%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP034^1 : 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 : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:01 AM UTC 2026
% Result : Theorem 2.78s 1.85s
% Output : Refutation 2.78s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 3
% Syntax : Number of formulae : 14 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 26 ( 8 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 104 ( 6 ~; 0 |; 0 &; 98 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 101 ( 98 usr; 8 con; 0-4 aty)
% Number of variables : 0 ( 0 ^; 0 !; 0 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
binary1439146945Tree_a: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_8,type,
int: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(func_def_0,type,
binary1439146945Tree_a: $tType ).
thf(func_def_1,type,
set_a: $tType ).
thf(func_def_2,type,
int: $tType ).
thf(func_def_3,type,
a: $tType ).
thf(func_def_4,type,
binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_5,type,
binary476621312_Tip_a: binary1439146945Tree_a ).
thf(func_def_6,type,
binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_7,type,
binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
thf(func_def_8,type,
binary1804682569move_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_9,type,
binary339557810e_rm_a: ( a > int ) > binary1439146945Tree_a > a ).
thf(func_def_10,type,
binary945792244etOf_a: binary1439146945Tree_a > set_a ).
thf(func_def_11,type,
binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
thf(func_def_12,type,
binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
thf(func_def_13,type,
binary1217730267_wrm_a: ( a > int ) > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_14,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(func_def_15,type,
sup_sup_o: $o > $o > $o ).
thf(func_def_16,type,
sup_sup_int: int > int > int ).
thf(func_def_17,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(func_def_18,type,
bot_bot_a_o: a > $o ).
thf(func_def_20,type,
bot_bot_set_a: set_a ).
thf(func_def_21,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(func_def_22,type,
ord_less_int: int > int > $o ).
thf(func_def_23,type,
ord_less_set_a: set_a > set_a > $o ).
thf(func_def_24,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(func_def_25,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_26,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_27,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_28,type,
insert_a: a > set_a > set_a ).
thf(func_def_29,type,
the_elem_a: set_a > a ).
thf(func_def_30,type,
member_a: a > set_a > $o ).
thf(func_def_31,type,
h: a > int ).
thf(func_def_32,type,
l: a ).
thf(func_def_33,type,
t1: binary1439146945Tree_a ).
thf(func_def_34,type,
t2: binary1439146945Tree_a ).
thf(func_def_35,type,
x: a ).
thf(func_def_50,type,
sP0: set_a > set_a > a > a > $o ).
thf(func_def_51,type,
sP1: set_a > set_a > a > a > $o ).
thf(func_def_52,type,
sP2: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_53,type,
sP3: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_54,type,
sP4: a > set_a > set_a > $o ).
thf(func_def_55,type,
sP5: set_a > a > set_a > $o ).
thf(func_def_56,type,
sP6: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_57,type,
sP7: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_58,type,
sK8: int > int ).
thf(func_def_59,type,
sK9: set_a > a > set_a ).
thf(func_def_60,type,
sK10: int > ( int > $o ) > int > int ).
thf(func_def_61,type,
sK11: int > int > ( int > $o ) > int ).
thf(func_def_62,type,
sK12: int > int ).
thf(func_def_63,type,
sK13: int > int ).
thf(func_def_64,type,
sK14: set_a > a ).
thf(func_def_65,type,
sK15: int > int ).
thf(func_def_66,type,
sK16: ( a > $o ) > set_a > set_a > a ).
thf(func_def_67,type,
sK17: ( a > $o ) > set_a > a ).
thf(func_def_68,type,
sK18: ( a > $o ) > set_a > a ).
thf(func_def_69,type,
sK19: set_a > set_a > a > a > set_a ).
thf(func_def_70,type,
sK20: binary1439146945Tree_a > a ).
thf(func_def_71,type,
sK21: binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_72,type,
sK22: binary1439146945Tree_a > binary1439146945Tree_a ).
thf(func_def_73,type,
sK23: set_a > ( a > $o ) > a ).
thf(func_def_74,type,
sK24: ( a > $o ) > set_a > a ).
thf(func_def_75,type,
sK25: set_a > ( a > $o ) > set_a > a ).
thf(func_def_76,type,
sK26: set_a > set_a > a ).
thf(func_def_77,type,
sK27: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_78,type,
sK28: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_79,type,
sK29: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_80,type,
sK30: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_81,type,
sK31: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_82,type,
sK32: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_83,type,
sK33: int > int ).
thf(func_def_84,type,
sK34: a > set_a > set_a ).
thf(func_def_85,type,
sK35: int > int ).
thf(func_def_86,type,
sK36: ( int > int > int ) > int ).
thf(func_def_87,type,
sK37: ( int > int > int ) > int ).
thf(func_def_88,type,
sK38: ( int > int > int ) > int ).
thf(func_def_89,type,
sK39: ( int > int > int ) > int ).
thf(func_def_90,type,
sK40: ( int > int > int ) > int ).
thf(func_def_91,type,
sK41: ( int > int > int ) > int ).
thf(func_def_92,type,
sK42: ( int > int > int ) > int ).
thf(func_def_93,type,
sK43: int > int ).
thf(func_def_94,type,
sK44: ( int > a > $o ) > int ).
thf(func_def_95,type,
sK45: ( int > a > $o ) > int ).
thf(func_def_96,type,
sK46: ( binary1439146945Tree_a > $o ) > binary1439146945Tree_a ).
thf(func_def_97,type,
sK47: ( binary1439146945Tree_a > $o ) > binary1439146945Tree_a ).
thf(func_def_98,type,
sK48: ( binary1439146945Tree_a > $o ) > a ).
thf(func_def_99,type,
sK49: ( ( a > $o ) > int ) > a > $o ).
thf(func_def_100,type,
sK50: ( ( a > $o ) > int ) > a > $o ).
thf(func_def_101,type,
sK51: int > int ).
thf(func_def_102,type,
sK52: a > ( a > int ) > binary1439146945Tree_a > a ).
thf(func_def_103,type,
sK53: binary1439146945Tree_a > a > ( a > int ) > a ).
thf(func_def_104,type,
sK54: int > int ).
thf(func_def_105,type,
sK55: int > int ).
thf(func_def_106,type,
sK56: set_a > a ).
thf(func_def_107,type,
sK57: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_108,type,
sK58: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_109,type,
sK59: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_110,type,
sK60: int > int ).
thf(func_def_111,type,
sK61: set_a > a ).
thf(func_def_112,type,
sK62: int > int ).
thf(func_def_113,type,
sK63: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_114,type,
sK64: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_115,type,
sK65: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_117,type,
ph67:
!>[X0: $tType] : X0 ).
thf(f1730,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1729,f1569]) ).
thf(f1569,plain,
( ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f801]) ).
thf(f801,plain,
( ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f800]) ).
thf(f800,plain,
ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ),
file('/export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2THF_16811.p',fact_0_hLess) ).
thf(f1729,plain,
( ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) )
!= $true ),
inference(forward_demodulation,[status(thm)],[f1370,f1548]) ).
thf(f1548,plain,
( ( binary339557810e_rm_a @ h @ t2 )
= ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ),
inference(cnf_transformation,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( binary339557810e_rm_a @ h @ t2 )
= ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2THF_16811.p',fact_1_rm__res) ).
thf(f1370,plain,
( ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f908]) ).
thf(f908,plain,
( ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) )
!= $true ),
inference(flattening,[status(thm)],[f363]) ).
thf(f363,plain,
( ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f362]) ).
thf(f362,plain,
~ ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ),
inference(rectify,[status(thm)],[f354]) ).
thf(f354,negated_conjecture,
~ ( ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f353]) ).
thf(f353,conjecture,
ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2THF_16811.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP034^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n011.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:23:47 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP034^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer BinaryTree problem prob_562__3255854_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : BinaryTree/prob_562__3255854_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % 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
% 0.19/0.46 % Syntax : Number of formulae : 389 ( 192 unt; 36 typ; 0 def)
% 0.19/0.46 % Number of atoms : 955 ( 354 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 8 ( 2 avg)
% 0.19/0.46 % Number of connectives : 2500 ( 83 ~; 18 |; 75 &;2043 @)
% 0.19/0.46 % ( 0 <=>; 281 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 18 ( 6 avg)
% 0.19/0.46 % Number of types : 5 ( 4 usr)
% 0.19/0.46 % Number of type conns : 314 ( 314 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 34 ( 32 usr; 8 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 944 ( 112 ^; 801 !; 31 ?; 944 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:47.772
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (4)
% 0.19/0.46 thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
% 0.19/0.46 binary1439146945Tree_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 set_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Int__Oint,type,
% 0.19/0.46 int: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_tf__a,type,
% 0.19/0.46 a: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (32)
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001tf__a,type,
% 0.19/0.46 binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001tf__a,type,
% 0.19/0.46 binary476621312_Tip_a: binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Obinsert_001tf__a,type,
% 0.19/0.46 binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Omemb_001tf__a,type,
% 0.19/0.46 binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oremove_001tf__a,type,
% 0.19/0.46 binary1804682569move_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Orm_001tf__a,type,
% 0.19/0.46 binary339557810e_rm_a: ( a > int ) > binary1439146945Tree_a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001tf__a,type,
% 0.19/0.46 binary945792244etOf_a: binary1439146945Tree_a > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001tf__a,type,
% 0.19/0.46 binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001tf__a,type,
% 0.19/0.46 binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Owrm_001tf__a,type,
% 0.19/0.46 binary1217730267_wrm_a: ( a > int ) > binary1439146945Tree_a > binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
% 0.19/0.46 sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
% 0.19/0.46 sup_sup_o: $o > $o > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 0.19/0.46 sup_sup_int: int > int > int ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 sup_sup_set_a: set_a > set_a > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 0.19/0.46 bot_bot_a_o: a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 0.19/0.46 bot_bot_o: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 bot_bot_set_a: set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
% 0.19/0.46 ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.19/0.46 ord_less_int: int > int > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 ord_less_set_a: set_a > set_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
% 0.19/0.46 ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.19/0.46 ord_less_eq_int: int > int > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.19/0.46 collect_a: ( a > $o ) > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Oinsert_001tf__a,type,
% 0.19/0.46 insert_a: a > set_a > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 0.19/0.46 the_elem_a: set_a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001tf__a,type,
% 0.19/0.46 member_a: a > set_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_h,type,
% 0.19/0.46 h: a > int ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_l____,type,
% 0.19/0.46 l: a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_t1____,type,
% 0.19/0.46 t1: binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_t2____,type,
% 0.19/0.46 t2: binary1439146945Tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_x____,type,
% 0.19/0.46 x: a ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (352)
% 0.19/0.46 thf(fact_0_hLess,axiom,
% 0.19/0.46 ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % hLess
% 0.19/0.46 thf(fact_1_rm__res,axiom,
% 0.19/0.46 ( ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
% 0.19/0.46 = ( binary339557810e_rm_a @ h @ t2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % rm_res
% 0.19/0.46 thf(fact_2_t2nTip,axiom,
% 0.19/0.46 t2 != binary476621312_Tip_a ).
% 0.19/0.46
% 0.19/0.46 % t2nTip
% 0.19/0.46 thf(fact_3_Tree_Oinject,axiom,
% 0.19/0.46 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a,Y21: binary1439146945Tree_a,Y22: a,Y23: binary1439146945Tree_a] :
% 0.19/0.46 ( ( ( binary717961607le_T_a @ X21 @ X22 @ X23 )
% 0.19/0.46 = ( binary717961607le_T_a @ Y21 @ Y22 @ Y23 ) )
% 0.19/0.46 = ( ( X21 = Y21 )
% 0.19/0.46 & ( X22 = Y22 )
% 0.19/0.46 & ( X23 = Y23 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Tree.inject
% 0.19/0.46 thf(fact_4_s,axiom,
% 0.19/0.46 binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% 0.19/0.46
% 0.19/0.46 % s
% 0.19/0.46 thf(fact_5_wrm__res,axiom,
% 0.19/0.46 ( ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
% 0.19/0.46 = ( binary717961607le_T_a @ t1 @ x @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % wrm_res
% 0.19/0.46 thf(fact_6_ldef,axiom,
% 0.19/0.46 member_a @ l @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ldef
% 0.19/0.46 thf(fact_7_s1,axiom,
% 0.19/0.46 binary1721989714Tree_a @ h @ t1 ).
% 0.19/0.46
% 0.19/0.46 % s1
% 0.19/0.46 thf(fact_8_s2,axiom,
% 0.19/0.46 binary1721989714Tree_a @ h @ t2 ).
