%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP028^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n009.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 08:54:58 AM UTC 2026
% Result : Theorem 3.02s 1.67s
% Output : Refutation 3.02s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 6
% Syntax : Number of formulae : 33 ( 13 unt; 0 typ; 2 def)
% Number of atoms : 221 ( 62 equ; 0 cnn)
% Maximal formula atoms : 8 ( 6 avg)
% Number of connectives : 275 ( 43 ~; 31 |; 18 &; 168 @)
% ( 2 <=>; 13 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 141 ( 138 usr; 15 con; 0-5 aty)
% Number of variables : 12 ( 0 ^; 8 !; 4 ?; 12 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_a: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_7,type,
e: $tType ).
thf(type_def_9,type,
produc715398134_a_a_e: $tType ).
thf(type_def_10,type,
set_Pr204296108_a_a_e: $tType ).
thf(type_def_11,type,
set_set_a: $tType ).
thf(type_def_12,type,
set_se821189730_a_a_e: $tType ).
thf(type_def_13,type,
set_set_set_a: $tType ).
thf(type_def_14,type,
set_a_e: $tType ).
thf(type_def_15,type,
set_Pr305929312_set_a: $tType ).
thf(type_def_16,type,
set_Pr1710391469_a_a_e: $tType ).
thf(type_def_17,type,
set_Pr2045688071_set_a: $tType ).
thf(type_def_18,type,
produc1965228714_set_a: $tType ).
thf(type_def_19,type,
produc169085773_a_a_e: $tType ).
thf(type_def_20,type,
produc1691597095_set_a: $tType ).
thf(func_def_0,type,
set_Pr1710391469_a_a_e: $tType ).
thf(func_def_1,type,
produc169085773_a_a_e: $tType ).
thf(func_def_2,type,
set_se821189730_a_a_e: $tType ).
thf(func_def_3,type,
set_Pr2045688071_set_a: $tType ).
thf(func_def_4,type,
set_Pr204296108_a_a_e: $tType ).
thf(func_def_5,type,
set_Pr305929312_set_a: $tType ).
thf(func_def_6,type,
produc1691597095_set_a: $tType ).
thf(func_def_7,type,
produc715398134_a_a_e: $tType ).
thf(func_def_8,type,
produc1965228714_set_a: $tType ).
thf(func_def_9,type,
set_set_set_a: $tType ).
thf(func_def_10,type,
set_set_a: $tType ).
thf(func_def_11,type,
set_a_e: $tType ).
thf(func_def_12,type,
set_a: $tType ).
thf(func_def_13,type,
e: $tType ).
thf(func_def_14,type,
a: $tType ).
thf(func_def_15,type,
bNF_fs1024561536_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > produc715398134_a_a_e > produc715398134_a_a_e ).
thf(func_def_16,type,
bNF_pi1175744974_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > set_a > ( a > e ) > a > e ).
thf(func_def_17,type,
bNF_pi929741261_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > set_a > ( a > e ) > set_a ).
thf(func_def_18,type,
bNF_sn375433437_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > produc715398134_a_a_e > produc715398134_a_a_e ).
thf(func_def_19,type,
bNF_Gr2072183031_a_a_e: set_Pr204296108_a_a_e > ( produc715398134_a_a_e > set_a ) > ( produc715398134_a_a_e > a > e ) > set_Pr204296108_a_a_e ).
thf(func_def_20,type,
bNF_Gr1872575811_a_a_e: set_set_a > ( set_a > set_a ) > ( set_a > a > e ) > set_Pr204296108_a_a_e ).
thf(func_def_21,type,
finite33065488_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e ).
thf(func_def_22,type,
finite_Fpow_set_a: set_set_a > set_set_set_a ).
thf(func_def_23,type,
finite_Fpow_a: set_a > set_set_a ).
thf(func_def_24,type,
fun_in_rel_set_a_a_e: set_Pr204296108_a_a_e > set_a > ( a > e ) > $o ).
thf(func_def_25,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(func_def_26,type,
ord_less_eq_set_a_e: set_a_e > set_a_e > $o ).
thf(func_def_27,type,
ord_le802356416_set_a: set_Pr305929312_set_a > set_Pr305929312_set_a > $o ).
thf(func_def_28,type,
ord_le700723212_a_a_e: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > $o ).
thf(func_def_29,type,
ord_le2103863998_set_a: set_set_set_a > set_set_set_a > $o ).
thf(func_def_30,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_31,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_32,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(func_def_33,type,
top_to2000253225_a_e_o: produc715398134_a_a_e > $o ).
thf(func_def_34,type,
top_top_set_a_o: set_a > $o ).
thf(func_def_35,type,
top_to970499581_a_a_e: set_Pr1710391469_a_a_e ).
thf(func_def_36,type,
top_to1601855068_a_a_e: set_Pr204296108_a_a_e ).
thf(func_def_37,type,
top_to144004695_set_a: set_Pr2045688071_set_a ).
thf(func_def_38,type,
top_top_set_set_a: set_set_a ).
thf(func_def_39,type,
top_top_set_a: set_a ).
thf(func_def_40,type,
produc479206876_set_a: ( a > e ) > set_a > produc1965228714_set_a ).
thf(func_def_41,type,
produc1134456688_a_a_e: set_a > ( a > e ) > produc715398134_a_a_e ).
thf(func_def_42,type,
produc952037978_a_e_o: ( produc715398134_a_a_e > $o ) > set_a > ( a > e ) > $o ).
thf(func_def_43,type,
produc387505142_set_a: ( produc715398134_a_a_e > set_a ) > ( produc715398134_a_a_e > set_a ) > produc169085773_a_a_e > produc1691597095_set_a ).
thf(func_def_44,type,
produc1443219402_e_a_e: ( set_a > set_a ) > ( ( a > e ) > a > e ) > produc715398134_a_a_e > produc715398134_a_a_e ).
thf(func_def_45,type,
produc1341862587_a_e_o: ( set_a > ( a > e ) > $o ) > produc715398134_a_a_e > $o ).
thf(func_def_46,type,
produc1096319049_a_a_e: ( set_a > ( a > e ) > set_Pr204296108_a_a_e ) > produc715398134_a_a_e > set_Pr204296108_a_a_e ).
thf(func_def_47,type,
produc1620938593_set_a: ( set_a > ( a > e ) > set_set_a ) > produc715398134_a_a_e > set_set_a ).
thf(func_def_48,type,
produc385678576_set_a: produc1965228714_set_a > a > e ).
thf(func_def_49,type,
produc1040928388_a_a_e: produc715398134_a_a_e > set_a ).
thf(func_def_50,type,
produc1809800494_set_a: produc1965228714_set_a > set_a ).
thf(func_def_51,type,
produc317566658_a_a_e: produc715398134_a_a_e > a > e ).
thf(func_def_52,type,
produc100464700_set_a: produc1965228714_set_a > produc715398134_a_a_e ).
thf(func_def_53,type,
produc755714512_a_a_e: produc715398134_a_a_e > produc1965228714_set_a ).
thf(func_def_54,type,
domain_set_a_a_e: set_Pr204296108_a_a_e > set_set_a ).
thf(func_def_55,type,
range_set_a_a_e: set_Pr204296108_a_a_e > set_a_e ).
thf(func_def_56,type,
converse_a_e_set_a: set_Pr305929312_set_a > set_Pr204296108_a_a_e ).
thf(func_def_57,type,
converse_set_a_a_e: set_Pr204296108_a_a_e > set_Pr305929312_set_a ).
thf(func_def_58,type,
relcom658603713_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > set_a > ( a > e ) > $o ).
thf(func_def_59,type,
relcom779348378_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > set_a > ( a > e ) > $o ).
thf(func_def_60,type,
collec1369693387_a_a_e: ( produc715398134_a_a_e > $o ) > set_Pr204296108_a_a_e ).
thf(func_def_61,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_62,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_63,type,
pow_Pr328302361_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e ).
thf(func_def_64,type,
pow_set_a: set_set_a > set_set_set_a ).
thf(func_def_65,type,
pow_a: set_a > set_set_a ).
thf(func_def_66,type,
image_a_e_a: ( ( a > e ) > a ) > set_a_e > set_a ).
thf(func_def_67,type,
image_376979367_a_a_e: ( produc1965228714_set_a > produc715398134_a_a_e ) > set_Pr305929312_set_a > set_Pr204296108_a_a_e ).
thf(func_def_68,type,
image_2101504031_set_a: ( produc169085773_a_a_e > produc1691597095_set_a ) > set_Pr1710391469_a_a_e > set_Pr2045688071_set_a ).
thf(func_def_69,type,
image_1149879910_e_a_e: ( produc715398134_a_a_e > a > e ) > set_Pr204296108_a_a_e > set_a_e ).
thf(func_def_70,type,
image_1875450791_set_a: ( produc715398134_a_a_e > produc1965228714_set_a ) > set_Pr204296108_a_a_e > set_Pr305929312_set_a ).
thf(func_def_71,type,
image_625620211_a_a_e: ( produc715398134_a_a_e > produc715398134_a_a_e ) > set_Pr204296108_a_a_e > set_Pr204296108_a_a_e ).
thf(func_def_72,type,
image_64240033_set_a: ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > set_set_a ).
thf(func_def_73,type,
image_2116492353_a_e_a: ( produc715398134_a_a_e > a ) > set_Pr204296108_a_a_e > set_a ).
thf(func_def_74,type,
image_1441218763_set_a: ( set_Pr204296108_a_a_e > set_set_a ) > set_se821189730_a_a_e > set_set_set_a ).
thf(func_def_75,type,
image_1956910981_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(func_def_76,type,
image_402591269_set_a: ( set_set_a > set_a ) > set_set_set_a > set_set_a ).
thf(func_def_77,type,
image_set_set_a_a: ( set_set_a > a ) > set_set_set_a > set_a ).
thf(func_def_78,type,
image_set_a_a_e: ( set_a > a > e ) > set_set_a > set_a_e ).
thf(func_def_79,type,
image_210900495_a_a_e: ( set_a > produc715398134_a_a_e ) > set_set_a > set_Pr204296108_a_a_e ).
thf(func_def_80,type,
image_1930509541_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(func_def_81,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(func_def_82,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(func_def_83,type,
image_a_a_e: ( a > a > e ) > set_a > set_a_e ).
thf(func_def_84,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
thf(func_def_85,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(func_def_86,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(func_def_87,type,
member_a_e: ( a > e ) > set_a_e > $o ).
thf(func_def_88,type,
member360729153_set_a: produc1965228714_set_a > set_Pr305929312_set_a > $o ).
thf(func_def_89,type,
member1258382221_a_a_e: produc715398134_a_a_e > set_Pr204296108_a_a_e > $o ).
thf(func_def_90,type,
member1437121347_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e > $o ).
thf(func_def_91,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(func_def_92,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_93,type,
member_a: a > set_a > $o ).
thf(func_def_94,type,
c: set_Pr204296108_a_a_e ).
thf(func_def_95,type,
a1: set_a ).
thf(func_def_96,type,
a2: set_a ).
thf(func_def_97,type,
a3: set_a ).
thf(func_def_98,type,
b1: a > e ).
thf(func_def_99,type,
b2: a > e ).
thf(func_def_100,type,
b3: a > e ).
thf(func_def_113,type,
sK0: ( produc715398134_a_a_e > $o ) > a > e ).
thf(func_def_114,type,
sK1: ( produc715398134_a_a_e > $o ) > set_a ).
thf(func_def_115,type,
sK2: produc715398134_a_a_e > a > e ).
thf(func_def_116,type,
sK3: produc715398134_a_a_e > set_a ).
thf(func_def_117,type,
sK4: ( produc715398134_a_a_e > set_a ) > set_a > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_118,type,
sK5: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > a > e ).
thf(func_def_119,type,
sK6: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > set_a ).
thf(func_def_120,type,
sK7: ( produc715398134_a_a_e > set_a ) > set_set_a > set_Pr204296108_a_a_e > set_Pr204296108_a_a_e ).
thf(func_def_121,type,
sK8: produc715398134_a_a_e > set_a > a > e ).
thf(func_def_122,type,
sK9: ( produc715398134_a_a_e > $o ) > set_a ).
thf(func_def_123,type,
sK10: ( produc715398134_a_a_e > $o ) > a > e ).
thf(func_def_124,type,
sK11: ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > set_set_a > produc715398134_a_a_e ).
thf(func_def_125,type,
sK12: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_126,type,
sK13: set_set_a > set_set_a > set_a ).
thf(func_def_127,type,
sK14: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_128,type,
sK15: set_set_a > set_set_a > set_a ).
thf(func_def_129,type,
sK16: set_Pr204296108_a_a_e > ( set_a > $o ) > ( produc715398134_a_a_e > set_a ) > set_a ).
thf(func_def_130,type,
sK17: set_set_a > set_set_a > set_a ).
thf(func_def_131,type,
sK18: set_a ).
thf(func_def_132,type,
sK19: ( produc715398134_a_a_e > set_a ) > ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_133,type,
sK20: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_134,type,
sK21: ( set_a > set_a ) > set_a ).
thf(func_def_135,type,
sK22: ( set_a > set_a ) > set_a ).
thf(func_def_136,type,
sK23: ( set_a > set_a ) > set_a ).
thf(func_def_137,type,
sK24: ( set_a > set_a ) > set_a ).
thf(func_def_138,type,
sK25: set_a > ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_139,type,
sK26: ( set_a > set_a ) > set_a ).
thf(func_def_140,type,
sK27: ( set_a > set_a ) > set_a ).
thf(func_def_141,type,
sK28: set_Pr204296108_a_a_e > set_set_a > ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e ).
thf(func_def_142,type,
sK29: ( set_a > set_a ) > set_a ).
thf(func_def_143,type,
sK30: ( set_a > set_a ) > set_a ).
thf(func_def_144,type,
sK31: ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > set_set_a > produc715398134_a_a_e ).
thf(func_def_145,type,
sK32: ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > set_set_a > produc715398134_a_a_e ).
thf(func_def_146,type,
sK33: ( produc715398134_a_a_e > set_a ) > ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_147,type,
sK34: ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > ( produc715398134_a_a_e > set_a ) > produc715398134_a_a_e ).
thf(func_def_148,type,
sK35: ( set_a > set_set_a ) > set_a ).
thf(func_def_149,type,
sK36: ( set_a > set_set_a ) > set_a ).
thf(func_def_150,type,
sK37: ( produc715398134_a_a_e > set_a ) > set_Pr204296108_a_a_e > ( set_a > $o ) > produc715398134_a_a_e ).
thf(func_def_151,type,
sK38: ( set_set_a > set_a ) > set_set_a ).
thf(func_def_152,type,
sK39: ( set_set_a > set_a ) > set_set_a ).
thf(func_def_153,type,
sK40: set_set_a > set_set_a > set_a ).
thf(func_def_154,type,
sK41: ( produc715398134_a_a_e > set_a ) > ( set_set_a > $o ) > set_Pr204296108_a_a_e > set_set_a ).
thf(func_def_155,type,
sK42: ( produc715398134_a_a_e > set_a ) > ( set_set_a > $o ) > set_Pr204296108_a_a_e > set_Pr204296108_a_a_e ).
thf(func_def_156,type,
sK43: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e ).
thf(func_def_157,type,
sK44: produc715398134_a_a_e > set_a ).
thf(func_def_158,type,
sK45: produc715398134_a_a_e > a > e ).
thf(func_def_159,type,
sK46: ( set_set_a > set_set_a ) > set_set_a ).
thf(func_def_160,type,
sK47: ( set_set_a > set_set_a ) > set_set_a ).
thf(func_def_162,type,
ph49:
!>[X0: $tType] : X0 ).
thf(f1540,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1470,f1471,f1509]) ).
thf(f1509,plain,
~ spl48_1,
inference(avatar_contradiction_clause,[status(thm)],[f1508]) ).
thf(f1508,plain,
( $false
| ~ spl48_1 ),
inference(subsumption_resolution,[status(thm)],[f1507,f1318]) ).
thf(f1318,plain,
( ( ord_less_eq_set_a @ a3 @ sK18 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1198]) ).
thf(f1198,plain,
( ( ( ord_less_eq_set_a @ a1 @ sK18 )
= $true )
& ( ( member_set_a @ sK18 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
= $true )
& ( ( ord_less_eq_set_a @ a3 @ sK18 )
= $true )
& ( ( ord_less_eq_set_a @ a2 @ sK18 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK18])],[f426,f1197]) ).
thf(f1197,plain,
( ? [X0: set_a] :
( ( ( ord_less_eq_set_a @ a1 @ X0 )
= $true )
& ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
= $true )
& ( $true
= ( ord_less_eq_set_a @ a3 @ X0 ) )
& ( ( ord_less_eq_set_a @ a2 @ X0 )
= $true ) )
=> ( ( ( ord_less_eq_set_a @ a1 @ sK18 )
= $true )
& ( ( member_set_a @ sK18 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
= $true )
& ( ( ord_less_eq_set_a @ a3 @ sK18 )
= $true )
& ( ( ord_less_eq_set_a @ a2 @ sK18 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f426,plain,
? [X0: set_a] :
( ( ( ord_less_eq_set_a @ a1 @ X0 )
= $true )
& ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
= $true )
& ( $true
= ( ord_less_eq_set_a @ a3 @ X0 ) )
& ( ( ord_less_eq_set_a @ a2 @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f425]) ).
thf(f425,plain,
? [X0: set_a] :
( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
& ( ord_less_eq_set_a @ a2 @ X0 )
& ( ord_less_eq_set_a @ a3 @ X0 )
& ( ord_less_eq_set_a @ a1 @ X0 ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
? [X2: set_a] :
( ( member_set_a @ X2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
& ( ord_less_eq_set_a @ a2 @ X2 )
& ( ord_less_eq_set_a @ a3 @ X2 )
& ( ord_less_eq_set_a @ a1 @ X2 ) ),
file('/export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2THF_17883.p',fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062) ).
thf(f1507,plain,
( ( ( ord_less_eq_set_a @ a3 @ sK18 )
!= $true )
| ~ spl48_1 ),
inference(subsumption_resolution,[status(thm)],[f1506,f1320]) ).
thf(f1320,plain,
( ( ord_less_eq_set_a @ a1 @ sK18 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1198]) ).
thf(f1506,plain,
( ( ( ord_less_eq_set_a @ a1 @ sK18 )
!= $true )
| ( ( ord_less_eq_set_a @ a3 @ sK18 )
!= $true )
| ~ spl48_1 ),
inference(subsumption_resolution,[status(thm)],[f1498,f1317]) ).
thf(f1317,plain,
( ( ord_less_eq_set_a @ a2 @ sK18 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1198]) ).
thf(f1498,plain,
( ( ( ord_less_eq_set_a @ a2 @ sK18 )
!= $true )
| ( ( ord_less_eq_set_a @ a3 @ sK18 )
!= $true )
| ( ( ord_less_eq_set_a @ a1 @ sK18 )
!= $true )
| ~ spl48_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1489]) ).
thf(f1489,plain,
( ( ( ord_less_eq_set_a @ a2 @ sK18 )
!= $true )
| ( ( ord_less_eq_set_a @ a3 @ sK18 )
!= $true )
| ( ( ord_less_eq_set_a @ a1 @ sK18 )
!= $true )
| ( $true != $true )
| ~ spl48_1 ),
inference(superposition,[status(thm)],[f1465,f1319]) ).
thf(f1319,plain,
( ( member_set_a @ sK18 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1198]) ).
thf(f1465,plain,
( ! [X0: set_a] :
( ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
!= $true )
| ( ( ord_less_eq_set_a @ a1 @ X0 )
!= $true )
| ( ( ord_less_eq_set_a @ a2 @ X0 )
!= $true )
| ( $true
!= ( ord_less_eq_set_a @ a3 @ X0 ) ) )
| ~ spl48_1 ),
inference(avatar_component_clause,[status(thm)],[f1464]) ).
thf(f1464,definition,
( spl48_1
<=> ! [X0: set_a] :
( ( ( ord_less_eq_set_a @ a2 @ X0 )
!= $true )
| ( $true
!= ( ord_less_eq_set_a @ a3 @ X0 ) )
| ( ( ord_less_eq_set_a @ a1 @ X0 )
!= $true )
| ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl48_1])],[avatar_definition]) ).
thf(f1471,plain,
~ spl48_2,
inference(avatar_split_clause,[status(thm)],[f1331,f1467]) ).
thf(f1467,definition,
( spl48_2
<=> ( thesis = $true ) ),
introduced(definition,[new_symbols(naming,[spl48_2])],[avatar_definition]) ).
thf(f1331,plain,
thesis != $true,
inference(cnf_transformation,[status(thm)],[f993]) ).
thf(f993,plain,
thesis != $true,
inference(flattening,[status(thm)],[f608]) ).
thf(f608,plain,
thesis != $true,
inference(fool_elimination,[status(thm)],[f607]) ).
thf(f607,plain,
~ thesis,
inference(rectify,[status(thm)],[f355]) ).
thf(f355,negated_conjecture,
~ thesis,
inference(negated_conjecture,[status(cth)],[f354]) ).
thf(f354,conjecture,
thesis,
file('/export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2THF_17883.p',conj_1) ).
thf(f1470,plain,
( spl48_1
| spl48_2 ),
inference(avatar_split_clause,[status(thm)],[f1340,f1467,f1464]) ).
thf(f1340,plain,
! [X0: set_a] :
( ( ( ord_less_eq_set_a @ a2 @ X0 )
!= $true )
| ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
!= $true )
| ( thesis = $true )
| ( ( ord_less_eq_set_a @ a1 @ X0 )
!= $true )
| ( $true
!= ( ord_less_eq_set_a @ a3 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1133]) ).
thf(f1133,plain,
! [X0: set_a] :
( ( thesis = $true )
| ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
!= $true )
| ( ( ord_less_eq_set_a @ a2 @ X0 )
!= $true )
| ( ( ord_less_eq_set_a @ a1 @ X0 )
!= $true )
| ( $true
!= ( ord_less_eq_set_a @ a3 @ X0 ) ) ),
inference(flattening,[status(thm)],[f1132]) ).
thf(f1132,plain,
! [X0: set_a] :
( ( thesis = $true )
| ( $true
!= ( ord_less_eq_set_a @ a3 @ X0 ) )
| ( ( ord_less_eq_set_a @ a2 @ X0 )
!= $true )
| ( ( ord_less_eq_set_a @ a1 @ X0 )
!= $true )
| ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f732]) ).
thf(f732,plain,
! [X0: set_a] :
( ( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
= $true )
=> ( ( ( ord_less_eq_set_a @ a1 @ X0 )
= $true )
=> ( ( ( ord_less_eq_set_a @ a2 @ X0 )
= $true )
=> ( ( $true
= ( ord_less_eq_set_a @ a3 @ X0 ) )
=> ( thesis = $true ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f731]) ).
thf(f731,plain,
! [X0: set_a] :
( ( member_set_a @ X0 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
=> ( ( ord_less_eq_set_a @ a1 @ X0 )
=> ( ( ord_less_eq_set_a @ a2 @ X0 )
=> ( ( ord_less_eq_set_a @ a3 @ X0 )
=> thesis ) ) ) ),
inference(rectify,[status(thm)],[f353]) ).
thf(f353,axiom,
! [X66: set_a] :
( ( member_set_a @ X66 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
=> ( ( ord_less_eq_set_a @ a1 @ X66 )
=> ( ( ord_less_eq_set_a @ a2 @ X66 )
=> ( ( ord_less_eq_set_a @ a3 @ X66 )
=> thesis ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2THF_17883.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : ITP028^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.35 % Computer : n009.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Feb 24 08:23:40 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.48 ---- Original DTF file ---
% 0.31/0.48 thf(spec,logic,$$dhol).
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 % File : ITP028^1 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.48 % Domain : Interactive Theorem Proving
% 0.31/0.48 % Problem : Sledgehammer Algebra8 problem prob_962__6421504_1
% 0.31/0.48 % Version : Especial.
% 0.31/0.48 % English :
% 0.31/0.48
% 0.31/0.48 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.48 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.48 % Source : [Des21]
% 0.31/0.48 % Names : Algebra8/prob_962__6421504_1 [Des21]
% 0.31/0.48
% 0.31/0.48 % Status : Theorem
% 0.31/0.48 % 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
% 0.31/0.48 % Syntax : Number of formulae : 456 ( 120 unt; 102 typ; 0 def)
% 0.31/0.48 % Number of atoms : 1002 ( 288 equ; 0 cnn)
% 0.31/0.48 % Maximal formula atoms : 8 ( 2 avg)
% 0.31/0.48 % Number of connectives : 2874 ( 78 ~; 10 |; 54 &;2255 @)
% 0.31/0.48 % ( 0 <=>; 477 =>; 0 <=; 0 <~>)
% 0.31/0.48 % Maximal formula depth : 16 ( 7 avg)
% 0.31/0.48 % Number of types : 16 ( 15 usr)
% 0.31/0.48 % Number of type conns : 618 ( 618 >; 0 *; 0 +; 0 <<)
% 0.31/0.48 % Number of symbols : 90 ( 87 usr; 12 con; 0-5 aty)
% 0.31/0.48 % Number of variables : 1103 ( 97 ^; 962 !; 44 ?;1103 :)
% 0.31/0.48 % SPC : TH0_THM_EQU_NAR
% 0.31/0.48
% 0.31/0.48 % Comments : This file was generated by Sledgehammer 2021-02-23 15:45:55.322
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 % Could-be-implicit typings (15)
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J_J,type,
% 0.31/0.48 set_Pr1710391469_a_a_e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
% 0.31/0.48 produc169085773_a_a_e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J_J,type,
% 0.31/0.48 set_se821189730_a_a_e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 0.31/0.48 set_Pr2045688071_set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
% 0.31/0.48 set_Pr204296108_a_a_e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 0.31/0.48 set_Pr305929312_set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.31/0.48 produc1691597095_set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
% 0.31/0.48 produc715398134_a_a_e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.31/0.48 produc1965228714_set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.31/0.48 set_set_set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.31/0.48 set_set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__e_J_J,type,
% 0.31/0.48 set_a_e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.31/0.48 set_a: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_tf__e,type,
% 0.31/0.48 e: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_n_tf__a,type,
% 0.31/0.48 a: $tType ).
% 0.31/0.48
% 0.31/0.48 % Explicit typings (87)
% 0.31/0.48 thf(sy_c_BNF__Def_OfstOp_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 bNF_fs1024561536_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > produc715398134_a_a_e > produc715398134_a_a_e ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_BNF__Def_Opick__middlep_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 bNF_pi1175744974_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > set_a > ( a > e ) > a > e ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_BNF__Def_Opick__middlep_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 bNF_pi929741261_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > set_a > ( a > e ) > set_a ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_BNF__Def_OsndOp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 bNF_sn375433437_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > produc715398134_a_a_e > produc715398134_a_a_e ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 bNF_Gr2072183031_a_a_e: set_Pr204296108_a_a_e > ( produc715398134_a_a_e > set_a ) > ( produc715398134_a_a_e > a > e ) > set_Pr204296108_a_a_e ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 bNF_Gr1872575811_a_a_e: set_set_a > ( set_a > set_a ) > ( set_a > a > e ) > set_Pr204296108_a_a_e ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
% 0.31/0.48 finite33065488_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
% 0.31/0.48 finite_Fpow_set_a: set_set_a > set_set_set_a ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Finite__Set_OFpow_001tf__a,type,
% 0.31/0.48 finite_Fpow_a: set_a > set_set_a ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Fun__Def_Oin__rel_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 0.31/0.48 fun_in_rel_set_a_a_e: set_Pr204296108_a_a_e > set_a > ( a > e ) > $o ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
% 0.31/0.48 ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__e_J_J,type,
% 0.31/0.48 ord_less_eq_set_a_e: set_a_e > set_a_e > $o ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 0.31/0.48 ord_le802356416_set_a: set_Pr305929312_set_a > set_Pr305929312_set_a > $o ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
% 0.31/0.48 ord_le700723212_a_a_e: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > $o ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.31/0.48 ord_le2103863998_set_a: set_set_set_a > set_set_set_a > $o ).
