/1.57 % (27955)Instruction limit reached!
% 2.69/1.57 % (27955)------------------------------
% 2.69/1.57 % (27955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (27955)Termination reason: Unknown
% 2.69/1.57 % (27955)Termination phase: shuffling
% 2.69/1.57
% 2.69/1.57 % (27955)Memory used [KB]: 2302
% 2.69/1.57 % (27955)Time elapsed: 0.012 s
% 2.69/1.57 % (27955)Instructions burned: 39 (million)
% 2.69/1.57 % (27955)------------------------------
% 2.69/1.57 % (27955)------------------------------
% 2.69/1.57 % (27957)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_27726 for (2999ds/3Mi)
% 2.69/1.58 % (27958)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_27726 for (2999ds/1041Mi)
% 2.69/1.58 % (27957)Instruction limit reached!
% 2.69/1.58 % (27957)------------------------------
% 2.69/1.58 % (27957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (27957)Termination reason: Unknown
% 2.69/1.58 % (27957)Termination phase: shuffling
% 2.69/1.58
% 2.69/1.58 % (27957)Memory used [KB]: 1535
% 2.69/1.58 % (27957)Time elapsed: 0.004 s
% 2.69/1.58 % (27957)Instructions burned: 3 (million)
% 2.69/1.58 % (27957)------------------------------
% 2.69/1.58 % (27957)------------------------------
% 2.69/1.58 % (27960)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/16Mi)
% 2.69/1.58 % (27961)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_27726 for (2999ds/3Mi)
% 2.69/1.58 % (27961)Instruction limit reached!
% 2.69/1.58 % (27961)------------------------------
% 2.69/1.58 % (27961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (27961)Termination reason: Unknown
% 2.69/1.58 % (27961)Termination phase: shuffling
% 2.69/1.58
% 2.69/1.58 % (27961)Memory used [KB]: 1535
% 2.69/1.58 % (27961)Time elapsed: 0.002 s
% 2.69/1.58 % (27961)Instructions burned: 3 (million)
% 2.69/1.58 % (27961)------------------------------
% 2.69/1.58 % (27961)------------------------------
% 2.69/1.58 % (27962)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_27726 for (2999ds/3Mi)
% 2.69/1.58 % (27962)Instruction limit reached!
% 2.69/1.58 % (27962)------------------------------
% 2.69/1.58 % (27962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (27962)Termination reason: Unknown
% 2.69/1.58 % (27962)Termination phase: shuffling
% 2.69/1.58
% 2.69/1.58 % (27962)Memory used [KB]: 1535
% 2.69/1.58 % (27962)Time elapsed: 0.002 s
% 2.69/1.58 % (27962)Instructions burned: 3 (million)
% 2.69/1.58 % (27962)------------------------------
% 2.69/1.58 % (27962)------------------------------
% 2.69/1.59 % (27960)Instruction limit reached!
% 2.69/1.59 % (27960)------------------------------
% 2.69/1.59 % (27960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (27960)Termination reason: Unknown
% 2.69/1.59 % (27960)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (27960)Memory used [KB]: 1791
% 2.69/1.59 % (27960)Time elapsed: 0.007 s
% 2.69/1.59 % (27960)Instructions burned: 18 (million)
% 2.69/1.59 % (27960)------------------------------
% 2.69/1.59 % (27960)------------------------------
% 2.69/1.59 % (27964)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/3Mi)
% 2.69/1.59 % (27964)Instruction limit reached!
% 2.69/1.59 % (27964)------------------------------
% 2.69/1.59 % (27964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (27964)Termination reason: Unknown
% 2.69/1.59 % (27964)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (27964)Memory used [KB]: 1535
% 2.69/1.59 % (27964)Time elapsed: 0.002 s
% 2.69/1.59 % (27964)Instructions burned: 3 (million)
% 2.69/1.59 % (27964)------------------------------
% 2.69/1.59 % (27964)------------------------------
% 2.69/1.59 % (27963)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/7Mi)
% 2.69/1.59 % (27965)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/4Mi)
% 2.69/1.60 % (27965)Instruction limit reached!