% 0.19/0.46
% 0.19/0.46 % s2
% 0.19/0.46 thf(fact_9_binsert_Osimps_I2_J,axiom,
% 0.19/0.46 ! [H: a > int,E: a,X: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
% 0.19/0.46 ( ( ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
% 0.19/0.46 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.46 = ( binary717961607le_T_a @ ( binary1226383794sert_a @ H @ E @ T1 ) @ X @ T2 ) ) )
% 0.19/0.46 & ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
% 0.19/0.46 => ( ( ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
% 0.19/0.46 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.46 = ( binary717961607le_T_a @ T1 @ X @ ( binary1226383794sert_a @ H @ E @ T2 ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
% 0.19/0.46 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.46 = ( binary717961607le_T_a @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % binsert.simps(2)
% 0.19/0.46 thf(fact_10_rm_Osimps,axiom,
% 0.19/0.46 ! [T2: binary1439146945Tree_a,H: a > int,T1: binary1439146945Tree_a,X: a] :
% 0.19/0.46 ( ( ( T2 = binary476621312_Tip_a )
% 0.19/0.46 => ( ( binary339557810e_rm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.46 = X ) )
% 0.19/0.46 & ( ( T2 != binary476621312_Tip_a )
% 0.19/0.46 => ( ( binary339557810e_rm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.46 = ( binary339557810e_rm_a @ H @ T2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rm.simps
% 0.19/0.46 thf(fact_11_minf_I7_J,axiom,
% 0.19/0.46 ! [T: int] :
% 0.19/0.46 ? [Z: int] :
% 0.19/0.46 ! [X2: int] :
% 0.19/0.46 ( ( ord_less_int @ X2 @ Z )
% 0.19/0.46 => ~ ( ord_less_int @ T @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(7)
% 0.19/0.46 thf(fact_12_minf_I5_J,axiom,
% 0.19/0.46 ! [T: int] :
% 0.19/0.46 ? [Z: int] :
% 0.19/0.46 ! [X2: int] :
% 0.19/0.46 ( ( ord_less_int @ X2 @ Z )
% 0.19/0.46 => ( ord_less_int @ X2 @ T ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(5)
% 0.19/0.46 thf(fact_13_minf_I4_J,axiom,
% 0.19/0.46 ! [T: int] :
% 0.19/0.46 ? [Z: int] :
% 0.19/0.46 ! [X2: int] :
% 0.19/0.46 ( ( ord_less_int @ X2 @ Z )
% 0.19/0.47 => ( X2 != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(4)
% 0.19/0.47 thf(fact_14_minf_I3_J,axiom,
% 0.19/0.47 ! [T: int] :
% 0.19/0.47 ? [Z: int] :
% 0.19/0.47 ! [X2: int] :
% 0.19/0.47 ( ( ord_less_int @ X2 @ Z )
% 0.19/0.47 => ( X2 != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(3)
% 0.19/0.47 thf(fact_15_minf_I2_J,axiom,
% 0.19/0.47 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 0.19/0.47 ( ? [Z2: int] :
% 0.19/0.47 ! [X3: int] :
% 0.19/0.47 ( ( ord_less_int @ X3 @ Z2 )
% 0.19/0.47 => ( ( P @ X3 )
% 0.19/0.47 = ( P2 @ X3 ) ) )
% 0.19/0.47 => ( ? [Z2: int] :
% 0.19/0.47 ! [X3: int] :
% 0.19/0.47 ( ( ord_less_int @ X3 @ Z2 )
% 0.19/0.47 => ( ( Q @ X3 )
% 0.19/0.47 = ( Q2 @ X3 ) ) )
% 0.19/0.47 => ? [Z: int] :
% 0.19/0.47 ! [X2: int] :
% 0.19/0.47 ( ( ord_less_int @ X2 @ Z )
% 0.19/0.47 => ( ( ( P @ X2 )
% 0.19/0.47 | ( Q @ X2 ) )
% 0.19/0.47 = ( ( P2 @ X2 )
% 0.19/0.47 | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(2)
% 0.19/0.47 thf(fact_16_sortedTree_Osimps_I1_J,axiom,
% 0.19/0.47 ! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% 0.19/0.47
% 0.19/0.47 % sortedTree.simps(1)
% 0.19/0.47 thf(fact_17_wrm__sort,axiom,
% 0.19/0.47 ! [T: binary1439146945Tree_a,H: a > int] :
% 0.19/0.47 ( ( ( T != binary476621312_Tip_a )
% 0.19/0.47 & ( binary1721989714Tree_a @ H @ T ) )
% 0.19/0.47 => ( binary1721989714Tree_a @ H @ ( binary1217730267_wrm_a @ H @ T ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % wrm_sort
% 0.19/0.47 thf(fact_18_binsert__sorted,axiom,
% 0.19/0.47 ! [H: a > int,T: binary1439146945Tree_a,X: a] :
% 0.19/0.47 ( ( binary1721989714Tree_a @ H @ T )
% 0.19/0.47 => ( binary1721989714Tree_a @ H @ ( binary1226383794sert_a @ H @ X @ T ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % binsert_sorted
% 0.19/0.47 thf(fact_19_rm__set,axiom,
% 0.19/0.47 ! [T: binary1439146945Tree_a,H: a > int] :
% 0.19/0.47 ( ( ( T != binary476621312_Tip_a )
% 0.19/0.47 & ( binary1721989714Tree_a @ H @ T ) )
% 0.19/0.47 => ( member_a @ ( binary339557810e_rm_a @ H @ T ) @ ( binary945792244etOf_a @ T ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rm_set
% 0.19/0.47 thf(fact_20_binsert_Osimps_I1_J,axiom,
% 0.19/0.47 ! [H: a > int,E: a] :
% 0.19/0.47 ( ( binary1226383794sert_a @ H @ E @ binary476621312_Tip_a )
% 0.19/0.47 = ( binary717961607le_T_a @ binary476621312_Tip_a @ E @ binary476621312_Tip_a ) ) ).
% 0.19/0.47
% 0.19/0.47 % binsert.simps(1)
% 0.19/0.47 thf(fact_21_wrm_Osimps,axiom,
% 0.19/0.47 ! [T2: binary1439146945Tree_a,H: a > int,T1: binary1439146945Tree_a,X: a] :
% 0.19/0.47 ( ( ( T2 = binary476621312_Tip_a )
% 0.19/0.47 => ( ( binary1217730267_wrm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.47 = T1 ) )
% 0.19/0.47 & ( ( T2 != binary476621312_Tip_a )
% 0.19/0.47 => ( ( binary1217730267_wrm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.47 = ( binary717961607le_T_a @ T1 @ X @ ( binary1217730267_wrm_a @ H @ T2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % wrm.simps
% 0.19/0.47 thf(fact_22_sortedTree_Osimps_I2_J,axiom,
% 0.19/0.47 ! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 0.19/0.47 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.19/0.47 = ( ( binary1721989714Tree_a @ H @ T1 )
% 0.19/0.47 & ! [X4: a] :
% 0.19/0.47 ( ( member_a @ X4 @ ( binary945792244etOf_a @ T1 ) )
% 0.19/0.47 => ( ord_less_int @ ( H @ X4 ) @ ( H @ X ) ) )
% 0.19/0.47 & ! [X4: a] :
% 0.19/0.47 ( ( member_a @ X4 @ ( binary945792244etOf_a @ T2 ) )
% 0.19/0.47 => ( ord_less_int @ ( H @ X ) @ ( H @ X4 ) ) )
% 0.19/0.47 & ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % sortedTree.simps(2)
% 0.31/0.47 thf(fact_23_Tree_Oexhaust,axiom,
% 0.31/0.47 ! [Y: binary1439146945Tree_a] :
% 0.31/0.47 ( ( Y != binary476621312_Tip_a )
% 0.31/0.47 => ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Tree.exhaust
% 0.31/0.47 thf(fact_24_Tree_Oinduct,axiom,
% 0.31/0.47 ! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
% 0.31/0.47 ( ( P @ binary476621312_Tip_a )
% 0.31/0.47 => ( ! [X1: binary1439146945Tree_a,X24: a,X32: binary1439146945Tree_a] :
% 0.31/0.47 ( ( P @ X1 )
% 0.31/0.47 => ( ( P @ X32 )
% 0.31/0.47 => ( P @ ( binary717961607le_T_a @ X1 @ X24 @ X32 ) ) ) )
% 0.31/0.47 => ( P @ Tree ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Tree.induct
% 0.31/0.47 thf(fact_25_Tree_Odistinct_I1_J,axiom,
% 0.31/0.47 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 0.31/0.47 ( binary476621312_Tip_a
% 0.31/0.47 != ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Tree.distinct(1)
% 0.31/0.47 thf(fact_26_sortLemmaR,axiom,
% 0.31/0.47 ! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 0.31/0.47 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.31/0.47 => ( binary1721989714Tree_a @ H @ T2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sortLemmaR
% 0.31/0.47 thf(fact_27_sortLemmaL,axiom,
% 0.31/0.47 ! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 0.31/0.47 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.31/0.47 => ( binary1721989714Tree_a @ H @ T1 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sortLemmaL
% 0.31/0.47 thf(fact_28_pinf_I1_J,axiom,
% 0.31/0.47 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 0.31/0.47 ( ? [Z2: int] :
% 0.31/0.47 ! [X3: int] :
% 0.31/0.47 ( ( ord_less_int @ Z2 @ X3 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z2: int] :
% 0.31/0.47 ! [X3: int] :
% 0.31/0.47 ( ( ord_less_int @ Z2 @ X3 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ( ( ( P @ X2 )
% 0.31/0.47 & ( Q @ X2 ) )
% 0.31/0.47 = ( ( P2 @ X2 )
% 0.31/0.47 & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(1)
% 0.31/0.47 thf(fact_29_pinf_I2_J,axiom,
% 0.31/0.47 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 0.31/0.47 ( ? [Z2: int] :
% 0.31/0.47 ! [X3: int] :
% 0.31/0.47 ( ( ord_less_int @ Z2 @ X3 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z2: int] :
% 0.31/0.47 ! [X3: int] :
% 0.31/0.47 ( ( ord_less_int @ Z2 @ X3 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ( ( ( P @ X2 )
% 0.31/0.47 | ( Q @ X2 ) )
% 0.31/0.47 = ( ( P2 @ X2 )
% 0.31/0.47 | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(2)
% 0.31/0.47 thf(fact_30_pinf_I3_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ( X2 != T ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(3)
% 0.31/0.47 thf(fact_31_pinf_I4_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ( X2 != T ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(4)
% 0.31/0.47 thf(fact_32_pinf_I5_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ~ ( ord_less_int @ X2 @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(5)
% 0.31/0.47 thf(fact_33_pinf_I7_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ( ord_less_int @ T @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(7)
% 0.31/0.47 thf(fact_34_minf_I1_J,axiom,
% 0.31/0.47 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 0.31/0.47 ( ? [Z2: int] :
% 0.31/0.47 ! [X3: int] :
% 0.31/0.47 ( ( ord_less_int @ X3 @ Z2 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z2: int] :
% 0.31/0.47 ! [X3: int] :
% 0.31/0.47 ( ( ord_less_int @ X3 @ Z2 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ X2 @ Z )
% 0.31/0.47 => ( ( ( P @ X2 )
% 0.31/0.47 & ( Q @ X2 ) )
% 0.31/0.47 = ( ( P2 @ X2 )
% 0.31/0.47 & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(1)
% 0.31/0.47 thf(fact_35_h1,axiom,
% 0.31/0.47 ( ( ( t1 != binary476621312_Tip_a )
% 0.31/0.47 & ( binary1721989714Tree_a @ h @ t1 ) )
% 0.31/0.47 => ! [X2: a] :
% 0.31/0.47 ( ( member_a @ X2 @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t1 ) ) )
% 0.31/0.47 => ( ord_less_int @ ( h @ X2 ) @ ( h @ ( binary339557810e_rm_a @ h @ t1 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % h1
% 0.31/0.47 thf(fact_36_h2,axiom,
% 0.31/0.47 ( ( ( t2 != binary476621312_Tip_a )
% 0.31/0.47 & ( binary1721989714Tree_a @ h @ t2 ) )
% 0.31/0.47 => ! [X2: a] :
% 0.31/0.47 ( ( member_a @ X2 @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t2 ) ) )
% 0.31/0.47 => ( ord_less_int @ ( h @ X2 ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % h2
% 0.31/0.47 thf(fact_37_memb__spec,axiom,
% 0.31/0.47 ! [H: a > int,T: binary1439146945Tree_a,X: a] :
% 0.31/0.47 ( ( binary1721989714Tree_a @ H @ T )
% 0.31/0.47 => ( ( binary2053421120memb_a @ H @ X @ T )
% 0.31/0.47 = ( member_a @ X @ ( binary945792244etOf_a @ T ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % memb_spec
% 0.31/0.47 thf(fact_38_sorted__distinct__pred__def,axiom,
% 0.31/0.47 ( binary670562003pred_a
% 0.31/0.47 = ( ^ [H2: a > int,A: a,B: a,T3: binary1439146945Tree_a] :
% 0.31/0.47 ( ( ( binary1721989714Tree_a @ H2 @ T3 )
% 0.31/0.47 & ( member_a @ A @ ( binary945792244etOf_a @ T3 ) )
% 0.31/0.47 & ( member_a @ B @ ( binary945792244etOf_a @ T3 ) )
% 0.31/0.47 & ( ( H2 @ A )
% 0.31/0.47 = ( H2 @ B ) ) )
% 0.31/0.47 => ( A = B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sorted_distinct_pred_def
% 0.31/0.47 thf(fact_39_l__scope,axiom,
% 0.31/0.47 member_a @ l @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ x @ bot_bot_set_a ) @ ( binary945792244etOf_a @ t1 ) ) @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % l_scope
% 0.31/0.47 thf(fact_40_wrm__set1,axiom,
% 0.31/0.47 ! [T: binary1439146945Tree_a,H: a > int] :
% 0.31/0.47 ( ( ( T != binary476621312_Tip_a )
% 0.31/0.47 & ( binary1721989714Tree_a @ H @ T ) )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ H @ T ) ) @ ( binary945792244etOf_a @ T ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % wrm_set1
% 0.31/0.47 thf(fact_41_remove_Osimps_I1_J,axiom,
% 0.31/0.47 ! [H: a > int,E: a] :
% 0.31/0.47 ( ( binary1804682569move_a @ H @ E @ binary476621312_Tip_a )
% 0.31/0.47 = binary476621312_Tip_a ) ).