% 0.31/0.48
% 0.31/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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% 0.31/0.48
% 0.31/0.48 % Relevant facts (352)
% 0.31/0.48 thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
% 0.31/0.48 ! [X: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X @ c )
% 0.31/0.48 => ! [Xa: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ Xa @ c )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ X ) @ ( produc1040928388_a_a_e @ Xa ) )
% 0.31/0.48 | ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ Xa ) @ ( produc1040928388_a_a_e @ X ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % \<open>\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a\<close>
% 0.31/0.48 thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
% 0.31/0.48 ? [X2: set_a] :
% 0.31/0.48 ( ( member_set_a @ X2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
% 0.31/0.48 & ( ord_less_eq_set_a @ a1 @ X2 )
% 0.31/0.48 & ( ord_less_eq_set_a @ a2 @ X2 )
% 0.31/0.48 & ( ord_less_eq_set_a @ a3 @ X2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % \<open>\<exists>a\<in>fst ` C. a1 \<subseteq> a \<and> a2 \<subseteq> a \<and> a3 \<subseteq> a\<close>
% 0.31/0.48 thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 0.31/0.48 ( ( ord_less_eq_set_a @ a1 @ a2 )
% 0.31/0.48 | ( ord_less_eq_set_a @ a2 @ a1 ) ) ).
% 0.31/0.48
% 0.31/0.48 % \<open>a1 \<subseteq> a2 \<or> a2 \<subseteq> a1\<close>
% 0.31/0.48 thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 0.31/0.48 ( ( ord_less_eq_set_a @ a1 @ a3 )
% 0.31/0.48 | ( ord_less_eq_set_a @ a3 @ a1 ) ) ).
% 0.31/0.48
% 0.31/0.48 % \<open>a1 \<subseteq> a3 \<or> a3 \<subseteq> a1\<close>
% 0.31/0.48 thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
% 0.31/0.48 ( ( ord_less_eq_set_a @ a2 @ a3 )
% 0.31/0.48 | ( ord_less_eq_set_a @ a3 @ a2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % \<open>a2 \<subseteq> a3 \<or> a3 \<subseteq> a2\<close>
% 0.31/0.48 thf(fact_5_A_I3_J,axiom,
% 0.31/0.48 member_set_a @ a3 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% 0.31/0.48
% 0.31/0.48 % A(3)
% 0.31/0.48 thf(fact_6_A_I2_J,axiom,
% 0.31/0.48 member_set_a @ a2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% 0.31/0.48
% 0.31/0.48 % A(2)
% 0.31/0.48 thf(fact_7_A_I1_J,axiom,
% 0.31/0.48 member_set_a @ a1 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% 0.31/0.48
% 0.31/0.48 % A(1)
% 0.31/0.48 thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
% 0.31/0.48 member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a1 @ b1 ) @ c ).
% 0.31/0.48
% 0.31/0.48 % \<open>(a1, b1) \<in> C\<close>
% 0.31/0.48 thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
% 0.31/0.48 member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a2 @ b2 ) @ c ).
% 0.31/0.48
% 0.31/0.48 % \<open>(a2, b2) \<in> C\<close>
% 0.31/0.48 thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
% 0.31/0.48 member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a3 @ b3 ) @ c ).
% 0.31/0.48
% 0.31/0.48 % \<open>(a3, b3) \<in> C\<close>
% 0.31/0.48 thf(fact_11_subsetI,axiom,
% 0.31/0.48 ! [A: set_a_e,B: set_a_e] :
% 0.31/0.48 ( ! [X2: a > e] :
% 0.31/0.48 ( ( member_a_e @ X2 @ A )
% 0.31/0.48 => ( member_a_e @ X2 @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a_e @ A @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_12_subsetI,axiom,
% 0.31/0.48 ! [A: set_Pr305929312_set_a,B: set_Pr305929312_set_a] :
% 0.31/0.48 ( ! [X2: produc1965228714_set_a] :
% 0.31/0.48 ( ( member360729153_set_a @ X2 @ A )
% 0.31/0.48 => ( member360729153_set_a @ X2 @ B ) )
% 0.31/0.48 => ( ord_le802356416_set_a @ A @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_13_subsetI,axiom,
% 0.31/0.48 ! [A: set_set_set_a,B: set_set_set_a] :
% 0.31/0.48 ( ! [X2: set_set_a] :
% 0.31/0.48 ( ( member_set_set_a @ X2 @ A )
% 0.31/0.48 => ( member_set_set_a @ X2 @ B ) )
% 0.31/0.48 => ( ord_le2103863998_set_a @ A @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_14_subsetI,axiom,
% 0.31/0.48 ! [A: set_set_a,B: set_set_a] :
% 0.31/0.48 ( ! [X2: set_a] :
% 0.31/0.48 ( ( member_set_a @ X2 @ A )
% 0.31/0.48 => ( member_set_a @ X2 @ B ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ A @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_15_subsetI,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.48 => ( member1258382221_a_a_e @ X2 @ B ) )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ A @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_16_subsetI,axiom,
% 0.31/0.48 ! [A: set_a,B: set_a] :
% 0.31/0.48 ( ! [X2: a] :
% 0.31/0.48 ( ( member_a @ X2 @ A )
% 0.31/0.48 => ( member_a @ X2 @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % subsetI
% 0.31/0.48 thf(fact_17_subset__antisym,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ B @ A )
% 0.31/0.48 => ( A = B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_antisym
% 0.31/0.48 thf(fact_18_subset__antisym,axiom,
% 0.31/0.48 ! [A: set_set_a,B: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ B @ A )
% 0.31/0.48 => ( A = B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_antisym
% 0.31/0.48 thf(fact_19_subset__antisym,axiom,
% 0.31/0.48 ! [A: set_set_set_a,B: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A @ B )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ B @ A )
% 0.31/0.48 => ( A = B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_antisym
% 0.31/0.48 thf(fact_20_subset__antisym,axiom,
% 0.31/0.48 ! [A: set_a,B: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B @ A )
% 0.31/0.48 => ( A = B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_antisym
% 0.31/0.48 thf(fact_21_image__eqI,axiom,
% 0.31/0.48 ! [B2: set_a,F: set_a > set_a,X3: set_a,A: set_set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_22_image__eqI,axiom,
% 0.31/0.48 ! [B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e,X3: set_a,A: set_set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_23_image__eqI,axiom,
% 0.31/0.48 ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.48 => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_24_image__eqI,axiom,
% 0.31/0.48 ! [B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.48 => ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_25_image__eqI,axiom,
% 0.31/0.48 ! [B2: a,F: a > a,X3: a,A: set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_a @ X3 @ A )
% 0.31/0.48 => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_26_image__eqI,axiom,
% 0.31/0.48 ! [B2: a,F: set_a > a,X3: set_a,A: set_set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_27_image__eqI,axiom,
% 0.31/0.48 ! [B2: set_a,F: a > set_a,X3: a,A: set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_a @ X3 @ A )
% 0.31/0.48 => ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_28_image__eqI,axiom,
% 0.31/0.48 ! [B2: a > e,F: a > a > e,X3: a,A: set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_a @ X3 @ A )
% 0.31/0.48 => ( member_a_e @ B2 @ ( image_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_29_image__eqI,axiom,
% 0.31/0.48 ! [B2: a,F: ( a > e ) > a,X3: a > e,A: set_a_e] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_a_e @ X3 @ A )
% 0.31/0.48 => ( member_a @ B2 @ ( image_a_e_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_30_image__eqI,axiom,
% 0.31/0.48 ! [B2: a > e,F: set_a > a > e,X3: set_a,A: set_set_a] :
% 0.31/0.48 ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( member_a_e @ B2 @ ( image_set_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_eqI
% 0.31/0.48 thf(fact_31_order__refl,axiom,
% 0.31/0.48 ! [X3: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ X3 @ X3 ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_32_order__refl,axiom,
% 0.31/0.48 ! [X3: set_set_a] : ( ord_le318720350_set_a @ X3 @ X3 ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_33_order__refl,axiom,
% 0.31/0.48 ! [X3: set_set_set_a] : ( ord_le2103863998_set_a @ X3 @ X3 ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_34_order__refl,axiom,
% 0.31/0.48 ! [X3: $o > set_a] : ( ord_less_eq_o_set_a @ X3 @ X3 ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_35_order__refl,axiom,
% 0.31/0.48 ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_36_image__mono,axiom,
% 0.31/0.48 ! [A: set_a,B: set_a,F: a > a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_37_image__mono,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ ( image_64240033_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_38_image__mono,axiom,
% 0.31/0.48 ! [A: set_a,B: set_a,F: a > set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_39_image__mono,axiom,
% 0.31/0.48 ! [A: set_set_a,B: set_set_a,F: set_a > a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_40_image__mono,axiom,
% 0.31/0.48 ! [A: set_a,B: set_a,F: a > set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( image_a_set_set_a @ F @ A ) @ ( image_a_set_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_41_image__mono,axiom,
% 0.31/0.48 ! [A: set_set_a,B: set_set_a,F: set_a > set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_42_image__mono,axiom,
% 0.31/0.48 ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A @ B )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_set_set_a_a @ F @ A ) @ ( image_set_set_a_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_43_image__mono,axiom,
% 0.31/0.48 ! [A: set_set_a,B: set_set_a,F: set_a > set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( image_1930509541_set_a @ F @ A ) @ ( image_1930509541_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_44_image__mono,axiom,
% 0.31/0.48 ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A @ B )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_402591269_set_a @ F @ A ) @ ( image_402591269_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_45_image__mono,axiom,
% 0.31/0.48 ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A @ B )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( image_1956910981_set_a @ F @ A ) @ ( image_1956910981_set_a @ F @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_mono
% 0.31/0.48 thf(fact_46_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_set_a,F: set_a > a,B: set_a] :
% 0.31/0.48 ( ! [X2: set_a] :
% 0.31/0.48 ( ( member_set_a @ X2 @ A )
% 0.31/0.48 => ( member_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_47_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
% 0.31/0.48 ( ! [X2: set_a] :
% 0.31/0.48 ( ( member_set_a @ X2 @ A )
% 0.31/0.48 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_48_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > a,B: set_a] :
% 0.31/0.48 ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.48 => ( member_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_2116492353_a_e_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_49_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ! [X2: set_a] :
% 0.31/0.48 ( ( member_set_a @ X2 @ A )
% 0.31/0.48 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_50_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a] :
% 0.31/0.48 ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.48 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_51_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.48 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ ( image_625620211_a_a_e @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_52_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_a,F: a > a,B: set_a] :
% 0.31/0.48 ( ! [X2: a] :
% 0.31/0.48 ( ( member_a @ X2 @ A )
% 0.31/0.48 => ( member_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_53_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_a,F: a > set_a,B: set_set_a] :
% 0.31/0.48 ( ! [X2: a] :
% 0.31/0.48 ( ( member_a @ X2 @ A )
% 0.31/0.48 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_54_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_a,F: a > a > e,B: set_a_e] :
% 0.31/0.48 ( ! [X2: a] :
% 0.31/0.48 ( ( member_a @ X2 @ A )
% 0.31/0.48 => ( member_a_e @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a_e @ ( image_a_a_e @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_55_image__subsetI,axiom,
% 0.31/0.48 ! [A: set_a_e,F: ( a > e ) > a,B: set_a] :
% 0.31/0.48 ( ! [X2: a > e] :
% 0.31/0.48 ( ( member_a_e @ X2 @ A )
% 0.31/0.48 => ( member_a @ ( F @ X2 ) @ B ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( image_a_e_a @ F @ A ) @ B ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subsetI
% 0.31/0.48 thf(fact_56_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_a,F: a > a,A: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_a_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_57_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_64240033_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_58_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_a,F: set_a > a,A: set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_set_a_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_59_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_a,F: a > set_a,A: set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_a_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_60_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_set_set_a_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_61_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_set_a_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_62_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_a_set_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_63_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_402591269_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_64_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_1930509541_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_65_subset__imageE,axiom,
% 0.31/0.48 ! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
% 0.31/0.48 => ~ ! [C: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ C @ A )
% 0.31/0.48 => ( B
% 0.31/0.48 != ( image_1956910981_set_a @ F @ C ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_imageE
% 0.31/0.48 thf(fact_66_image__subset__iff,axiom,
% 0.31/0.48 ! [F: set_a > a,A: set_set_a,B: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B )
% 0.31/0.48 = ( ! [X4: set_a] :
% 0.31/0.48 ( ( member_set_a @ X4 @ A )
% 0.31/0.48 => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subset_iff
% 0.31/0.48 thf(fact_67_image__subset__iff,axiom,
% 0.31/0.48 ! [F: a > a,A: set_a,B: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
% 0.31/0.48 = ( ! [X4: a] :
% 0.31/0.48 ( ( member_a @ X4 @ A )
% 0.31/0.48 => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subset_iff
% 0.31/0.48 thf(fact_68_image__subset__iff,axiom,
% 0.31/0.48 ! [F: set_a > produc715398134_a_a_e,A: set_set_a,B: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B )
% 0.31/0.48 = ( ! [X4: set_a] :
% 0.31/0.48 ( ( member_set_a @ X4 @ A )
% 0.31/0.48 => ( member1258382221_a_a_e @ ( F @ X4 ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subset_iff
% 0.31/0.48 thf(fact_69_image__subset__iff,axiom,
% 0.31/0.48 ! [F: set_a > set_a,A: set_set_a,B: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B )
% 0.31/0.48 = ( ! [X4: set_a] :
% 0.31/0.48 ( ( member_set_a @ X4 @ A )
% 0.31/0.48 => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subset_iff
% 0.31/0.48 thf(fact_70_image__subset__iff,axiom,
% 0.31/0.48 ! [F: set_Pr204296108_a_a_e > set_set_a,A: set_se821189730_a_a_e,B: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ F @ A ) @ B )
% 0.31/0.48 = ( ! [X4: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( member1437121347_a_a_e @ X4 @ A )
% 0.31/0.48 => ( member_set_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subset_iff
% 0.31/0.48 thf(fact_71_image__subset__iff,axiom,
% 0.31/0.48 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
% 0.31/0.48 = ( ! [X4: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X4 @ A )
% 0.31/0.48 => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_subset_iff
% 0.31/0.48 thf(fact_72_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_a,F: a > a,A: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_a_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_73_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_64240033_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_74_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_a,F: set_a > a,A: set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_75_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_a,F: a > set_a,A: set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_76_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_set_set_a_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_77_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_78_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_a_set_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_79_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_402591269_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_80_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_1930509541_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_81_subset__image__iff,axiom,
% 0.31/0.48 ! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [AA: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ AA @ A )
% 0.31/0.48 & ( B
% 0.31/0.48 = ( image_1956910981_set_a @ F @ AA ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % subset_image_iff
% 0.31/0.48 thf(fact_82_PairE__lemma,axiom,
% 0.31/0.48 ! [P: produc1965228714_set_a] :
% 0.31/0.48 ? [X2: a > e,Y: set_a] :
% 0.31/0.48 ( P
% 0.31/0.48 = ( produc479206876_set_a @ X2 @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % PairE_lemma
% 0.31/0.48 thf(fact_83_PairE__lemma,axiom,
% 0.31/0.48 ! [P: produc715398134_a_a_e] :
% 0.31/0.48 ? [X2: set_a,Y: a > e] :
% 0.31/0.48 ( P
% 0.31/0.48 = ( produc1134456688_a_a_e @ X2 @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % PairE_lemma
% 0.31/0.48 thf(fact_84_dual__order_Oantisym,axiom,
% 0.31/0.48 ! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.antisym
% 0.31/0.48 thf(fact_85_dual__order_Oantisym,axiom,
% 0.31/0.48 ! [B2: set_set_a,A2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.antisym
% 0.31/0.48 thf(fact_86_dual__order_Oantisym,axiom,
% 0.31/0.48 ! [B2: set_set_set_a,A2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.antisym
% 0.31/0.48 thf(fact_87_dual__order_Oantisym,axiom,
% 0.31/0.48 ! [B2: $o > set_a,A2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.antisym
% 0.31/0.48 thf(fact_88_dual__order_Oantisym,axiom,
% 0.31/0.48 ! [B2: set_a,A2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.antisym
% 0.31/0.48 thf(fact_89_dual__order_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ B3 @ A3 )
% 0.31/0.48 & ( ord_le700723212_a_a_e @ A3 @ B3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.eq_iff
% 0.31/0.48 thf(fact_90_dual__order_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B3 @ A3 )
% 0.31/0.48 & ( ord_le318720350_set_a @ A3 @ B3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.eq_iff
% 0.31/0.48 thf(fact_91_dual__order_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B3 @ A3 )
% 0.31/0.48 & ( ord_le2103863998_set_a @ A3 @ B3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.eq_iff
% 0.31/0.48 thf(fact_92_dual__order_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: $o > set_a,B3: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ B3 @ A3 )
% 0.31/0.48 & ( ord_less_eq_o_set_a @ A3 @ B3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.eq_iff
% 0.31/0.48 thf(fact_93_dual__order_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_a,B3: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B3 @ A3 )
% 0.31/0.48 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.eq_iff
% 0.31/0.48 thf(fact_94_dual__order_Otrans,axiom,
% 0.31/0.48 ! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ C2 @ B2 )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ C2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.trans
% 0.31/0.48 thf(fact_95_dual__order_Otrans,axiom,
% 0.31/0.48 ! [B2: set_set_a,A2: set_set_a,C2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ C2 @ B2 )
% 0.31/0.48 => ( ord_le318720350_set_a @ C2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.trans
% 0.31/0.48 thf(fact_96_dual__order_Otrans,axiom,
% 0.31/0.48 ! [B2: set_set_set_a,A2: set_set_set_a,C2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ C2 @ B2 )
% 0.31/0.48 => ( ord_le2103863998_set_a @ C2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.trans
% 0.31/0.48 thf(fact_97_dual__order_Otrans,axiom,
% 0.31/0.48 ! [B2: $o > set_a,A2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ C2 @ B2 )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ C2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.trans
% 0.31/0.48 thf(fact_98_dual__order_Otrans,axiom,
% 0.31/0.48 ! [B2: set_a,A2: set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ C2 @ B2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.trans
% 0.31/0.48 thf(fact_99_dual__order_Orefl,axiom,
% 0.31/0.48 ! [A2: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ A2 @ A2 ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.refl
% 0.31/0.48 thf(fact_100_dual__order_Orefl,axiom,
% 0.31/0.48 ! [A2: set_set_a] : ( ord_le318720350_set_a @ A2 @ A2 ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.refl
% 0.31/0.48 thf(fact_101_dual__order_Orefl,axiom,
% 0.31/0.48 ! [A2: set_set_set_a] : ( ord_le2103863998_set_a @ A2 @ A2 ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.refl
% 0.31/0.48 thf(fact_102_dual__order_Orefl,axiom,
% 0.31/0.48 ! [A2: $o > set_a] : ( ord_less_eq_o_set_a @ A2 @ A2 ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.refl
% 0.31/0.48 thf(fact_103_dual__order_Orefl,axiom,
% 0.31/0.48 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.31/0.48
% 0.31/0.48 % dual_order.refl
% 0.31/0.48 thf(fact_104_order__trans,axiom,
% 0.31/0.48 ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e,Z2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ Y3 @ Z2 )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ X3 @ Z2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_trans
% 0.31/0.48 thf(fact_105_order__trans,axiom,
% 0.31/0.48 ! [X3: set_set_a,Y3: set_set_a,Z2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ Y3 @ Z2 )
% 0.31/0.48 => ( ord_le318720350_set_a @ X3 @ Z2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_trans
% 0.31/0.48 thf(fact_106_order__trans,axiom,
% 0.31/0.48 ! [X3: set_set_set_a,Y3: set_set_set_a,Z2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ Y3 @ Z2 )
% 0.31/0.48 => ( ord_le2103863998_set_a @ X3 @ Z2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_trans
% 0.31/0.48 thf(fact_107_order__trans,axiom,
% 0.31/0.48 ! [X3: $o > set_a,Y3: $o > set_a,Z2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ Y3 @ Z2 )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ X3 @ Z2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_trans
% 0.31/0.48 thf(fact_108_order__trans,axiom,
% 0.31/0.48 ! [X3: set_a,Y3: set_a,Z2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ Y3 @ Z2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_trans
% 0.31/0.48 thf(fact_109_order__class_Oorder_Oantisym,axiom,
% 0.31/0.48 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ B2 @ A2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.antisym
% 0.31/0.48 thf(fact_110_order__class_Oorder_Oantisym,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ B2 @ A2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.antisym
% 0.31/0.48 thf(fact_111_order__class_Oorder_Oantisym,axiom,
% 0.31/0.48 ! [A2: set_set_set_a,B2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ B2 @ A2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.antisym
% 0.31/0.48 thf(fact_112_order__class_Oorder_Oantisym,axiom,
% 0.31/0.48 ! [A2: $o > set_a,B2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ A2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.antisym
% 0.31/0.48 thf(fact_113_order__class_Oorder_Oantisym,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.48 => ( A2 = B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.antisym
% 0.31/0.48 thf(fact_114_ord__le__eq__trans,axiom,
% 0.31/0.48 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 0.31/0.48 => ( ( B2 = C2 )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_trans
% 0.31/0.48 thf(fact_115_ord__le__eq__trans,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( B2 = C2 )
% 0.31/0.48 => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_trans
% 0.31/0.48 thf(fact_116_ord__le__eq__trans,axiom,
% 0.31/0.48 ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( B2 = C2 )
% 0.31/0.48 => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_trans
% 0.31/0.48 thf(fact_117_ord__le__eq__trans,axiom,
% 0.31/0.48 ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( B2 = C2 )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_trans
% 0.31/0.48 thf(fact_118_ord__le__eq__trans,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( B2 = C2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_trans
% 0.31/0.48 thf(fact_119_ord__eq__le__trans,axiom,
% 0.31/0.48 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( A2 = B2 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ B2 @ C2 )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_trans
% 0.31/0.48 thf(fact_120_ord__eq__le__trans,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
% 0.31/0.48 ( ( A2 = B2 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_trans
% 0.31/0.48 thf(fact_121_ord__eq__le__trans,axiom,
% 0.31/0.48 ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
% 0.31/0.48 ( ( A2 = B2 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_trans
% 0.31/0.48 thf(fact_122_ord__eq__le__trans,axiom,
% 0.31/0.48 ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( A2 = B2 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_trans
% 0.31/0.48 thf(fact_123_ord__eq__le__trans,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,C2: set_a] :
% 0.31/0.48 ( ( A2 = B2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_trans
% 0.31/0.48 thf(fact_124_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ A3 @ B3 )
% 0.31/0.48 & ( ord_le700723212_a_a_e @ B3 @ A3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.eq_iff
% 0.31/0.48 thf(fact_125_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A3 @ B3 )
% 0.31/0.48 & ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.eq_iff
% 0.31/0.48 thf(fact_126_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A3 @ B3 )
% 0.31/0.48 & ( ord_le2103863998_set_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.eq_iff
% 0.31/0.48 thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: $o > set_a,B3: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ A3 @ B3 )
% 0.31/0.48 & ( ord_less_eq_o_set_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.eq_iff
% 0.31/0.48 thf(fact_128_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [A3: set_a,B3: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.48 & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_class.order.eq_iff
% 0.31/0.48 thf(fact_129_antisym__conv,axiom,
% 0.31/0.48 ! [Y3: set_Pr204296108_a_a_e,X3: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
% 0.31/0.48 = ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym_conv
% 0.31/0.48 thf(fact_130_antisym__conv,axiom,
% 0.31/0.48 ! [Y3: set_set_a,X3: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ X3 @ Y3 )
% 0.31/0.48 = ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym_conv
% 0.31/0.48 thf(fact_131_antisym__conv,axiom,
% 0.31/0.48 ! [Y3: set_set_set_a,X3: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ X3 @ Y3 )
% 0.31/0.48 = ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym_conv
% 0.31/0.48 thf(fact_132_antisym__conv,axiom,
% 0.31/0.48 ! [Y3: $o > set_a,X3: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
% 0.31/0.48 = ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym_conv
% 0.31/0.48 thf(fact_133_antisym__conv,axiom,
% 0.31/0.48 ! [Y3: set_a,X3: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 0.31/0.48 = ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym_conv
% 0.31/0.48 thf(fact_134_order_Otrans,axiom,
% 0.31/0.48 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ B2 @ C2 )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order.trans
% 0.31/0.48 thf(fact_135_order_Otrans,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order.trans
% 0.31/0.48 thf(fact_136_order_Otrans,axiom,
% 0.31/0.48 ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order.trans
% 0.31/0.48 thf(fact_137_order_Otrans,axiom,
% 0.31/0.48 ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order.trans
% 0.31/0.48 thf(fact_138_order_Otrans,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order.trans
% 0.31/0.48 thf(fact_139_eq__refl,axiom,
% 0.31/0.48 ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( X3 = Y3 )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ X3 @ Y3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_refl
% 0.31/0.48 thf(fact_140_eq__refl,axiom,
% 0.31/0.48 ! [X3: set_set_a,Y3: set_set_a] :
% 0.31/0.48 ( ( X3 = Y3 )
% 0.31/0.48 => ( ord_le318720350_set_a @ X3 @ Y3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_refl
% 0.31/0.48 thf(fact_141_eq__refl,axiom,
% 0.31/0.48 ! [X3: set_set_set_a,Y3: set_set_set_a] :
% 0.31/0.48 ( ( X3 = Y3 )
% 0.31/0.48 => ( ord_le2103863998_set_a @ X3 @ Y3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_refl
% 0.31/0.48 thf(fact_142_eq__refl,axiom,
% 0.31/0.48 ! [X3: $o > set_a,Y3: $o > set_a] :
% 0.31/0.48 ( ( X3 = Y3 )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ X3 @ Y3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_refl
% 0.31/0.48 thf(fact_143_eq__refl,axiom,
% 0.31/0.48 ! [X3: set_a,Y3: set_a] :
% 0.31/0.48 ( ( X3 = Y3 )
% 0.31/0.48 => ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_refl
% 0.31/0.48 thf(fact_144_antisym,axiom,
% 0.31/0.48 ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
% 0.31/0.48 => ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym
% 0.31/0.48 thf(fact_145_antisym,axiom,
% 0.31/0.48 ! [X3: set_set_a,Y3: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_le318720350_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym
% 0.31/0.48 thf(fact_146_antisym,axiom,
% 0.31/0.48 ! [X3: set_set_set_a,Y3: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_le2103863998_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym
% 0.31/0.48 thf(fact_147_antisym,axiom,
% 0.31/0.48 ! [X3: $o > set_a,Y3: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym
% 0.31/0.48 thf(fact_148_antisym,axiom,
% 0.31/0.48 ! [X3: set_a,Y3: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ Y3 @ X3 )
% 0.31/0.48 => ( X3 = Y3 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % antisym
% 0.31/0.48 thf(fact_149_eq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [X4: set_Pr204296108_a_a_e,Y4: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( ord_le700723212_a_a_e @ X4 @ Y4 )
% 0.31/0.48 & ( ord_le700723212_a_a_e @ Y4 @ X4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_iff
% 0.31/0.48 thf(fact_150_eq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [X4: set_set_a,Y4: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X4 @ Y4 )
% 0.31/0.48 & ( ord_le318720350_set_a @ Y4 @ X4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_iff
% 0.31/0.48 thf(fact_151_eq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [X4: set_set_set_a,Y4: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ X4 @ Y4 )
% 0.31/0.48 & ( ord_le2103863998_set_a @ Y4 @ X4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_iff
% 0.31/0.48 thf(fact_152_eq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [X4: $o > set_a,Y4: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X4 @ Y4 )
% 0.31/0.48 & ( ord_less_eq_o_set_a @ Y4 @ X4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_iff
% 0.31/0.48 thf(fact_153_eq__iff,axiom,
% 0.31/0.48 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 0.31/0.48 = ( ^ [X4: set_a,Y4: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 0.31/0.48 & ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % eq_iff
% 0.31/0.48 thf(fact_154_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_155_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,F: set_a > set_set_a,C2: set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_156_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_set_a,Y: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_157_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,F: set_a > set_set_set_a,C2: set_set_set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_158_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,F: set_a > $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_159_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C2: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_set_a,Y: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_160_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_set_set_a,Y: set_set_set_a] :
% 0.31/0.48 ( ( ord_le2103863998_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_161_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: $o > set_a,B2: $o > set_a,F: ( $o > set_a ) > set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_162_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_set_a,C2: set_set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_set_a,Y: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_163_ord__le__eq__subst,axiom,
% 0.31/0.48 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ( F @ B2 )
% 0.31/0.48 = C2 )
% 0.31/0.48 => ( ! [X2: set_set_a,Y: set_set_a] :
% 0.31/0.48 ( ( ord_le318720350_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_le_eq_subst
% 0.31/0.48 thf(fact_164_ord__eq__le__subst,axiom,
% 0.31/0.48 ! [A2: set_Pr204296108_a_a_e,F: ( $o > set_a ) > set_Pr204296108_a_a_e,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( F @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le700723212_a_a_e @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_subst
% 0.31/0.48 thf(fact_165_ord__eq__le__subst,axiom,
% 0.31/0.48 ! [A2: set_set_a,F: ( $o > set_a ) > set_set_a,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( F @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le318720350_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_subst
% 0.31/0.48 thf(fact_166_ord__eq__le__subst,axiom,
% 0.31/0.48 ! [A2: set_set_set_a,F: ( $o > set_a ) > set_set_set_a,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( F @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_le2103863998_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_subst
% 0.31/0.48 thf(fact_167_ord__eq__le__subst,axiom,
% 0.31/0.48 ! [A2: $o > set_a,F: ( $o > set_a ) > $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( F @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 0.31/0.48 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_o_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_subst
% 0.31/0.48 thf(fact_168_ord__eq__le__subst,axiom,
% 0.31/0.48 ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
% 0.31/0.48 ( ( A2
% 0.31/0.48 = ( F @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ord_eq_le_subst
% 0.31/0.48 thf(fact_169_order__subst2,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_subst2
% 0.31/0.48 thf(fact_170_order__subst1,axiom,
% 0.31/0.48 ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 0.31/0.48 => ( ! [X2: set_a,Y: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.48 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_subst1
% 0.31/0.48 thf(fact_171_mem__Collect__eq,axiom,
% 0.31/0.48 ! [A2: set_a,P2: set_a > $o] :
% 0.31/0.48 ( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
% 0.31/0.48 = ( P2 @ A2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % mem_Collect_eq
% 0.31/0.48 thf(fact_172_mem__Collect__eq,axiom,
% 0.31/0.48 ! [A2: produc715398134_a_a_e,P2: produc715398134_a_a_e > $o] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ A2 @ ( collec1369693387_a_a_e @ P2 ) )
% 0.31/0.48 = ( P2 @ A2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % mem_Collect_eq
% 0.31/0.48 thf(fact_173_Collect__mem__eq,axiom,
% 0.31/0.48 ! [A: set_set_a] :
% 0.31/0.48 ( ( collect_set_a
% 0.31/0.48 @ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
% 0.31/0.48 = A ) ).