% 2.69/1.60 % (27965)------------------------------
% 2.69/1.60 % (27965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.60 % (27965)Termination reason: Unknown
% 2.69/1.60 % (27965)Termination phase: shuffling
% 2.69/1.60
% 2.69/1.60 % (27965)Memory used [KB]: 1535
% 2.69/1.60 % (27965)Time elapsed: 0.003 s
% 2.69/1.60 % (27965)Instructions burned: 6 (million)
% 2.69/1.60 % (27965)------------------------------
% 2.69/1.60 % (27965)------------------------------
% 2.74/1.60 % (27966)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27726 for (2999ds/18Mi)
% 2.74/1.60 % (27963)Instruction limit reached!
% 2.74/1.60 % (27963)------------------------------
% 2.74/1.60 % (27963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.60 % (27963)Termination reason: Unknown
% 2.74/1.60 % (27963)Termination phase: shuffling
% 2.74/1.60
% 2.74/1.60 % (27963)Memory used [KB]: 1535
% 2.74/1.60 % (27963)Time elapsed: 0.007 s
% 2.74/1.60 % (27963)Instructions burned: 7 (million)
% 2.74/1.60 % (27963)------------------------------
% 2.74/1.60 % (27963)------------------------------
% 2.74/1.60 % (27952)First to succeed.
% 2.74/1.60 % (27952)Refutation found. Thanks to Tanya!
% 2.74/1.60 % SZS status Theorem for DTF2THF_27726
% 2.74/1.60 % SZS output start Proof for DTF2THF_27726
% 2.74/1.60 thf(type_def_5, type, set_Product_prod_a_a: $tType).
% 2.74/1.60 thf(type_def_6, type, a: $tType).
% 2.74/1.60 thf(type_def_7, type, set_a: $tType).
% 2.74/1.60 thf(type_def_9, type, set_Pr2045688071_set_a: $tType).
% 2.74/1.60 thf(type_def_10, type, set_se1596668135od_a_a: $tType).
% 2.74/1.60 thf(type_def_11, type, set_set_set_a: $tType).
% 2.74/1.60 thf(type_def_12, type, set_set_a: $tType).
% 2.74/1.60 thf(type_def_13, type, set_Pr1948701895od_a_a: $tType).
% 2.74/1.60 thf(type_def_14, type, set_Pr1295299783od_a_a: $tType).
% 2.74/1.60 thf(type_def_15, type, product_prod_a_a: $tType).
% 2.74/1.60 thf(type_def_16, type, produc1691597095_set_a: $tType).
% 2.74/1.60 thf(type_def_17, type, produc1299534078_set_a: $tType).
% 2.74/1.60 thf(type_def_18, type, produc1472671646_set_a: $tType).
% 2.74/1.60 thf(type_def_19, type, produc386552094_set_a: $tType).
% 2.74/1.60 thf(type_def_20, type, produc275558334_set_a: $tType).
% 2.74/1.60 thf(type_def_21, type, produc398359056_set_a: $tType).
% 2.74/1.60 thf(type_def_22, type, produc270245360od_a_a: $tType).
% 2.74/1.60 thf(type_def_23, type, produc407913767_set_a: $tType).
% 2.74/1.60 thf(type_def_24, type, produc522143687_set_a: $tType).
% 2.74/1.60 thf(type_def_25, type, produc304417648_set_a: $tType).
% 2.74/1.60 thf(type_def_26, type, produc376942416od_a_a: $tType).
% 2.74/1.60 thf(type_def_27, type, produc1083127431_set_a: $tType).
% 2.74/1.60 thf(type_def_28, type, produc1572603623od_a_a: $tType).
% 2.74/1.60 thf(type_def_29, type, produc1990712039od_a_a: $tType).
% 2.74/1.60 thf(func_def_0, type, set_Pr1295299783od_a_a: $tType).
% 2.74/1.60 thf(func_def_1, type, produc1990712039od_a_a: $tType).