% 0.31/0.47
% 0.31/0.47 % remove.simps(1)
% 0.31/0.47 thf(fact_42_mem__Collect__eq,axiom,
% 0.31/0.47 ! [A2: a,P: a > $o] :
% 0.31/0.47 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 0.31/0.47 = ( P @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % mem_Collect_eq
% 0.31/0.47 thf(fact_43_Collect__mem__eq,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
% 0.31/0.47 = A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mem_eq
% 0.31/0.47 thf(fact_44_Collect__cong,axiom,
% 0.31/0.47 ! [P: a > $o,Q: a > $o] :
% 0.31/0.47 ( ! [X3: a] :
% 0.31/0.47 ( ( P @ X3 )
% 0.31/0.47 = ( Q @ X3 ) )
% 0.31/0.47 => ( ( collect_a @ P )
% 0.31/0.47 = ( collect_a @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_cong
% 0.31/0.47 thf(fact_45_linorder__neqE__linordered__idom,axiom,
% 0.31/0.47 ! [X: int,Y: int] :
% 0.31/0.47 ( ( X != Y )
% 0.31/0.47 => ( ~ ( ord_less_int @ X @ Y )
% 0.31/0.47 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_neqE_linordered_idom
% 0.31/0.47 thf(fact_46_complete__interval,axiom,
% 0.31/0.47 ! [A2: int,B2: int,P: int > $o] :
% 0.31/0.47 ( ( ord_less_int @ A2 @ B2 )
% 0.31/0.47 => ( ( P @ A2 )
% 0.31/0.47 => ( ~ ( P @ B2 )
% 0.31/0.47 => ? [C: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ A2 @ C )
% 0.31/0.47 & ( ord_less_eq_int @ C @ B2 )
% 0.31/0.47 & ! [X2: int] :
% 0.31/0.47 ( ( ( ord_less_eq_int @ A2 @ X2 )
% 0.31/0.47 & ( ord_less_int @ X2 @ C ) )
% 0.31/0.47 => ( P @ X2 ) )
% 0.31/0.47 & ! [D: int] :
% 0.31/0.47 ( ! [X3: int] :
% 0.31/0.47 ( ( ( ord_less_eq_int @ A2 @ X3 )
% 0.31/0.47 & ( ord_less_int @ X3 @ D ) )
% 0.31/0.47 => ( P @ X3 ) )
% 0.31/0.47 => ( ord_less_eq_int @ D @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % complete_interval
% 0.31/0.47 thf(fact_47_setOf_Osimps_I2_J,axiom,
% 0.31/0.47 ! [T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 0.31/0.47 ( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 0.31/0.47 = ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % setOf.simps(2)
% 0.31/0.47 thf(fact_48_minf_I8_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ X2 @ Z )
% 0.31/0.47 => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(8)
% 0.31/0.47 thf(fact_49_minf_I6_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ X2 @ Z )
% 0.31/0.47 => ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(6)
% 0.31/0.47 thf(fact_50_pinf_I8_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(8)
% 0.31/0.47 thf(fact_51_pinf_I6_J,axiom,
% 0.31/0.47 ! [T: int] :
% 0.31/0.47 ? [Z: int] :
% 0.31/0.47 ! [X2: int] :
% 0.31/0.47 ( ( ord_less_int @ Z @ X2 )
% 0.31/0.47 => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(6)
% 0.31/0.47 thf(fact_52_sorted__distinct,axiom,
% 0.31/0.47 ! [H: a > int,A2: a,B2: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A2 @ B2 @ T ) ).
% 0.31/0.47
% 0.31/0.47 % sorted_distinct
% 0.31/0.47 thf(fact_53_setOf_Osimps_I1_J,axiom,
% 0.31/0.47 ( ( binary945792244etOf_a @ binary476621312_Tip_a )
% 0.31/0.47 = bot_bot_set_a ) ).
% 0.31/0.47
% 0.31/0.47 % setOf.simps(1)
% 0.31/0.47 thf(fact_54_singleton__insert__inj__eq_H,axiom,
% 0.31/0.47 ! [A2: a,A3: set_a,B2: a] :
% 0.31/0.47 ( ( ( insert_a @ A2 @ A3 )
% 0.31/0.47 = ( insert_a @ B2 @ bot_bot_set_a ) )
% 0.31/0.47 = ( ( A2 = B2 )
% 0.31/0.47 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_insert_inj_eq'
% 0.31/0.47 thf(fact_55_singleton__insert__inj__eq,axiom,
% 0.31/0.47 ! [B2: a,A2: a,A3: set_a] :
% 0.31/0.47 ( ( ( insert_a @ B2 @ bot_bot_set_a )
% 0.31/0.47 = ( insert_a @ A2 @ A3 ) )
% 0.31/0.47 = ( ( A2 = B2 )
% 0.31/0.47 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_insert_inj_eq
% 0.31/0.47 thf(fact_56_singleton__conv2,axiom,
% 0.31/0.47 ! [A2: a] :
% 0.31/0.47 ( ( collect_a
% 0.31/0.47 @ ( ^ [Y2: a,Z3: a] : ( Y2 = Z3 )
% 0.31/0.47 @ A2 ) )
% 0.31/0.47 = ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_conv2
% 0.31/0.47 thf(fact_57_singleton__conv,axiom,
% 0.31/0.47 ! [A2: a] :
% 0.31/0.47 ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] : ( X4 = A2 ) )
% 0.31/0.47 = ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_conv
% 0.31/0.47 thf(fact_58_Un__insert__right,axiom,
% 0.31/0.47 ! [A3: set_a,A2: a,B3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A3 @ ( insert_a @ A2 @ B3 ) )
% 0.31/0.47 = ( insert_a @ A2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_insert_right
% 0.31/0.47 thf(fact_59_Un__insert__left,axiom,
% 0.31/0.47 ! [A2: a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
% 0.31/0.47 = ( insert_a @ A2 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_insert_left
% 0.31/0.47 thf(fact_60_Un__subset__iff,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
% 0.31/0.47 = ( ( ord_less_eq_set_a @ A3 @ C2 )
% 0.31/0.47 & ( ord_less_eq_set_a @ B3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_subset_iff
% 0.31/0.47 thf(fact_61_Un__empty,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 0.31/0.47 = bot_bot_set_a )
% 0.31/0.47 = ( ( A3 = bot_bot_set_a )
% 0.31/0.47 & ( B3 = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_empty
% 0.31/0.47 thf(fact_62_empty__iff,axiom,
% 0.31/0.47 ! [C3: a] :
% 0.31/0.47 ~ ( member_a @ C3 @ bot_bot_set_a ) ).
% 0.31/0.47
% 0.31/0.47 % empty_iff
% 0.31/0.47 thf(fact_63_all__not__in__conv,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ( ! [X4: a] :
% 0.31/0.47 ~ ( member_a @ X4 @ A3 ) )
% 0.31/0.47 = ( A3 = bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % all_not_in_conv
% 0.31/0.47 thf(fact_64_Collect__empty__eq,axiom,
% 0.31/0.47 ! [P: a > $o] :
% 0.31/0.47 ( ( ( collect_a @ P )
% 0.31/0.47 = bot_bot_set_a )
% 0.31/0.47 = ( ! [X4: a] :
% 0.31/0.47 ~ ( P @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_empty_eq
% 0.31/0.47 thf(fact_65_empty__Collect__eq,axiom,
% 0.31/0.47 ! [P: a > $o] :
% 0.31/0.47 ( ( bot_bot_set_a
% 0.31/0.47 = ( collect_a @ P ) )
% 0.31/0.47 = ( ! [X4: a] :
% 0.31/0.47 ~ ( P @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % empty_Collect_eq
% 0.31/0.47 thf(fact_66_subsetI,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ! [X3: a] :
% 0.31/0.47 ( ( member_a @ X3 @ A3 )
% 0.31/0.47 => ( member_a @ X3 @ B3 ) )
% 0.31/0.47 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetI
% 0.31/0.47 thf(fact_67_subset__antisym,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ B3 @ A3 )
% 0.31/0.47 => ( A3 = B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_antisym
% 0.31/0.47 thf(fact_68_psubsetI,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( A3 != B3 )
% 0.31/0.47 => ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetI
% 0.31/0.47 thf(fact_69_insertCI,axiom,
% 0.31/0.47 ! [A2: a,B3: set_a,B2: a] :
% 0.31/0.47 ( ( ~ ( member_a @ A2 @ B3 )
% 0.31/0.47 => ( A2 = B2 ) )
% 0.31/0.47 => ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insertCI
% 0.31/0.47 thf(fact_70_insert__iff,axiom,
% 0.31/0.47 ! [A2: a,B2: a,A3: set_a] :
% 0.31/0.47 ( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
% 0.31/0.47 = ( ( A2 = B2 )
% 0.31/0.47 | ( member_a @ A2 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_iff
% 0.31/0.47 thf(fact_71_insert__absorb2,axiom,
% 0.31/0.47 ! [X: a,A3: set_a] :
% 0.31/0.47 ( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
% 0.31/0.47 = ( insert_a @ X @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_absorb2
% 0.31/0.47 thf(fact_72_UnCI,axiom,
% 0.31/0.47 ! [C3: a,B3: set_a,A3: set_a] :
% 0.31/0.47 ( ( ~ ( member_a @ C3 @ B3 )
% 0.31/0.47 => ( member_a @ C3 @ A3 ) )
% 0.31/0.47 => ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % UnCI
% 0.31/0.47 thf(fact_73_Un__iff,axiom,
% 0.31/0.47 ! [C3: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 = ( ( member_a @ C3 @ A3 )
% 0.31/0.47 | ( member_a @ C3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_iff
% 0.31/0.47 thf(fact_74_subset__empty,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 0.31/0.47 = ( A3 = bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_empty
% 0.31/0.47 thf(fact_75_empty__subsetI,axiom,
% 0.31/0.47 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 0.31/0.47
% 0.31/0.47 % empty_subsetI
% 0.31/0.47 thf(fact_76_singletonI,axiom,
% 0.31/0.47 ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % singletonI
% 0.31/0.47 thf(fact_77_insert__subset,axiom,
% 0.31/0.47 ! [X: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B3 )
% 0.31/0.47 = ( ( member_a @ X @ B3 )
% 0.31/0.47 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_subset
% 0.31/0.47 thf(fact_78_less__eq__set__def,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ord_less_eq_a_o
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A4 )
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_eq_set_def
% 0.31/0.47 thf(fact_79_emptyE,axiom,
% 0.31/0.47 ! [A2: a] :
% 0.31/0.47 ~ ( member_a @ A2 @ bot_bot_set_a ) ).
% 0.31/0.47
% 0.31/0.47 % emptyE
% 0.31/0.47 thf(fact_80_equals0D,axiom,
% 0.31/0.47 ! [A3: set_a,A2: a] :
% 0.31/0.47 ( ( A3 = bot_bot_set_a )
% 0.31/0.47 => ~ ( member_a @ A2 @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % equals0D
% 0.31/0.47 thf(fact_81_equals0I,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ! [Y3: a] :
% 0.31/0.47 ~ ( member_a @ Y3 @ A3 )
% 0.31/0.47 => ( A3 = bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % equals0I
% 0.31/0.47 thf(fact_82_ex__in__conv,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
% 0.31/0.47 = ( A3 != bot_bot_set_a ) ) ).