% 0.31/0.48
% 0.31/0.48 % Collect_mem_eq
% 0.31/0.48 thf(fact_174_Collect__mem__eq,axiom,
% 0.31/0.48 ! [A: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( collec1369693387_a_a_e
% 0.31/0.48 @ ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ A ) )
% 0.31/0.48 = A ) ).
% 0.31/0.48
% 0.31/0.48 % Collect_mem_eq
% 0.31/0.48 thf(fact_175_rev__image__eqI,axiom,
% 0.31/0.48 ! [X3: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
% 0.31/0.48 ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev_image_eqI
% 0.31/0.48 thf(fact_176_rev__image__eqI,axiom,
% 0.31/0.48 ! [X3: set_a,A: set_set_a,B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e] :
% 0.31/0.48 ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev_image_eqI
% 0.31/0.48 thf(fact_177_rev__image__eqI,axiom,
% 0.31/0.48 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: set_a,F: produc715398134_a_a_e > set_a] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.48 => ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev_image_eqI
% 0.31/0.48 thf(fact_178_rev__image__eqI,axiom,
% 0.31/0.48 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.48 => ( ( B2
% 0.31/0.48 = ( F @ X3 ) )
% 0.31/0.48 => ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % rev_image_eqI
% 0.31/0.48 thf(fact_179_ball__imageD,axiom,
% 0.31/0.48 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
% 0.31/0.48 ( ! [X2: set_a] :
% 0.31/0.48 ( ( member_set_a @ X2 @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.48 => ( P2 @ X2 ) )
% 0.31/0.48 => ! [X: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X @ A )
% 0.31/0.48 => ( P2 @ ( F @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ball_imageD
% 0.31/0.48 thf(fact_180_image__cong,axiom,
% 0.31/0.48 ! [M: set_Pr204296108_a_a_e,N: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
% 0.31/0.48 ( ( M = N )
% 0.31/0.48 => ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X2 @ N )
% 0.31/0.48 => ( ( F @ X2 )
% 0.31/0.48 = ( G @ X2 ) ) )
% 0.31/0.48 => ( ( image_64240033_set_a @ F @ M )
% 0.31/0.48 = ( image_64240033_set_a @ G @ N ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_cong
% 0.31/0.48 thf(fact_181_bex__imageD,axiom,
% 0.31/0.48 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
% 0.31/0.48 ( ? [X: set_a] :
% 0.31/0.48 ( ( member_set_a @ X @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.48 & ( P2 @ X ) )
% 0.31/0.48 => ? [X2: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.48 & ( P2 @ ( F @ X2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % bex_imageD
% 0.31/0.48 thf(fact_182_image__iff,axiom,
% 0.31/0.48 ! [Z2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 0.31/0.48 ( ( member_set_a @ Z2 @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.48 = ( ? [X4: produc715398134_a_a_e] :
% 0.31/0.48 ( ( member1258382221_a_a_e @ X4 @ A )
% 0.31/0.48 & ( Z2
% 0.31/0.48 = ( F @ X4 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % image_iff
% 0.31/0.48 thf(fact_183_imageI,axiom,
% 0.31/0.48 ! [X3: set_a,A: set_set_a,F: set_a > set_a] :
% 0.31/0.48 ( ( member_set_a @ X3 @ A )
% 0.31/0.48 => ( member_set_a @ ( F @ X3 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % imageI
% 0.31/0.49 thf(fact_184_imageI,axiom,
% 0.31/0.49 ! [X3: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ X3 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % imageI
% 0.31/0.49 thf(fact_185_imageI,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % imageI
% 0.31/0.49 thf(fact_186_imageI,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % imageI
% 0.31/0.49 thf(fact_187_Collect__mono__iff,axiom,
% 0.31/0.49 ! [P2: a > $o,Q: a > $o] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
% 0.31/0.49 = ( ! [X4: a] :
% 0.31/0.49 ( ( P2 @ X4 )
% 0.31/0.49 => ( Q @ X4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Collect_mono_iff
% 0.31/0.49 thf(fact_188_set__eq__subset,axiom,
% 0.31/0.49 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 0.31/0.49 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 0.31/0.49 & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % set_eq_subset
% 0.31/0.49 thf(fact_189_subset__trans,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a,C3: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ B @ C3 )
% 0.31/0.49 => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_trans
% 0.31/0.49 thf(fact_190_Collect__mono,axiom,
% 0.31/0.49 ! [P2: a > $o,Q: a > $o] :
% 0.31/0.49 ( ! [X2: a] :
% 0.31/0.49 ( ( P2 @ X2 )
% 0.31/0.49 => ( Q @ X2 ) )
% 0.31/0.49 => ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Collect_mono
% 0.31/0.49 thf(fact_191_subset__iff,axiom,
% 0.31/0.49 ( ord_le318720350_set_a
% 0.31/0.49 = ( ^ [A4: set_set_a,B4: set_set_a] :
% 0.31/0.49 ! [T: set_a] :
% 0.31/0.49 ( ( member_set_a @ T @ A4 )
% 0.31/0.49 => ( member_set_a @ T @ B4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_iff
% 0.31/0.49 thf(fact_192_subset__iff,axiom,
% 0.31/0.49 ( ord_le700723212_a_a_e
% 0.31/0.49 = ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
% 0.31/0.49 ! [T: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ T @ A4 )
% 0.31/0.49 => ( member1258382221_a_a_e @ T @ B4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_iff
% 0.31/0.49 thf(fact_193_subset__iff,axiom,
% 0.31/0.49 ( ord_less_eq_set_a
% 0.31/0.49 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.49 ! [T: a] :
% 0.31/0.49 ( ( member_a @ T @ A4 )
% 0.31/0.49 => ( member_a @ T @ B4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_iff
% 0.31/0.49 thf(fact_194_equalityD2,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( ord_less_eq_set_a @ B @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 % equalityD2
% 0.31/0.49 thf(fact_195_subset__eq,axiom,
% 0.31/0.49 ( ord_le318720350_set_a
% 0.31/0.49 = ( ^ [A4: set_set_a,B4: set_set_a] :
% 0.31/0.49 ! [X4: set_a] :
% 0.31/0.49 ( ( member_set_a @ X4 @ A4 )
% 0.31/0.49 => ( member_set_a @ X4 @ B4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_eq
% 0.31/0.49 thf(fact_196_subset__eq,axiom,
% 0.31/0.49 ( ord_le700723212_a_a_e
% 0.31/0.49 = ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
% 0.31/0.49 ! [X4: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X4 @ A4 )
% 0.31/0.49 => ( member1258382221_a_a_e @ X4 @ B4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_eq
% 0.31/0.49 thf(fact_197_subset__eq,axiom,
% 0.31/0.49 ( ord_less_eq_set_a
% 0.31/0.49 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.49 ! [X4: a] :
% 0.31/0.49 ( ( member_a @ X4 @ A4 )
% 0.31/0.49 => ( member_a @ X4 @ B4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_eq
% 0.31/0.49 thf(fact_198_equalityE,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ~ ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % equalityE
% 0.31/0.49 thf(fact_199_subsetD,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a,C2: set_a] :
% 0.31/0.49 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 => ( ( member_set_a @ C2 @ A )
% 0.31/0.49 => ( member_set_a @ C2 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subsetD
% 0.31/0.49 thf(fact_200_subsetD,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ C2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ C2 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subsetD
% 0.31/0.49 thf(fact_201_subsetD,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a,C2: a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( ( member_a @ C2 @ A )
% 0.31/0.49 => ( member_a @ C2 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subsetD
% 0.31/0.49 thf(fact_202_in__mono,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 0.31/0.49 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 => ( ( member_set_a @ X3 @ A )
% 0.31/0.49 => ( member_set_a @ X3 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % in_mono
% 0.31/0.49 thf(fact_203_in__mono,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ X3 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % in_mono
% 0.31/0.49 thf(fact_204_in__mono,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a,X3: a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( ( member_a @ X3 @ A )
% 0.31/0.49 => ( member_a @ X3 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % in_mono
% 0.31/0.49 thf(fact_205_prod_Oinject,axiom,
% 0.31/0.49 ! [X1: set_a,X22: a > e,Y1: set_a,Y22: a > e] :
% 0.31/0.49 ( ( ( produc1134456688_a_a_e @ X1 @ X22 )
% 0.31/0.49 = ( produc1134456688_a_a_e @ Y1 @ Y22 ) )
% 0.31/0.49 = ( ( X1 = Y1 )
% 0.31/0.49 & ( X22 = Y22 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod.inject
% 0.31/0.49 thf(fact_206_old_Oprod_Oinject,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
% 0.31/0.49 ( ( ( produc1134456688_a_a_e @ A2 @ B2 )
% 0.31/0.49 = ( produc1134456688_a_a_e @ A5 @ B5 ) )
% 0.31/0.49 = ( ( A2 = A5 )
% 0.31/0.49 & ( B2 = B5 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.inject
% 0.31/0.49 thf(fact_207_eq__fst__iff,axiom,
% 0.31/0.49 ! [A2: set_a,P: produc715398134_a_a_e] :
% 0.31/0.49 ( ( A2
% 0.31/0.49 = ( produc1040928388_a_a_e @ P ) )
% 0.31/0.49 = ( ? [B3: a > e] :
% 0.31/0.49 ( P
% 0.31/0.49 = ( produc1134456688_a_a_e @ A2 @ B3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % eq_fst_iff
% 0.31/0.49 thf(fact_208_fst__conv,axiom,
% 0.31/0.49 ! [X1: set_a,X22: a > e] :
% 0.31/0.49 ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
% 0.31/0.49 = X1 ) ).
% 0.31/0.49
% 0.31/0.49 % fst_conv
% 0.31/0.49 thf(fact_209_fst__eqD,axiom,
% 0.31/0.49 ! [X3: set_a,Y3: a > e,A2: set_a] :
% 0.31/0.49 ( ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 0.31/0.49 = A2 )
% 0.31/0.49 => ( X3 = A2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % fst_eqD
% 0.31/0.49 thf(fact_210_fstI,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
% 0.31/0.49 ( ( X3
% 0.31/0.49 = ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
% 0.31/0.49 => ( ( produc1040928388_a_a_e @ X3 )
% 0.31/0.49 = Y3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % fstI
% 0.31/0.49 thf(fact_211_all__subset__image,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_set_a > $o] :
% 0.31/0.49 ( ( ! [B4: set_set_a] :
% 0.31/0.49 ( ( ord_le318720350_set_a @ B4 @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.49 => ( P2 @ B4 ) ) )
% 0.31/0.49 = ( ! [B4: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( ord_le700723212_a_a_e @ B4 @ A )
% 0.31/0.49 => ( P2 @ ( image_64240033_set_a @ F @ B4 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % all_subset_image
% 0.31/0.49 thf(fact_212_all__subset__image,axiom,
% 0.31/0.49 ! [F: a > a,A: set_a,P2: set_a > $o] :
% 0.31/0.49 ( ( ! [B4: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
% 0.31/0.49 => ( P2 @ B4 ) ) )
% 0.31/0.49 = ( ! [B4: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ B4 @ A )
% 0.31/0.49 => ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % all_subset_image
% 0.31/0.49 thf(fact_213_Greatest__equality,axiom,
% 0.31/0.49 ! [P2: set_a > $o,X3: set_a] :
% 0.31/0.49 ( ( P2 @ X3 )
% 0.31/0.49 => ( ! [Y: set_a] :
% 0.31/0.49 ( ( P2 @ Y )
% 0.31/0.49 => ( ord_less_eq_set_a @ Y @ X3 ) )
% 0.31/0.49 => ( ( order_Greatest_set_a @ P2 )
% 0.31/0.49 = X3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Greatest_equality
% 0.31/0.49 thf(fact_214_GreatestI2__order,axiom,
% 0.31/0.49 ! [P2: set_a > $o,X3: set_a,Q: set_a > $o] :
% 0.31/0.49 ( ( P2 @ X3 )
% 0.31/0.49 => ( ! [Y: set_a] :
% 0.31/0.49 ( ( P2 @ Y )
% 0.31/0.49 => ( ord_less_eq_set_a @ Y @ X3 ) )
% 0.31/0.49 => ( ! [X2: set_a] :
% 0.31/0.49 ( ( P2 @ X2 )
% 0.31/0.49 => ( ! [Y5: set_a] :
% 0.31/0.49 ( ( P2 @ Y5 )
% 0.31/0.49 => ( ord_less_eq_set_a @ Y5 @ X2 ) )
% 0.31/0.49 => ( Q @ X2 ) ) )
% 0.31/0.49 => ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % GreatestI2_order
% 0.31/0.49 thf(fact_215_old_Oprod_Oinducts,axiom,
% 0.31/0.49 ! [P2: produc715398134_a_a_e > $o,Prod: produc715398134_a_a_e] :
% 0.31/0.49 ( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
% 0.31/0.49 => ( P2 @ Prod ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.inducts
% 0.31/0.49 thf(fact_216_old_Oprod_Oexhaust,axiom,
% 0.31/0.49 ! [Y3: produc715398134_a_a_e] :
% 0.31/0.49 ~ ! [A6: set_a,B6: a > e] :
% 0.31/0.49 ( Y3
% 0.31/0.49 != ( produc1134456688_a_a_e @ A6 @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.exhaust
% 0.31/0.49 thf(fact_217_Pair__inject,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
% 0.31/0.49 ( ( ( produc1134456688_a_a_e @ A2 @ B2 )
% 0.31/0.49 = ( produc1134456688_a_a_e @ A5 @ B5 ) )
% 0.31/0.49 => ~ ( ( A2 = A5 )
% 0.31/0.49 => ( B2 != B5 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Pair_inject
% 0.31/0.49 thf(fact_218_prod__cases,axiom,
% 0.31/0.49 ! [P2: produc715398134_a_a_e > $o,P: produc715398134_a_a_e] :
% 0.31/0.49 ( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
% 0.31/0.49 => ( P2 @ P ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases
% 0.31/0.49 thf(fact_219_image__Fpow__mono,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 0.31/0.49 ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
% 0.31/0.49 => ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( finite33065488_a_a_e @ A ) ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image_Fpow_mono
% 0.31/0.49 thf(fact_220_not__sub,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a] :
% 0.31/0.49 ( ~ ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 => ? [A6: set_a] :
% 0.31/0.49 ( ( member_set_a @ A6 @ A )
% 0.31/0.49 & ~ ( member_set_a @ A6 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_sub
% 0.31/0.49 thf(fact_221_not__sub,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ~ ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 => ? [A6: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ A6 @ A )
% 0.31/0.49 & ~ ( member1258382221_a_a_e @ A6 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_sub
% 0.31/0.49 thf(fact_222_not__sub,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ~ ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ? [A6: a] :
% 0.31/0.49 ( ( member_a @ A6 @ A )
% 0.31/0.49 & ~ ( member_a @ A6 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_sub
% 0.31/0.49 thf(fact_223_eqsets__sub,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 % eqsets_sub
% 0.31/0.49 thf(fact_224_sub__which1,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 0.31/0.49 ( ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 | ( ord_le318720350_set_a @ B @ A ) )
% 0.31/0.49 => ( ( member_set_a @ X3 @ A )
% 0.31/0.49 => ( ~ ( member_set_a @ X3 @ B )
% 0.31/0.49 => ( ord_le318720350_set_a @ B @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sub_which1
% 0.31/0.49 thf(fact_225_sub__which1,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 | ( ord_le700723212_a_a_e @ B @ A ) )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( ~ ( member1258382221_a_a_e @ X3 @ B )
% 0.31/0.49 => ( ord_le700723212_a_a_e @ B @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sub_which1
% 0.31/0.49 thf(fact_226_sub__which1,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a,X3: a] :
% 0.31/0.49 ( ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 | ( ord_less_eq_set_a @ B @ A ) )
% 0.31/0.49 => ( ( member_a @ X3 @ A )
% 0.31/0.49 => ( ~ ( member_a @ X3 @ B )
% 0.31/0.49 => ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sub_which1
% 0.31/0.49 thf(fact_227_Fpow__mono,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( ord_le318720350_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Fpow_mono
% 0.31/0.49 thf(fact_228_le__convert,axiom,
% 0.31/0.49 ! [A2: set_a,B2: set_a,C2: set_a] :
% 0.31/0.49 ( ( A2 = B2 )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ A2 @ C2 )
% 0.31/0.49 => ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_convert
% 0.31/0.49 thf(fact_229_ge__convert,axiom,
% 0.31/0.49 ! [A2: set_a,B2: set_a,C2: set_a] :
% 0.31/0.49 ( ( A2 = B2 )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ C2 @ A2 )
% 0.31/0.49 => ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % ge_convert
% 0.31/0.49 thf(fact_230_mem__in__image3,axiom,
% 0.31/0.49 ! [B2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) )
% 0.31/0.49 => ? [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.49 & ( B2
% 0.31/0.49 = ( F @ X2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image3
% 0.31/0.49 thf(fact_231_mem__in__image2,axiom,
% 0.31/0.49 ! [A2: set_a,A: set_set_a,F: set_a > set_a] :
% 0.31/0.49 ( ( member_set_a @ A2 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image2
% 0.31/0.49 thf(fact_232_mem__in__image2,axiom,
% 0.31/0.49 ! [A2: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ A2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image2
% 0.31/0.49 thf(fact_233_mem__in__image2,axiom,
% 0.31/0.49 ! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ A2 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image2
% 0.31/0.49 thf(fact_234_mem__in__image2,axiom,
% 0.31/0.49 ! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ A2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image2
% 0.31/0.49 thf(fact_235_mem__in__image1,axiom,
% 0.31/0.49 ! [A: set_set_a,F: set_a > set_a,B: set_set_a,A2: set_a] :
% 0.31/0.49 ( ! [X2: set_a] :
% 0.31/0.49 ( ( member_set_a @ X2 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 0.31/0.49 => ( ( member_set_a @ A2 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image1
% 0.31/0.49 thf(fact_236_mem__in__image1,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a,A2: produc715398134_a_a_e] :
% 0.31/0.49 ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ A2 @ A )
% 0.31/0.49 => ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image1
% 0.31/0.49 thf(fact_237_mem__in__image1,axiom,
% 0.31/0.49 ! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: set_a] :
% 0.31/0.49 ( ! [X2: set_a] :
% 0.31/0.49 ( ( member_set_a @ X2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 0.31/0.49 => ( ( member_set_a @ A2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image1
% 0.31/0.49 thf(fact_238_mem__in__image1,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: produc715398134_a_a_e] :
% 0.31/0.49 ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ A2 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_in_image1
% 0.31/0.49 thf(fact_239_proper__subset,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 0.31/0.49 ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 => ( ~ ( member_set_a @ X3 @ A )
% 0.31/0.49 => ( ( member_set_a @ X3 @ B )
% 0.31/0.49 => ( A != B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % proper_subset
% 0.31/0.49 thf(fact_240_proper__subset,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 => ( ~ ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ X3 @ B )
% 0.31/0.49 => ( A != B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % proper_subset
% 0.31/0.49 thf(fact_241_proper__subset,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a,X3: a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( ~ ( member_a @ X3 @ A )
% 0.31/0.49 => ( ( member_a @ X3 @ B )
% 0.31/0.49 => ( A != B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % proper_subset
% 0.31/0.49 thf(fact_242_not__subseteq,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a] :
% 0.31/0.49 ( ~ ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 => ? [X2: set_a] :
% 0.31/0.49 ( ( member_set_a @ X2 @ A )
% 0.31/0.49 & ~ ( member_set_a @ X2 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_subseteq
% 0.31/0.49 thf(fact_243_not__subseteq,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ~ ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 => ? [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.49 & ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_subseteq
% 0.31/0.49 thf(fact_244_not__subseteq,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ~ ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ? [X2: a] :
% 0.31/0.49 ( ( member_a @ X2 @ A )
% 0.31/0.49 & ~ ( member_a @ X2 @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_subseteq
% 0.31/0.49 thf(fact_245_subset__self,axiom,
% 0.31/0.49 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.31/0.49
% 0.31/0.49 % subset_self
% 0.31/0.49 thf(fact_246_sets__not__eq,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a] :
% 0.31/0.49 ( ( A != B )
% 0.31/0.49 => ( ( ord_le318720350_set_a @ B @ A )
% 0.31/0.49 => ? [X2: set_a] :
% 0.31/0.49 ( ( member_set_a @ X2 @ A )
% 0.31/0.49 & ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sets_not_eq
% 0.31/0.49 thf(fact_247_sets__not__eq,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( A != B )
% 0.31/0.49 => ( ( ord_le700723212_a_a_e @ B @ A )
% 0.31/0.49 => ? [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.49 & ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sets_not_eq
% 0.31/0.49 thf(fact_248_sets__not__eq,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( A != B )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ B @ A )
% 0.31/0.49 => ? [X2: a] :
% 0.31/0.49 ( ( member_a @ X2 @ A )
% 0.31/0.49 & ~ ( member_a @ X2 @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sets_not_eq
% 0.31/0.49 thf(fact_249_sub__which2,axiom,
% 0.31/0.49 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 0.31/0.49 ( ( ( ord_le318720350_set_a @ A @ B )
% 0.31/0.49 | ( ord_le318720350_set_a @ B @ A ) )
% 0.31/0.49 => ( ~ ( member_set_a @ X3 @ A )
% 0.31/0.49 => ( ( member_set_a @ X3 @ B )
% 0.31/0.49 => ( ord_le318720350_set_a @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sub_which2
% 0.31/0.49 thf(fact_250_sub__which2,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ( ord_le700723212_a_a_e @ A @ B )
% 0.31/0.49 | ( ord_le700723212_a_a_e @ B @ A ) )
% 0.31/0.49 => ( ~ ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ X3 @ B )
% 0.31/0.49 => ( ord_le700723212_a_a_e @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sub_which2
% 0.31/0.49 thf(fact_251_sub__which2,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a,X3: a] :
% 0.31/0.49 ( ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 | ( ord_less_eq_set_a @ B @ A ) )
% 0.31/0.49 => ( ~ ( member_a @ X3 @ A )
% 0.31/0.49 => ( ( member_a @ X3 @ B )
% 0.31/0.49 => ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sub_which2
% 0.31/0.49 thf(fact_252_image__Pow__mono,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 0.31/0.49 ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
% 0.31/0.49 => ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) ) @ ( pow_set_a @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image_Pow_mono
% 0.31/0.49 thf(fact_253_Sup_OSUP__cong,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Sup: set_set_a > set_a] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ B )
% 0.31/0.49 => ( ( C3 @ X2 )
% 0.31/0.49 = ( D @ X2 ) ) )
% 0.31/0.49 => ( ( Sup @ ( image_64240033_set_a @ C3 @ A ) )
% 0.31/0.49 = ( Sup @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Sup.SUP_cong
% 0.31/0.49 thf(fact_254_Inf_OINF__cong,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Inf: set_set_a > set_a] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( ! [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X2 @ B )
% 0.31/0.49 => ( ( C3 @ X2 )
% 0.31/0.49 = ( D @ X2 ) ) )
% 0.31/0.49 => ( ( Inf @ ( image_64240033_set_a @ C3 @ A ) )
% 0.31/0.49 = ( Inf @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inf.INF_cong
% 0.31/0.49 thf(fact_255_subrelI,axiom,
% 0.31/0.49 ! [R: set_Pr204296108_a_a_e,S: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ! [X2: set_a,Y: a > e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ S ) )
% 0.31/0.49 => ( ord_le700723212_a_a_e @ R @ S ) ) ).