% 2.74/1.60 thf(func_def_2, type, produc398359056_set_a: $tType).
% 2.74/1.60 thf(func_def_3, type, produc1299534078_set_a: $tType).
% 2.74/1.60 thf(func_def_4, type, set_Pr1948701895od_a_a: $tType).
% 2.74/1.60 thf(func_def_5, type, produc304417648_set_a: $tType).
% 2.74/1.60 thf(func_def_6, type, produc270245360od_a_a: $tType).
% 2.74/1.60 thf(func_def_7, type, produc386552094_set_a: $tType).
% 2.74/1.60 thf(func_def_8, type, produc1472671646_set_a: $tType).
% 2.74/1.60 thf(func_def_9, type, produc1572603623od_a_a: $tType).
% 2.74/1.60 thf(func_def_10, type, produc376942416od_a_a: $tType).
% 2.74/1.60 thf(func_def_11, type, produc275558334_set_a: $tType).
% 2.74/1.60 thf(func_def_12, type, produc407913767_set_a: $tType).
% 2.74/1.60 thf(func_def_13, type, set_Pr2045688071_set_a: $tType).
% 2.74/1.60 thf(func_def_14, type, produc1083127431_set_a: $tType).
% 2.74/1.60 thf(func_def_15, type, produc522143687_set_a: $tType).
% 2.74/1.60 thf(func_def_16, type, set_se1596668135od_a_a: $tType).
% 2.74/1.60 thf(func_def_17, type, produc1691597095_set_a: $tType).
% 2.74/1.60 thf(func_def_18, type, set_Product_prod_a_a: $tType).
% 2.74/1.60 thf(func_def_19, type, set_set_set_a: $tType).
% 2.74/1.60 thf(func_def_20, type, product_prod_a_a: $tType).
% 2.74/1.60 thf(func_def_21, type, set_set_a: $tType).
% 2.74/1.60 thf(func_def_22, type, set_a: $tType).
% 2.74/1.60 thf(func_def_23, type, a: $tType).
% 2.74/1.60 thf(func_def_24, type, bNF_Ca850758276_set_a: set_Product_prod_a_a > (a > set_a) > $o).
% 2.74/1.60 thf(func_def_25, type, finite283787856_set_a: set_Pr2045688071_set_a > $o).
% 2.74/1.60 thf(func_def_26, type, finite179568208od_a_a: set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_27, type, finite1145471536od_a_a: set_se1596668135od_a_a > $o).
% 2.74/1.60 thf(func_def_28, type, finite1606323175_set_a: set_set_set_a > $o).
% 2.74/1.60 thf(func_def_29, type, finite_finite_set_a: set_set_a > $o).
% 2.74/1.60 thf(func_def_30, type, finite_finite_a: set_a > $o).
% 2.74/1.60 thf(func_def_31, type, inf_in920712437od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_32, type, inf_inf_set_a: set_a > set_a > set_a).
% 2.74/1.60 thf(func_def_33, type, sup_su615935707od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_34, type, sup_sup_set_a: set_a > set_a > set_a).
% 2.74/1.60 thf(func_def_35, type, lattic1600076923_set_a: set_set_a > set_a).
% 2.74/1.60 thf(func_def_36, type, lattic501550101_set_a: set_set_a > set_a).
% 2.74/1.60 thf(func_def_37, type, order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.74/1.60 thf(func_def_38, type, order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_39, type, order_816746613_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.74/1.60 thf(func_def_40, type, order_preorder_on_a: set_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_41, type, order_under_a: set_Product_prod_a_a > a > set_a).
% 2.74/1.60 thf(func_def_42, type, bot_bo1293121322_a_a_o: product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_43, type, bot_bot_a_o: a > $o).
% 2.74/1.60 thf(func_def_44, type, bot_bo1519632275_set_a: produc1691597095_set_a).
% 2.74/1.60 thf(func_def_45, type, bot_bo1116205875od_a_a: set_Pr1948701895od_a_a).