% 0.31/0.47
% 0.31/0.47 % ex_in_conv
% 0.31/0.47 thf(fact_83_not__psubset__empty,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% 0.31/0.47
% 0.31/0.47 % not_psubset_empty
% 0.31/0.47 thf(fact_84_in__mono,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,X: a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( member_a @ X @ A3 )
% 0.31/0.47 => ( member_a @ X @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_mono
% 0.31/0.47 thf(fact_85_subsetD,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C3: a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( member_a @ C3 @ A3 )
% 0.31/0.47 => ( member_a @ C3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetD
% 0.31/0.47 thf(fact_86_equalityE,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( A3 = B3 )
% 0.31/0.47 => ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % equalityE
% 0.31/0.47 thf(fact_87_subset__eq,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ! [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ A4 )
% 0.31/0.47 => ( member_a @ X4 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_eq
% 0.31/0.47 thf(fact_88_equalityD1,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( A3 = B3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % equalityD1
% 0.31/0.47 thf(fact_89_equalityD2,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( A3 = B3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % equalityD2
% 0.31/0.47 thf(fact_90_subset__iff,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ! [T3: a] :
% 0.31/0.47 ( ( member_a @ T3 @ A4 )
% 0.31/0.47 => ( member_a @ T3 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_iff
% 0.31/0.47 thf(fact_91_subset__refl,axiom,
% 0.31/0.47 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% 0.31/0.47
% 0.31/0.47 % subset_refl
% 0.31/0.47 thf(fact_92_Collect__mono,axiom,
% 0.31/0.47 ! [P: a > $o,Q: a > $o] :
% 0.31/0.47 ( ! [X3: a] :
% 0.31/0.47 ( ( P @ X3 )
% 0.31/0.47 => ( Q @ X3 ) )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono
% 0.31/0.47 thf(fact_93_subset__trans,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.31/0.47 => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_trans
% 0.31/0.47 thf(fact_94_set__eq__subset,axiom,
% 0.31/0.47 ( ( ^ [Y2: set_a,Z3: set_a] : ( Y2 = Z3 ) )
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 0.31/0.47 & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % set_eq_subset
% 0.31/0.47 thf(fact_95_Collect__mono__iff,axiom,
% 0.31/0.47 ! [P: a > $o,Q: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.31/0.47 = ( ! [X4: a] :
% 0.31/0.47 ( ( P @ X4 )
% 0.31/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono_iff
% 0.31/0.47 thf(fact_96_subset__iff__psubset__eq,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ( ord_less_set_a @ A4 @ B4 )
% 0.31/0.47 | ( A4 = B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_iff_psubset_eq
% 0.31/0.47 thf(fact_97_subset__psubset__trans,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_set_a @ B3 @ C2 )
% 0.31/0.47 => ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_psubset_trans
% 0.31/0.47 thf(fact_98_subset__not__subset__eq,axiom,
% 0.31/0.47 ( ord_less_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 0.31/0.47 & ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_not_subset_eq
% 0.31/0.47 thf(fact_99_psubset__subset__trans,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.31/0.47 => ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubset_subset_trans
% 0.31/0.47 thf(fact_100_psubset__imp__subset,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubset_imp_subset
% 0.31/0.47 thf(fact_101_psubset__eq,axiom,
% 0.31/0.47 ( ord_less_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 0.31/0.47 & ( A4 != B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubset_eq
% 0.31/0.47 thf(fact_102_psubsetE,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.31/0.47 => ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetE
% 0.31/0.47 thf(fact_103_insertE,axiom,
% 0.31/0.47 ! [A2: a,B2: a,A3: set_a] :
% 0.31/0.47 ( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
% 0.31/0.47 => ( ( A2 != B2 )
% 0.31/0.47 => ( member_a @ A2 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insertE
% 0.31/0.47 thf(fact_104_insertI1,axiom,
% 0.31/0.47 ! [A2: a,B3: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % insertI1
% 0.31/0.47 thf(fact_105_insertI2,axiom,
% 0.31/0.47 ! [A2: a,B3: set_a,B2: a] :
% 0.31/0.47 ( ( member_a @ A2 @ B3 )
% 0.31/0.47 => ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insertI2
% 0.31/0.47 thf(fact_106_Set_Oset__insert,axiom,
% 0.31/0.47 ! [X: a,A3: set_a] :
% 0.31/0.47 ( ( member_a @ X @ A3 )
% 0.31/0.47 => ~ ! [B5: set_a] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( insert_a @ X @ B5 ) )
% 0.31/0.47 => ( member_a @ X @ B5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Set.set_insert
% 0.31/0.47 thf(fact_107_insert__ident,axiom,
% 0.31/0.47 ! [X: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ~ ( member_a @ X @ A3 )
% 0.31/0.47 => ( ~ ( member_a @ X @ B3 )
% 0.31/0.47 => ( ( ( insert_a @ X @ A3 )
% 0.31/0.47 = ( insert_a @ X @ B3 ) )
% 0.31/0.47 = ( A3 = B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_ident
% 0.31/0.47 thf(fact_108_insert__absorb,axiom,
% 0.31/0.47 ! [A2: a,A3: set_a] :
% 0.31/0.47 ( ( member_a @ A2 @ A3 )
% 0.31/0.47 => ( ( insert_a @ A2 @ A3 )
% 0.31/0.47 = A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_absorb
% 0.31/0.47 thf(fact_109_insert__eq__iff,axiom,
% 0.31/0.47 ! [A2: a,A3: set_a,B2: a,B3: set_a] :
% 0.31/0.47 ( ~ ( member_a @ A2 @ A3 )
% 0.31/0.47 => ( ~ ( member_a @ B2 @ B3 )
% 0.31/0.47 => ( ( ( insert_a @ A2 @ A3 )
% 0.31/0.47 = ( insert_a @ B2 @ B3 ) )
% 0.31/0.47 = ( ( ( A2 = B2 )
% 0.31/0.47 => ( A3 = B3 ) )
% 0.31/0.47 & ( ( A2 != B2 )
% 0.31/0.47 => ? [C4: set_a] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( insert_a @ B2 @ C4 ) )
% 0.31/0.47 & ~ ( member_a @ B2 @ C4 )
% 0.31/0.47 & ( B3
% 0.31/0.47 = ( insert_a @ A2 @ C4 ) )
% 0.31/0.47 & ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_eq_iff
% 0.31/0.47 thf(fact_110_insert__commute,axiom,
% 0.31/0.47 ! [X: a,Y: a,A3: set_a] :
% 0.31/0.47 ( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
% 0.31/0.47 = ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_commute
% 0.31/0.47 thf(fact_111_mk__disjoint__insert,axiom,
% 0.31/0.47 ! [A2: a,A3: set_a] :
% 0.31/0.47 ( ( member_a @ A2 @ A3 )
% 0.31/0.47 => ? [B5: set_a] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( insert_a @ A2 @ B5 ) )
% 0.31/0.47 & ~ ( member_a @ A2 @ B5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % mk_disjoint_insert
% 0.31/0.47 thf(fact_112_UnE,axiom,
% 0.31/0.47 ! [C3: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 => ( ~ ( member_a @ C3 @ A3 )
% 0.31/0.47 => ( member_a @ C3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % UnE
% 0.31/0.47 thf(fact_113_UnI1,axiom,
% 0.31/0.47 ! [C3: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( member_a @ C3 @ A3 )
% 0.31/0.47 => ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % UnI1
% 0.31/0.47 thf(fact_114_UnI2,axiom,
% 0.31/0.47 ! [C3: a,B3: set_a,A3: set_a] :
% 0.31/0.47 ( ( member_a @ C3 @ B3 )
% 0.31/0.47 => ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % UnI2
% 0.31/0.47 thf(fact_115_bex__Un,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,P: a > $o] :
% 0.31/0.47 ( ( ? [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 & ( P @ X4 ) ) )
% 0.31/0.47 = ( ? [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ A3 )
% 0.31/0.47 & ( P @ X4 ) )
% 0.31/0.47 | ? [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ B3 )
% 0.31/0.47 & ( P @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bex_Un
% 0.31/0.47 thf(fact_116_ball__Un,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,P: a > $o] :
% 0.31/0.47 ( ( ! [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 => ( P @ X4 ) ) )
% 0.31/0.47 = ( ! [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ A3 )
% 0.31/0.47 => ( P @ X4 ) )
% 0.31/0.47 & ! [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ B3 )
% 0.31/0.47 => ( P @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ball_Un
% 0.31/0.47 thf(fact_117_Un__assoc,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
% 0.31/0.47 = ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_assoc
% 0.31/0.47 thf(fact_118_Un__absorb,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A3 @ A3 )
% 0.31/0.47 = A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Un_absorb
% 0.31/0.47 thf(fact_119_Un__commute,axiom,
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_commute
% 0.31/0.47 thf(fact_120_Un__left__absorb,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 = ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_left_absorb
% 0.31/0.47 thf(fact_121_Un__left__commute,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) )
% 0.31/0.47 = ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_left_commute
% 0.31/0.47 thf(fact_122_empty__def,axiom,
% 0.31/0.47 ( bot_bot_set_a
% 0.31/0.47 = ( collect_a
% 0.31/0.47 @ ^ [X4: a] : $false ) ) ).
% 0.31/0.47
% 0.31/0.47 % empty_def
% 0.31/0.47 thf(fact_123_Collect__subset,axiom,
% 0.31/0.47 ! [A3: set_a,P: a > $o] :
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 @ ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ A3 )
% 0.31/0.47 & ( P @ X4 ) ) )
% 0.31/0.47 @ A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_subset
% 0.31/0.47 thf(fact_124_insert__compr,axiom,
% 0.31/0.47 ( insert_a
% 0.31/0.47 = ( ^ [A: a,B4: set_a] :
% 0.31/0.47 ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( X4 = A )
% 0.31/0.47 | ( member_a @ X4 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_compr
% 0.31/0.47 thf(fact_125_insert__Collect,axiom,
% 0.31/0.47 ! [A2: a,P: a > $o] :
% 0.31/0.47 ( ( insert_a @ A2 @ ( collect_a @ P ) )
% 0.31/0.47 = ( collect_a
% 0.31/0.47 @ ^ [U: a] :
% 0.31/0.47 ( ( U != A2 )
% 0.31/0.47 => ( P @ U ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_Collect
% 0.31/0.47 thf(fact_126_Un__def,axiom,
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( member_a @ X4 @ A4 )
% 0.31/0.47 | ( member_a @ X4 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_def
% 0.31/0.47 thf(fact_127_Collect__disj__eq,axiom,
% 0.31/0.47 ! [P: a > $o,Q: a > $o] :
% 0.31/0.47 ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( P @ X4 )
% 0.31/0.47 | ( Q @ X4 ) ) )
% 0.31/0.47 = ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_disj_eq
% 0.31/0.47 thf(fact_128_singletonD,axiom,
% 0.31/0.47 ! [B2: a,A2: a] :
% 0.31/0.47 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 0.31/0.47 => ( B2 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % singletonD
% 0.31/0.47 thf(fact_129_singleton__iff,axiom,
% 0.31/0.47 ! [B2: a,A2: a] :
% 0.31/0.47 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 0.31/0.47 = ( B2 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_iff
% 0.31/0.47 thf(fact_130_doubleton__eq__iff,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C3: a,D2: a] :
% 0.31/0.47 ( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
% 0.31/0.47 = ( insert_a @ C3 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
% 0.31/0.47 = ( ( ( A2 = C3 )
% 0.31/0.47 & ( B2 = D2 ) )
% 0.31/0.47 | ( ( A2 = D2 )
% 0.31/0.47 & ( B2 = C3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % doubleton_eq_iff
% 0.31/0.47 thf(fact_131_insert__not__empty,axiom,
% 0.31/0.47 ! [A2: a,A3: set_a] :
% 0.31/0.47 ( ( insert_a @ A2 @ A3 )
% 0.31/0.47 != bot_bot_set_a ) ).
% 0.31/0.47
% 0.31/0.47 % insert_not_empty
% 0.31/0.47 thf(fact_132_singleton__inject,axiom,
% 0.31/0.47 ! [A2: a,B2: a] :
% 0.31/0.47 ( ( ( insert_a @ A2 @ bot_bot_set_a )
% 0.31/0.47 = ( insert_a @ B2 @ bot_bot_set_a ) )
% 0.31/0.47 => ( A2 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_inject
% 0.31/0.47 thf(fact_133_insert__mono,axiom,
% 0.31/0.47 ! [C2: set_a,D3: set_a,A2: a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ C2 @ D3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_mono
% 0.31/0.47 thf(fact_134_subset__insert,axiom,
% 0.31/0.47 ! [X: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ~ ( member_a @ X @ A3 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B3 ) )
% 0.31/0.47 = ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insert
% 0.31/0.47 thf(fact_135_subset__insertI,axiom,
% 0.31/0.47 ! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A2 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insertI
% 0.31/0.47 thf(fact_136_subset__insertI2,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,B2: a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insertI2
% 0.31/0.47 thf(fact_137_Un__empty__left,axiom,
% 0.31/0.47 ! [B3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
% 0.31/0.47 = B3 ) ).
% 0.31/0.47
% 0.31/0.47 % Un_empty_left
% 0.31/0.47 thf(fact_138_Un__empty__right,axiom,
% 0.31/0.47 ! [A3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
% 0.31/0.47 = A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Un_empty_right
% 0.31/0.47 thf(fact_139_Un__mono,axiom,
% 0.31/0.47 ! [A3: set_a,C2: set_a,B3: set_a,D3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ C2 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ B3 @ D3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ C2 @ D3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_mono
% 0.31/0.47 thf(fact_140_Un__least,axiom,
% 0.31/0.47 ! [A3: set_a,C2: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ C2 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_least
% 0.31/0.47 thf(fact_141_Un__upper1,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_upper1
% 0.31/0.47 thf(fact_142_Un__upper2,axiom,
% 0.31/0.47 ! [B3: set_a,A3: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_upper2
% 0.31/0.47 thf(fact_143_Un__absorb1,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( sup_sup_set_a @ A3 @ B3 )
% 0.31/0.47 = B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_absorb1
% 0.31/0.47 thf(fact_144_Un__absorb2,axiom,
% 0.31/0.47 ! [B3: set_a,A3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ B3 @ A3 )
% 0.31/0.47 => ( ( sup_sup_set_a @ A3 @ B3 )
% 0.31/0.47 = A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_absorb2
% 0.31/0.47 thf(fact_145_subset__UnE,axiom,
% 0.31/0.47 ! [C2: set_a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 => ~ ! [A5: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A5 @ A3 )
% 0.31/0.47 => ! [B6: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ B6 @ B3 )
% 0.31/0.47 => ( C2
% 0.31/0.47 != ( sup_sup_set_a @ A5 @ B6 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_UnE
% 0.31/0.47 thf(fact_146_subset__Un__eq,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A4 @ B4 )
% 0.31/0.47 = B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_Un_eq
% 0.31/0.47 thf(fact_147_Collect__conv__if,axiom,
% 0.31/0.47 ! [P: a > $o,A2: a] :
% 0.31/0.47 ( ( ( P @ A2 )