% 0.31/0.49
% 0.31/0.49 % subrelI
% 0.31/0.49 thf(fact_256_ssubst__Pair__rhs,axiom,
% 0.31/0.49 ! [R: set_a,S: a > e,R2: set_Pr204296108_a_a_e,S2: a > e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S ) @ R2 )
% 0.31/0.49 => ( ( S2 = S )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S2 ) @ R2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % ssubst_Pair_rhs
% 0.31/0.49 thf(fact_257_PowI,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % PowI
% 0.31/0.49 thf(fact_258_Pow__iff,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( member_set_a @ A @ ( pow_a @ B ) )
% 0.31/0.49 = ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 % Pow_iff
% 0.31/0.49 thf(fact_259_Pow__top,axiom,
% 0.31/0.49 ! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 % Pow_top
% 0.31/0.49 thf(fact_260_PowD,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( member_set_a @ A @ ( pow_a @ B ) )
% 0.31/0.49 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 % PowD
% 0.31/0.49 thf(fact_261_Pow__mono,axiom,
% 0.31/0.49 ! [A: set_a,B: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ A @ B )
% 0.31/0.49 => ( ord_le318720350_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Pow_mono
% 0.31/0.49 thf(fact_262_image__Pow__surj,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 0.31/0.49 ( ( ( image_64240033_set_a @ F @ A )
% 0.31/0.49 = B )
% 0.31/0.49 => ( ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) )
% 0.31/0.49 = ( pow_set_a @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image_Pow_surj
% 0.31/0.49 thf(fact_263_le__rel__bool__arg__iff,axiom,
% 0.31/0.49 ( ord_less_eq_o_set_a
% 0.31/0.49 = ( ^ [X5: $o > set_a,Y6: $o > set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.31/0.49 & ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_rel_bool_arg_iff
% 0.31/0.49 thf(fact_264_curryI,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
% 0.31/0.49 ( ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
% 0.31/0.49 => ( produc952037978_a_e_o @ F @ A2 @ B2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % curryI
% 0.31/0.49 thf(fact_265_curryD,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
% 0.31/0.49 ( ( produc952037978_a_e_o @ F @ A2 @ B2 )
% 0.31/0.49 => ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % curryD
% 0.31/0.49 thf(fact_266_curryE,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
% 0.31/0.49 ( ( produc952037978_a_e_o @ F @ A2 @ B2 )
% 0.31/0.49 => ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % curryE
% 0.31/0.49 thf(fact_267_fst__eq__Domain,axiom,
% 0.31/0.49 ! [R2: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( image_64240033_set_a @ produc1040928388_a_a_e @ R2 )
% 0.31/0.49 = ( domain_set_a_a_e @ R2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % fst_eq_Domain
% 0.31/0.49 thf(fact_268_Domain__fst,axiom,
% 0.31/0.49 ( domain_set_a_a_e
% 0.31/0.49 = ( image_64240033_set_a @ produc1040928388_a_a_e ) ) ).
% 0.31/0.49
% 0.31/0.49 % Domain_fst
% 0.31/0.49 thf(fact_269_RangeE,axiom,
% 0.31/0.49 ! [B2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) )
% 0.31/0.49 => ~ ! [A6: set_a] :
% 0.31/0.49 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B2 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % RangeE
% 0.31/0.49 thf(fact_270_Range__iff,axiom,
% 0.31/0.49 ! [A2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
% 0.31/0.49 = ( ? [Y4: set_a] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y4 @ A2 ) @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Range_iff
% 0.31/0.49 thf(fact_271_Range_Ocases,axiom,
% 0.31/0.49 ! [A2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
% 0.31/0.49 => ~ ! [A6: set_a] :
% 0.31/0.49 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ A2 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % Range.cases
% 0.31/0.49 thf(fact_272_Range_Osimps,axiom,
% 0.31/0.49 ! [A2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
% 0.31/0.49 = ( ? [A3: set_a,B3: a > e] :
% 0.31/0.49 ( ( A2 = B3 )
% 0.31/0.49 & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Range.simps
% 0.31/0.49 thf(fact_273_Range_Ointros,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
% 0.31/0.49 => ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Range.intros
% 0.31/0.49 thf(fact_274_Range_Oinducts,axiom,
% 0.31/0.49 ! [X3: a > e,R: set_Pr204296108_a_a_e,P2: ( a > e ) > $o] :
% 0.31/0.49 ( ( member_a_e @ X3 @ ( range_set_a_a_e @ R ) )
% 0.31/0.49 => ( ! [A6: set_a,B6: a > e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
% 0.31/0.49 => ( P2 @ B6 ) )
% 0.31/0.49 => ( P2 @ X3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Range.inducts
% 0.31/0.49 thf(fact_275_DomainE,axiom,
% 0.31/0.49 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 0.31/0.49 => ~ ! [B6: a > e] :
% 0.31/0.49 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % DomainE
% 0.31/0.49 thf(fact_276_Domain__iff,axiom,
% 0.31/0.49 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 0.31/0.49 = ( ? [Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ Y4 ) @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Domain_iff
% 0.31/0.49 thf(fact_277_Domain_Ocases,axiom,
% 0.31/0.49 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 0.31/0.49 => ~ ! [B6: a > e] :
% 0.31/0.49 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % Domain.cases
% 0.31/0.49 thf(fact_278_Domain_Osimps,axiom,
% 0.31/0.49 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 0.31/0.49 = ( ? [A3: set_a,B3: a > e] :
% 0.31/0.49 ( ( A2 = A3 )
% 0.31/0.49 & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Domain.simps
% 0.31/0.49 thf(fact_279_Domain_ODomainI,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
% 0.31/0.49 => ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Domain.DomainI
% 0.31/0.49 thf(fact_280_Domain_Oinducts,axiom,
% 0.31/0.49 ! [X3: set_a,R: set_Pr204296108_a_a_e,P2: set_a > $o] :
% 0.31/0.49 ( ( member_set_a @ X3 @ ( domain_set_a_a_e @ R ) )
% 0.31/0.49 => ( ! [A6: set_a,B6: a > e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
% 0.31/0.49 => ( P2 @ A6 ) )
% 0.31/0.49 => ( P2 @ X3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Domain.inducts
% 0.31/0.49 thf(fact_281_pair__in__swap__image,axiom,
% 0.31/0.49 ! [Y3: a > e,X3: set_a,A: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ Y3 @ X3 ) @ ( image_1875450791_set_a @ produc755714512_a_a_e @ A ) )
% 0.31/0.49 = ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 % pair_in_swap_image
% 0.31/0.49 thf(fact_282_pair__in__swap__image,axiom,
% 0.31/0.49 ! [Y3: set_a,X3: a > e,A: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y3 @ X3 ) @ ( image_376979367_a_a_e @ produc100464700_set_a @ A ) )
% 0.31/0.49 = ( member360729153_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 % pair_in_swap_image
% 0.31/0.49 thf(fact_283_swap__simp,axiom,
% 0.31/0.49 ! [X3: a > e,Y3: set_a] :
% 0.31/0.49 ( ( produc100464700_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) )
% 0.31/0.49 = ( produc1134456688_a_a_e @ Y3 @ X3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % swap_simp
% 0.31/0.49 thf(fact_284_swap__simp,axiom,
% 0.31/0.49 ! [X3: set_a,Y3: a > e] :
% 0.31/0.49 ( ( produc755714512_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 0.31/0.49 = ( produc479206876_set_a @ Y3 @ X3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % swap_simp
% 0.31/0.49 thf(fact_285_converse__iff,axiom,
% 0.31/0.49 ! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
% 0.31/0.49 = ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse_iff
% 0.31/0.49 thf(fact_286_converse__iff,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
% 0.31/0.49 = ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse_iff
% 0.31/0.49 thf(fact_287_converse_Oinducts,axiom,
% 0.31/0.49 ! [X1: a > e,X22: set_a,R: set_Pr204296108_a_a_e,P2: ( a > e ) > set_a > $o] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ X1 @ X22 ) @ ( converse_set_a_a_e @ R ) )
% 0.31/0.49 => ( ! [A6: set_a,B6: a > e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
% 0.31/0.49 => ( P2 @ B6 @ A6 ) )
% 0.31/0.49 => ( P2 @ X1 @ X22 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.inducts
% 0.31/0.49 thf(fact_288_converse_Oinducts,axiom,
% 0.31/0.49 ! [X1: set_a,X22: a > e,R: set_Pr305929312_set_a,P2: set_a > ( a > e ) > $o] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) @ ( converse_a_e_set_a @ R ) )
% 0.31/0.49 => ( ! [A6: a > e,B6: set_a] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ A6 @ B6 ) @ R )
% 0.31/0.49 => ( P2 @ B6 @ A6 ) )
% 0.31/0.49 => ( P2 @ X1 @ X22 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.inducts
% 0.31/0.49 thf(fact_289_converse_Ointros,axiom,
% 0.31/0.49 ! [A2: a > e,B2: set_a,R: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ R )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ ( converse_a_e_set_a @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.intros
% 0.31/0.49 thf(fact_290_converse_Ointros,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
% 0.31/0.49 => ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ ( converse_set_a_a_e @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.intros
% 0.31/0.49 thf(fact_291_converse_Osimps,axiom,
% 0.31/0.49 ! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
% 0.31/0.49 = ( ? [A3: set_a,B3: a > e] :
% 0.31/0.49 ( ( A1 = B3 )
% 0.31/0.49 & ( A22 = A3 )
% 0.31/0.49 & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.simps
% 0.31/0.49 thf(fact_292_converse_Osimps,axiom,
% 0.31/0.49 ! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
% 0.31/0.49 = ( ? [A3: a > e,B3: set_a] :
% 0.31/0.49 ( ( A1 = B3 )
% 0.31/0.49 & ( A22 = A3 )
% 0.31/0.49 & ( member360729153_set_a @ ( produc479206876_set_a @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.simps
% 0.31/0.49 thf(fact_293_converse_Ocases,axiom,
% 0.31/0.49 ! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A22 @ A1 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.cases
% 0.31/0.49 thf(fact_294_converse_Ocases,axiom,
% 0.31/0.49 ! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
% 0.31/0.49 => ( member360729153_set_a @ ( produc479206876_set_a @ A22 @ A1 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % converse.cases
% 0.31/0.49 thf(fact_295_converseE,axiom,
% 0.31/0.49 ! [Yx: produc1965228714_set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member360729153_set_a @ Yx @ ( converse_set_a_a_e @ R ) )
% 0.31/0.49 => ~ ! [X2: set_a,Y: a > e] :
% 0.31/0.49 ( ( Yx
% 0.31/0.49 = ( produc479206876_set_a @ Y @ X2 ) )
% 0.31/0.49 => ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converseE
% 0.31/0.49 thf(fact_296_converseE,axiom,
% 0.31/0.49 ! [Yx: produc715398134_a_a_e,R: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ Yx @ ( converse_a_e_set_a @ R ) )
% 0.31/0.49 => ~ ! [X2: a > e,Y: set_a] :
% 0.31/0.49 ( ( Yx
% 0.31/0.49 = ( produc1134456688_a_a_e @ Y @ X2 ) )
% 0.31/0.49 => ~ ( member360729153_set_a @ ( produc479206876_set_a @ X2 @ Y ) @ R ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % converseE
% 0.31/0.49 thf(fact_297_converseD,axiom,
% 0.31/0.49 ! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % converseD
% 0.31/0.49 thf(fact_298_converseD,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
% 0.31/0.49 => ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).
% 0.31/0.49
% 0.31/0.49 % converseD
% 0.31/0.49 thf(fact_299_prod_Oswap__def,axiom,
% 0.31/0.49 ( produc100464700_set_a
% 0.31/0.49 = ( ^ [P3: produc1965228714_set_a] : ( produc1134456688_a_a_e @ ( produc1809800494_set_a @ P3 ) @ ( produc385678576_set_a @ P3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod.swap_def
% 0.31/0.49 thf(fact_300_prod_Oswap__def,axiom,
% 0.31/0.49 ( produc755714512_a_a_e
% 0.31/0.49 = ( ^ [P3: produc715398134_a_a_e] : ( produc479206876_set_a @ ( produc317566658_a_a_e @ P3 ) @ ( produc1040928388_a_a_e @ P3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod.swap_def
% 0.31/0.49 thf(fact_301_map__prod__imageI,axiom,
% 0.31/0.49 ! [A2: set_a,B2: a > e,R2: set_Pr204296108_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R2 )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % map_prod_imageI
% 0.31/0.49 thf(fact_302_UNIV__I,axiom,
% 0.31/0.49 ! [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_I
% 0.31/0.49 thf(fact_303_UNIV__I,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X3 @ top_to1601855068_a_a_e ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_I
% 0.31/0.49 thf(fact_304_map__prod__simp,axiom,
% 0.31/0.49 ! [F: set_a > set_a,G: ( a > e ) > a > e,A2: set_a,B2: a > e] :
% 0.31/0.49 ( ( produc1443219402_e_a_e @ F @ G @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
% 0.31/0.49 = ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % map_prod_simp
% 0.31/0.49 thf(fact_305_fst__map__prod,axiom,
% 0.31/0.49 ! [F: set_a > set_a,G: ( a > e ) > a > e,X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( produc1040928388_a_a_e @ ( produc1443219402_e_a_e @ F @ G @ X3 ) )
% 0.31/0.49 = ( F @ ( produc1040928388_a_a_e @ X3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % fst_map_prod
% 0.31/0.49 thf(fact_306_prod_Ocollapse,axiom,
% 0.31/0.49 ! [Prod: produc715398134_a_a_e] :
% 0.31/0.49 ( ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) )
% 0.31/0.49 = Prod ) ).
% 0.31/0.49
% 0.31/0.49 % prod.collapse
% 0.31/0.49 thf(fact_307_fst__swap,axiom,
% 0.31/0.49 ! [X3: produc1965228714_set_a] :
% 0.31/0.49 ( ( produc1040928388_a_a_e @ ( produc100464700_set_a @ X3 ) )
% 0.31/0.49 = ( produc1809800494_set_a @ X3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % fst_swap
% 0.31/0.49 thf(fact_308_snd__swap,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( produc1809800494_set_a @ ( produc755714512_a_a_e @ X3 ) )
% 0.31/0.49 = ( produc1040928388_a_a_e @ X3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % snd_swap
% 0.31/0.49 thf(fact_309_range__fst,axiom,
% 0.31/0.49 ( ( image_64240033_set_a @ produc1040928388_a_a_e @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a ) ).
% 0.31/0.49
% 0.31/0.49 % range_fst
% 0.31/0.49 thf(fact_310_rangeI,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] : ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ).
% 0.31/0.49
% 0.31/0.49 % rangeI
% 0.31/0.49 thf(fact_311_range__eqI,axiom,
% 0.31/0.49 ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] :
% 0.31/0.49 ( ( B2
% 0.31/0.49 = ( F @ X3 ) )
% 0.31/0.49 => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % range_eqI
% 0.31/0.49 thf(fact_312_surj__def,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a] :
% 0.31/0.49 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a )
% 0.31/0.49 = ( ! [Y4: set_a] :
% 0.31/0.49 ? [X4: produc715398134_a_a_e] :
% 0.31/0.49 ( Y4
% 0.31/0.49 = ( F @ X4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % surj_def
% 0.31/0.49 thf(fact_313_surjI,axiom,
% 0.31/0.49 ! [G: produc715398134_a_a_e > set_a,F: set_a > produc715398134_a_a_e] :
% 0.31/0.49 ( ! [X2: set_a] :
% 0.31/0.49 ( ( G @ ( F @ X2 ) )
% 0.31/0.49 = X2 )
% 0.31/0.49 => ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a ) ) ).
% 0.31/0.49
% 0.31/0.49 % surjI
% 0.31/0.49 thf(fact_314_surjE,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
% 0.31/0.49 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a )
% 0.31/0.49 => ~ ! [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( Y3
% 0.31/0.49 != ( F @ X2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % surjE
% 0.31/0.49 thf(fact_315_surjD,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
% 0.31/0.49 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a )
% 0.31/0.49 => ? [X2: produc715398134_a_a_e] :
% 0.31/0.49 ( Y3
% 0.31/0.49 = ( F @ X2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % surjD
% 0.31/0.49 thf(fact_316_map__prod__surj,axiom,
% 0.31/0.49 ! [F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
% 0.31/0.49 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a )
% 0.31/0.49 => ( ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
% 0.31/0.49 = top_top_set_set_a )
% 0.31/0.49 => ( ( image_2101504031_set_a @ ( produc387505142_set_a @ F @ G ) @ top_to970499581_a_a_e )
% 0.31/0.49 = top_to144004695_set_a ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % map_prod_surj
% 0.31/0.49 thf(fact_317_top_Oextremum__uniqueI,axiom,
% 0.31/0.49 ! [A2: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
% 0.31/0.49 => ( A2 = top_top_set_a ) ) ).
% 0.31/0.49
% 0.31/0.49 % top.extremum_uniqueI
% 0.31/0.49 thf(fact_318_top_Oextremum__unique,axiom,
% 0.31/0.49 ! [A2: set_a] :
% 0.31/0.49 ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
% 0.31/0.49 = ( A2 = top_top_set_a ) ) ).
% 0.31/0.49
% 0.31/0.49 % top.extremum_unique
% 0.31/0.49 thf(fact_319_top__greatest,axiom,
% 0.31/0.49 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% 0.31/0.49
% 0.31/0.49 % top_greatest
% 0.31/0.49 thf(fact_320_subset__UNIV,axiom,
% 0.31/0.49 ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% 0.31/0.49
% 0.31/0.49 % subset_UNIV
% 0.31/0.49 thf(fact_321_sndI,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
% 0.31/0.49 ( ( X3
% 0.31/0.49 = ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
% 0.31/0.49 => ( ( produc317566658_a_a_e @ X3 )
% 0.31/0.49 = Z2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sndI
% 0.31/0.49 thf(fact_322_snd__eqD,axiom,
% 0.31/0.49 ! [X3: set_a,Y3: a > e,A2: a > e] :
% 0.31/0.49 ( ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 0.31/0.49 = A2 )
% 0.31/0.49 => ( Y3 = A2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % snd_eqD
% 0.31/0.49 thf(fact_323_snd__conv,axiom,
% 0.31/0.49 ! [X1: set_a,X22: a > e] :
% 0.31/0.49 ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
% 0.31/0.49 = X22 ) ).
% 0.31/0.49
% 0.31/0.49 % snd_conv
% 0.31/0.49 thf(fact_324_eq__snd__iff,axiom,
% 0.31/0.49 ! [B2: a > e,P: produc715398134_a_a_e] :
% 0.31/0.49 ( ( B2
% 0.31/0.49 = ( produc317566658_a_a_e @ P ) )
% 0.31/0.49 = ( ? [A3: set_a] :
% 0.31/0.49 ( P
% 0.31/0.49 = ( produc1134456688_a_a_e @ A3 @ B2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % eq_snd_iff
% 0.31/0.49 thf(fact_325_prod__eq__iff,axiom,
% 0.31/0.49 ( ( ^ [Y2: produc715398134_a_a_e,Z: produc715398134_a_a_e] : ( Y2 = Z ) )
% 0.31/0.49 = ( ^ [S3: produc715398134_a_a_e,T: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ( produc1040928388_a_a_e @ S3 )
% 0.31/0.49 = ( produc1040928388_a_a_e @ T ) )
% 0.31/0.49 & ( ( produc317566658_a_a_e @ S3 )
% 0.31/0.49 = ( produc317566658_a_a_e @ T ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_eq_iff
% 0.31/0.49 thf(fact_326_prod_Oexpand,axiom,
% 0.31/0.49 ! [Prod: produc715398134_a_a_e,Prod2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ( ( produc1040928388_a_a_e @ Prod )
% 0.31/0.49 = ( produc1040928388_a_a_e @ Prod2 ) )
% 0.31/0.49 & ( ( produc317566658_a_a_e @ Prod )
% 0.31/0.49 = ( produc317566658_a_a_e @ Prod2 ) ) )
% 0.31/0.49 => ( Prod = Prod2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod.expand
% 0.31/0.49 thf(fact_327_prod__eqI,axiom,
% 0.31/0.49 ! [P: produc715398134_a_a_e,Q2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( ( produc1040928388_a_a_e @ P )
% 0.31/0.49 = ( produc1040928388_a_a_e @ Q2 ) )
% 0.31/0.49 => ( ( ( produc317566658_a_a_e @ P )
% 0.31/0.49 = ( produc317566658_a_a_e @ Q2 ) )
% 0.31/0.49 => ( P = Q2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_eqI
% 0.31/0.49 thf(fact_328_UNIV__eq__I,axiom,
% 0.31/0.49 ! [A: set_set_a] :
% 0.31/0.49 ( ! [X2: set_a] : ( member_set_a @ X2 @ A )
% 0.31/0.49 => ( top_top_set_set_a = A ) ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_eq_I
% 0.31/0.49 thf(fact_329_UNIV__eq__I,axiom,
% 0.31/0.49 ! [A: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ! [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ A )
% 0.31/0.49 => ( top_to1601855068_a_a_e = A ) ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_eq_I
% 0.31/0.49 thf(fact_330_UNIV__witness,axiom,
% 0.31/0.49 ? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_witness
% 0.31/0.49 thf(fact_331_UNIV__witness,axiom,
% 0.31/0.49 ? [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ top_to1601855068_a_a_e ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_witness
% 0.31/0.49 thf(fact_332_surjective__pairing,axiom,
% 0.31/0.49 ! [T2: produc715398134_a_a_e] :
% 0.31/0.49 ( T2
% 0.31/0.49 = ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ T2 ) @ ( produc317566658_a_a_e @ T2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % surjective_pairing
% 0.31/0.49 thf(fact_333_prod_Oexhaust__sel,axiom,
% 0.31/0.49 ! [Prod: produc715398134_a_a_e] :
% 0.31/0.49 ( Prod
% 0.31/0.49 = ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod.exhaust_sel
% 0.31/0.49 thf(fact_334_prod__fun__imageE,axiom,
% 0.31/0.49 ! [C2: produc715398134_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e,R2: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ C2 @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) )
% 0.31/0.49 => ~ ! [X2: set_a,Y: a > e] :
% 0.31/0.49 ( ( C2
% 0.31/0.49 = ( produc1134456688_a_a_e @ ( F @ X2 ) @ ( G @ Y ) ) )
% 0.31/0.49 => ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_fun_imageE
% 0.31/0.49 thf(fact_335_exI__realizer,axiom,
% 0.31/0.49 ! [P2: ( a > e ) > set_a > $o,Y3: a > e,X3: set_a] :
% 0.31/0.49 ( ( P2 @ Y3 @ X3 )
% 0.31/0.49 => ( P2 @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % exI_realizer
% 0.31/0.49 thf(fact_336_conjI__realizer,axiom,
% 0.31/0.49 ! [P2: set_a > $o,P: set_a,Q: ( a > e ) > $o,Q2: a > e] :
% 0.31/0.49 ( ( P2 @ P )
% 0.31/0.49 => ( ( Q @ Q2 )
% 0.31/0.49 => ( ( P2 @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) )
% 0.31/0.49 & ( Q @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % conjI_realizer
% 0.31/0.49 thf(fact_337_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.31/0.49 ! [P2: set_a > ( a > e ) > $o,X3: set_a,Y3: a > e,A2: produc715398134_a_a_e] :
% 0.31/0.49 ( ( P2 @ X3 @ Y3 )
% 0.31/0.49 => ( ( A2
% 0.31/0.49 = ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 0.31/0.49 => ( P2 @ ( produc1040928388_a_a_e @ A2 ) @ ( produc317566658_a_a_e @ A2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % BNF_Greatest_Fixpoint.subst_Pair
% 0.31/0.49 thf(fact_338_top__empty__eq,axiom,
% 0.31/0.49 ( top_top_set_a_o
% 0.31/0.49 = ( ^ [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % top_empty_eq
% 0.31/0.49 thf(fact_339_top__empty__eq,axiom,
% 0.31/0.49 ( top_to2000253225_a_e_o
% 0.31/0.49 = ( ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ top_to1601855068_a_a_e ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % top_empty_eq
% 0.31/0.49 thf(fact_340_exE__realizer_H,axiom,
% 0.31/0.49 ! [P2: ( a > e ) > set_a > $o,P: produc715398134_a_a_e] :
% 0.31/0.49 ( ( P2 @ ( produc317566658_a_a_e @ P ) @ ( produc1040928388_a_a_e @ P ) )
% 0.31/0.49 => ~ ! [X2: set_a,Y: a > e] :
% 0.31/0.49 ~ ( P2 @ Y @ X2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % exE_realizer'
% 0.31/0.49 thf(fact_341_Collect__split__mono__strong,axiom,
% 0.31/0.49 ! [X6: set_set_a,A: set_Pr204296108_a_a_e,Y7: set_a_e,P2: set_a > ( a > e ) > $o,Q: set_a > ( a > e ) > $o] :
% 0.31/0.49 ( ( X6
% 0.31/0.49 = ( image_64240033_set_a @ produc1040928388_a_a_e @ A ) )
% 0.31/0.49 => ( ( Y7
% 0.31/0.49 = ( image_1149879910_e_a_e @ produc317566658_a_a_e @ A ) )
% 0.31/0.49 => ( ! [X2: set_a] :
% 0.31/0.49 ( ( member_set_a @ X2 @ X6 )
% 0.31/0.49 => ! [Xa2: a > e] :
% 0.31/0.49 ( ( member_a_e @ Xa2 @ Y7 )
% 0.31/0.49 => ( ( P2 @ X2 @ Xa2 )
% 0.31/0.49 => ( Q @ X2 @ Xa2 ) ) ) )
% 0.31/0.49 => ( ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) )
% 0.31/0.49 => ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Collect_split_mono_strong
% 0.31/0.49 thf(fact_342_sndOp__def,axiom,
% 0.31/0.49 ( bNF_sn375433437_a_a_e
% 0.31/0.49 = ( ^ [P4: set_a > set_a > $o,Q3: set_a > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( bNF_pi929741261_a_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sndOp_def
% 0.31/0.49 thf(fact_343_Product__Type_OCollect__case__prodD,axiom,
% 0.31/0.49 ! [X3: produc715398134_a_a_e,A: set_a > ( a > e ) > $o] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ X3 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ A ) ) )
% 0.31/0.49 => ( A @ ( produc1040928388_a_a_e @ X3 ) @ ( produc317566658_a_a_e @ X3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Product_Type.Collect_case_prodD
% 0.31/0.49 thf(fact_344_fstOp__def,axiom,
% 0.31/0.49 ( bNF_fs1024561536_e_a_e
% 0.31/0.49 = ( ^ [P4: set_a > ( a > e ) > $o,Q3: ( a > e ) > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Ac ) @ ( bNF_pi1175744974_e_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % fstOp_def
% 0.31/0.49 thf(fact_345_mem__case__prodE,axiom,
% 0.31/0.49 ! [Z2: set_a,C2: set_a > ( a > e ) > set_set_a,P: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member_set_a @ Z2 @ ( produc1620938593_set_a @ C2 @ P ) )
% 0.31/0.49 => ~ ! [X2: set_a,Y: a > e] :
% 0.31/0.49 ( ( P
% 0.31/0.49 = ( produc1134456688_a_a_e @ X2 @ Y ) )
% 0.31/0.49 => ~ ( member_set_a @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_case_prodE
% 0.31/0.49 thf(fact_346_mem__case__prodE,axiom,
% 0.31/0.49 ! [Z2: produc715398134_a_a_e,C2: set_a > ( a > e ) > set_Pr204296108_a_a_e,P: produc715398134_a_a_e] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ Z2 @ ( produc1096319049_a_a_e @ C2 @ P ) )
% 0.31/0.49 => ~ ! [X2: set_a,Y: a > e] :
% 0.31/0.49 ( ( P
% 0.31/0.49 = ( produc1134456688_a_a_e @ X2 @ Y ) )
% 0.31/0.49 => ~ ( member1258382221_a_a_e @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_case_prodE
% 0.31/0.49 thf(fact_347_in__rel__def,axiom,
% 0.31/0.49 ( fun_in_rel_set_a_a_e
% 0.31/0.49 = ( ^ [R3: set_Pr204296108_a_a_e,X4: set_a,Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X4 @ Y4 ) @ R3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % in_rel_def
% 0.31/0.49 thf(fact_348_fstOp__in,axiom,
% 0.31/0.49 ! [Ac2: produc715398134_a_a_e,P2: set_a > ( a > e ) > $o,Q: ( a > e ) > ( a > e ) > $o] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom658603713_e_a_e @ P2 @ Q ) ) ) )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( bNF_fs1024561536_e_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % fstOp_in
% 0.31/0.49 thf(fact_349_sndOp__in,axiom,
% 0.31/0.49 ! [Ac2: produc715398134_a_a_e,P2: set_a > set_a > $o,Q: set_a > ( a > e ) > $o] :
% 0.31/0.49 ( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom779348378_a_a_e @ P2 @ Q ) ) ) )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( bNF_sn375433437_a_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sndOp_in
% 0.31/0.49 thf(fact_350_image2__eqI,axiom,
% 0.31/0.49 ! [B2: set_a,F: set_a > set_a,X3: set_a,C2: a > e,G: set_a > a > e,A: set_set_a] :
% 0.31/0.49 ( ( B2
% 0.31/0.49 = ( F @ X3 ) )
% 0.31/0.49 => ( ( C2
% 0.31/0.49 = ( G @ X3 ) )
% 0.31/0.49 => ( ( member_set_a @ X3 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr1872575811_a_a_e @ A @ F @ G ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image2_eqI
% 0.31/0.49 thf(fact_351_image2__eqI,axiom,
% 0.31/0.49 ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,C2: a > e,G: produc715398134_a_a_e > a > e,A: set_Pr204296108_a_a_e] :
% 0.31/0.49 ( ( B2
% 0.31/0.49 = ( F @ X3 ) )
% 0.31/0.49 => ( ( C2
% 0.31/0.49 = ( G @ X3 ) )
% 0.31/0.49 => ( ( member1258382221_a_a_e @ X3 @ A )
% 0.31/0.49 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr2072183031_a_a_e @ A @ F @ G ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image2_eqI
% 0.31/0.49
% 0.31/0.49 % Conjectures (2)
% 0.31/0.49 thf(conj_0,hypothesis,
% 0.31/0.49 ! [A7: set_a] :
% 0.31/0.49 ( ( member_set_a @ A7 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ a1 @ A7 )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ a2 @ A7 )
% 0.31/0.49 => ( ( ord_less_eq_set_a @ a3 @ A7 )
% 0.31/0.49 => thesis ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(conj_1,conjecture,
% 0.31/0.49 thesis ).