% 2.74/1.60 thf(func_def_46, type, bot_bo671027059_set_a: set_Pr2045688071_set_a).
% 2.74/1.60 thf(func_def_47, type, bot_bo2131659635od_a_a: set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_48, type, bot_bo1424703315od_a_a: set_se1596668135od_a_a).
% 2.74/1.60 thf(func_def_49, type, bot_bo1264989578_set_a: set_set_set_a).
% 2.74/1.60 thf(func_def_50, type, bot_bot_set_set_a: set_set_a).
% 2.74/1.60 thf(func_def_51, type, bot_bot_set_a: set_a).
% 2.74/1.60 thf(func_def_52, type, ord_le2039441566_set_a: produc1299534078_set_a > produc1299534078_set_a > $o).
% 2.74/1.60 thf(func_def_53, type, ord_le1729087294_set_a: produc1472671646_set_a > produc1472671646_set_a > $o).
% 2.74/1.60 thf(func_def_54, type, ord_le642967742_set_a: produc386552094_set_a > produc386552094_set_a > $o).
% 2.74/1.60 thf(func_def_55, type, ord_le2095302494_set_a: produc275558334_set_a > produc275558334_set_a > $o).
% 2.74/1.60 thf(func_def_56, type, ord_le1138266544_set_a: produc398359056_set_a > produc398359056_set_a > $o).
% 2.74/1.60 thf(func_def_57, type, ord_le526661008od_a_a: produc270245360od_a_a > produc270245360od_a_a > $o).
% 2.74/1.60 thf(func_def_58, type, ord_le723713223_set_a: produc407913767_set_a > produc407913767_set_a > $o).
% 2.74/1.60 thf(func_def_59, type, ord_le1201700711_set_a: produc522143687_set_a > produc522143687_set_a > $o).
% 2.74/1.60 thf(func_def_60, type, ord_le560833296_set_a: produc304417648_set_a > produc304417648_set_a > $o).
% 2.74/1.60 thf(func_def_61, type, ord_le49202928od_a_a: produc376942416od_a_a > produc376942416od_a_a > $o).
% 2.74/1.60 thf(func_def_62, type, ord_le1762684455_set_a: produc1083127431_set_a > produc1083127431_set_a > $o).
% 2.74/1.60 thf(func_def_63, type, ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o).
% 2.74/1.60 thf(func_def_64, type, ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_65, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_66, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 2.74/1.60 thf(func_def_67, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.74/1.60 thf(func_def_68, type, order_Greatest_set_a: (set_a > $o) > set_a).
% 2.74/1.60 thf(func_def_69, type, prefer1867335523od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 2.74/1.60 thf(func_def_70, type, prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_71, type, prefer270414042_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a).
% 2.74/1.60 thf(func_def_72, type, prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a).
% 2.74/1.60 thf(func_def_73, type, prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 2.74/1.60 thf(func_def_74, type, prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_75, type, prefer1203288218_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a).
% 2.74/1.60 thf(func_def_76, type, prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a).
% 2.74/1.60 thf(func_def_77, type, prefer926864306od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 2.74/1.60 thf(func_def_78, type, prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_79, type, prefer1671525929_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a).
% 2.74/1.60 thf(func_def_80, type, prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a).
% 2.74/1.60 thf(func_def_81, type, prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.74/1.60 thf(func_def_82, type, prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_83, type, prefer1710114912_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.74/1.60 thf(func_def_84, type, prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_85, type, prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.74/1.60 thf(func_def_86, type, prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_87, type, prefer484977842_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.74/1.60 thf(func_def_88, type, prefer719835986ence_a: set_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_89, type, prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_90, type, produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a).
% 2.74/1.60 thf(func_def_91, type, produc759028718_set_a: produc1691597095_set_a > set_set_a > produc1299534078_set_a).
% 2.74/1.60 thf(func_def_92, type, produc1719072398_set_a: produc1691597095_set_a > set_a > produc1472671646_set_a).