% 0.31/0.47 => ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( X4 = A2 )
% 0.31/0.47 & ( P @ X4 ) ) )
% 0.31/0.47 = ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 0.31/0.47 & ( ~ ( P @ A2 )
% 0.31/0.47 => ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( X4 = A2 )
% 0.31/0.47 & ( P @ X4 ) ) )
% 0.31/0.47 = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_conv_if
% 0.31/0.47 thf(fact_148_Collect__conv__if2,axiom,
% 0.31/0.47 ! [P: a > $o,A2: a] :
% 0.31/0.47 ( ( ( P @ A2 )
% 0.31/0.47 => ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( A2 = X4 )
% 0.31/0.47 & ( P @ X4 ) ) )
% 0.31/0.47 = ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 0.31/0.47 & ( ~ ( P @ A2 )
% 0.31/0.47 => ( ( collect_a
% 0.31/0.47 @ ^ [X4: a] :
% 0.31/0.47 ( ( A2 = X4 )
% 0.31/0.47 & ( P @ X4 ) ) )
% 0.31/0.47 = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_conv_if2
% 0.31/0.47 thf(fact_149_insert__def,axiom,
% 0.31/0.47 ( insert_a
% 0.31/0.47 = ( ^ [A: a] :
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 @ ( collect_a
% 0.31/0.47 @ ^ [X4: a] : ( X4 = A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_def
% 0.31/0.47 thf(fact_150_subset__singletonD,axiom,
% 0.31/0.47 ! [A3: set_a,X: a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.47 => ( ( A3 = bot_bot_set_a )
% 0.31/0.47 | ( A3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_singletonD
% 0.31/0.47 thf(fact_151_subset__singleton__iff,axiom,
% 0.31/0.47 ! [X5: set_a,A2: a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 0.31/0.47 = ( ( X5 = bot_bot_set_a )
% 0.31/0.47 | ( X5
% 0.31/0.47 = ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_singleton_iff
% 0.31/0.47 thf(fact_152_insert__is__Un,axiom,
% 0.31/0.47 ( insert_a
% 0.31/0.47 = ( ^ [A: a] : ( sup_sup_set_a @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_is_Un
% 0.31/0.47 thf(fact_153_Un__singleton__iff,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,X: a] :
% 0.31/0.47 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.47 = ( ( ( A3 = bot_bot_set_a )
% 0.31/0.47 & ( B3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) ) )
% 0.31/0.47 | ( ( A3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.47 & ( B3 = bot_bot_set_a ) )
% 0.31/0.47 | ( ( A3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.47 & ( B3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Un_singleton_iff
% 0.31/0.47 thf(fact_154_singleton__Un__iff,axiom,
% 0.31/0.47 ! [X: a,A3: set_a,B3: set_a] :
% 0.31/0.47 ( ( ( insert_a @ X @ bot_bot_set_a )
% 0.31/0.47 = ( sup_sup_set_a @ A3 @ B3 ) )
% 0.31/0.47 = ( ( ( A3 = bot_bot_set_a )
% 0.31/0.47 & ( B3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) ) )
% 0.31/0.47 | ( ( A3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.47 & ( B3 = bot_bot_set_a ) )
% 0.31/0.47 | ( ( A3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.47 & ( B3
% 0.31/0.47 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % singleton_Un_iff
% 0.31/0.47 thf(fact_155_sup__bot__left,axiom,
% 0.31/0.47 ! [X: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ bot_bot_a_o @ X )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot_left
% 0.31/0.47 thf(fact_156_sup__bot__left,axiom,
% 0.31/0.47 ! [X: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ bot_bot_set_a @ X )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot_left
% 0.31/0.47 thf(fact_157_sup__bot__right,axiom,
% 0.31/0.47 ! [X: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X @ bot_bot_a_o )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot_right
% 0.31/0.47 thf(fact_158_sup__bot__right,axiom,
% 0.31/0.47 ! [X: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X @ bot_bot_set_a )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot_right
% 0.31/0.47 thf(fact_159_bot__eq__sup__iff,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o] :
% 0.31/0.47 ( ( bot_bot_a_o
% 0.31/0.47 = ( sup_sup_a_o @ X @ Y ) )
% 0.31/0.47 = ( ( X = bot_bot_a_o )
% 0.31/0.47 & ( Y = bot_bot_a_o ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_eq_sup_iff
% 0.31/0.47 thf(fact_160_bot__eq__sup__iff,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a] :
% 0.31/0.47 ( ( bot_bot_set_a
% 0.31/0.47 = ( sup_sup_set_a @ X @ Y ) )
% 0.31/0.47 = ( ( X = bot_bot_set_a )
% 0.31/0.47 & ( Y = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_eq_sup_iff
% 0.31/0.47 thf(fact_161_sup__eq__bot__iff,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o] :
% 0.31/0.47 ( ( ( sup_sup_a_o @ X @ Y )
% 0.31/0.47 = bot_bot_a_o )
% 0.31/0.47 = ( ( X = bot_bot_a_o )
% 0.31/0.47 & ( Y = bot_bot_a_o ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_eq_bot_iff
% 0.31/0.47 thf(fact_162_sup__eq__bot__iff,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a] :
% 0.31/0.47 ( ( ( sup_sup_set_a @ X @ Y )
% 0.31/0.47 = bot_bot_set_a )
% 0.31/0.47 = ( ( X = bot_bot_set_a )
% 0.31/0.47 & ( Y = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_eq_bot_iff
% 0.31/0.47 thf(fact_163_sup__bot_Oeq__neutr__iff,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( ( sup_sup_a_o @ A2 @ B2 )
% 0.31/0.47 = bot_bot_a_o )
% 0.31/0.47 = ( ( A2 = bot_bot_a_o )
% 0.31/0.47 & ( B2 = bot_bot_a_o ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.eq_neutr_iff
% 0.31/0.47 thf(fact_164_sup__bot_Oeq__neutr__iff,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( ( sup_sup_set_a @ A2 @ B2 )
% 0.31/0.47 = bot_bot_set_a )
% 0.31/0.47 = ( ( A2 = bot_bot_set_a )
% 0.31/0.47 & ( B2 = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.eq_neutr_iff
% 0.31/0.47 thf(fact_165_sup__bot_Oleft__neutral,axiom,
% 0.31/0.47 ! [A2: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.left_neutral
% 0.31/0.47 thf(fact_166_sup__bot_Oleft__neutral,axiom,
% 0.31/0.47 ! [A2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.left_neutral
% 0.31/0.47 thf(fact_167_sup__bot_Oneutr__eq__iff,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( bot_bot_a_o
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ B2 ) )
% 0.31/0.47 = ( ( A2 = bot_bot_a_o )
% 0.31/0.47 & ( B2 = bot_bot_a_o ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.neutr_eq_iff
% 0.31/0.47 thf(fact_168_sup__bot_Oneutr__eq__iff,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( bot_bot_set_a
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ B2 ) )
% 0.31/0.47 = ( ( A2 = bot_bot_set_a )
% 0.31/0.47 & ( B2 = bot_bot_set_a ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.neutr_eq_iff
% 0.31/0.47 thf(fact_169_sup_Oright__idem,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ B2 )
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.right_idem
% 0.31/0.47 thf(fact_170_sup_Oright__idem,axiom,
% 0.31/0.47 ! [A2: int,B2: int] :
% 0.31/0.47 ( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ B2 )
% 0.31/0.47 = ( sup_sup_int @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.right_idem
% 0.31/0.47 thf(fact_171_sup_Oright__idem,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.right_idem
% 0.31/0.47 thf(fact_172_sup__left__idem,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
% 0.31/0.47 = ( sup_sup_a_o @ X @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_left_idem
% 0.31/0.47 thf(fact_173_sup__left__idem,axiom,
% 0.31/0.47 ! [X: int,Y: int] :
% 0.31/0.47 ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
% 0.31/0.47 = ( sup_sup_int @ X @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_left_idem
% 0.31/0.47 thf(fact_174_sup__left__idem,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
% 0.31/0.47 = ( sup_sup_set_a @ X @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_left_idem
% 0.31/0.47 thf(fact_175_sup_Oleft__idem,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ A2 @ B2 ) )
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.left_idem
% 0.31/0.47 thf(fact_176_sup_Oleft__idem,axiom,
% 0.31/0.47 ! [A2: int,B2: int] :
% 0.31/0.47 ( ( sup_sup_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) )
% 0.31/0.47 = ( sup_sup_int @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.left_idem
% 0.31/0.47 thf(fact_177_sup_Oleft__idem,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.left_idem
% 0.31/0.47 thf(fact_178_sup__idem,axiom,
% 0.31/0.47 ! [X: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X @ X )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_idem
% 0.31/0.47 thf(fact_179_sup__idem,axiom,
% 0.31/0.47 ! [X: int] :
% 0.31/0.47 ( ( sup_sup_int @ X @ X )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_idem
% 0.31/0.47 thf(fact_180_sup__idem,axiom,
% 0.31/0.47 ! [X: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X @ X )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % sup_idem
% 0.31/0.47 thf(fact_181_sup_Oidem,axiom,
% 0.31/0.47 ! [A2: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ A2 @ A2 )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup.idem
% 0.31/0.47 thf(fact_182_sup_Oidem,axiom,
% 0.31/0.47 ! [A2: int] :
% 0.31/0.47 ( ( sup_sup_int @ A2 @ A2 )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup.idem
% 0.31/0.47 thf(fact_183_sup_Oidem,axiom,
% 0.31/0.47 ! [A2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A2 @ A2 )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup.idem
% 0.31/0.47 thf(fact_184_sup__apply,axiom,
% 0.31/0.47 ( sup_sup_a_o
% 0.31/0.47 = ( ^ [F: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_apply
% 0.31/0.47 thf(fact_185_sup_Obounded__iff,axiom,
% 0.31/0.47 ! [B2: int,C3: int,A2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
% 0.31/0.47 = ( ( ord_less_eq_int @ B2 @ A2 )
% 0.31/0.47 & ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.bounded_iff
% 0.31/0.47 thf(fact_186_sup_Obounded__iff,axiom,
% 0.31/0.47 ! [B2: a > $o,C3: a > $o,A2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
% 0.31/0.47 = ( ( ord_less_eq_a_o @ B2 @ A2 )
% 0.31/0.47 & ( ord_less_eq_a_o @ C3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.bounded_iff
% 0.31/0.47 thf(fact_187_sup_Obounded__iff,axiom,
% 0.31/0.47 ! [B2: set_a,C3: set_a,A2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
% 0.31/0.47 = ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.47 & ( ord_less_eq_set_a @ C3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.bounded_iff
% 0.31/0.47 thf(fact_188_le__sup__iff,axiom,
% 0.31/0.47 ! [X: int,Y: int,Z4: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( ( ord_less_eq_int @ X @ Z4 )
% 0.31/0.47 & ( ord_less_eq_int @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_sup_iff
% 0.31/0.47 thf(fact_189_le__sup__iff,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( ( ord_less_eq_a_o @ X @ Z4 )
% 0.31/0.47 & ( ord_less_eq_a_o @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_sup_iff
% 0.31/0.47 thf(fact_190_le__sup__iff,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a,Z4: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( ( ord_less_eq_set_a @ X @ Z4 )
% 0.31/0.47 & ( ord_less_eq_set_a @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_sup_iff
% 0.31/0.47 thf(fact_191_sup__bot_Oright__neutral,axiom,
% 0.31/0.47 ! [A2: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.right_neutral
% 0.31/0.47 thf(fact_192_sup__bot_Oright__neutral,axiom,
% 0.31/0.47 ! [A2: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
% 0.31/0.47 = A2 ) ).
% 0.31/0.47
% 0.31/0.47 % sup_bot.right_neutral
% 0.31/0.47 thf(fact_193_less__set__def,axiom,
% 0.31/0.47 ( ord_less_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( ord_less_a_o
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A4 )
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_set_def
% 0.31/0.47 thf(fact_194_psubset__trans,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C2: set_a] :
% 0.31/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_set_a @ B3 @ C2 )
% 0.31/0.47 => ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubset_trans
% 0.31/0.47 thf(fact_195_psubsetD,axiom,
% 0.31/0.47 ! [A3: set_a,B3: set_a,C3: a] :
% 0.31/0.47 ( ( ord_less_set_a @ A3 @ B3 )
% 0.31/0.47 => ( ( member_a @ C3 @ A3 )
% 0.31/0.47 => ( member_a @ C3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetD
% 0.31/0.47 thf(fact_196_bot__set__def,axiom,
% 0.31/0.47 ( bot_bot_set_a
% 0.31/0.47 = ( collect_a @ bot_bot_a_o ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_set_def
% 0.31/0.47 thf(fact_197_sup__set__def,axiom,
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 = ( ^ [A4: set_a,B4: set_a] :
% 0.31/0.47 ( collect_a
% 0.31/0.47 @ ( sup_sup_a_o
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A4 )
% 0.31/0.47 @ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_set_def
% 0.31/0.47 thf(fact_198_sup__left__commute,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) )
% 0.31/0.47 = ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_left_commute
% 0.31/0.47 thf(fact_199_sup__left__commute,axiom,
% 0.31/0.47 ! [X: int,Y: int,Z4: int] :
% 0.31/0.47 ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) )
% 0.31/0.47 = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_left_commute
% 0.31/0.47 thf(fact_200_sup__left__commute,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a,Z4: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
% 0.31/0.47 = ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_left_commute
% 0.31/0.47 thf(fact_201_sup_Oleft__commute,axiom,
% 0.31/0.47 ! [B2: a > $o,A2: a > $o,C3: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ B2 @ ( sup_sup_a_o @ A2 @ C3 ) )
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.left_commute
% 0.31/0.47 thf(fact_202_sup_Oleft__commute,axiom,
% 0.31/0.47 ! [B2: int,A2: int,C3: int] :
% 0.31/0.47 ( ( sup_sup_int @ B2 @ ( sup_sup_int @ A2 @ C3 ) )
% 0.31/0.47 = ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.left_commute
% 0.31/0.47 thf(fact_203_sup_Oleft__commute,axiom,
% 0.31/0.47 ! [B2: set_a,A2: set_a,C3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C3 ) )
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.left_commute
% 0.31/0.47 thf(fact_204_sup__commute,axiom,
% 0.31/0.47 ( sup_sup_a_o
% 0.31/0.47 = ( ^ [X4: a > $o,Y4: a > $o] : ( sup_sup_a_o @ Y4 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_commute
% 0.31/0.47 thf(fact_205_sup__commute,axiom,
% 0.31/0.47 ( sup_sup_int
% 0.31/0.47 = ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_commute
% 0.31/0.47 thf(fact_206_sup__commute,axiom,
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 = ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_commute
% 0.31/0.47 thf(fact_207_sup_Ocommute,axiom,
% 0.31/0.47 ( sup_sup_a_o
% 0.31/0.47 = ( ^ [A: a > $o,B: a > $o] : ( sup_sup_a_o @ B @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.commute
% 0.31/0.47 thf(fact_208_sup_Ocommute,axiom,
% 0.31/0.47 ( sup_sup_int
% 0.31/0.47 = ( ^ [A: int,B: int] : ( sup_sup_int @ B @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.commute
% 0.31/0.47 thf(fact_209_sup_Ocommute,axiom,
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 = ( ^ [A: set_a,B: set_a] : ( sup_sup_set_a @ B @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.commute