% 0.31/0.49
% 0.31/0.49 %------------------------------------------------------------------------------
% 0.31/0.49 ------------------------
% 2.91/1.52 ---- Embedded in THF ---
% 2.91/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.91/1.52 %%% Generated on Tue Feb 24 08:23:41 EST 2026
% 2.91/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.91/1.52 %%% Logic specification used:
% 2.91/1.52 %%% thf(spec, logic, $$dhol).
% 2.91/1.52
% 2.91/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2DTF17883.p
% See solution above
% 2.91/1.52 ------------------------
% 2.91/1.58 ---- Cleaned THF ---
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J_J,type,
% 2.91/1.58 set_Pr1710391469_a_a_e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
% 2.91/1.58 produc169085773_a_a_e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J_J,type,
% 2.91/1.58 set_se821189730_a_a_e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 2.91/1.58 set_Pr2045688071_set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
% 2.91/1.58 set_Pr204296108_a_a_e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 2.91/1.58 set_Pr305929312_set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 2.91/1.58 produc1691597095_set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
% 2.91/1.58 produc715398134_a_a_e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J,type,
% 2.91/1.58 produc1965228714_set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.91/1.58 set_set_set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.91/1.58 set_set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__e_J_J,type,
% 2.91/1.58 set_a_e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.91/1.58 set_a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_tf__e,type,
% 2.91/1.58 e: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(ty_n_tf__a,type,
% 2.91/1.58 a: $tType ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_BNF__Def_OfstOp_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 bNF_fs1024561536_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > produc715398134_a_a_e > produc715398134_a_a_e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_BNF__Def_Opick__middlep_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 bNF_pi1175744974_e_a_e: ( set_a > ( a > e ) > $o ) > ( ( a > e ) > ( a > e ) > $o ) > set_a > ( a > e ) > a > e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_BNF__Def_Opick__middlep_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 bNF_pi929741261_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > set_a > ( a > e ) > set_a ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_BNF__Def_OsndOp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 bNF_sn375433437_a_a_e: ( set_a > set_a > $o ) > ( set_a > ( a > e ) > $o ) > produc715398134_a_a_e > produc715398134_a_a_e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 bNF_Gr2072183031_a_a_e: set_Pr204296108_a_a_e > ( produc715398134_a_a_e > set_a ) > ( produc715398134_a_a_e > a > e ) > set_Pr204296108_a_a_e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 bNF_Gr1872575811_a_a_e: set_set_a > ( set_a > set_a ) > ( set_a > a > e ) > set_Pr204296108_a_a_e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Finite__Set_OFpow_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
% 2.91/1.58 finite33065488_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
% 2.91/1.58 finite_Fpow_set_a: set_set_a > set_set_set_a ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Finite__Set_OFpow_001tf__a,type,
% 2.91/1.58 finite_Fpow_a: set_a > set_set_a ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Fun__Def_Oin__rel_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
% 2.91/1.58 fun_in_rel_set_a_a_e: set_Pr204296108_a_a_e > set_a > ( a > e ) > $o ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
% 2.91/1.58 ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__e_J_J,type,
% 2.91/1.58 ord_less_eq_set_a_e: set_a_e > set_a_e > $o ).
% 2.91/1.58
% 2.91/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J_J,type,
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% 2.91/1.58 member_a: a > set_a > $o ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_C,type,
% 2.91/1.58 c: set_Pr204296108_a_a_e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_a1,type,
% 2.91/1.58 a1: set_a ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_a2,type,
% 2.91/1.58 a2: set_a ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_a3,type,
% 2.91/1.58 a3: set_a ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_b1,type,
% 2.91/1.58 b1: a > e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_b2,type,
% 2.91/1.58 b2: a > e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_b3,type,
% 2.91/1.58 b3: a > e ).
% 2.91/1.58
% 2.91/1.58 thf(sy_v_thesis____,type,
% 2.91/1.58 thesis: $o ).
% 2.91/1.58
% 2.91/1.58 thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
% 2.91/1.58 ! [X: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X @ c )
% 2.91/1.58 => ! [Xa: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ Xa @ c )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ X ) @ ( produc1040928388_a_a_e @ Xa ) )
% 2.91/1.58 | ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ Xa ) @ ( produc1040928388_a_a_e @ X ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
% 2.91/1.58 ? [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
% 2.91/1.58 & ( ord_less_eq_set_a @ a1 @ X2 )
% 2.91/1.58 & ( ord_less_eq_set_a @ a2 @ X2 )
% 2.91/1.58 & ( ord_less_eq_set_a @ a3 @ X2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 2.91/1.58 ( ( ord_less_eq_set_a @ a1 @ a2 )
% 2.91/1.58 | ( ord_less_eq_set_a @ a2 @ a1 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 2.91/1.58 ( ( ord_less_eq_set_a @ a1 @ a3 )
% 2.91/1.58 | ( ord_less_eq_set_a @ a3 @ a1 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
% 2.91/1.58 ( ( ord_less_eq_set_a @ a2 @ a3 )
% 2.91/1.58 | ( ord_less_eq_set_a @ a3 @ a2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_5_A_I3_J,axiom,
% 2.91/1.58 member_set_a @ a3 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_6_A_I2_J,axiom,
% 2.91/1.58 member_set_a @ a2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_7_A_I1_J,axiom,
% 2.91/1.58 member_set_a @ a1 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
% 2.91/1.58 member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a1 @ b1 ) @ c ).
% 2.91/1.58
% 2.91/1.58 thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
% 2.91/1.58 member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a2 @ b2 ) @ c ).
% 2.91/1.58
% 2.91/1.58 thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
% 2.91/1.58 member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a3 @ b3 ) @ c ).
% 2.91/1.58
% 2.91/1.58 thf(fact_11_subsetI,axiom,
% 2.91/1.58 ! [A: set_a_e,B: set_a_e] :
% 2.91/1.58 ( ! [X2: a > e] :
% 2.91/1.58 ( ( member_a_e @ X2 @ A )
% 2.91/1.58 => ( member_a_e @ X2 @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a_e @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_12_subsetI,axiom,
% 2.91/1.58 ! [A: set_Pr305929312_set_a,B: set_Pr305929312_set_a] :
% 2.91/1.58 ( ! [X2: produc1965228714_set_a] :
% 2.91/1.58 ( ( member360729153_set_a @ X2 @ A )
% 2.91/1.58 => ( member360729153_set_a @ X2 @ B ) )
% 2.91/1.58 => ( ord_le802356416_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_13_subsetI,axiom,
% 2.91/1.58 ! [A: set_set_set_a,B: set_set_set_a] :
% 2.91/1.58 ( ! [X2: set_set_a] :
% 2.91/1.58 ( ( member_set_set_a @ X2 @ A )
% 2.91/1.58 => ( member_set_set_a @ X2 @ B ) )
% 2.91/1.58 => ( ord_le2103863998_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_14_subsetI,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( member_set_a @ X2 @ B ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_15_subsetI,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ X2 @ B ) )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_16_subsetI,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ! [X2: a] :
% 2.91/1.58 ( ( member_a @ X2 @ A )
% 2.91/1.58 => ( member_a @ X2 @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_17_subset__antisym,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ B @ A )
% 2.91/1.58 => ( A = B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_18_subset__antisym,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ B @ A )
% 2.91/1.58 => ( A = B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_19_subset__antisym,axiom,
% 2.91/1.58 ! [A: set_set_set_a,B: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A @ B )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ B @ A )
% 2.91/1.58 => ( A = B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_20_subset__antisym,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B @ A )
% 2.91/1.58 => ( A = B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_21_image__eqI,axiom,
% 2.91/1.58 ! [B2: set_a,F: set_a > set_a,X3: set_a,A: set_set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_22_image__eqI,axiom,
% 2.91/1.58 ! [B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e,X3: set_a,A: set_set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_23_image__eqI,axiom,
% 2.91/1.58 ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_24_image__eqI,axiom,
% 2.91/1.58 ! [B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_25_image__eqI,axiom,
% 2.91/1.58 ! [B2: a,F: a > a,X3: a,A: set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_a @ X3 @ A )
% 2.91/1.58 => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_26_image__eqI,axiom,
% 2.91/1.58 ! [B2: a,F: set_a > a,X3: set_a,A: set_set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_27_image__eqI,axiom,
% 2.91/1.58 ! [B2: set_a,F: a > set_a,X3: a,A: set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_a @ X3 @ A )
% 2.91/1.58 => ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_28_image__eqI,axiom,
% 2.91/1.58 ! [B2: a > e,F: a > a > e,X3: a,A: set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_a @ X3 @ A )
% 2.91/1.58 => ( member_a_e @ B2 @ ( image_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_29_image__eqI,axiom,
% 2.91/1.58 ! [B2: a,F: ( a > e ) > a,X3: a > e,A: set_a_e] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_a_e @ X3 @ A )
% 2.91/1.58 => ( member_a @ B2 @ ( image_a_e_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_30_image__eqI,axiom,
% 2.91/1.58 ! [B2: a > e,F: set_a > a > e,X3: set_a,A: set_set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member_a_e @ B2 @ ( image_set_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_31_order__refl,axiom,
% 2.91/1.58 ! [X3: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ X3 @ X3 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_32_order__refl,axiom,
% 2.91/1.58 ! [X3: set_set_a] : ( ord_le318720350_set_a @ X3 @ X3 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_33_order__refl,axiom,
% 2.91/1.58 ! [X3: set_set_set_a] : ( ord_le2103863998_set_a @ X3 @ X3 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_34_order__refl,axiom,
% 2.91/1.58 ! [X3: $o > set_a] : ( ord_less_eq_o_set_a @ X3 @ X3 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_35_order__refl,axiom,
% 2.91/1.58 ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_36_image__mono,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,F: a > a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_37_image__mono,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ ( image_64240033_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_38_image__mono,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,F: a > set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_39_image__mono,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,F: set_a > a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_40_image__mono,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,F: a > set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( image_a_set_set_a @ F @ A ) @ ( image_a_set_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_41_image__mono,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,F: set_a > set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_42_image__mono,axiom,
% 2.91/1.58 ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A @ B )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_set_set_a_a @ F @ A ) @ ( image_set_set_a_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_43_image__mono,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,F: set_a > set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( image_1930509541_set_a @ F @ A ) @ ( image_1930509541_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_44_image__mono,axiom,
% 2.91/1.58 ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_402591269_set_a @ F @ A ) @ ( image_402591269_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_45_image__mono,axiom,
% 2.91/1.58 ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A @ B )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( image_1956910981_set_a @ F @ A ) @ ( image_1956910981_set_a @ F @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_46_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_set_a,F: set_a > a,B: set_a] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( member_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_47_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_48_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > a,B: set_a] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( member_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_2116492353_a_e_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_49_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_50_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_51_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ ( image_625620211_a_a_e @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_52_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_a,F: a > a,B: set_a] :
% 2.91/1.58 ( ! [X2: a] :
% 2.91/1.58 ( ( member_a @ X2 @ A )
% 2.91/1.58 => ( member_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_53_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_a,F: a > set_a,B: set_set_a] :
% 2.91/1.58 ( ! [X2: a] :
% 2.91/1.58 ( ( member_a @ X2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_54_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_a,F: a > a > e,B: set_a_e] :
% 2.91/1.58 ( ! [X2: a] :
% 2.91/1.58 ( ( member_a @ X2 @ A )
% 2.91/1.58 => ( member_a_e @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a_e @ ( image_a_a_e @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_55_image__subsetI,axiom,
% 2.91/1.58 ! [A: set_a_e,F: ( a > e ) > a,B: set_a] :
% 2.91/1.58 ( ! [X2: a > e] :
% 2.91/1.58 ( ( member_a_e @ X2 @ A )
% 2.91/1.58 => ( member_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( image_a_e_a @ F @ A ) @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_56_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_a,F: a > a,A: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_a_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_57_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_64240033_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_58_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_a,F: set_a > a,A: set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_set_a_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_59_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_a,F: a > set_a,A: set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_a_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_60_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_set_set_a_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_61_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_set_a_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_62_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_a_set_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_63_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_402591269_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_64_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_1930509541_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_65_subset__imageE,axiom,
% 2.91/1.58 ! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
% 2.91/1.58 => ~ ! [C: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ C @ A )
% 2.91/1.58 => ( B
% 2.91/1.58 != ( image_1956910981_set_a @ F @ C ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_66_image__subset__iff,axiom,
% 2.91/1.58 ! [F: set_a > a,A: set_set_a,B: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B )
% 2.91/1.58 = ( ! [X4: set_a] :
% 2.91/1.58 ( ( member_set_a @ X4 @ A )
% 2.91/1.58 => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_67_image__subset__iff,axiom,
% 2.91/1.58 ! [F: a > a,A: set_a,B: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
% 2.91/1.58 = ( ! [X4: a] :
% 2.91/1.58 ( ( member_a @ X4 @ A )
% 2.91/1.58 => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_68_image__subset__iff,axiom,
% 2.91/1.58 ! [F: set_a > produc715398134_a_a_e,A: set_set_a,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B )
% 2.91/1.58 = ( ! [X4: set_a] :
% 2.91/1.58 ( ( member_set_a @ X4 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X4 ) @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_69_image__subset__iff,axiom,
% 2.91/1.58 ! [F: set_a > set_a,A: set_set_a,B: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B )
% 2.91/1.58 = ( ! [X4: set_a] :
% 2.91/1.58 ( ( member_set_a @ X4 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_70_image__subset__iff,axiom,
% 2.91/1.58 ! [F: set_Pr204296108_a_a_e > set_set_a,A: set_se821189730_a_a_e,B: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ F @ A ) @ B )
% 2.91/1.58 = ( ! [X4: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member1437121347_a_a_e @ X4 @ A )
% 2.91/1.58 => ( member_set_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_71_image__subset__iff,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
% 2.91/1.58 = ( ! [X4: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X4 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_72_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_a,F: a > a,A: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_a_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_73_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_64240033_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_74_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_a,F: set_a > a,A: set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_75_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_a,F: a > set_a,A: set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_76_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_set_set_a_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_77_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_78_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_a_set_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_79_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_402591269_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_80_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_1930509541_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_81_subset__image__iff,axiom,
% 2.91/1.58 ! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [AA: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ AA @ A )
% 2.91/1.58 & ( B
% 2.91/1.58 = ( image_1956910981_set_a @ F @ AA ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_82_PairE__lemma,axiom,
% 2.91/1.58 ! [P: produc1965228714_set_a] :
% 2.91/1.58 ? [X2: a > e,Y: set_a] :
% 2.91/1.58 ( P
% 2.91/1.58 = ( produc479206876_set_a @ X2 @ Y ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_83_PairE__lemma,axiom,
% 2.91/1.58 ! [P: produc715398134_a_a_e] :
% 2.91/1.58 ? [X2: set_a,Y: a > e] :
% 2.91/1.58 ( P
% 2.91/1.58 = ( produc1134456688_a_a_e @ X2 @ Y ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_84_dual__order_Oantisym,axiom,
% 2.91/1.58 ! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_85_dual__order_Oantisym,axiom,
% 2.91/1.58 ! [B2: set_set_a,A2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_86_dual__order_Oantisym,axiom,
% 2.91/1.58 ! [B2: set_set_set_a,A2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_87_dual__order_Oantisym,axiom,
% 2.91/1.58 ! [B2: $o > set_a,A2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_88_dual__order_Oantisym,axiom,
% 2.91/1.58 ! [B2: set_a,A2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_89_dual__order_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ B3 @ A3 )
% 2.91/1.58 & ( ord_le700723212_a_a_e @ A3 @ B3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_90_dual__order_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B3 @ A3 )
% 2.91/1.58 & ( ord_le318720350_set_a @ A3 @ B3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_91_dual__order_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B3 @ A3 )
% 2.91/1.58 & ( ord_le2103863998_set_a @ A3 @ B3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_92_dual__order_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: $o > set_a,B3: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ B3 @ A3 )
% 2.91/1.58 & ( ord_less_eq_o_set_a @ A3 @ B3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_93_dual__order_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_a,B3: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B3 @ A3 )
% 2.91/1.58 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_94_dual__order_Otrans,axiom,
% 2.91/1.58 ! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ C2 @ B2 )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ C2 @ A2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_95_dual__order_Otrans,axiom,
% 2.91/1.58 ! [B2: set_set_a,A2: set_set_a,C2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ C2 @ B2 )
% 2.91/1.58 => ( ord_le318720350_set_a @ C2 @ A2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_96_dual__order_Otrans,axiom,
% 2.91/1.58 ! [B2: set_set_set_a,A2: set_set_set_a,C2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ C2 @ B2 )
% 2.91/1.58 => ( ord_le2103863998_set_a @ C2 @ A2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_97_dual__order_Otrans,axiom,
% 2.91/1.58 ! [B2: $o > set_a,A2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ C2 @ B2 )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ C2 @ A2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_98_dual__order_Otrans,axiom,
% 2.91/1.58 ! [B2: set_a,A2: set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ C2 @ B2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_99_dual__order_Orefl,axiom,
% 2.91/1.58 ! [A2: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ A2 @ A2 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_100_dual__order_Orefl,axiom,
% 2.91/1.58 ! [A2: set_set_a] : ( ord_le318720350_set_a @ A2 @ A2 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_101_dual__order_Orefl,axiom,
% 2.91/1.58 ! [A2: set_set_set_a] : ( ord_le2103863998_set_a @ A2 @ A2 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_102_dual__order_Orefl,axiom,
% 2.91/1.58 ! [A2: $o > set_a] : ( ord_less_eq_o_set_a @ A2 @ A2 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_103_dual__order_Orefl,axiom,
% 2.91/1.58 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_104_order__trans,axiom,
% 2.91/1.58 ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e,Z2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ Y3 @ Z2 )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ X3 @ Z2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_105_order__trans,axiom,
% 2.91/1.58 ! [X3: set_set_a,Y3: set_set_a,Z2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ Y3 @ Z2 )
% 2.91/1.58 => ( ord_le318720350_set_a @ X3 @ Z2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_106_order__trans,axiom,
% 2.91/1.58 ! [X3: set_set_set_a,Y3: set_set_set_a,Z2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ Y3 @ Z2 )
% 2.91/1.58 => ( ord_le2103863998_set_a @ X3 @ Z2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_107_order__trans,axiom,
% 2.91/1.58 ! [X3: $o > set_a,Y3: $o > set_a,Z2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ Y3 @ Z2 )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ X3 @ Z2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_108_order__trans,axiom,
% 2.91/1.58 ! [X3: set_a,Y3: set_a,Z2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ Y3 @ Z2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_109_order__class_Oorder_Oantisym,axiom,
% 2.91/1.58 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ B2 @ A2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_110_order__class_Oorder_Oantisym,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ B2 @ A2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_111_order__class_Oorder_Oantisym,axiom,
% 2.91/1.58 ! [A2: set_set_set_a,B2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ B2 @ A2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_112_order__class_Oorder_Oantisym,axiom,
% 2.91/1.58 ! [A2: $o > set_a,B2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ A2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_113_order__class_Oorder_Oantisym,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.91/1.58 => ( A2 = B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_114_ord__le__eq__trans,axiom,
% 2.91/1.58 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 2.91/1.58 => ( ( B2 = C2 )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_115_ord__le__eq__trans,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( B2 = C2 )
% 2.91/1.58 => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_116_ord__le__eq__trans,axiom,
% 2.91/1.58 ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( B2 = C2 )
% 2.91/1.58 => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_117_ord__le__eq__trans,axiom,
% 2.91/1.58 ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( B2 = C2 )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_118_ord__le__eq__trans,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( B2 = C2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_119_ord__eq__le__trans,axiom,
% 2.91/1.58 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ B2 @ C2 )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_120_ord__eq__le__trans,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_121_ord__eq__le__trans,axiom,
% 2.91/1.58 ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_122_ord__eq__le__trans,axiom,
% 2.91/1.58 ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_123_ord__eq__le__trans,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_124_order__class_Oorder_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A3 @ B3 )
% 2.91/1.58 & ( ord_le700723212_a_a_e @ B3 @ A3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_125_order__class_Oorder_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_set_a,B3: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A3 @ B3 )
% 2.91/1.58 & ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_126_order__class_Oorder_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A3 @ B3 )
% 2.91/1.58 & ( ord_le2103863998_set_a @ B3 @ A3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: $o > set_a,B3: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ A3 @ B3 )
% 2.91/1.58 & ( ord_less_eq_o_set_a @ B3 @ A3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_128_order__class_Oorder_Oeq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A3: set_a,B3: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.91/1.58 & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_129_antisym__conv,axiom,
% 2.91/1.58 ! [Y3: set_Pr204296108_a_a_e,X3: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
% 2.91/1.58 = ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_130_antisym__conv,axiom,
% 2.91/1.58 ! [Y3: set_set_a,X3: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ X3 @ Y3 )
% 2.91/1.58 = ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_131_antisym__conv,axiom,
% 2.91/1.58 ! [Y3: set_set_set_a,X3: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ X3 @ Y3 )
% 2.91/1.58 = ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_132_antisym__conv,axiom,
% 2.91/1.58 ! [Y3: $o > set_a,X3: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
% 2.91/1.58 = ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_133_antisym__conv,axiom,
% 2.91/1.58 ! [Y3: set_a,X3: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 2.91/1.58 = ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_134_order_Otrans,axiom,
% 2.91/1.58 ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ B2 @ C2 )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_135_order_Otrans,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_136_order_Otrans,axiom,
% 2.91/1.58 ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_137_order_Otrans,axiom,
% 2.91/1.58 ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_138_order_Otrans,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_139_eq__refl,axiom,
% 2.91/1.58 ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( X3 = Y3 )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ X3 @ Y3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_140_eq__refl,axiom,
% 2.91/1.58 ! [X3: set_set_a,Y3: set_set_a] :
% 2.91/1.58 ( ( X3 = Y3 )
% 2.91/1.58 => ( ord_le318720350_set_a @ X3 @ Y3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_141_eq__refl,axiom,
% 2.91/1.58 ! [X3: set_set_set_a,Y3: set_set_set_a] :
% 2.91/1.58 ( ( X3 = Y3 )
% 2.91/1.58 => ( ord_le2103863998_set_a @ X3 @ Y3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_142_eq__refl,axiom,
% 2.91/1.58 ! [X3: $o > set_a,Y3: $o > set_a] :
% 2.91/1.58 ( ( X3 = Y3 )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ X3 @ Y3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_143_eq__refl,axiom,
% 2.91/1.58 ! [X3: set_a,Y3: set_a] :
% 2.91/1.58 ( ( X3 = Y3 )
% 2.91/1.58 => ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_144_antisym,axiom,
% 2.91/1.58 ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
% 2.91/1.58 => ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_145_antisym,axiom,
% 2.91/1.58 ! [X3: set_set_a,Y3: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_146_antisym,axiom,
% 2.91/1.58 ! [X3: set_set_set_a,Y3: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_le2103863998_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_147_antisym,axiom,