% 2.74/1.60 thf(func_def_93, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 2.74/1.60 thf(func_def_94, type, produc1599887374_set_a: set_Product_prod_a_a > set_set_a > produc386552094_set_a).
% 2.74/1.60 thf(func_def_95, type, produc194376366_set_a: set_Product_prod_a_a > set_a > produc275558334_set_a).
% 2.74/1.60 thf(func_def_96, type, produc760262016_set_a: set_set_a > produc1691597095_set_a > produc398359056_set_a).
% 2.74/1.60 thf(func_def_97, type, produc1618062688od_a_a: set_set_a > set_Product_prod_a_a > produc270245360od_a_a).
% 2.74/1.60 thf(func_def_98, type, produc2102896407_set_a: set_set_a > set_set_a > produc407913767_set_a).
% 2.74/1.60 thf(func_def_99, type, produc32105911_set_a: set_set_a > set_a > produc522143687_set_a).
% 2.74/1.60 thf(func_def_100, type, produc272677600_set_a: set_a > produc1691597095_set_a > produc304417648_set_a).
% 2.74/1.60 thf(func_def_101, type, produc1397177024od_a_a: set_a > set_Product_prod_a_a > produc376942416od_a_a).
% 2.74/1.60 thf(func_def_102, type, produc1560024183_set_a: set_a > set_set_a > produc1083127431_set_a).
% 2.74/1.60 thf(func_def_103, type, produc1928581911_set_a: set_a > set_a > produc1691597095_set_a).
% 2.74/1.60 thf(func_def_104, type, product_Pair_a_a: a > a > product_prod_a_a).
% 2.74/1.60 thf(func_def_105, type, id_on_1182990020od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a).
% 2.74/1.60 thf(func_def_106, type, id_on_a: set_a > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_107, type, refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_108, type, refl_on_a: set_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_109, type, total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_110, type, total_on_a: set_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_111, type, trans_a: set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_112, type, collec1635618130od_a_a: (produc1572603623od_a_a > $o) > set_Pr1948701895od_a_a).
% 2.74/1.60 thf(func_def_113, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_114, type, collect_set_a: (set_a > $o) > set_set_a).
% 2.74/1.60 thf(func_def_115, type, collect_a: (a > $o) > set_a).
% 2.74/1.60 thf(func_def_116, type, is_empty_a: set_a > $o).
% 2.74/1.60 thf(func_def_117, type, member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o).
% 2.74/1.60 thf(func_def_118, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 2.74/1.60 thf(func_def_119, type, member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o).
% 2.74/1.60 thf(func_def_120, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 2.74/1.60 thf(func_def_121, type, member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o).
% 2.74/1.60 thf(func_def_122, type, member_set_set_a: set_set_a > set_set_set_a > $o).
% 2.74/1.60 thf(func_def_123, type, member_set_a: set_a > set_set_a > $o).
% 2.74/1.60 thf(func_def_124, type, member_a: a > set_a > $o).
% 2.74/1.60 thf(func_def_125, type, carrier: set_a).
% 2.74/1.60 thf(func_def_126, type, relation: set_Product_prod_a_a).
% 2.74/1.60 thf(func_def_127, type, x: a).
% 2.74/1.60 thf(func_def_128, type, y: a).
% 2.74/1.60 thf(func_def_143, type, sK0: (product_prod_a_a > $o) > a).
% 2.74/1.60 thf(func_def_144, type, sK1: (product_prod_a_a > $o) > a).
% 2.74/1.60 thf(func_def_145, type, sK2: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_146, type, sK3: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_147, type, sK4: set_a > a).
% 2.74/1.60 thf(func_def_148, type, sK5: (product_prod_a_a > $o) > a).
% 2.74/1.60 thf(func_def_149, type, sK6: (product_prod_a_a > $o) > a).
% 2.74/1.60 thf(func_def_150, type, sK7: set_a > a).
% 2.74/1.60 thf(func_def_151, type, sK8: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_152, type, sK9: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_153, type, sK10: product_prod_a_a > a).