% 0.31/0.47 thf(fact_210_sup__assoc,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_assoc
% 0.31/0.47 thf(fact_211_sup__assoc,axiom,
% 0.31/0.47 ! [X: int,Y: int,Z4: int] :
% 0.31/0.47 ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_assoc
% 0.31/0.47 thf(fact_212_sup__assoc,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a,Z4: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_assoc
% 0.31/0.47 thf(fact_213_sup_Oassoc,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o,C3: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ C3 )
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.assoc
% 0.31/0.47 thf(fact_214_sup_Oassoc,axiom,
% 0.31/0.47 ! [A2: int,B2: int,C3: int] :
% 0.31/0.47 ( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ C3 )
% 0.31/0.47 = ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.assoc
% 0.31/0.47 thf(fact_215_sup_Oassoc,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a,C3: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C3 )
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.assoc
% 0.31/0.47 thf(fact_216_boolean__algebra__cancel_Osup2,axiom,
% 0.31/0.47 ! [B3: a > $o,K: a > $o,B2: a > $o,A2: a > $o] :
% 0.31/0.47 ( ( B3
% 0.31/0.47 = ( sup_sup_a_o @ K @ B2 ) )
% 0.31/0.47 => ( ( sup_sup_a_o @ A2 @ B3 )
% 0.31/0.47 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % boolean_algebra_cancel.sup2
% 0.31/0.47 thf(fact_217_boolean__algebra__cancel_Osup2,axiom,
% 0.31/0.47 ! [B3: int,K: int,B2: int,A2: int] :
% 0.31/0.47 ( ( B3
% 0.31/0.47 = ( sup_sup_int @ K @ B2 ) )
% 0.31/0.47 => ( ( sup_sup_int @ A2 @ B3 )
% 0.31/0.47 = ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % boolean_algebra_cancel.sup2
% 0.31/0.47 thf(fact_218_boolean__algebra__cancel_Osup2,axiom,
% 0.31/0.47 ! [B3: set_a,K: set_a,B2: set_a,A2: set_a] :
% 0.31/0.47 ( ( B3
% 0.31/0.47 = ( sup_sup_set_a @ K @ B2 ) )
% 0.31/0.47 => ( ( sup_sup_set_a @ A2 @ B3 )
% 0.31/0.47 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % boolean_algebra_cancel.sup2
% 0.31/0.47 thf(fact_219_boolean__algebra__cancel_Osup1,axiom,
% 0.31/0.47 ! [A3: a > $o,K: a > $o,A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( sup_sup_a_o @ K @ A2 ) )
% 0.31/0.47 => ( ( sup_sup_a_o @ A3 @ B2 )
% 0.31/0.47 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % boolean_algebra_cancel.sup1
% 0.31/0.47 thf(fact_220_boolean__algebra__cancel_Osup1,axiom,
% 0.31/0.47 ! [A3: int,K: int,A2: int,B2: int] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( sup_sup_int @ K @ A2 ) )
% 0.31/0.47 => ( ( sup_sup_int @ A3 @ B2 )
% 0.31/0.47 = ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % boolean_algebra_cancel.sup1
% 0.31/0.47 thf(fact_221_boolean__algebra__cancel_Osup1,axiom,
% 0.31/0.47 ! [A3: set_a,K: set_a,A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( sup_sup_set_a @ K @ A2 ) )
% 0.31/0.47 => ( ( sup_sup_set_a @ A3 @ B2 )
% 0.31/0.47 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % boolean_algebra_cancel.sup1
% 0.31/0.47 thf(fact_222_sup__fun__def,axiom,
% 0.31/0.47 ( sup_sup_a_o
% 0.31/0.47 = ( ^ [F: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_fun_def
% 0.31/0.47 thf(fact_223_inf__sup__aci_I5_J,axiom,
% 0.31/0.47 ( sup_sup_a_o
% 0.31/0.47 = ( ^ [X4: a > $o,Y4: a > $o] : ( sup_sup_a_o @ Y4 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(5)
% 0.31/0.47 thf(fact_224_inf__sup__aci_I5_J,axiom,
% 0.31/0.47 ( sup_sup_int
% 0.31/0.47 = ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(5)
% 0.31/0.47 thf(fact_225_inf__sup__aci_I5_J,axiom,
% 0.31/0.47 ( sup_sup_set_a
% 0.31/0.47 = ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(5)
% 0.31/0.47 thf(fact_226_inf__sup__aci_I6_J,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(6)
% 0.31/0.47 thf(fact_227_inf__sup__aci_I6_J,axiom,
% 0.31/0.47 ! [X: int,Y: int,Z4: int] :
% 0.31/0.47 ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(6)
% 0.31/0.47 thf(fact_228_inf__sup__aci_I6_J,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a,Z4: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
% 0.31/0.47 = ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(6)
% 0.31/0.47 thf(fact_229_inf__sup__aci_I7_J,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) )
% 0.31/0.47 = ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(7)
% 0.31/0.47 thf(fact_230_inf__sup__aci_I7_J,axiom,
% 0.31/0.47 ! [X: int,Y: int,Z4: int] :
% 0.31/0.47 ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) )
% 0.31/0.47 = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(7)
% 0.31/0.47 thf(fact_231_inf__sup__aci_I7_J,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a,Z4: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
% 0.31/0.47 = ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(7)
% 0.31/0.47 thf(fact_232_inf__sup__aci_I8_J,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
% 0.31/0.47 = ( sup_sup_a_o @ X @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(8)
% 0.31/0.47 thf(fact_233_inf__sup__aci_I8_J,axiom,
% 0.31/0.47 ! [X: int,Y: int] :
% 0.31/0.47 ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
% 0.31/0.47 = ( sup_sup_int @ X @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(8)
% 0.31/0.47 thf(fact_234_inf__sup__aci_I8_J,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
% 0.31/0.47 = ( sup_sup_set_a @ X @ Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_aci(8)
% 0.31/0.47 thf(fact_235_sup_OcoboundedI2,axiom,
% 0.31/0.47 ! [C3: int,B2: int,A2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ C3 @ B2 )
% 0.31/0.47 => ( ord_less_eq_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.coboundedI2
% 0.31/0.47 thf(fact_236_sup_OcoboundedI2,axiom,
% 0.31/0.47 ! [C3: a > $o,B2: a > $o,A2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ C3 @ B2 )
% 0.31/0.47 => ( ord_less_eq_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.coboundedI2
% 0.31/0.47 thf(fact_237_sup_OcoboundedI2,axiom,
% 0.31/0.47 ! [C3: set_a,B2: set_a,A2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ C3 @ B2 )
% 0.31/0.47 => ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.coboundedI2
% 0.31/0.47 thf(fact_238_sup_OcoboundedI1,axiom,
% 0.31/0.47 ! [C3: int,A2: int,B2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ C3 @ A2 )
% 0.31/0.47 => ( ord_less_eq_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.coboundedI1
% 0.31/0.47 thf(fact_239_sup_OcoboundedI1,axiom,
% 0.31/0.47 ! [C3: a > $o,A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ C3 @ A2 )
% 0.31/0.47 => ( ord_less_eq_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.coboundedI1
% 0.31/0.47 thf(fact_240_sup_OcoboundedI1,axiom,
% 0.31/0.47 ! [C3: set_a,A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ C3 @ A2 )
% 0.31/0.47 => ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.coboundedI1
% 0.31/0.47 thf(fact_241_sup_Oabsorb__iff2,axiom,
% 0.31/0.47 ( ord_less_eq_int
% 0.31/0.47 = ( ^ [A: int,B: int] :
% 0.31/0.47 ( ( sup_sup_int @ A @ B )
% 0.31/0.47 = B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb_iff2
% 0.31/0.47 thf(fact_242_sup_Oabsorb__iff2,axiom,
% 0.31/0.47 ( ord_less_eq_a_o
% 0.31/0.47 = ( ^ [A: a > $o,B: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ A @ B )
% 0.31/0.47 = B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb_iff2
% 0.31/0.47 thf(fact_243_sup_Oabsorb__iff2,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [A: set_a,B: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A @ B )
% 0.31/0.47 = B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb_iff2
% 0.31/0.47 thf(fact_244_sup_Oabsorb__iff1,axiom,
% 0.31/0.47 ( ord_less_eq_int
% 0.31/0.47 = ( ^ [B: int,A: int] :
% 0.31/0.47 ( ( sup_sup_int @ A @ B )
% 0.31/0.47 = A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb_iff1
% 0.31/0.47 thf(fact_245_sup_Oabsorb__iff1,axiom,
% 0.31/0.47 ( ord_less_eq_a_o
% 0.31/0.47 = ( ^ [B: a > $o,A: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ A @ B )
% 0.31/0.47 = A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb_iff1
% 0.31/0.47 thf(fact_246_sup_Oabsorb__iff1,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [B: set_a,A: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ A @ B )
% 0.31/0.47 = A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb_iff1
% 0.31/0.47 thf(fact_247_sup_Ocobounded2,axiom,
% 0.31/0.47 ! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.cobounded2
% 0.31/0.47 thf(fact_248_sup_Ocobounded2,axiom,
% 0.31/0.47 ! [B2: a > $o,A2: a > $o] : ( ord_less_eq_a_o @ B2 @ ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.cobounded2
% 0.31/0.47 thf(fact_249_sup_Ocobounded2,axiom,
% 0.31/0.47 ! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.cobounded2
% 0.31/0.47 thf(fact_250_sup_Ocobounded1,axiom,
% 0.31/0.47 ! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.cobounded1
% 0.31/0.47 thf(fact_251_sup_Ocobounded1,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] : ( ord_less_eq_a_o @ A2 @ ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.cobounded1
% 0.31/0.47 thf(fact_252_sup_Ocobounded1,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.cobounded1
% 0.31/0.47 thf(fact_253_sup_Oorder__iff,axiom,
% 0.31/0.47 ( ord_less_eq_int
% 0.31/0.47 = ( ^ [B: int,A: int] :
% 0.31/0.47 ( A
% 0.31/0.47 = ( sup_sup_int @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.order_iff
% 0.31/0.47 thf(fact_254_sup_Oorder__iff,axiom,
% 0.31/0.47 ( ord_less_eq_a_o
% 0.31/0.47 = ( ^ [B: a > $o,A: a > $o] :
% 0.31/0.47 ( A
% 0.31/0.47 = ( sup_sup_a_o @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.order_iff
% 0.31/0.47 thf(fact_255_sup_Oorder__iff,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [B: set_a,A: set_a] :
% 0.31/0.47 ( A
% 0.31/0.47 = ( sup_sup_set_a @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.order_iff
% 0.31/0.47 thf(fact_256_sup_OboundedI,axiom,
% 0.31/0.47 ! [B2: int,A2: int,C3: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq_int @ C3 @ A2 )
% 0.31/0.47 => ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.boundedI
% 0.31/0.47 thf(fact_257_sup_OboundedI,axiom,
% 0.31/0.47 ! [B2: a > $o,A2: a > $o,C3: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq_a_o @ C3 @ A2 )
% 0.31/0.47 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.boundedI
% 0.31/0.47 thf(fact_258_sup_OboundedI,axiom,
% 0.31/0.47 ! [B2: set_a,A2: set_a,C3: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ C3 @ A2 )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.boundedI
% 0.31/0.47 thf(fact_259_sup_OboundedE,axiom,
% 0.31/0.47 ! [B2: int,C3: int,A2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
% 0.31/0.47 => ~ ( ( ord_less_eq_int @ B2 @ A2 )
% 0.31/0.47 => ~ ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.boundedE
% 0.31/0.47 thf(fact_260_sup_OboundedE,axiom,
% 0.31/0.47 ! [B2: a > $o,C3: a > $o,A2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
% 0.31/0.47 => ~ ( ( ord_less_eq_a_o @ B2 @ A2 )
% 0.31/0.47 => ~ ( ord_less_eq_a_o @ C3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.boundedE
% 0.31/0.47 thf(fact_261_sup_OboundedE,axiom,
% 0.31/0.47 ! [B2: set_a,C3: set_a,A2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
% 0.31/0.47 => ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.47 => ~ ( ord_less_eq_set_a @ C3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.boundedE
% 0.31/0.47 thf(fact_262_sup__absorb2,axiom,
% 0.31/0.47 ! [X: int,Y: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ X @ Y )
% 0.31/0.47 => ( ( sup_sup_int @ X @ Y )
% 0.31/0.47 = Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_absorb2
% 0.31/0.47 thf(fact_263_sup__absorb2,axiom,
% 0.31/0.47 ! [X: a > $o,Y: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ X @ Y )
% 0.31/0.47 => ( ( sup_sup_a_o @ X @ Y )
% 0.31/0.47 = Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_absorb2
% 0.31/0.47 thf(fact_264_sup__absorb2,axiom,
% 0.31/0.47 ! [X: set_a,Y: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ X @ Y )
% 0.31/0.47 => ( ( sup_sup_set_a @ X @ Y )
% 0.31/0.47 = Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_absorb2
% 0.31/0.47 thf(fact_265_sup__absorb1,axiom,
% 0.31/0.47 ! [Y: int,X: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ Y @ X )
% 0.31/0.47 => ( ( sup_sup_int @ X @ Y )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_absorb1
% 0.31/0.47 thf(fact_266_sup__absorb1,axiom,
% 0.31/0.47 ! [Y: a > $o,X: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ Y @ X )
% 0.31/0.47 => ( ( sup_sup_a_o @ X @ Y )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_absorb1
% 0.31/0.47 thf(fact_267_sup__absorb1,axiom,
% 0.31/0.47 ! [Y: set_a,X: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ Y @ X )
% 0.31/0.47 => ( ( sup_sup_set_a @ X @ Y )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_absorb1
% 0.31/0.47 thf(fact_268_sup_Oabsorb2,axiom,
% 0.31/0.47 ! [A2: int,B2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ A2 @ B2 )
% 0.31/0.47 => ( ( sup_sup_int @ A2 @ B2 )
% 0.31/0.47 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb2
% 0.31/0.47 thf(fact_269_sup_Oabsorb2,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ A2 @ B2 )
% 0.31/0.47 => ( ( sup_sup_a_o @ A2 @ B2 )
% 0.31/0.47 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb2
% 0.31/0.47 thf(fact_270_sup_Oabsorb2,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.31/0.47 => ( ( sup_sup_set_a @ A2 @ B2 )
% 0.31/0.47 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb2
% 0.31/0.47 thf(fact_271_sup_Oabsorb1,axiom,
% 0.31/0.47 ! [B2: int,A2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ B2 @ A2 )
% 0.31/0.47 => ( ( sup_sup_int @ A2 @ B2 )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb1
% 0.31/0.47 thf(fact_272_sup_Oabsorb1,axiom,
% 0.31/0.47 ! [B2: a > $o,A2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ B2 @ A2 )
% 0.31/0.47 => ( ( sup_sup_a_o @ A2 @ B2 )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb1
% 0.31/0.47 thf(fact_273_sup_Oabsorb1,axiom,
% 0.31/0.47 ! [B2: set_a,A2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.47 => ( ( sup_sup_set_a @ A2 @ B2 )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.absorb1
% 0.31/0.47 thf(fact_274_sup__unique,axiom,
% 0.31/0.47 ! [F2: int > int > int,X: int,Y: int] :
% 0.31/0.47 ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
% 0.31/0.47 => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 0.31/0.47 => ( ! [X3: int,Y3: int,Z: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.31/0.47 => ( ( ord_less_eq_int @ Z @ X3 )
% 0.31/0.47 => ( ord_less_eq_int @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 0.31/0.47 => ( ( sup_sup_int @ X @ Y )
% 0.31/0.47 = ( F2 @ X @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_unique
% 0.31/0.47 thf(fact_275_sup__unique,axiom,
% 0.31/0.47 ! [F2: ( a > $o ) > ( a > $o ) > a > $o,X: a > $o,Y: a > $o] :
% 0.31/0.47 ( ! [X3: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ X3 @ ( F2 @ X3 @ Y3 ) )