% 2.91/1.58 ! [X3: $o > set_a,Y3: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_148_antisym,axiom,
% 2.91/1.58 ! [X3: set_a,Y3: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ Y3 @ X3 )
% 2.91/1.58 => ( X3 = Y3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_149_eq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [X4: set_Pr204296108_a_a_e,Y4: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ X4 @ Y4 )
% 2.91/1.58 & ( ord_le700723212_a_a_e @ Y4 @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_150_eq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [X4: set_set_a,Y4: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X4 @ Y4 )
% 2.91/1.58 & ( ord_le318720350_set_a @ Y4 @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_151_eq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [X4: set_set_set_a,Y4: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ X4 @ Y4 )
% 2.91/1.58 & ( ord_le2103863998_set_a @ Y4 @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_152_eq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [X4: $o > set_a,Y4: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X4 @ Y4 )
% 2.91/1.58 & ( ord_less_eq_o_set_a @ Y4 @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_153_eq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [X4: set_a,Y4: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X4 @ Y4 )
% 2.91/1.58 & ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_154_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_155_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,F: set_a > set_set_a,C2: set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_156_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_set_a,Y: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_157_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,F: set_a > set_set_set_a,C2: set_set_set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_158_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,F: set_a > $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_159_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C2: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_set_a,Y: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_160_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_set_set_a,Y: set_set_set_a] :
% 2.91/1.58 ( ( ord_le2103863998_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_161_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: $o > set_a,B2: $o > set_a,F: ( $o > set_a ) > set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_162_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_set_a,C2: set_set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_set_a,Y: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_163_ord__le__eq__subst,axiom,
% 2.91/1.58 ! [A2: set_set_a,B2: set_set_a,F: set_set_a > $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ( F @ B2 )
% 2.91/1.58 = C2 )
% 2.91/1.58 => ( ! [X2: set_set_a,Y: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_164_ord__eq__le__subst,axiom,
% 2.91/1.58 ! [A2: set_Pr204296108_a_a_e,F: ( $o > set_a ) > set_Pr204296108_a_a_e,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( A2
% 2.91/1.58 = ( F @ B2 ) )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_165_ord__eq__le__subst,axiom,
% 2.91/1.58 ! [A2: set_set_a,F: ( $o > set_a ) > set_set_a,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( A2
% 2.91/1.58 = ( F @ B2 ) )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le318720350_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_166_ord__eq__le__subst,axiom,
% 2.91/1.58 ! [A2: set_set_set_a,F: ( $o > set_a ) > set_set_set_a,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( A2
% 2.91/1.58 = ( F @ B2 ) )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_le2103863998_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_167_ord__eq__le__subst,axiom,
% 2.91/1.58 ! [A2: $o > set_a,F: ( $o > set_a ) > $o > set_a,B2: $o > set_a,C2: $o > set_a] :
% 2.91/1.58 ( ( A2
% 2.91/1.58 = ( F @ B2 ) )
% 2.91/1.58 => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ! [X2: $o > set_a,Y: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_o_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_o_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_168_ord__eq__le__subst,axiom,
% 2.91/1.58 ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( A2
% 2.91/1.58 = ( F @ B2 ) )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_169_order__subst2,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_170_order__subst1,axiom,
% 2.91/1.58 ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 2.91/1.58 => ( ! [X2: set_a,Y: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_171_mem__Collect__eq,axiom,
% 2.91/1.58 ! [A2: set_a,P2: set_a > $o] :
% 2.91/1.58 ( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
% 2.91/1.58 = ( P2 @ A2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_172_mem__Collect__eq,axiom,
% 2.91/1.58 ! [A2: produc715398134_a_a_e,P2: produc715398134_a_a_e > $o] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ A2 @ ( collec1369693387_a_a_e @ P2 ) )
% 2.91/1.58 = ( P2 @ A2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_173_Collect__mem__eq,axiom,
% 2.91/1.58 ! [A: set_set_a] :
% 2.91/1.58 ( ( collect_set_a
% 2.91/1.58 @ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
% 2.91/1.58 = A ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_174_Collect__mem__eq,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( collec1369693387_a_a_e
% 2.91/1.58 @ ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ A ) )
% 2.91/1.58 = A ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_175_rev__image__eqI,axiom,
% 2.91/1.58 ! [X3: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
% 2.91/1.58 ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_176_rev__image__eqI,axiom,
% 2.91/1.58 ! [X3: set_a,A: set_set_a,B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_177_rev__image__eqI,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: set_a,F: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_178_rev__image__eqI,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_179_ball__imageD,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 => ( P2 @ X2 ) )
% 2.91/1.58 => ! [X: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X @ A )
% 2.91/1.58 => ( P2 @ ( F @ X ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_180_image__cong,axiom,
% 2.91/1.58 ! [M: set_Pr204296108_a_a_e,N: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( M = N )
% 2.91/1.58 => ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ N )
% 2.91/1.58 => ( ( F @ X2 )
% 2.91/1.58 = ( G @ X2 ) ) )
% 2.91/1.58 => ( ( image_64240033_set_a @ F @ M )
% 2.91/1.58 = ( image_64240033_set_a @ G @ N ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_181_bex__imageD,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
% 2.91/1.58 ( ? [X: set_a] :
% 2.91/1.58 ( ( member_set_a @ X @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 & ( P2 @ X ) )
% 2.91/1.58 => ? [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 & ( P2 @ ( F @ X2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_182_image__iff,axiom,
% 2.91/1.58 ! [Z2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ Z2 @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 = ( ? [X4: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X4 @ A )
% 2.91/1.58 & ( Z2
% 2.91/1.58 = ( F @ X4 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_183_imageI,axiom,
% 2.91/1.58 ! [X3: set_a,A: set_set_a,F: set_a > set_a] :
% 2.91/1.58 ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X3 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_184_imageI,axiom,
% 2.91/1.58 ! [X3: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_185_imageI,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_186_imageI,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_187_Collect__mono__iff,axiom,
% 2.91/1.58 ! [P2: a > $o,Q: a > $o] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
% 2.91/1.58 = ( ! [X4: a] :
% 2.91/1.58 ( ( P2 @ X4 )
% 2.91/1.58 => ( Q @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_188_set__eq__subset,axiom,
% 2.91/1.58 ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [A4: set_a,B4: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 2.91/1.58 & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_189_subset__trans,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,C3: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B @ C3 )
% 2.91/1.58 => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_190_Collect__mono,axiom,
% 2.91/1.58 ! [P2: a > $o,Q: a > $o] :
% 2.91/1.58 ( ! [X2: a] :
% 2.91/1.58 ( ( P2 @ X2 )
% 2.91/1.58 => ( Q @ X2 ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_191_subset__iff,axiom,
% 2.91/1.58 ( ord_le318720350_set_a
% 2.91/1.58 = ( ^ [A4: set_set_a,B4: set_set_a] :
% 2.91/1.58 ! [T: set_a] :
% 2.91/1.58 ( ( member_set_a @ T @ A4 )
% 2.91/1.58 => ( member_set_a @ T @ B4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_192_subset__iff,axiom,
% 2.91/1.58 ( ord_le700723212_a_a_e
% 2.91/1.58 = ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
% 2.91/1.58 ! [T: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ T @ A4 )
% 2.91/1.58 => ( member1258382221_a_a_e @ T @ B4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_193_subset__iff,axiom,
% 2.91/1.58 ( ord_less_eq_set_a
% 2.91/1.58 = ( ^ [A4: set_a,B4: set_a] :
% 2.91/1.58 ! [T: a] :
% 2.91/1.58 ( ( member_a @ T @ A4 )
% 2.91/1.58 => ( member_a @ T @ B4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_194_equalityD2,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( A = B )
% 2.91/1.58 => ( ord_less_eq_set_a @ B @ A ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_195_subset__eq,axiom,
% 2.91/1.58 ( ord_le318720350_set_a
% 2.91/1.58 = ( ^ [A4: set_set_a,B4: set_set_a] :
% 2.91/1.58 ! [X4: set_a] :
% 2.91/1.58 ( ( member_set_a @ X4 @ A4 )
% 2.91/1.58 => ( member_set_a @ X4 @ B4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_196_subset__eq,axiom,
% 2.91/1.58 ( ord_le700723212_a_a_e
% 2.91/1.58 = ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
% 2.91/1.58 ! [X4: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X4 @ A4 )
% 2.91/1.58 => ( member1258382221_a_a_e @ X4 @ B4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_197_subset__eq,axiom,
% 2.91/1.58 ( ord_less_eq_set_a
% 2.91/1.58 = ( ^ [A4: set_a,B4: set_a] :
% 2.91/1.58 ! [X4: a] :
% 2.91/1.58 ( ( member_a @ X4 @ A4 )
% 2.91/1.58 => ( member_a @ X4 @ B4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_198_equalityE,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( A = B )
% 2.91/1.58 => ~ ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_199_subsetD,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,C2: set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ( member_set_a @ C2 @ A )
% 2.91/1.58 => ( member_set_a @ C2 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_200_subsetD,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ C2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ C2 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_201_subsetD,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,C2: a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ( member_a @ C2 @ A )
% 2.91/1.58 => ( member_a @ C2 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_202_in__mono,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member_set_a @ X3 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_203_in__mono,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ X3 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_204_in__mono,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,X3: a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ( member_a @ X3 @ A )
% 2.91/1.58 => ( member_a @ X3 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_205_prod_Oinject,axiom,
% 2.91/1.58 ! [X1: set_a,X22: a > e,Y1: set_a,Y22: a > e] :
% 2.91/1.58 ( ( ( produc1134456688_a_a_e @ X1 @ X22 )
% 2.91/1.58 = ( produc1134456688_a_a_e @ Y1 @ Y22 ) )
% 2.91/1.58 = ( ( X1 = Y1 )
% 2.91/1.58 & ( X22 = Y22 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_206_old_Oprod_Oinject,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
% 2.91/1.58 ( ( ( produc1134456688_a_a_e @ A2 @ B2 )
% 2.91/1.58 = ( produc1134456688_a_a_e @ A5 @ B5 ) )
% 2.91/1.58 = ( ( A2 = A5 )
% 2.91/1.58 & ( B2 = B5 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_207_eq__fst__iff,axiom,
% 2.91/1.58 ! [A2: set_a,P: produc715398134_a_a_e] :
% 2.91/1.58 ( ( A2
% 2.91/1.58 = ( produc1040928388_a_a_e @ P ) )
% 2.91/1.58 = ( ? [B3: a > e] :
% 2.91/1.58 ( P
% 2.91/1.58 = ( produc1134456688_a_a_e @ A2 @ B3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_208_fst__conv,axiom,
% 2.91/1.58 ! [X1: set_a,X22: a > e] :
% 2.91/1.58 ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
% 2.91/1.58 = X1 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_209_fst__eqD,axiom,
% 2.91/1.58 ! [X3: set_a,Y3: a > e,A2: set_a] :
% 2.91/1.58 ( ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 2.91/1.58 = A2 )
% 2.91/1.58 => ( X3 = A2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_210_fstI,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
% 2.91/1.58 ( ( X3
% 2.91/1.58 = ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
% 2.91/1.58 => ( ( produc1040928388_a_a_e @ X3 )
% 2.91/1.58 = Y3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_211_all__subset__image,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_set_a > $o] :
% 2.91/1.58 ( ( ! [B4: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ B4 @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 => ( P2 @ B4 ) ) )
% 2.91/1.58 = ( ! [B4: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ B4 @ A )
% 2.91/1.58 => ( P2 @ ( image_64240033_set_a @ F @ B4 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_212_all__subset__image,axiom,
% 2.91/1.58 ! [F: a > a,A: set_a,P2: set_a > $o] :
% 2.91/1.58 ( ( ! [B4: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
% 2.91/1.58 => ( P2 @ B4 ) ) )
% 2.91/1.58 = ( ! [B4: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ B4 @ A )
% 2.91/1.58 => ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_213_Greatest__equality,axiom,
% 2.91/1.58 ! [P2: set_a > $o,X3: set_a] :
% 2.91/1.58 ( ( P2 @ X3 )
% 2.91/1.58 => ( ! [Y: set_a] :
% 2.91/1.58 ( ( P2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ Y @ X3 ) )
% 2.91/1.58 => ( ( order_Greatest_set_a @ P2 )
% 2.91/1.58 = X3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_214_GreatestI2__order,axiom,
% 2.91/1.58 ! [P2: set_a > $o,X3: set_a,Q: set_a > $o] :
% 2.91/1.58 ( ( P2 @ X3 )
% 2.91/1.58 => ( ! [Y: set_a] :
% 2.91/1.58 ( ( P2 @ Y )
% 2.91/1.58 => ( ord_less_eq_set_a @ Y @ X3 ) )
% 2.91/1.58 => ( ! [X2: set_a] :
% 2.91/1.58 ( ( P2 @ X2 )
% 2.91/1.58 => ( ! [Y5: set_a] :
% 2.91/1.58 ( ( P2 @ Y5 )
% 2.91/1.58 => ( ord_less_eq_set_a @ Y5 @ X2 ) )
% 2.91/1.58 => ( Q @ X2 ) ) )
% 2.91/1.58 => ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_215_old_Oprod_Oinducts,axiom,
% 2.91/1.58 ! [P2: produc715398134_a_a_e > $o,Prod: produc715398134_a_a_e] :
% 2.91/1.58 ( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
% 2.91/1.58 => ( P2 @ Prod ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_216_old_Oprod_Oexhaust,axiom,
% 2.91/1.58 ! [Y3: produc715398134_a_a_e] :
% 2.91/1.58 ~ ! [A6: set_a,B6: a > e] :
% 2.91/1.58 ( Y3
% 2.91/1.58 != ( produc1134456688_a_a_e @ A6 @ B6 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_217_Pair__inject,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
% 2.91/1.58 ( ( ( produc1134456688_a_a_e @ A2 @ B2 )
% 2.91/1.58 = ( produc1134456688_a_a_e @ A5 @ B5 ) )
% 2.91/1.58 => ~ ( ( A2 = A5 )
% 2.91/1.58 => ( B2 != B5 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_218_prod__cases,axiom,
% 2.91/1.58 ! [P2: produc715398134_a_a_e > $o,P: produc715398134_a_a_e] :
% 2.91/1.58 ( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
% 2.91/1.58 => ( P2 @ P ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_219_image__Fpow__mono,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( finite33065488_a_a_e @ A ) ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_220_not__sub,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a] :
% 2.91/1.58 ( ~ ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ? [A6: set_a] :
% 2.91/1.58 ( ( member_set_a @ A6 @ A )
% 2.91/1.58 & ~ ( member_set_a @ A6 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_221_not__sub,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ~ ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ? [A6: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ A6 @ A )
% 2.91/1.58 & ~ ( member1258382221_a_a_e @ A6 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_222_not__sub,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ~ ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ? [A6: a] :
% 2.91/1.58 ( ( member_a @ A6 @ A )
% 2.91/1.58 & ~ ( member_a @ A6 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_223_eqsets__sub,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( A = B )
% 2.91/1.58 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_224_sub__which1,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 2.91/1.58 ( ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 | ( ord_le318720350_set_a @ B @ A ) )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( ~ ( member_set_a @ X3 @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ B @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_225_sub__which1,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 | ( ord_le700723212_a_a_e @ B @ A ) )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( ~ ( member1258382221_a_a_e @ X3 @ B )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ B @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_226_sub__which1,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,X3: a] :
% 2.91/1.58 ( ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 | ( ord_less_eq_set_a @ B @ A ) )
% 2.91/1.58 => ( ( member_a @ X3 @ A )
% 2.91/1.58 => ( ~ ( member_a @ X3 @ B )
% 2.91/1.58 => ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_227_Fpow__mono,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_228_le__convert,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ A2 @ C2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_229_ge__convert,axiom,
% 2.91/1.58 ! [A2: set_a,B2: set_a,C2: set_a] :
% 2.91/1.58 ( ( A2 = B2 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ C2 @ A2 )
% 2.91/1.58 => ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_230_mem__in__image3,axiom,
% 2.91/1.58 ! [B2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) )
% 2.91/1.58 => ? [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 & ( B2
% 2.91/1.58 = ( F @ X2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_231_mem__in__image2,axiom,
% 2.91/1.58 ! [A2: set_a,A: set_set_a,F: set_a > set_a] :
% 2.91/1.58 ( ( member_set_a @ A2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_232_mem__in__image2,axiom,
% 2.91/1.58 ! [A2: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ A2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_233_mem__in__image2,axiom,
% 2.91/1.58 ! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ A2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_234_mem__in__image2,axiom,
% 2.91/1.58 ! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ A2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_235_mem__in__image1,axiom,
% 2.91/1.58 ! [A: set_set_a,F: set_a > set_a,B: set_set_a,A2: set_a] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ( member_set_a @ A2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_236_mem__in__image1,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a,A2: produc715398134_a_a_e] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ A2 @ A )
% 2.91/1.58 => ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_237_mem__in__image1,axiom,
% 2.91/1.58 ! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: set_a] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ( member_set_a @ A2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_238_mem__in__image1,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: produc715398134_a_a_e] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ A2 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_239_proper__subset,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ( ~ ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( ( member_set_a @ X3 @ B )
% 2.91/1.58 => ( A != B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_240_proper__subset,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ( ~ ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ B )
% 2.91/1.58 => ( A != B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_241_proper__subset,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,X3: a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ~ ( member_a @ X3 @ A )
% 2.91/1.58 => ( ( member_a @ X3 @ B )
% 2.91/1.58 => ( A != B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_242_not__subseteq,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a] :
% 2.91/1.58 ( ~ ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 => ? [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 & ~ ( member_set_a @ X2 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_243_not__subseteq,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ~ ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 => ? [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 & ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_244_not__subseteq,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ~ ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ? [X2: a] :
% 2.91/1.58 ( ( member_a @ X2 @ A )
% 2.91/1.58 & ~ ( member_a @ X2 @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_245_subset__self,axiom,
% 2.91/1.58 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_246_sets__not__eq,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a] :
% 2.91/1.58 ( ( A != B )
% 2.91/1.58 => ( ( ord_le318720350_set_a @ B @ A )
% 2.91/1.58 => ? [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ A )
% 2.91/1.58 & ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_247_sets__not__eq,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( A != B )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ B @ A )
% 2.91/1.58 => ? [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 & ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_248_sets__not__eq,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( A != B )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ B @ A )
% 2.91/1.58 => ? [X2: a] :
% 2.91/1.58 ( ( member_a @ X2 @ A )
% 2.91/1.58 & ~ ( member_a @ X2 @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_249_sub__which2,axiom,
% 2.91/1.58 ! [A: set_set_a,B: set_set_a,X3: set_a] :
% 2.91/1.58 ( ( ( ord_le318720350_set_a @ A @ B )
% 2.91/1.58 | ( ord_le318720350_set_a @ B @ A ) )
% 2.91/1.58 => ( ~ ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( ( member_set_a @ X3 @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ A @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_250_sub__which2,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ( ord_le700723212_a_a_e @ A @ B )
% 2.91/1.58 | ( ord_le700723212_a_a_e @ B @ A ) )
% 2.91/1.58 => ( ~ ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ B )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_251_sub__which2,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a,X3: a] :
% 2.91/1.58 ( ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 | ( ord_less_eq_set_a @ B @ A ) )
% 2.91/1.58 => ( ~ ( member_a @ X3 @ A )
% 2.91/1.58 => ( ( member_a @ X3 @ B )
% 2.91/1.58 => ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_252_image__Pow__mono,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 2.91/1.58 ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
% 2.91/1.58 => ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) ) @ ( pow_set_a @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_253_Sup_OSUP__cong,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Sup: set_set_a > set_a] :
% 2.91/1.58 ( ( A = B )
% 2.91/1.58 => ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ B )
% 2.91/1.58 => ( ( C3 @ X2 )
% 2.91/1.58 = ( D @ X2 ) ) )
% 2.91/1.58 => ( ( Sup @ ( image_64240033_set_a @ C3 @ A ) )
% 2.91/1.58 = ( Sup @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_254_Inf_OINF__cong,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Inf: set_set_a > set_a] :
% 2.91/1.58 ( ( A = B )
% 2.91/1.58 => ( ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X2 @ B )
% 2.91/1.58 => ( ( C3 @ X2 )
% 2.91/1.58 = ( D @ X2 ) ) )
% 2.91/1.58 => ( ( Inf @ ( image_64240033_set_a @ C3 @ A ) )
% 2.91/1.58 = ( Inf @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_255_subrelI,axiom,
% 2.91/1.58 ! [R: set_Pr204296108_a_a_e,S: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ! [X2: set_a,Y: a > e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ S ) )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ R @ S ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_256_ssubst__Pair__rhs,axiom,
% 2.91/1.58 ! [R: set_a,S: a > e,R2: set_Pr204296108_a_a_e,S2: a > e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S ) @ R2 )
% 2.91/1.58 => ( ( S2 = S )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S2 ) @ R2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_257_PowI,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_258_Pow__iff,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( member_set_a @ A @ ( pow_a @ B ) )
% 2.91/1.58 = ( ord_less_eq_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_259_Pow__top,axiom,
% 2.91/1.58 ! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_260_PowD,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( member_set_a @ A @ ( pow_a @ B ) )
% 2.91/1.58 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_261_Pow__mono,axiom,
% 2.91/1.58 ! [A: set_a,B: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ A @ B )
% 2.91/1.58 => ( ord_le318720350_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_262_image__Pow__surj,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
% 2.91/1.58 ( ( ( image_64240033_set_a @ F @ A )
% 2.91/1.58 = B )
% 2.91/1.58 => ( ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) )
% 2.91/1.58 = ( pow_set_a @ B ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_263_le__rel__bool__arg__iff,axiom,
% 2.91/1.58 ( ord_less_eq_o_set_a
% 2.91/1.58 = ( ^ [X5: $o > set_a,Y6: $o > set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.91/1.58 & ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_264_curryI,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
% 2.91/1.58 ( ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
% 2.91/1.58 => ( produc952037978_a_e_o @ F @ A2 @ B2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_265_curryD,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
% 2.91/1.58 ( ( produc952037978_a_e_o @ F @ A2 @ B2 )
% 2.91/1.58 => ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_266_curryE,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
% 2.91/1.58 ( ( produc952037978_a_e_o @ F @ A2 @ B2 )
% 2.91/1.58 => ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_267_fst__eq__Domain,axiom,
% 2.91/1.58 ! [R2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( image_64240033_set_a @ produc1040928388_a_a_e @ R2 )
% 2.91/1.58 = ( domain_set_a_a_e @ R2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_268_Domain__fst,axiom,
% 2.91/1.58 ( domain_set_a_a_e
% 2.91/1.58 = ( image_64240033_set_a @ produc1040928388_a_a_e ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_269_RangeE,axiom,
% 2.91/1.58 ! [B2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) )
% 2.91/1.58 => ~ ! [A6: set_a] :
% 2.91/1.58 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B2 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_270_Range__iff,axiom,
% 2.91/1.58 ! [A2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
% 2.91/1.58 = ( ? [Y4: set_a] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y4 @ A2 ) @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_271_Range_Ocases,axiom,
% 2.91/1.58 ! [A2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
% 2.91/1.58 => ~ ! [A6: set_a] :
% 2.91/1.58 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ A2 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_272_Range_Osimps,axiom,
% 2.91/1.58 ! [A2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
% 2.91/1.58 = ( ? [A3: set_a,B3: a > e] :
% 2.91/1.58 ( ( A2 = B3 )
% 2.91/1.58 & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_273_Range_Ointros,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
% 2.91/1.58 => ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_274_Range_Oinducts,axiom,
% 2.91/1.58 ! [X3: a > e,R: set_Pr204296108_a_a_e,P2: ( a > e ) > $o] :
% 2.91/1.58 ( ( member_a_e @ X3 @ ( range_set_a_a_e @ R ) )
% 2.91/1.58 => ( ! [A6: set_a,B6: a > e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
% 2.91/1.58 => ( P2 @ B6 ) )
% 2.91/1.58 => ( P2 @ X3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_275_DomainE,axiom,
% 2.91/1.58 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 2.91/1.58 => ~ ! [B6: a > e] :
% 2.91/1.58 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_276_Domain__iff,axiom,
% 2.91/1.58 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 2.91/1.58 = ( ? [Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ Y4 ) @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_277_Domain_Ocases,axiom,
% 2.91/1.58 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 2.91/1.58 => ~ ! [B6: a > e] :
% 2.91/1.58 ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_278_Domain_Osimps,axiom,
% 2.91/1.58 ! [A2: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
% 2.91/1.58 = ( ? [A3: set_a,B3: a > e] :
% 2.91/1.58 ( ( A2 = A3 )
% 2.91/1.58 & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_279_Domain_ODomainI,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
% 2.91/1.58 => ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_280_Domain_Oinducts,axiom,
% 2.91/1.58 ! [X3: set_a,R: set_Pr204296108_a_a_e,P2: set_a > $o] :
% 2.91/1.58 ( ( member_set_a @ X3 @ ( domain_set_a_a_e @ R ) )
% 2.91/1.58 => ( ! [A6: set_a,B6: a > e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
% 2.91/1.58 => ( P2 @ A6 ) )