% 2.74/1.60 thf(func_def_154, type, sK11: product_prod_a_a > a).
% 2.74/1.60 thf(func_def_155, type, sK12: product_prod_a_a > a).
% 2.74/1.60 thf(func_def_156, type, sK13: product_prod_a_a > a).
% 2.74/1.60 thf(func_def_157, type, sK14: set_a > a).
% 2.74/1.60 thf(func_def_158, type, sK15: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_159, type, sK16: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_160, type, sK17: set_a > set_a > a).
% 2.74/1.60 thf(func_def_161, type, sK18: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_162, type, sK19: (set_a > set_a) > set_a).
% 2.74/1.60 thf(func_def_163, type, sK20: set_a > a).
% 2.74/1.60 thf(func_def_165, type, ph22: !>[X0: $tType]:(X0)).
% 2.74/1.60 thf(f1315,plain,(
% 2.74/1.60 $false),
% 2.74/1.60 inference(subsumption_resolution,[status(thm)],[f1273,f1255])).
% 2.74/1.60 thf(f1255,plain,(
% 2.74/1.60 ( ! [X0 : set_a] : (($true = (ord_less_eq_set_a @ X0 @ X0))) )),
% 2.74/1.60 inference(cnf_transformation,[status(thm)],[f909])).
% 2.74/1.60 thf(f909,plain,(
% 2.74/1.60 ! [X0 : set_a] : ($true = (ord_less_eq_set_a @ X0 @ X0))),
% 2.74/1.60 inference(fool_elimination,[status(thm)],[f908])).
% 2.74/1.60 thf(f908,plain,(
% 2.74/1.60 ! [X0 : set_a] : (ord_less_eq_set_a @ X0 @ X0)),
% 2.74/1.60 inference(rectify,[status(thm)],[f196])).
% 2.74/1.60 thf(f196,axiom,(
% 2.74/1.60 ! [X13 : set_a] : (ord_less_eq_set_a @ X13 @ X13)),
% 2.74/1.60 file('/export/starexec/sandbox2/tmp/tmp.b3Te47zhCv/DTF2THF_27726.p',fact_195_subset__refl)).
% 2.74/1.60 thf(f1273,plain,(
% 2.74/1.60 ((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true)),
% 2.74/1.60 inference(cnf_transformation,[status(thm)],[f1066])).
% 2.74/1.60 thf(f1066,plain,(
% 2.74/1.60 ((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true) & ((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true)),
% 2.74/1.60 inference(ennf_transformation,[status(thm)],[f767])).
% 2.74/1.60 thf(f767,plain,(
% 2.74/1.60 ~(((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) = $true) | ((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) = $true))),
% 2.74/1.60 inference(fool_elimination,[status(thm)],[f766])).
% 2.74/1.60 thf(f766,plain,(
% 2.74/1.60 ~((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) | (ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)))),
% 2.74/1.60 inference(rectify,[status(thm)],[f356])).
% 2.74/1.60 thf(f356,negated_conjecture,(
% 2.74/1.60 ~((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) | (ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)))),
% 2.74/1.60 inference(negated_conjecture,[status(cth)],[f355])).
% 2.74/1.60 thf(f355,conjecture,(
% 2.74/1.60 (ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) | (ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation))),
% 2.74/1.60 file('/export/starexec/sandbox2/tmp/tmp.b3Te47zhCv/DTF2THF_27726.p',conj_0)).
% 2.74/1.60 % SZS output end Proof for DTF2THF_27726
% 2.74/1.60 % (27952)------------------------------
% 2.74/1.60 % (27952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.60 % (27952)Termination reason: Refutation
% 2.74/1.60
% 2.74/1.60 % (27952)Memory used [KB]: 6780
% 2.74/1.60 % (27952)Time elapsed: 0.048 s
% 2.74/1.60 % (27952)Instructions burned: 78 (million)
% 2.74/1.60 % (27952)------------------------------
% 2.74/1.60 % (27952)------------------------------
% 2.74/1.60 % (27946)Success in time 0.081 s
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