% 0.31/0.47 => ( ! [X3: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 0.31/0.47 => ( ! [X3: a > $o,Y3: a > $o,Z: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ Y3 @ X3 )
% 0.31/0.47 => ( ( ord_less_eq_a_o @ Z @ X3 )
% 0.31/0.47 => ( ord_less_eq_a_o @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 0.31/0.47 => ( ( sup_sup_a_o @ X @ Y )
% 0.31/0.47 = ( F2 @ X @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_unique
% 0.31/0.47 thf(fact_276_sup__unique,axiom,
% 0.31/0.47 ! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
% 0.31/0.47 ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F2 @ X3 @ Y3 ) )
% 0.31/0.47 => ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 0.31/0.47 => ( ! [X3: set_a,Y3: set_a,Z: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ Y3 @ X3 )
% 0.31/0.47 => ( ( ord_less_eq_set_a @ Z @ X3 )
% 0.31/0.47 => ( ord_less_eq_set_a @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 0.31/0.47 => ( ( sup_sup_set_a @ X @ Y )
% 0.31/0.47 = ( F2 @ X @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup_unique
% 0.31/0.47 thf(fact_277_sup_OorderI,axiom,
% 0.31/0.47 ! [A2: int,B2: int] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( sup_sup_int @ A2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq_int @ B2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.orderI
% 0.31/0.47 thf(fact_278_sup_OorderI,axiom,
% 0.31/0.47 ! [A2: a > $o,B2: a > $o] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq_a_o @ B2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.orderI
% 0.31/0.47 thf(fact_279_sup_OorderI,axiom,
% 0.31/0.47 ! [A2: set_a,B2: set_a] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.orderI
% 0.31/0.47 thf(fact_280_sup_OorderE,axiom,
% 0.31/0.47 ! [B2: int,A2: int] :
% 0.31/0.47 ( ( ord_less_eq_int @ B2 @ A2 )
% 0.31/0.47 => ( A2
% 0.31/0.47 = ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.orderE
% 0.31/0.47 thf(fact_281_sup_OorderE,axiom,
% 0.31/0.47 ! [B2: a > $o,A2: a > $o] :
% 0.31/0.47 ( ( ord_less_eq_a_o @ B2 @ A2 )
% 0.31/0.47 => ( A2
% 0.31/0.47 = ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.orderE
% 0.31/0.47 thf(fact_282_sup_OorderE,axiom,
% 0.31/0.47 ! [B2: set_a,A2: set_a] :
% 0.31/0.47 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.31/0.47 => ( A2
% 0.31/0.47 = ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sup.orderE
% 0.31/0.47 thf(fact_283_le__iff__sup,axiom,
% 0.31/0.47 ( ord_less_eq_int
% 0.31/0.47 = ( ^ [X4: int,Y4: int] :
% 0.31/0.47 ( ( sup_sup_int @ X4 @ Y4 )
% 0.31/0.47 = Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_iff_sup
% 0.31/0.47 thf(fact_284_le__iff__sup,axiom,
% 0.31/0.47 ( ord_less_eq_a_o
% 0.31/0.47 = ( ^ [X4: a > $o,Y4: a > $o] :
% 0.31/0.47 ( ( sup_sup_a_o @ X4 @ Y4 )
% 0.31/0.47 = Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_iff_sup
% 0.31/0.47 thf(fact_285_le__iff__sup,axiom,
% 0.31/0.47 ( ord_less_eq_set_a
% 0.31/0.47 = ( ^ [X4: set_a,Y4: set_a] :
% 0.31/0.47 ( ( sup_sup_set_a @ X4 @ Y4 )
% 0.31/0.47 = Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_iff_sup
% 0.31/0.48 thf(fact_286_sup__least,axiom,
% 0.31/0.48 ! [Y: int,X: int,Z4: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ Y @ X )
% 0.31/0.48 => ( ( ord_less_eq_int @ Z4 @ X )
% 0.31/0.48 => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z4 ) @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_least
% 0.31/0.48 thf(fact_287_sup__least,axiom,
% 0.31/0.48 ! [Y: a > $o,X: a > $o,Z4: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ Y @ X )
% 0.31/0.48 => ( ( ord_less_eq_a_o @ Z4 @ X )
% 0.31/0.48 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ Y @ Z4 ) @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_least
% 0.31/0.48 thf(fact_288_sup__least,axiom,
% 0.31/0.48 ! [Y: set_a,X: set_a,Z4: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ Y @ X )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ Z4 @ X )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_least
% 0.31/0.48 thf(fact_289_sup__mono,axiom,
% 0.31/0.48 ! [A2: int,C3: int,B2: int,D2: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ A2 @ C3 )
% 0.31/0.48 => ( ( ord_less_eq_int @ B2 @ D2 )
% 0.31/0.48 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C3 @ D2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_mono
% 0.31/0.48 thf(fact_290_sup__mono,axiom,
% 0.31/0.48 ! [A2: a > $o,C3: a > $o,B2: a > $o,D2: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ A2 @ C3 )
% 0.31/0.48 => ( ( ord_less_eq_a_o @ B2 @ D2 )
% 0.31/0.48 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ ( sup_sup_a_o @ C3 @ D2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_mono
% 0.31/0.48 thf(fact_291_sup__mono,axiom,
% 0.31/0.48 ! [A2: set_a,C3: set_a,B2: set_a,D2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ C3 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ D2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_mono
% 0.31/0.48 thf(fact_292_sup_Omono,axiom,
% 0.31/0.48 ! [C3: int,A2: int,D2: int,B2: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ C3 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_int @ D2 @ B2 )
% 0.31/0.48 => ( ord_less_eq_int @ ( sup_sup_int @ C3 @ D2 ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.mono
% 0.31/0.48 thf(fact_293_sup_Omono,axiom,
% 0.31/0.48 ! [C3: a > $o,A2: a > $o,D2: a > $o,B2: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ C3 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_a_o @ D2 @ B2 )
% 0.31/0.48 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ C3 @ D2 ) @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.mono
% 0.31/0.48 thf(fact_294_sup_Omono,axiom,
% 0.31/0.48 ! [C3: set_a,A2: set_a,D2: set_a,B2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ C3 @ A2 )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ D2 @ B2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C3 @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.mono
% 0.31/0.48 thf(fact_295_le__supI2,axiom,
% 0.31/0.48 ! [X: int,B2: int,A2: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ X @ B2 )
% 0.31/0.48 => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI2
% 0.31/0.48 thf(fact_296_le__supI2,axiom,
% 0.31/0.48 ! [X: a > $o,B2: a > $o,A2: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ X @ B2 )
% 0.31/0.48 => ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI2
% 0.31/0.48 thf(fact_297_le__supI2,axiom,
% 0.31/0.48 ! [X: set_a,B2: set_a,A2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X @ B2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI2
% 0.31/0.48 thf(fact_298_le__supI1,axiom,
% 0.31/0.48 ! [X: int,A2: int,B2: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ X @ A2 )
% 0.31/0.48 => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI1
% 0.31/0.48 thf(fact_299_le__supI1,axiom,
% 0.31/0.48 ! [X: a > $o,A2: a > $o,B2: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ X @ A2 )
% 0.31/0.48 => ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI1
% 0.31/0.48 thf(fact_300_le__supI1,axiom,
% 0.31/0.48 ! [X: set_a,A2: set_a,B2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ X @ A2 )
% 0.31/0.48 => ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI1
% 0.31/0.48 thf(fact_301_sup__ge2,axiom,
% 0.31/0.48 ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_ge2
% 0.31/0.48 thf(fact_302_sup__ge2,axiom,
% 0.31/0.48 ! [Y: a > $o,X: a > $o] : ( ord_less_eq_a_o @ Y @ ( sup_sup_a_o @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_ge2
% 0.31/0.48 thf(fact_303_sup__ge2,axiom,
% 0.31/0.48 ! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_ge2
% 0.31/0.48 thf(fact_304_sup__ge1,axiom,
% 0.31/0.48 ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_ge1
% 0.31/0.48 thf(fact_305_sup__ge1,axiom,
% 0.31/0.48 ! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_ge1
% 0.31/0.48 thf(fact_306_sup__ge1,axiom,
% 0.31/0.48 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup_ge1
% 0.31/0.48 thf(fact_307_le__supI,axiom,
% 0.31/0.48 ! [A2: int,X: int,B2: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ A2 @ X )
% 0.31/0.48 => ( ( ord_less_eq_int @ B2 @ X )
% 0.31/0.48 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI
% 0.31/0.48 thf(fact_308_le__supI,axiom,
% 0.31/0.48 ! [A2: a > $o,X: a > $o,B2: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ A2 @ X )
% 0.31/0.48 => ( ( ord_less_eq_a_o @ B2 @ X )
% 0.31/0.48 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI
% 0.31/0.48 thf(fact_309_le__supI,axiom,
% 0.31/0.48 ! [A2: set_a,X: set_a,B2: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ A2 @ X )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ X )
% 0.31/0.48 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supI
% 0.31/0.48 thf(fact_310_le__supE,axiom,
% 0.31/0.48 ! [A2: int,B2: int,X: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X )
% 0.31/0.48 => ~ ( ( ord_less_eq_int @ A2 @ X )
% 0.31/0.48 => ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supE
% 0.31/0.48 thf(fact_311_le__supE,axiom,
% 0.31/0.48 ! [A2: a > $o,B2: a > $o,X: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ X )
% 0.31/0.48 => ~ ( ( ord_less_eq_a_o @ A2 @ X )
% 0.31/0.48 => ~ ( ord_less_eq_a_o @ B2 @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supE
% 0.31/0.48 thf(fact_312_le__supE,axiom,
% 0.31/0.48 ! [A2: set_a,B2: set_a,X: set_a] :
% 0.31/0.48 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
% 0.31/0.48 => ~ ( ( ord_less_eq_set_a @ A2 @ X )
% 0.31/0.48 => ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % le_supE
% 0.31/0.48 thf(fact_313_inf__sup__ord_I3_J,axiom,
% 0.31/0.48 ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % inf_sup_ord(3)
% 0.31/0.48 thf(fact_314_inf__sup__ord_I3_J,axiom,
% 0.31/0.48 ! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % inf_sup_ord(3)
% 0.31/0.48 thf(fact_315_inf__sup__ord_I3_J,axiom,
% 0.31/0.48 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % inf_sup_ord(3)
% 0.31/0.48 thf(fact_316_inf__sup__ord_I4_J,axiom,
% 0.31/0.48 ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % inf_sup_ord(4)
% 0.31/0.48 thf(fact_317_inf__sup__ord_I4_J,axiom,
% 0.31/0.48 ! [Y: a > $o,X: a > $o] : ( ord_less_eq_a_o @ Y @ ( sup_sup_a_o @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % inf_sup_ord(4)
% 0.31/0.48 thf(fact_318_inf__sup__ord_I4_J,axiom,
% 0.31/0.48 ! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% 0.31/0.48
% 0.31/0.48 % inf_sup_ord(4)
% 0.31/0.48 thf(fact_319_less__supI1,axiom,
% 0.31/0.48 ! [X: a > $o,A2: a > $o,B2: a > $o] :
% 0.31/0.48 ( ( ord_less_a_o @ X @ A2 )
% 0.31/0.48 => ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_supI1
% 0.31/0.48 thf(fact_320_less__supI1,axiom,
% 0.31/0.48 ! [X: set_a,A2: set_a,B2: set_a] :
% 0.31/0.48 ( ( ord_less_set_a @ X @ A2 )
% 0.31/0.48 => ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_supI1
% 0.31/0.48 thf(fact_321_less__supI1,axiom,
% 0.31/0.48 ! [X: int,A2: int,B2: int] :
% 0.31/0.48 ( ( ord_less_int @ X @ A2 )
% 0.31/0.48 => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_supI1
% 0.31/0.48 thf(fact_322_less__supI2,axiom,
% 0.31/0.48 ! [X: a > $o,B2: a > $o,A2: a > $o] :
% 0.31/0.48 ( ( ord_less_a_o @ X @ B2 )
% 0.31/0.48 => ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_supI2
% 0.31/0.48 thf(fact_323_less__supI2,axiom,
% 0.31/0.48 ! [X: set_a,B2: set_a,A2: set_a] :
% 0.31/0.48 ( ( ord_less_set_a @ X @ B2 )
% 0.31/0.48 => ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_supI2
% 0.31/0.48 thf(fact_324_less__supI2,axiom,
% 0.31/0.48 ! [X: int,B2: int,A2: int] :
% 0.31/0.48 ( ( ord_less_int @ X @ B2 )
% 0.31/0.48 => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % less_supI2
% 0.31/0.48 thf(fact_325_sup_Ostrict__boundedE,axiom,
% 0.31/0.48 ! [B2: a > $o,C3: a > $o,A2: a > $o] :
% 0.31/0.48 ( ( ord_less_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
% 0.31/0.48 => ~ ( ( ord_less_a_o @ B2 @ A2 )
% 0.31/0.48 => ~ ( ord_less_a_o @ C3 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_boundedE
% 0.31/0.48 thf(fact_326_sup_Ostrict__boundedE,axiom,
% 0.31/0.48 ! [B2: set_a,C3: set_a,A2: set_a] :
% 0.31/0.48 ( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
% 0.31/0.48 => ~ ( ( ord_less_set_a @ B2 @ A2 )
% 0.31/0.48 => ~ ( ord_less_set_a @ C3 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_boundedE
% 0.31/0.48 thf(fact_327_sup_Ostrict__boundedE,axiom,
% 0.31/0.48 ! [B2: int,C3: int,A2: int] :
% 0.31/0.48 ( ( ord_less_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
% 0.31/0.48 => ~ ( ( ord_less_int @ B2 @ A2 )
% 0.31/0.48 => ~ ( ord_less_int @ C3 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_boundedE
% 0.31/0.48 thf(fact_328_sup_Ostrict__order__iff,axiom,
% 0.31/0.48 ( ord_less_a_o
% 0.31/0.48 = ( ^ [B: a > $o,A: a > $o] :
% 0.31/0.48 ( ( A
% 0.31/0.48 = ( sup_sup_a_o @ A @ B ) )
% 0.31/0.48 & ( A != B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_order_iff
% 0.31/0.48 thf(fact_329_sup_Ostrict__order__iff,axiom,
% 0.31/0.48 ( ord_less_set_a
% 0.31/0.48 = ( ^ [B: set_a,A: set_a] :
% 0.31/0.48 ( ( A
% 0.31/0.48 = ( sup_sup_set_a @ A @ B ) )
% 0.31/0.48 & ( A != B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_order_iff
% 0.31/0.48 thf(fact_330_sup_Ostrict__order__iff,axiom,
% 0.31/0.48 ( ord_less_int
% 0.31/0.48 = ( ^ [B: int,A: int] :
% 0.31/0.48 ( ( A
% 0.31/0.48 = ( sup_sup_int @ A @ B ) )
% 0.31/0.48 & ( A != B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_order_iff
% 0.31/0.48 thf(fact_331_sup_Ostrict__coboundedI1,axiom,
% 0.31/0.48 ! [C3: a > $o,A2: a > $o,B2: a > $o] :
% 0.31/0.48 ( ( ord_less_a_o @ C3 @ A2 )
% 0.31/0.48 => ( ord_less_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_coboundedI1
% 0.31/0.48 thf(fact_332_sup_Ostrict__coboundedI1,axiom,
% 0.31/0.48 ! [C3: set_a,A2: set_a,B2: set_a] :
% 0.31/0.48 ( ( ord_less_set_a @ C3 @ A2 )
% 0.31/0.48 => ( ord_less_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_coboundedI1
% 0.31/0.48 thf(fact_333_sup_Ostrict__coboundedI1,axiom,
% 0.31/0.48 ! [C3: int,A2: int,B2: int] :
% 0.31/0.48 ( ( ord_less_int @ C3 @ A2 )
% 0.31/0.48 => ( ord_less_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_coboundedI1
% 0.31/0.48 thf(fact_334_sup_Ostrict__coboundedI2,axiom,
% 0.31/0.48 ! [C3: a > $o,B2: a > $o,A2: a > $o] :
% 0.31/0.48 ( ( ord_less_a_o @ C3 @ B2 )
% 0.31/0.48 => ( ord_less_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_coboundedI2
% 0.31/0.48 thf(fact_335_sup_Ostrict__coboundedI2,axiom,
% 0.31/0.48 ! [C3: set_a,B2: set_a,A2: set_a] :
% 0.31/0.48 ( ( ord_less_set_a @ C3 @ B2 )
% 0.31/0.48 => ( ord_less_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_coboundedI2
% 0.31/0.48 thf(fact_336_sup_Ostrict__coboundedI2,axiom,
% 0.31/0.48 ! [C3: int,B2: int,A2: int] :
% 0.31/0.48 ( ( ord_less_int @ C3 @ B2 )
% 0.31/0.48 => ( ord_less_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup.strict_coboundedI2
% 0.31/0.48 thf(fact_337_predicate1I,axiom,
% 0.31/0.48 ! [P: a > $o,Q: a > $o] :
% 0.31/0.48 ( ! [X3: a] :
% 0.31/0.48 ( ( P @ X3 )
% 0.31/0.48 => ( Q @ X3 ) )
% 0.31/0.48 => ( ord_less_eq_a_o @ P @ Q ) ) ).