% 2.91/1.58 => ( P2 @ X3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_281_pair__in__swap__image,axiom,
% 2.91/1.58 ! [Y3: a > e,X3: set_a,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ Y3 @ X3 ) @ ( image_1875450791_set_a @ produc755714512_a_a_e @ A ) )
% 2.91/1.58 = ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) @ A ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_282_pair__in__swap__image,axiom,
% 2.91/1.58 ! [Y3: set_a,X3: a > e,A: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y3 @ X3 ) @ ( image_376979367_a_a_e @ produc100464700_set_a @ A ) )
% 2.91/1.58 = ( member360729153_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) @ A ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_283_swap__simp,axiom,
% 2.91/1.58 ! [X3: a > e,Y3: set_a] :
% 2.91/1.58 ( ( produc100464700_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) )
% 2.91/1.58 = ( produc1134456688_a_a_e @ Y3 @ X3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_284_swap__simp,axiom,
% 2.91/1.58 ! [X3: set_a,Y3: a > e] :
% 2.91/1.58 ( ( produc755714512_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 2.91/1.58 = ( produc479206876_set_a @ Y3 @ X3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_285_converse__iff,axiom,
% 2.91/1.58 ! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
% 2.91/1.58 = ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_286_converse__iff,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
% 2.91/1.58 = ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_287_converse_Oinducts,axiom,
% 2.91/1.58 ! [X1: a > e,X22: set_a,R: set_Pr204296108_a_a_e,P2: ( a > e ) > set_a > $o] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ X1 @ X22 ) @ ( converse_set_a_a_e @ R ) )
% 2.91/1.58 => ( ! [A6: set_a,B6: a > e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
% 2.91/1.58 => ( P2 @ B6 @ A6 ) )
% 2.91/1.58 => ( P2 @ X1 @ X22 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_288_converse_Oinducts,axiom,
% 2.91/1.58 ! [X1: set_a,X22: a > e,R: set_Pr305929312_set_a,P2: set_a > ( a > e ) > $o] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) @ ( converse_a_e_set_a @ R ) )
% 2.91/1.58 => ( ! [A6: a > e,B6: set_a] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ A6 @ B6 ) @ R )
% 2.91/1.58 => ( P2 @ B6 @ A6 ) )
% 2.91/1.58 => ( P2 @ X1 @ X22 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_289_converse_Ointros,axiom,
% 2.91/1.58 ! [A2: a > e,B2: set_a,R: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ R )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ ( converse_a_e_set_a @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_290_converse_Ointros,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
% 2.91/1.58 => ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ ( converse_set_a_a_e @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_291_converse_Osimps,axiom,
% 2.91/1.58 ! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
% 2.91/1.58 = ( ? [A3: set_a,B3: a > e] :
% 2.91/1.58 ( ( A1 = B3 )
% 2.91/1.58 & ( A22 = A3 )
% 2.91/1.58 & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_292_converse_Osimps,axiom,
% 2.91/1.58 ! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
% 2.91/1.58 = ( ? [A3: a > e,B3: set_a] :
% 2.91/1.58 ( ( A1 = B3 )
% 2.91/1.58 & ( A22 = A3 )
% 2.91/1.58 & ( member360729153_set_a @ ( produc479206876_set_a @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_293_converse_Ocases,axiom,
% 2.91/1.58 ! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A22 @ A1 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_294_converse_Ocases,axiom,
% 2.91/1.58 ! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
% 2.91/1.58 => ( member360729153_set_a @ ( produc479206876_set_a @ A22 @ A1 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_295_converseE,axiom,
% 2.91/1.58 ! [Yx: produc1965228714_set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member360729153_set_a @ Yx @ ( converse_set_a_a_e @ R ) )
% 2.91/1.58 => ~ ! [X2: set_a,Y: a > e] :
% 2.91/1.58 ( ( Yx
% 2.91/1.58 = ( produc479206876_set_a @ Y @ X2 ) )
% 2.91/1.58 => ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_296_converseE,axiom,
% 2.91/1.58 ! [Yx: produc715398134_a_a_e,R: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ Yx @ ( converse_a_e_set_a @ R ) )
% 2.91/1.58 => ~ ! [X2: a > e,Y: set_a] :
% 2.91/1.58 ( ( Yx
% 2.91/1.58 = ( produc1134456688_a_a_e @ Y @ X2 ) )
% 2.91/1.58 => ~ ( member360729153_set_a @ ( produc479206876_set_a @ X2 @ Y ) @ R ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_297_converseD,axiom,
% 2.91/1.58 ! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_298_converseD,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
% 2.91/1.58 => ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_299_prod_Oswap__def,axiom,
% 2.91/1.58 ( produc100464700_set_a
% 2.91/1.58 = ( ^ [P3: produc1965228714_set_a] : ( produc1134456688_a_a_e @ ( produc1809800494_set_a @ P3 ) @ ( produc385678576_set_a @ P3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_300_prod_Oswap__def,axiom,
% 2.91/1.58 ( produc755714512_a_a_e
% 2.91/1.58 = ( ^ [P3: produc715398134_a_a_e] : ( produc479206876_set_a @ ( produc317566658_a_a_e @ P3 ) @ ( produc1040928388_a_a_e @ P3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_301_map__prod__imageI,axiom,
% 2.91/1.58 ! [A2: set_a,B2: a > e,R2: set_Pr204296108_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R2 )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_302_UNIV__I,axiom,
% 2.91/1.58 ! [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_303_UNIV__I,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X3 @ top_to1601855068_a_a_e ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_304_map__prod__simp,axiom,
% 2.91/1.58 ! [F: set_a > set_a,G: ( a > e ) > a > e,A2: set_a,B2: a > e] :
% 2.91/1.58 ( ( produc1443219402_e_a_e @ F @ G @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
% 2.91/1.58 = ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_305_fst__map__prod,axiom,
% 2.91/1.58 ! [F: set_a > set_a,G: ( a > e ) > a > e,X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( produc1040928388_a_a_e @ ( produc1443219402_e_a_e @ F @ G @ X3 ) )
% 2.91/1.58 = ( F @ ( produc1040928388_a_a_e @ X3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_306_prod_Ocollapse,axiom,
% 2.91/1.58 ! [Prod: produc715398134_a_a_e] :
% 2.91/1.58 ( ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) )
% 2.91/1.58 = Prod ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_307_fst__swap,axiom,
% 2.91/1.58 ! [X3: produc1965228714_set_a] :
% 2.91/1.58 ( ( produc1040928388_a_a_e @ ( produc100464700_set_a @ X3 ) )
% 2.91/1.58 = ( produc1809800494_set_a @ X3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_308_snd__swap,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( produc1809800494_set_a @ ( produc755714512_a_a_e @ X3 ) )
% 2.91/1.58 = ( produc1040928388_a_a_e @ X3 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_309_range__fst,axiom,
% 2.91/1.58 ( ( image_64240033_set_a @ produc1040928388_a_a_e @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_310_rangeI,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] : ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_311_range__eqI,axiom,
% 2.91/1.58 ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_312_surj__def,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a )
% 2.91/1.58 = ( ! [Y4: set_a] :
% 2.91/1.58 ? [X4: produc715398134_a_a_e] :
% 2.91/1.58 ( Y4
% 2.91/1.58 = ( F @ X4 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_313_surjI,axiom,
% 2.91/1.58 ! [G: produc715398134_a_a_e > set_a,F: set_a > produc715398134_a_a_e] :
% 2.91/1.58 ( ! [X2: set_a] :
% 2.91/1.58 ( ( G @ ( F @ X2 ) )
% 2.91/1.58 = X2 )
% 2.91/1.58 => ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_314_surjE,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
% 2.91/1.58 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a )
% 2.91/1.58 => ~ ! [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( Y3
% 2.91/1.58 != ( F @ X2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_315_surjD,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
% 2.91/1.58 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a )
% 2.91/1.58 => ? [X2: produc715398134_a_a_e] :
% 2.91/1.58 ( Y3
% 2.91/1.58 = ( F @ X2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_316_map__prod__surj,axiom,
% 2.91/1.58 ! [F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
% 2.91/1.58 ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a )
% 2.91/1.58 => ( ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
% 2.91/1.58 = top_top_set_set_a )
% 2.91/1.58 => ( ( image_2101504031_set_a @ ( produc387505142_set_a @ F @ G ) @ top_to970499581_a_a_e )
% 2.91/1.58 = top_to144004695_set_a ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_317_top_Oextremum__uniqueI,axiom,
% 2.91/1.58 ! [A2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
% 2.91/1.58 => ( A2 = top_top_set_a ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_318_top_Oextremum__unique,axiom,
% 2.91/1.58 ! [A2: set_a] :
% 2.91/1.58 ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
% 2.91/1.58 = ( A2 = top_top_set_a ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_319_top__greatest,axiom,
% 2.91/1.58 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_320_subset__UNIV,axiom,
% 2.91/1.58 ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_321_sndI,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
% 2.91/1.58 ( ( X3
% 2.91/1.58 = ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
% 2.91/1.58 => ( ( produc317566658_a_a_e @ X3 )
% 2.91/1.58 = Z2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_322_snd__eqD,axiom,
% 2.91/1.58 ! [X3: set_a,Y3: a > e,A2: a > e] :
% 2.91/1.58 ( ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 2.91/1.58 = A2 )
% 2.91/1.58 => ( Y3 = A2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_323_snd__conv,axiom,
% 2.91/1.58 ! [X1: set_a,X22: a > e] :
% 2.91/1.58 ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
% 2.91/1.58 = X22 ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_324_eq__snd__iff,axiom,
% 2.91/1.58 ! [B2: a > e,P: produc715398134_a_a_e] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( produc317566658_a_a_e @ P ) )
% 2.91/1.58 = ( ? [A3: set_a] :
% 2.91/1.58 ( P
% 2.91/1.58 = ( produc1134456688_a_a_e @ A3 @ B2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_325_prod__eq__iff,axiom,
% 2.91/1.58 ( ( ^ [Y2: produc715398134_a_a_e,Z: produc715398134_a_a_e] : ( Y2 = Z ) )
% 2.91/1.58 = ( ^ [S3: produc715398134_a_a_e,T: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ( produc1040928388_a_a_e @ S3 )
% 2.91/1.58 = ( produc1040928388_a_a_e @ T ) )
% 2.91/1.58 & ( ( produc317566658_a_a_e @ S3 )
% 2.91/1.58 = ( produc317566658_a_a_e @ T ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_326_prod_Oexpand,axiom,
% 2.91/1.58 ! [Prod: produc715398134_a_a_e,Prod2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ( ( produc1040928388_a_a_e @ Prod )
% 2.91/1.58 = ( produc1040928388_a_a_e @ Prod2 ) )
% 2.91/1.58 & ( ( produc317566658_a_a_e @ Prod )
% 2.91/1.58 = ( produc317566658_a_a_e @ Prod2 ) ) )
% 2.91/1.58 => ( Prod = Prod2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_327_prod__eqI,axiom,
% 2.91/1.58 ! [P: produc715398134_a_a_e,Q2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( ( produc1040928388_a_a_e @ P )
% 2.91/1.58 = ( produc1040928388_a_a_e @ Q2 ) )
% 2.91/1.58 => ( ( ( produc317566658_a_a_e @ P )
% 2.91/1.58 = ( produc317566658_a_a_e @ Q2 ) )
% 2.91/1.58 => ( P = Q2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_328_UNIV__eq__I,axiom,
% 2.91/1.58 ! [A: set_set_a] :
% 2.91/1.58 ( ! [X2: set_a] : ( member_set_a @ X2 @ A )
% 2.91/1.58 => ( top_top_set_set_a = A ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_329_UNIV__eq__I,axiom,
% 2.91/1.58 ! [A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ! [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ A )
% 2.91/1.58 => ( top_to1601855068_a_a_e = A ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_330_UNIV__witness,axiom,
% 2.91/1.58 ? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_331_UNIV__witness,axiom,
% 2.91/1.58 ? [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ top_to1601855068_a_a_e ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_332_surjective__pairing,axiom,
% 2.91/1.58 ! [T2: produc715398134_a_a_e] :
% 2.91/1.58 ( T2
% 2.91/1.58 = ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ T2 ) @ ( produc317566658_a_a_e @ T2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_333_prod_Oexhaust__sel,axiom,
% 2.91/1.58 ! [Prod: produc715398134_a_a_e] :
% 2.91/1.58 ( Prod
% 2.91/1.58 = ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_334_prod__fun__imageE,axiom,
% 2.91/1.58 ! [C2: produc715398134_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e,R2: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ C2 @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) )
% 2.91/1.58 => ~ ! [X2: set_a,Y: a > e] :
% 2.91/1.58 ( ( C2
% 2.91/1.58 = ( produc1134456688_a_a_e @ ( F @ X2 ) @ ( G @ Y ) ) )
% 2.91/1.58 => ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R2 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_335_exI__realizer,axiom,
% 2.91/1.58 ! [P2: ( a > e ) > set_a > $o,Y3: a > e,X3: set_a] :
% 2.91/1.58 ( ( P2 @ Y3 @ X3 )
% 2.91/1.58 => ( P2 @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_336_conjI__realizer,axiom,
% 2.91/1.58 ! [P2: set_a > $o,P: set_a,Q: ( a > e ) > $o,Q2: a > e] :
% 2.91/1.58 ( ( P2 @ P )
% 2.91/1.58 => ( ( Q @ Q2 )
% 2.91/1.58 => ( ( P2 @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) )
% 2.91/1.58 & ( Q @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_337_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.91/1.58 ! [P2: set_a > ( a > e ) > $o,X3: set_a,Y3: a > e,A2: produc715398134_a_a_e] :
% 2.91/1.58 ( ( P2 @ X3 @ Y3 )
% 2.91/1.58 => ( ( A2
% 2.91/1.58 = ( produc1134456688_a_a_e @ X3 @ Y3 ) )
% 2.91/1.58 => ( P2 @ ( produc1040928388_a_a_e @ A2 ) @ ( produc317566658_a_a_e @ A2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_338_top__empty__eq,axiom,
% 2.91/1.58 ( top_top_set_a_o
% 2.91/1.58 = ( ^ [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_339_top__empty__eq,axiom,
% 2.91/1.58 ( top_to2000253225_a_e_o
% 2.91/1.58 = ( ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ top_to1601855068_a_a_e ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_340_exE__realizer_H,axiom,
% 2.91/1.58 ! [P2: ( a > e ) > set_a > $o,P: produc715398134_a_a_e] :
% 2.91/1.58 ( ( P2 @ ( produc317566658_a_a_e @ P ) @ ( produc1040928388_a_a_e @ P ) )
% 2.91/1.58 => ~ ! [X2: set_a,Y: a > e] :
% 2.91/1.58 ~ ( P2 @ Y @ X2 ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_341_Collect__split__mono__strong,axiom,
% 2.91/1.58 ! [X6: set_set_a,A: set_Pr204296108_a_a_e,Y7: set_a_e,P2: set_a > ( a > e ) > $o,Q: set_a > ( a > e ) > $o] :
% 2.91/1.58 ( ( X6
% 2.91/1.58 = ( image_64240033_set_a @ produc1040928388_a_a_e @ A ) )
% 2.91/1.58 => ( ( Y7
% 2.91/1.58 = ( image_1149879910_e_a_e @ produc317566658_a_a_e @ A ) )
% 2.91/1.58 => ( ! [X2: set_a] :
% 2.91/1.58 ( ( member_set_a @ X2 @ X6 )
% 2.91/1.58 => ! [Xa2: a > e] :
% 2.91/1.58 ( ( member_a_e @ Xa2 @ Y7 )
% 2.91/1.58 => ( ( P2 @ X2 @ Xa2 )
% 2.91/1.58 => ( Q @ X2 @ Xa2 ) ) ) )
% 2.91/1.58 => ( ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) )
% 2.91/1.58 => ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_342_sndOp__def,axiom,
% 2.91/1.58 ( bNF_sn375433437_a_a_e
% 2.91/1.58 = ( ^ [P4: set_a > set_a > $o,Q3: set_a > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( bNF_pi929741261_a_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_343_Product__Type_OCollect__case__prodD,axiom,
% 2.91/1.58 ! [X3: produc715398134_a_a_e,A: set_a > ( a > e ) > $o] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ X3 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ A ) ) )
% 2.91/1.58 => ( A @ ( produc1040928388_a_a_e @ X3 ) @ ( produc317566658_a_a_e @ X3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_344_fstOp__def,axiom,
% 2.91/1.58 ( bNF_fs1024561536_e_a_e
% 2.91/1.58 = ( ^ [P4: set_a > ( a > e ) > $o,Q3: ( a > e ) > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Ac ) @ ( bNF_pi1175744974_e_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_345_mem__case__prodE,axiom,
% 2.91/1.58 ! [Z2: set_a,C2: set_a > ( a > e ) > set_set_a,P: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member_set_a @ Z2 @ ( produc1620938593_set_a @ C2 @ P ) )
% 2.91/1.58 => ~ ! [X2: set_a,Y: a > e] :
% 2.91/1.58 ( ( P
% 2.91/1.58 = ( produc1134456688_a_a_e @ X2 @ Y ) )
% 2.91/1.58 => ~ ( member_set_a @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_346_mem__case__prodE,axiom,
% 2.91/1.58 ! [Z2: produc715398134_a_a_e,C2: set_a > ( a > e ) > set_Pr204296108_a_a_e,P: produc715398134_a_a_e] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ Z2 @ ( produc1096319049_a_a_e @ C2 @ P ) )
% 2.91/1.58 => ~ ! [X2: set_a,Y: a > e] :
% 2.91/1.58 ( ( P
% 2.91/1.58 = ( produc1134456688_a_a_e @ X2 @ Y ) )
% 2.91/1.58 => ~ ( member1258382221_a_a_e @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_347_in__rel__def,axiom,
% 2.91/1.58 ( fun_in_rel_set_a_a_e
% 2.91/1.58 = ( ^ [R3: set_Pr204296108_a_a_e,X4: set_a,Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X4 @ Y4 ) @ R3 ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_348_fstOp__in,axiom,
% 2.91/1.58 ! [Ac2: produc715398134_a_a_e,P2: set_a > ( a > e ) > $o,Q: ( a > e ) > ( a > e ) > $o] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom658603713_e_a_e @ P2 @ Q ) ) ) )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( bNF_fs1024561536_e_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_349_sndOp__in,axiom,
% 2.91/1.58 ! [Ac2: produc715398134_a_a_e,P2: set_a > set_a > $o,Q: set_a > ( a > e ) > $o] :
% 2.91/1.58 ( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom779348378_a_a_e @ P2 @ Q ) ) ) )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( bNF_sn375433437_a_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_350_image2__eqI,axiom,
% 2.91/1.58 ! [B2: set_a,F: set_a > set_a,X3: set_a,C2: a > e,G: set_a > a > e,A: set_set_a] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( C2
% 2.91/1.58 = ( G @ X3 ) )
% 2.91/1.58 => ( ( member_set_a @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr1872575811_a_a_e @ A @ F @ G ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(fact_351_image2__eqI,axiom,
% 2.91/1.58 ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,C2: a > e,G: produc715398134_a_a_e > a > e,A: set_Pr204296108_a_a_e] :
% 2.91/1.58 ( ( B2
% 2.91/1.58 = ( F @ X3 ) )
% 2.91/1.58 => ( ( C2
% 2.91/1.58 = ( G @ X3 ) )
% 2.91/1.58 => ( ( member1258382221_a_a_e @ X3 @ A )
% 2.91/1.58 => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr2072183031_a_a_e @ A @ F @ G ) ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(conj_0,hypothesis,
% 2.91/1.58 ! [A7: set_a] :
% 2.91/1.58 ( ( member_set_a @ A7 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ a1 @ A7 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ a2 @ A7 )
% 2.91/1.58 => ( ( ord_less_eq_set_a @ a3 @ A7 )
% 2.91/1.58 => thesis ) ) ) ) ).
% 2.91/1.58
% 2.91/1.58 thf(conj_1,conjecture,
% 2.91/1.58 thesis ).
% 2.91/1.58 ------------------------
% 3.02/1.61 % (18216)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/2Mi)
% 3.02/1.61 % (18217)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_17883 for (2999ds/2Mi)
% 3.02/1.61 % (18216)Instruction limit reached!
% 3.02/1.61 % (18216)------------------------------
% 3.02/1.61 % (18216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (18216)Termination reason: Unknown
% 3.02/1.61 % (18216)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (18216)Memory used [KB]: 1407
% 3.02/1.61 % (18216)Time elapsed: 0.002 s
% 3.02/1.61 % (18216)Instructions burned: 2 (million)
% 3.02/1.61 % (18216)------------------------------
% 3.02/1.61 % (18216)------------------------------
% 3.02/1.61 % (18217)Instruction limit reached!
% 3.02/1.61 % (18217)------------------------------
% 3.02/1.61 % (18217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (18217)Termination reason: Unknown
% 3.02/1.61 % (18217)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (18217)Memory used [KB]: 1407
% 3.02/1.61 % (18217)Time elapsed: 0.002 s
% 3.02/1.61 % (18217)Instructions burned: 2 (million)
% 3.02/1.61 % (18217)------------------------------
% 3.02/1.61 % (18217)------------------------------
% 3.02/1.61 % (18219)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/18Mi)
% 3.02/1.61 % (18219)Instruction limit reached!
% 3.02/1.61 % (18219)------------------------------
% 3.02/1.61 % (18219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (18219)Termination reason: Unknown
% 3.02/1.61 % (18219)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (18219)Memory used [KB]: 1791
% 3.02/1.61 % (18219)Time elapsed: 0.007 s
% 3.02/1.61 % (18219)Instructions burned: 18 (million)
% 3.02/1.61 % (18219)------------------------------
% 3.02/1.61 % (18219)------------------------------
% 3.02/1.61 % (18213)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/183Mi)
% 3.02/1.61 % (18214)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_17883 for (2999ds/4Mi)
% 3.02/1.61 % (18215)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_17883 for (2999ds/27Mi)
% 3.02/1.61 % (18218)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_17883 for (2999ds/275Mi)
% 3.02/1.62 % (18220)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/3Mi)
% 3.02/1.62 % (18221)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_17883 for (2999ds/37Mi)
% 3.02/1.62 % (18214)Instruction limit reached!
% 3.02/1.62 % (18214)------------------------------
% 3.02/1.62 % (18214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.62 % (18214)Termination reason: Unknown
% 3.02/1.62 % (18214)Termination phase: shuffling
% 3.02/1.62
% 3.02/1.62 % (18214)Memory used [KB]: 1535
% 3.02/1.62 % (18214)Time elapsed: 0.005 s
% 3.02/1.62 % (18214)Instructions burned: 5 (million)
% 3.02/1.62 % (18214)------------------------------
% 3.02/1.62 % (18214)------------------------------
% 3.02/1.62 % (18220)Instruction limit reached!
% 3.02/1.62 % (18220)------------------------------
% 3.02/1.62 % (18220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.62 % (18220)Termination reason: Unknown
% 3.02/1.62 % (18220)Termination phase: shuffling
% 3.02/1.62
% 3.02/1.62 % (18220)Memory used [KB]: 1535
% 3.02/1.62 % (18220)Time elapsed: 0.003 s
% 3.02/1.62 % (18220)Instructions burned: 5 (million)
% 3.02/1.62 % (18220)------------------------------
% 3.02/1.62 % (18220)------------------------------
% 3.02/1.62 % (18222)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_17883 for (2999ds/15Mi)
% 3.02/1.63 % (18221)Instruction limit reached!
% 3.02/1.63 % (18221)------------------------------
% 3.02/1.63 % (18221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18221)Termination reason: Unknown
% 3.02/1.63 % (18221)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (18221)Memory used [KB]: 2174
% 3.02/1.63 % (18221)Time elapsed: 0.012 s
% 3.02/1.63 % (18221)Instructions burned: 38 (million)
% 3.02/1.63 % (18221)------------------------------
% 3.02/1.63 % (18221)------------------------------
% 3.02/1.63 % (18224)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_17883 for (2999ds/1041Mi)
% 3.02/1.63 % (18222)Instruction limit reached!
% 3.02/1.63 % (18222)------------------------------
% 3.02/1.63 % (18222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18222)Termination reason: Unknown
% 3.02/1.63 % (18222)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (18222)Memory used [KB]: 1791
% 3.02/1.63 % (18222)Time elapsed: 0.007 s
% 3.02/1.63 % (18222)Instructions burned: 17 (million)
% 3.02/1.63 % (18222)------------------------------
% 3.02/1.63 % (18222)------------------------------
% 3.02/1.63 % (18215)Instruction limit reached!
% 3.02/1.63 % (18215)------------------------------
% 3.02/1.63 % (18215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18215)Termination reason: Unknown
% 3.02/1.63 % (18215)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (18215)Memory used [KB]: 2046
% 3.02/1.63 % (18215)Time elapsed: 0.019 s
% 3.02/1.63 % (18215)Instructions burned: 27 (million)
% 3.02/1.63 % (18215)------------------------------
% 3.02/1.63 % (18215)------------------------------
% 3.02/1.63 % (18223)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_17883 for (2999ds/3Mi)
% 3.02/1.63 % (18223)Instruction limit reached!
% 3.02/1.63 % (18223)------------------------------
% 3.02/1.63 % (18223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18223)Termination reason: Unknown
% 3.02/1.63 % (18223)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (18223)Memory used [KB]: 1407
% 3.02/1.63 % (18223)Time elapsed: 0.003 s
% 3.02/1.63 % (18223)Instructions burned: 3 (million)
% 3.02/1.63 % (18223)------------------------------
% 3.02/1.63 % (18223)------------------------------
% 3.02/1.64 % (18225)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/7Mi)
% 3.02/1.64 % (18226)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/16Mi)
% 3.02/1.64 % (18225)Instruction limit reached!
% 3.02/1.64 % (18225)------------------------------
% 3.02/1.64 % (18225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (18225)Termination reason: Unknown
% 3.02/1.64 % (18225)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (18225)Memory used [KB]: 1535
% 3.02/1.64 % (18225)Time elapsed: 0.004 s
% 3.02/1.64 % (18225)Instructions burned: 8 (million)
% 3.02/1.64 % (18225)------------------------------
% 3.02/1.64 % (18225)------------------------------
% 3.02/1.64 % (18228)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_17883 for (2999ds/3Mi)
% 3.02/1.64 % (18226)Instruction limit reached!
% 3.02/1.64 % (18226)------------------------------
% 3.02/1.64 % (18226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (18226)Termination reason: Unknown
% 3.02/1.64 % (18226)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (18226)Memory used [KB]: 1791
% 3.02/1.64 % (18226)Time elapsed: 0.007 s
% 3.02/1.64 % (18226)Instructions burned: 18 (million)
% 3.02/1.64 % (18226)------------------------------
% 3.02/1.64 % (18226)------------------------------
% 3.02/1.65 % (18228)Instruction limit reached!
% 3.02/1.65 % (18228)------------------------------
% 3.02/1.65 % (18228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18228)Termination reason: Unknown
% 3.02/1.65 % (18228)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18228)Memory used [KB]: 1535
% 3.02/1.65 % (18228)Time elapsed: 0.003 s
% 3.02/1.65 % (18228)Instructions burned: 5 (million)
% 3.02/1.65 % (18228)------------------------------
% 3.02/1.65 % (18228)------------------------------
% 3.02/1.65 % (18227)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_17883 for (2999ds/3Mi)
% 3.02/1.65 % (18229)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/7Mi)
% 3.02/1.65 % (18227)Instruction limit reached!
% 3.02/1.65 % (18227)------------------------------
% 3.02/1.65 % (18227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18227)Termination reason: Unknown
% 3.02/1.65 % (18227)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18227)Memory used [KB]: 1535
% 3.02/1.65 % (18227)Time elapsed: 0.004 s
% 3.02/1.65 % (18227)Instructions burned: 3 (million)
% 3.02/1.65 % (18227)------------------------------
% 3.02/1.65 % (18227)------------------------------
% 3.02/1.65 % (18229)Instruction limit reached!
% 3.02/1.65 % (18229)------------------------------
% 3.02/1.65 % (18229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18229)Termination reason: Unknown
% 3.02/1.65 % (18229)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18229)Memory used [KB]: 1663
% 3.02/1.65 % (18229)Time elapsed: 0.004 s
% 3.02/1.65 % (18229)Instructions burned: 9 (million)
% 3.02/1.65 % (18229)------------------------------
% 3.02/1.65 % (18229)------------------------------
% 3.02/1.65 % (18230)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/3Mi)
% 3.02/1.65 % (18230)Instruction limit reached!
% 3.02/1.65 % (18230)------------------------------
% 3.02/1.65 % (18230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18230)Termination reason: Unknown
% 3.02/1.65 % (18230)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18230)Memory used [KB]: 1535
% 3.02/1.65 % (18230)Time elapsed: 0.002 s
% 3.02/1.65 % (18230)Instructions burned: 3 (million)
% 3.02/1.65 % (18230)------------------------------
% 3.02/1.65 % (18230)------------------------------
% 3.02/1.66 % (18231)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/4Mi)
% 3.02/1.66 % (18231)Instruction limit reached!
% 3.02/1.66 % (18231)------------------------------
% 3.02/1.66 % (18231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.66 % (18231)Termination reason: Unknown
% 3.02/1.66 % (18231)Termination phase: shuffling
% 3.02/1.66
% 3.02/1.66 % (18231)Memory used [KB]: 1535
% 3.02/1.66 % (18231)Time elapsed: 0.003 s
% 3.02/1.66 % (18231)Instructions burned: 4 (million)
% 3.02/1.66 % (18231)------------------------------
% 3.02/1.66 % (18231)------------------------------
% 3.02/1.66 % (18233)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/710Mi)
% 3.02/1.66 % (18234)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/6Mi)
% 3.02/1.67 % (18234)Instruction limit reached!
% 3.02/1.67 % (18234)------------------------------
% 3.02/1.67 % (18234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (18234)Termination reason: Unknown
% 3.02/1.67 % (18234)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (18234)Memory used [KB]: 1535
% 3.02/1.67 % (18234)Time elapsed: 0.003 s
% 3.02/1.67 % (18234)Instructions burned: 6 (million)
% 3.02/1.67 % (18234)------------------------------
% 3.02/1.67 % (18234)------------------------------
% 3.02/1.67 % (18235)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/902Mi)
% 3.02/1.67 % (18218)First to succeed.
% 3.02/1.67 % (18232)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17883 for (2999ds/18Mi)
% 3.02/1.67 % (18218)Refutation found. Thanks to Tanya!
% 3.02/1.67 % SZS status Theorem for DTF2THF_17883
% 3.02/1.67 % SZS output start Proof for DTF2THF_17883
% 3.02/1.67 thf(type_def_5, type, set_a: $tType).
% 3.02/1.67 thf(type_def_6, type, a: $tType).
% 3.02/1.67 thf(type_def_7, type, e: $tType).
% 3.02/1.67 thf(type_def_9, type, produc715398134_a_a_e: $tType).
% 3.02/1.67 thf(type_def_10, type, set_Pr204296108_a_a_e: $tType).
% 3.02/1.67 thf(type_def_11, type, set_set_a: $tType).
% 3.02/1.67 thf(type_def_12, type, set_se821189730_a_a_e: $tType).
% 3.02/1.67 thf(type_def_13, type, set_set_set_a: $tType).
% 3.02/1.67 thf(type_def_14, type, set_a_e: $tType).
% 3.02/1.67 thf(type_def_15, type, set_Pr305929312_set_a: $tType).
% 3.02/1.67 thf(type_def_16, type, set_Pr1710391469_a_a_e: $tType).
% 3.02/1.67 thf(type_def_17, type, set_Pr2045688071_set_a: $tType).
% 3.02/1.67 thf(type_def_18, type, produc1965228714_set_a: $tType).
% 3.02/1.67 thf(type_def_19, type, produc169085773_a_a_e: $tType).
% 3.02/1.67 thf(type_def_20, type, produc1691597095_set_a: $tType).
% 3.02/1.67 thf(func_def_0, type, set_Pr1710391469_a_a_e: $tType).
% 3.02/1.67 thf(func_def_1, type, produc169085773_a_a_e: $tType).
% 3.02/1.67 thf(func_def_2, type, set_se821189730_a_a_e: $tType).
% 3.02/1.67 thf(func_def_3, type, set_Pr2045688071_set_a: $tType).
% 3.02/1.67 thf(func_def_4, type, set_Pr204296108_a_a_e: $tType).
% 3.02/1.67 thf(func_def_5, type, set_Pr305929312_set_a: $tType).
% 3.02/1.67 thf(func_def_6, type, produc1691597095_set_a: $tType).
% 3.02/1.67 thf(func_def_7, type, produc715398134_a_a_e: $tType).
% 3.02/1.67 thf(func_def_8, type, produc1965228714_set_a: $tType).
% 3.02/1.67 thf(func_def_9, type, set_set_set_a: $tType).
% 3.02/1.67 thf(func_def_10, type, set_set_a: $tType).
% 3.02/1.67 thf(func_def_11, type, set_a_e: $tType).
% 3.02/1.67 thf(func_def_12, type, set_a: $tType).
% 3.02/1.67 thf(func_def_13, type, e: $tType).
% 3.02/1.67 thf(func_def_14, type, a: $tType).
% 3.02/1.67 thf(func_def_15, type, bNF_fs1024561536_e_a_e: (set_a > (a > e) > $o) > ((a > e) > (a > e) > $o) > produc715398134_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_16, type, bNF_pi1175744974_e_a_e: (set_a > (a > e) > $o) > ((a > e) > (a > e) > $o) > set_a > (a > e) > a > e).
% 3.02/1.67 thf(func_def_17, type, bNF_pi929741261_a_a_e: (set_a > set_a > $o) > (set_a > (a > e) > $o) > set_a > (a > e) > set_a).
% 3.02/1.67 thf(func_def_18, type, bNF_sn375433437_a_a_e: (set_a > set_a > $o) > (set_a > (a > e) > $o) > produc715398134_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_19, type, bNF_Gr2072183031_a_a_e: set_Pr204296108_a_a_e > (produc715398134_a_a_e > set_a) > (produc715398134_a_a_e > a > e) > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_20, type, bNF_Gr1872575811_a_a_e: set_set_a > (set_a > set_a) > (set_a > a > e) > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_21, type, finite33065488_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e).
% 3.02/1.67 thf(func_def_22, type, finite_Fpow_set_a: set_set_a > set_set_set_a).
% 3.02/1.67 thf(func_def_23, type, finite_Fpow_a: set_a > set_set_a).
% 3.02/1.67 thf(func_def_24, type, fun_in_rel_set_a_a_e: set_Pr204296108_a_a_e > set_a > (a > e) > $o).
% 3.02/1.67 thf(func_def_25, type, ord_less_eq_o_set_a: ($o > set_a) > ($o > set_a) > $o).
% 3.02/1.67 thf(func_def_26, type, ord_less_eq_set_a_e: set_a_e > set_a_e > $o).
% 3.02/1.67 thf(func_def_27, type, ord_le802356416_set_a: set_Pr305929312_set_a > set_Pr305929312_set_a > $o).
% 3.02/1.67 thf(func_def_28, type, ord_le700723212_a_a_e: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > $o).
% 3.02/1.67 thf(func_def_29, type, ord_le2103863998_set_a: set_set_set_a > set_set_set_a > $o).
% 3.02/1.67 thf(func_def_30, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 3.02/1.67 thf(func_def_31, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.02/1.67 thf(func_def_32, type, order_Greatest_set_a: (set_a > $o) > set_a).
% 3.02/1.67 thf(func_def_33, type, top_to2000253225_a_e_o: produc715398134_a_a_e > $o).
% 3.02/1.67 thf(func_def_34, type, top_top_set_a_o: set_a > $o).
% 3.02/1.67 thf(func_def_35, type, top_to970499581_a_a_e: set_Pr1710391469_a_a_e).
% 3.02/1.67 thf(func_def_36, type, top_to1601855068_a_a_e: set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_37, type, top_to144004695_set_a: set_Pr2045688071_set_a).
% 3.02/1.67 thf(func_def_38, type, top_top_set_set_a: set_set_a).
% 3.02/1.67 thf(func_def_39, type, top_top_set_a: set_a).
% 3.02/1.67 thf(func_def_40, type, produc479206876_set_a: (a > e) > set_a > produc1965228714_set_a).
% 3.02/1.67 thf(func_def_41, type, produc1134456688_a_a_e: set_a > (a > e) > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_42, type, produc952037978_a_e_o: (produc715398134_a_a_e > $o) > set_a > (a > e) > $o).
% 3.02/1.67 thf(func_def_43, type, produc387505142_set_a: (produc715398134_a_a_e > set_a) > (produc715398134_a_a_e > set_a) > produc169085773_a_a_e > produc1691597095_set_a).
% 3.02/1.67 thf(func_def_44, type, produc1443219402_e_a_e: (set_a > set_a) > ((a > e) > a > e) > produc715398134_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_45, type, produc1341862587_a_e_o: (set_a > (a > e) > $o) > produc715398134_a_a_e > $o).
% 3.02/1.67 thf(func_def_46, type, produc1096319049_a_a_e: (set_a > (a > e) > set_Pr204296108_a_a_e) > produc715398134_a_a_e > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_47, type, produc1620938593_set_a: (set_a > (a > e) > set_set_a) > produc715398134_a_a_e > set_set_a).
% 3.02/1.67 thf(func_def_48, type, produc385678576_set_a: produc1965228714_set_a > a > e).
% 3.02/1.67 thf(func_def_49, type, produc1040928388_a_a_e: produc715398134_a_a_e > set_a).
% 3.02/1.67 thf(func_def_50, type, produc1809800494_set_a: produc1965228714_set_a > set_a).
% 3.02/1.67 thf(func_def_51, type, produc317566658_a_a_e: produc715398134_a_a_e > a > e).
% 3.02/1.67 thf(func_def_52, type, produc100464700_set_a: produc1965228714_set_a > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_53, type, produc755714512_a_a_e: produc715398134_a_a_e > produc1965228714_set_a).
% 3.02/1.67 thf(func_def_54, type, domain_set_a_a_e: set_Pr204296108_a_a_e > set_set_a).
% 3.02/1.67 thf(func_def_55, type, range_set_a_a_e: set_Pr204296108_a_a_e > set_a_e).
% 3.02/1.67 thf(func_def_56, type, converse_a_e_set_a: set_Pr305929312_set_a > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_57, type, converse_set_a_a_e: set_Pr204296108_a_a_e > set_Pr305929312_set_a).
% 3.02/1.67 thf(func_def_58, type, relcom658603713_e_a_e: (set_a > (a > e) > $o) > ((a > e) > (a > e) > $o) > set_a > (a > e) > $o).
% 3.02/1.67 thf(func_def_59, type, relcom779348378_a_a_e: (set_a > set_a > $o) > (set_a > (a > e) > $o) > set_a > (a > e) > $o).
% 3.02/1.67 thf(func_def_60, type, collec1369693387_a_a_e: (produc715398134_a_a_e > $o) > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_61, type, collect_set_a: (set_a > $o) > set_set_a).
% 3.02/1.67 thf(func_def_62, type, collect_a: (a > $o) > set_a).
% 3.02/1.67 thf(func_def_63, type, pow_Pr328302361_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e).
% 3.02/1.67 thf(func_def_64, type, pow_set_a: set_set_a > set_set_set_a).
% 3.02/1.67 thf(func_def_65, type, pow_a: set_a > set_set_a).
% 3.02/1.67 thf(func_def_66, type, image_a_e_a: ((a > e) > a) > set_a_e > set_a).
% 3.02/1.67 thf(func_def_67, type, image_376979367_a_a_e: (produc1965228714_set_a > produc715398134_a_a_e) > set_Pr305929312_set_a > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_68, type, image_2101504031_set_a: (produc169085773_a_a_e > produc1691597095_set_a) > set_Pr1710391469_a_a_e > set_Pr2045688071_set_a).
% 3.02/1.67 thf(func_def_69, type, image_1149879910_e_a_e: (produc715398134_a_a_e > a > e) > set_Pr204296108_a_a_e > set_a_e).
% 3.02/1.67 thf(func_def_70, type, image_1875450791_set_a: (produc715398134_a_a_e > produc1965228714_set_a) > set_Pr204296108_a_a_e > set_Pr305929312_set_a).
% 3.02/1.67 thf(func_def_71, type, image_625620211_a_a_e: (produc715398134_a_a_e > produc715398134_a_a_e) > set_Pr204296108_a_a_e > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_72, type, image_64240033_set_a: (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > set_set_a).
% 3.02/1.67 thf(func_def_73, type, image_2116492353_a_e_a: (produc715398134_a_a_e > a) > set_Pr204296108_a_a_e > set_a).
% 3.02/1.67 thf(func_def_74, type, image_1441218763_set_a: (set_Pr204296108_a_a_e > set_set_a) > set_se821189730_a_a_e > set_set_set_a).
% 3.02/1.67 thf(func_def_75, type, image_1956910981_set_a: (set_set_a > set_set_a) > set_set_set_a > set_set_set_a).
% 3.02/1.67 thf(func_def_76, type, image_402591269_set_a: (set_set_a > set_a) > set_set_set_a > set_set_a).
% 3.02/1.67 thf(func_def_77, type, image_set_set_a_a: (set_set_a > a) > set_set_set_a > set_a).
% 3.02/1.67 thf(func_def_78, type, image_set_a_a_e: (set_a > a > e) > set_set_a > set_a_e).
% 3.02/1.67 thf(func_def_79, type, image_210900495_a_a_e: (set_a > produc715398134_a_a_e) > set_set_a > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_80, type, image_1930509541_set_a: (set_a > set_set_a) > set_set_a > set_set_set_a).
% 3.02/1.67 thf(func_def_81, type, image_set_a_set_a: (set_a > set_a) > set_set_a > set_set_a).
% 3.02/1.67 thf(func_def_82, type, image_set_a_a: (set_a > a) > set_set_a > set_a).
% 3.02/1.67 thf(func_def_83, type, image_a_a_e: (a > a > e) > set_a > set_a_e).
% 3.02/1.67 thf(func_def_84, type, image_a_set_set_a: (a > set_set_a) > set_a > set_set_set_a).
% 3.02/1.67 thf(func_def_85, type, image_a_set_a: (a > set_a) > set_a > set_set_a).
% 3.02/1.67 thf(func_def_86, type, image_a_a: (a > a) > set_a > set_a).
% 3.02/1.67 thf(func_def_87, type, member_a_e: (a > e) > set_a_e > $o).
% 3.02/1.67 thf(func_def_88, type, member360729153_set_a: produc1965228714_set_a > set_Pr305929312_set_a > $o).
% 3.02/1.67 thf(func_def_89, type, member1258382221_a_a_e: produc715398134_a_a_e > set_Pr204296108_a_a_e > $o).
% 3.02/1.67 thf(func_def_90, type, member1437121347_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e > $o).
% 3.02/1.67 thf(func_def_91, type, member_set_set_a: set_set_a > set_set_set_a > $o).
% 3.02/1.67 thf(func_def_92, type, member_set_a: set_a > set_set_a > $o).
% 3.02/1.67 thf(func_def_93, type, member_a: a > set_a > $o).
% 3.02/1.67 thf(func_def_94, type, c: set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_95, type, a1: set_a).
% 3.02/1.67 thf(func_def_96, type, a2: set_a).
% 3.02/1.67 thf(func_def_97, type, a3: set_a).
% 3.02/1.67 thf(func_def_98, type, b1: a > e).
% 3.02/1.67 thf(func_def_99, type, b2: a > e).
% 3.02/1.67 thf(func_def_100, type, b3: a > e).
% 3.02/1.67 thf(func_def_113, type, sK0: (produc715398134_a_a_e > $o) > a > e).
% 3.02/1.67 thf(func_def_114, type, sK1: (produc715398134_a_a_e > $o) > set_a).
% 3.02/1.67 thf(func_def_115, type, sK2: produc715398134_a_a_e > a > e).
% 3.02/1.67 thf(func_def_116, type, sK3: produc715398134_a_a_e > set_a).
% 3.02/1.67 thf(func_def_117, type, sK4: (produc715398134_a_a_e > set_a) > set_a > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_118, type, sK5: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > a > e).
% 3.02/1.67 thf(func_def_119, type, sK6: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > set_a).
% 3.02/1.67 thf(func_def_120, type, sK7: (produc715398134_a_a_e > set_a) > set_set_a > set_Pr204296108_a_a_e > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_121, type, sK8: produc715398134_a_a_e > set_a > a > e).
% 3.02/1.67 thf(func_def_122, type, sK9: (produc715398134_a_a_e > $o) > set_a).
% 3.02/1.67 thf(func_def_123, type, sK10: (produc715398134_a_a_e > $o) > a > e).
% 3.02/1.67 thf(func_def_124, type, sK11: (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > set_set_a > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_125, type, sK12: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_126, type, sK13: set_set_a > set_set_a > set_a).
% 3.02/1.67 thf(func_def_127, type, sK14: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_128, type, sK15: set_set_a > set_set_a > set_a).
% 3.02/1.67 thf(func_def_129, type, sK16: set_Pr204296108_a_a_e > (set_a > $o) > (produc715398134_a_a_e > set_a) > set_a).
% 3.02/1.67 thf(func_def_130, type, sK17: set_set_a > set_set_a > set_a).
% 3.02/1.67 thf(func_def_131, type, sK18: set_a).
% 3.02/1.67 thf(func_def_132, type, sK19: (produc715398134_a_a_e > set_a) > (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_133, type, sK20: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_134, type, sK21: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_135, type, sK22: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_136, type, sK23: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_137, type, sK24: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_138, type, sK25: set_a > (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_139, type, sK26: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_140, type, sK27: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_141, type, sK28: set_Pr204296108_a_a_e > set_set_a > (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_142, type, sK29: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_143, type, sK30: (set_a > set_a) > set_a).
% 3.02/1.67 thf(func_def_144, type, sK31: (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > set_set_a > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_145, type, sK32: (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > set_set_a > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_146, type, sK33: (produc715398134_a_a_e > set_a) > (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_147, type, sK34: (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > (produc715398134_a_a_e > set_a) > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_148, type, sK35: (set_a > set_set_a) > set_a).
% 3.02/1.67 thf(func_def_149, type, sK36: (set_a > set_set_a) > set_a).
% 3.02/1.67 thf(func_def_150, type, sK37: (produc715398134_a_a_e > set_a) > set_Pr204296108_a_a_e > (set_a > $o) > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_151, type, sK38: (set_set_a > set_a) > set_set_a).
% 3.02/1.67 thf(func_def_152, type, sK39: (set_set_a > set_a) > set_set_a).
% 3.02/1.67 thf(func_def_153, type, sK40: set_set_a > set_set_a > set_a).
% 3.02/1.67 thf(func_def_154, type, sK41: (produc715398134_a_a_e > set_a) > (set_set_a > $o) > set_Pr204296108_a_a_e > set_set_a).
% 3.02/1.67 thf(func_def_155, type, sK42: (produc715398134_a_a_e > set_a) > (set_set_a > $o) > set_Pr204296108_a_a_e > set_Pr204296108_a_a_e).
% 3.02/1.67 thf(func_def_156, type, sK43: set_Pr204296108_a_a_e > set_Pr204296108_a_a_e > produc715398134_a_a_e).
% 3.02/1.67 thf(func_def_157, type, sK44: produc715398134_a_a_e > set_a).
% 3.02/1.67 thf(func_def_158, type, sK45: produc715398134_a_a_e > a > e).
% 3.02/1.67 thf(func_def_159, type, sK46: (set_set_a > set_set_a) > set_set_a).
% 3.02/1.67 thf(func_def_160, type, sK47: (set_set_a > set_set_a) > set_set_a).
% 3.02/1.67 thf(func_def_162, type, ph49: !>[X0: $tType]:(X0)).
% 3.02/1.67 thf(f1540,plain,(
% 3.02/1.67 $false),
% 3.02/1.67 inference(avatar_sat_refutation,[status(thm)],[f1470,f1471,f1509])).
% 3.02/1.67 thf(f1509,plain,(
% 3.02/1.67 ~spl48_1),
% 3.02/1.67 inference(avatar_contradiction_clause,[status(thm)],[f1508])).
% 3.02/1.67 thf(f1508,plain,(
% 3.02/1.67 $false | ~spl48_1),
% 3.02/1.67 inference(subsumption_resolution,[status(thm)],[f1507,f1318])).
% 3.02/1.67 thf(f1318,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a3 @ sK18) = $true)),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f1198])).
% 3.02/1.67 thf(f1198,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a1 @ sK18) = $true) & ((member_set_a @ sK18 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) = $true) & ((ord_less_eq_set_a @ a3 @ sK18) = $true) & ((ord_less_eq_set_a @ a2 @ sK18) = $true)),
% 3.02/1.67 inference(skolemisation,[status(esa),new_symbols(skolem,[sK18])],[f426,f1197])).
% 3.02/1.67 thf(f1197,plain,(
% 3.02/1.67 ? [X0 : set_a] : (((ord_less_eq_set_a @ a1 @ X0) = $true) & ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) = $true) & ($true = (ord_less_eq_set_a @ a3 @ X0)) & ((ord_less_eq_set_a @ a2 @ X0) = $true)) => (((ord_less_eq_set_a @ a1 @ sK18) = $true) & ((member_set_a @ sK18 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) = $true) & ((ord_less_eq_set_a @ a3 @ sK18) = $true) & ((ord_less_eq_set_a @ a2 @ sK18) = $true))),
% 3.02/1.67 introduced(definition,[],[choice_axiom])).
% 3.02/1.67 thf(f426,plain,(
% 3.02/1.67 ? [X0 : set_a] : (((ord_less_eq_set_a @ a1 @ X0) = $true) & ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) = $true) & ($true = (ord_less_eq_set_a @ a3 @ X0)) & ((ord_less_eq_set_a @ a2 @ X0) = $true))),
% 3.02/1.67 inference(fool_elimination,[status(thm)],[f425])).
% 3.02/1.67 thf(f425,plain,(
% 3.02/1.67 ? [X0 : set_a] : ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) & (ord_less_eq_set_a @ a2 @ X0) & (ord_less_eq_set_a @ a3 @ X0) & (ord_less_eq_set_a @ a1 @ X0))),
% 3.02/1.67 inference(rectify,[status(thm)],[f2])).
% 3.02/1.67 thf(f2,axiom,(
% 3.02/1.67 ? [X2 : set_a] : ((member_set_a @ X2 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) & (ord_less_eq_set_a @ a2 @ X2) & (ord_less_eq_set_a @ a3 @ X2) & (ord_less_eq_set_a @ a1 @ X2))),
% 3.02/1.67 file('/export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2THF_17883.p',fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062)).
% 3.02/1.67 thf(f1507,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a3 @ sK18) != $true) | ~spl48_1),
% 3.02/1.67 inference(subsumption_resolution,[status(thm)],[f1506,f1320])).
% 3.02/1.67 thf(f1320,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a1 @ sK18) = $true)),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f1198])).
% 3.02/1.67 thf(f1506,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a1 @ sK18) != $true) | ((ord_less_eq_set_a @ a3 @ sK18) != $true) | ~spl48_1),
% 3.02/1.67 inference(subsumption_resolution,[status(thm)],[f1498,f1317])).
% 3.02/1.67 thf(f1317,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a2 @ sK18) = $true)),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f1198])).
% 3.02/1.67 thf(f1498,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a2 @ sK18) != $true) | ((ord_less_eq_set_a @ a3 @ sK18) != $true) | ((ord_less_eq_set_a @ a1 @ sK18) != $true) | ~spl48_1),
% 3.02/1.67 inference(trivial_inequality_removal,[status(thm)],[f1489])).
% 3.02/1.67 thf(f1489,plain,(
% 3.02/1.67 ((ord_less_eq_set_a @ a2 @ sK18) != $true) | ((ord_less_eq_set_a @ a3 @ sK18) != $true) | ((ord_less_eq_set_a @ a1 @ sK18) != $true) | ($true != $true) | ~spl48_1),
% 3.02/1.67 inference(superposition,[status(thm)],[f1465,f1319])).
% 3.02/1.67 thf(f1319,plain,(
% 3.02/1.67 ((member_set_a @ sK18 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) = $true)),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f1198])).
% 3.02/1.67 thf(f1465,plain,(
% 3.02/1.67 ( ! [X0 : set_a] : (((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) != $true) | ((ord_less_eq_set_a @ a1 @ X0) != $true) | ((ord_less_eq_set_a @ a2 @ X0) != $true) | ($true != (ord_less_eq_set_a @ a3 @ X0))) ) | ~spl48_1),
% 3.02/1.67 inference(avatar_component_clause,[status(thm)],[f1464])).
% 3.02/1.67 thf(f1464,definition,(
% 3.02/1.67 spl48_1 <=> ! [X0 : set_a] : (((ord_less_eq_set_a @ a2 @ X0) != $true) | ($true != (ord_less_eq_set_a @ a3 @ X0)) | ((ord_less_eq_set_a @ a1 @ X0) != $true) | ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) != $true))),
% 3.02/1.67 introduced(definition,[new_symbols(naming,[spl48_1])],[avatar_definition])).
% 3.02/1.67 thf(f1471,plain,(
% 3.02/1.67 ~spl48_2),
% 3.02/1.67 inference(avatar_split_clause,[status(thm)],[f1331,f1467])).
% 3.02/1.67 thf(f1467,definition,(
% 3.02/1.67 spl48_2 <=> (thesis = $true)),
% 3.02/1.67 introduced(definition,[new_symbols(naming,[spl48_2])],[avatar_definition])).
% 3.02/1.67 thf(f1331,plain,(
% 3.02/1.67 (thesis != $true)),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f993])).
% 3.02/1.67 thf(f993,plain,(
% 3.02/1.67 (thesis != $true)),
% 3.02/1.67 inference(flattening,[status(thm)],[f608])).
% 3.02/1.67 thf(f608,plain,(
% 3.02/1.67 ~(thesis = $true)),
% 3.02/1.67 inference(fool_elimination,[status(thm)],[f607])).
% 3.02/1.67 thf(f607,plain,(
% 3.02/1.67 ~thesis),
% 3.02/1.67 inference(rectify,[status(thm)],[f355])).
% 3.02/1.67 thf(f355,negated_conjecture,(
% 3.02/1.67 ~thesis),
% 3.02/1.67 inference(negated_conjecture,[status(cth)],[f354])).
% 3.02/1.67 thf(f354,conjecture,(
% 3.02/1.67 thesis),
% 3.02/1.67 file('/export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2THF_17883.p',conj_1)).
% 3.02/1.67 thf(f1470,plain,(
% 3.02/1.67 spl48_1 | spl48_2),
% 3.02/1.67 inference(avatar_split_clause,[status(thm)],[f1340,f1467,f1464])).
% 3.02/1.67 thf(f1340,plain,(
% 3.02/1.67 ( ! [X0 : set_a] : (((ord_less_eq_set_a @ a2 @ X0) != $true) | ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) != $true) | (thesis = $true) | ((ord_less_eq_set_a @ a1 @ X0) != $true) | ($true != (ord_less_eq_set_a @ a3 @ X0))) )),
% 3.02/1.67 inference(cnf_transformation,[status(thm)],[f1133])).
% 3.02/1.67 thf(f1133,plain,(
% 3.02/1.67 ! [X0 : set_a] : ((thesis = $true) | ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) != $true) | ((ord_less_eq_set_a @ a2 @ X0) != $true) | ((ord_less_eq_set_a @ a1 @ X0) != $true) | ($true != (ord_less_eq_set_a @ a3 @ X0)))),
% 3.02/1.67 inference(flattening,[status(thm)],[f1132])).
% 3.02/1.67 thf(f1132,plain,(
% 3.02/1.67 ! [X0 : set_a] : (((((thesis = $true) | ($true != (ord_less_eq_set_a @ a3 @ X0))) | ((ord_less_eq_set_a @ a2 @ X0) != $true)) | ((ord_less_eq_set_a @ a1 @ X0) != $true)) | ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) != $true))),
% 3.02/1.67 inference(ennf_transformation,[status(thm)],[f732])).
% 3.02/1.67 thf(f732,plain,(
% 3.02/1.67 ! [X0 : set_a] : (((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) = $true) => (((ord_less_eq_set_a @ a1 @ X0) = $true) => (((ord_less_eq_set_a @ a2 @ X0) = $true) => (($true = (ord_less_eq_set_a @ a3 @ X0)) => (thesis = $true)))))),
% 3.02/1.67 inference(fool_elimination,[status(thm)],[f731])).
% 3.02/1.67 thf(f731,plain,(
% 3.02/1.67 ! [X0 : set_a] : ((member_set_a @ X0 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) => ((ord_less_eq_set_a @ a1 @ X0) => ((ord_less_eq_set_a @ a2 @ X0) => ((ord_less_eq_set_a @ a3 @ X0) => thesis))))),
% 3.02/1.67 inference(rectify,[status(thm)],[f353])).
% 3.02/1.67 thf(f353,axiom,(
% 3.02/1.67 ! [X66 : set_a] : ((member_set_a @ X66 @ (image_64240033_set_a @ produc1040928388_a_a_e @ c)) => ((ord_less_eq_set_a @ a1 @ X66) => ((ord_less_eq_set_a @ a2 @ X66) => ((ord_less_eq_set_a @ a3 @ X66) => thesis))))),
% 3.02/1.67 file('/export/starexec/sandbox/tmp/tmp.uqlUukaGoc/DTF2THF_17883.p',conj_0)).
% 3.02/1.67 % SZS output end Proof for DTF2THF_17883
% 3.02/1.67 % (18218)------------------------------
% 3.02/1.67 % (18218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (18218)Termination reason: Refutation
% 3.02/1.67
% 3.02/1.67 % (18218)Memory used [KB]: 6908
% 3.02/1.67 % (18218)Time elapsed: 0.056 s
% 3.02/1.67 % (18218)Instructions burned: 95 (million)
% 3.02/1.67 % (18218)------------------------------
% 3.02/1.67 % (18218)------------------------------
% 3.02/1.67 % (18212)Success in time 0.077 s
%------------------------------------------------------------------------------