% 0.31/0.48
% 0.31/0.48 % predicate1I
% 0.31/0.48 thf(fact_338_pred__subset__eq,axiom,
% 0.31/0.48 ! [R: set_a,S: set_a] :
% 0.31/0.48 ( ( ord_less_eq_a_o
% 0.31/0.48 @ ^ [X4: a] : ( member_a @ X4 @ R )
% 0.31/0.48 @ ^ [X4: a] : ( member_a @ X4 @ S ) )
% 0.31/0.48 = ( ord_less_eq_set_a @ R @ S ) ) ).
% 0.31/0.48
% 0.31/0.48 % pred_subset_eq
% 0.31/0.48 thf(fact_339_the__elem__eq,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.31/0.48 = X ) ).
% 0.31/0.48
% 0.31/0.48 % the_elem_eq
% 0.31/0.48 thf(fact_340_order__refl,axiom,
% 0.31/0.48 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_341_order__refl,axiom,
% 0.31/0.48 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_342_order__refl,axiom,
% 0.31/0.48 ! [X: a > $o] : ( ord_less_eq_a_o @ X @ X ) ).
% 0.31/0.48
% 0.31/0.48 % order_refl
% 0.31/0.48 thf(fact_343_sup1CI,axiom,
% 0.31/0.48 ! [B3: a > $o,X: a,A3: a > $o] :
% 0.31/0.48 ( ( ~ ( B3 @ X )
% 0.31/0.48 => ( A3 @ X ) )
% 0.31/0.48 => ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup1CI
% 0.31/0.48 thf(fact_344_bot__apply,axiom,
% 0.31/0.48 ( bot_bot_a_o
% 0.31/0.48 = ( ^ [X4: a] : bot_bot_o ) ) ).
% 0.31/0.48
% 0.31/0.48 % bot_apply
% 0.31/0.48 thf(fact_345_sup1I2,axiom,
% 0.31/0.48 ! [B3: a > $o,X: a,A3: a > $o] :
% 0.31/0.48 ( ( B3 @ X )
% 0.31/0.48 => ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup1I2
% 0.31/0.48 thf(fact_346_sup1I1,axiom,
% 0.31/0.48 ! [A3: a > $o,X: a,B3: a > $o] :
% 0.31/0.48 ( ( A3 @ X )
% 0.31/0.48 => ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup1I1
% 0.31/0.48 thf(fact_347_sup1E,axiom,
% 0.31/0.48 ! [A3: a > $o,B3: a > $o,X: a] :
% 0.31/0.48 ( ( sup_sup_a_o @ A3 @ B3 @ X )
% 0.31/0.48 => ( ~ ( A3 @ X )
% 0.31/0.48 => ( B3 @ X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sup1E
% 0.31/0.48 thf(fact_348_order__subst1,axiom,
% 0.31/0.48 ! [A2: int,F2: ( a > $o ) > int,B2: a > $o,C3: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_int @ A2 @ ( F2 @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_a_o @ B2 @ C3 )
% 0.31/0.48 => ( ! [X3: a > $o,Y3: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ X3 @ Y3 )
% 0.31/0.48 => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.48 => ( ord_less_eq_int @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_subst1
% 0.31/0.48 thf(fact_349_order__subst1,axiom,
% 0.31/0.48 ! [A2: a > $o,F2: set_a > a > $o,B2: set_a,C3: set_a] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_set_a @ B2 @ C3 )
% 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_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.48 => ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_subst1
% 0.31/0.48 thf(fact_350_order__subst1,axiom,
% 0.31/0.48 ! [A2: a > $o,F2: int > a > $o,B2: int,C3: int] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_int @ B2 @ C3 )
% 0.31/0.48 => ( ! [X3: int,Y3: int] :
% 0.31/0.48 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.31/0.48 => ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.48 => ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_subst1
% 0.31/0.48 thf(fact_351_order__subst1,axiom,
% 0.31/0.48 ! [A2: a > $o,F2: ( a > $o ) > a > $o,B2: a > $o,C3: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
% 0.31/0.48 => ( ( ord_less_eq_a_o @ B2 @ C3 )
% 0.31/0.48 => ( ! [X3: a > $o,Y3: a > $o] :
% 0.31/0.48 ( ( ord_less_eq_a_o @ X3 @ Y3 )
% 0.31/0.48 => ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.31/0.48 => ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % order_subst1
% 0.31/0.48
% 0.31/0.48 % Conjectures (1)
% 0.31/0.48 thf(conj_0,conjecture,
% 0.31/0.48 ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 2.56/1.78 ---- Embedded in THF ---
% 2.56/1.78 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.56/1.78 %%% Generated on Tue Feb 24 08:23:49 EST 2026
% 2.56/1.78 %%% using '$$dhol' embedding, version 1.3.0.
% 2.56/1.78 %%% Logic specification used:
% 2.56/1.78 %%% thf(spec, logic, $$dhol).
% 2.56/1.78
% 2.56/1.78 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2DTF16811.p
% See solution above
% 2.56/1.78 ------------------------
% 2.78/1.80 ---- Cleaned THF ---
% 2.78/1.80 thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
% 2.78/1.80 binary1439146945Tree_a: $tType ).
% 2.78/1.80
% 2.78/1.80 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.78/1.80 set_a: $tType ).
% 2.78/1.80
% 2.78/1.80 thf(ty_n_t__Int__Oint,type,
% 2.78/1.80 int: $tType ).
% 2.78/1.80
% 2.78/1.80 thf(ty_n_tf__a,type,
% 2.78/1.80 a: $tType ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001tf__a,type,
% 2.78/1.80 binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001tf__a,type,
% 2.78/1.80 binary476621312_Tip_a: binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Obinsert_001tf__a,type,
% 2.78/1.80 binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Omemb_001tf__a,type,
% 2.78/1.80 binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oremove_001tf__a,type,
% 2.78/1.80 binary1804682569move_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Orm_001tf__a,type,
% 2.78/1.80 binary339557810e_rm_a: ( a > int ) > binary1439146945Tree_a > a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001tf__a,type,
% 2.78/1.80 binary945792244etOf_a: binary1439146945Tree_a > set_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001tf__a,type,
% 2.78/1.80 binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001tf__a,type,
% 2.78/1.80 binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Owrm_001tf__a,type,
% 2.78/1.80 binary1217730267_wrm_a: ( a > int ) > binary1439146945Tree_a > binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
% 2.78/1.80 sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
% 2.78/1.80 sup_sup_o: $o > $o > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 2.78/1.80 sup_sup_int: int > int > int ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
% 2.78/1.80 sup_sup_set_a: set_a > set_a > set_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 2.78/1.80 bot_bot_a_o: a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 2.78/1.80 bot_bot_o: $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 2.78/1.80 bot_bot_set_a: set_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
% 2.78/1.80 ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 2.78/1.80 ord_less_int: int > int > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 2.78/1.80 ord_less_set_a: set_a > set_a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
% 2.78/1.80 ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.78/1.80 ord_less_eq_int: int > int > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.78/1.80 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.78/1.80 collect_a: ( a > $o ) > set_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Set_Oinsert_001tf__a,type,
% 2.78/1.80 insert_a: a > set_a > set_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 2.78/1.80 the_elem_a: set_a > a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_c_member_001tf__a,type,
% 2.78/1.80 member_a: a > set_a > $o ).
% 2.78/1.80
% 2.78/1.80 thf(sy_v_h,type,
% 2.78/1.80 h: a > int ).
% 2.78/1.80
% 2.78/1.80 thf(sy_v_l____,type,
% 2.78/1.80 l: a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_v_t1____,type,
% 2.78/1.80 t1: binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_v_t2____,type,
% 2.78/1.80 t2: binary1439146945Tree_a ).
% 2.78/1.80
% 2.78/1.80 thf(sy_v_x____,type,
% 2.78/1.80 x: a ).
% 2.78/1.80
% 2.78/1.80 thf(fact_0_hLess,axiom,
% 2.78/1.80 ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_1_rm__res,axiom,
% 2.78/1.80 ( ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
% 2.78/1.80 = ( binary339557810e_rm_a @ h @ t2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_2_t2nTip,axiom,
% 2.78/1.80 t2 != binary476621312_Tip_a ).
% 2.78/1.80
% 2.78/1.80 thf(fact_3_Tree_Oinject,axiom,
% 2.78/1.80 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a,Y21: binary1439146945Tree_a,Y22: a,Y23: binary1439146945Tree_a] :
% 2.78/1.80 ( ( ( binary717961607le_T_a @ X21 @ X22 @ X23 )
% 2.78/1.80 = ( binary717961607le_T_a @ Y21 @ Y22 @ Y23 ) )
% 2.78/1.80 = ( ( X21 = Y21 )
% 2.78/1.80 & ( X22 = Y22 )
% 2.78/1.80 & ( X23 = Y23 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_4_s,axiom,
% 2.78/1.80 binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_5_wrm__res,axiom,
% 2.78/1.80 ( ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
% 2.78/1.80 = ( binary717961607le_T_a @ t1 @ x @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_6_ldef,axiom,
% 2.78/1.80 member_a @ l @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_7_s1,axiom,
% 2.78/1.80 binary1721989714Tree_a @ h @ t1 ).
% 2.78/1.80
% 2.78/1.80 thf(fact_8_s2,axiom,
% 2.78/1.80 binary1721989714Tree_a @ h @ t2 ).
% 2.78/1.80
% 2.78/1.80 thf(fact_9_binsert_Osimps_I2_J,axiom,
% 2.78/1.80 ! [H: a > int,E: a,X: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
% 2.78/1.80 ( ( ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
% 2.78/1.80 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( binary717961607le_T_a @ ( binary1226383794sert_a @ H @ E @ T1 ) @ X @ T2 ) ) )
% 2.78/1.80 & ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
% 2.78/1.80 => ( ( ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
% 2.78/1.80 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( binary717961607le_T_a @ T1 @ X @ ( binary1226383794sert_a @ H @ E @ T2 ) ) ) )
% 2.78/1.80 & ( ~ ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
% 2.78/1.80 => ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( binary717961607le_T_a @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_10_rm_Osimps,axiom,
% 2.78/1.80 ! [T2: binary1439146945Tree_a,H: a > int,T1: binary1439146945Tree_a,X: a] :
% 2.78/1.80 ( ( ( T2 = binary476621312_Tip_a )
% 2.78/1.80 => ( ( binary339557810e_rm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = X ) )
% 2.78/1.80 & ( ( T2 != binary476621312_Tip_a )
% 2.78/1.80 => ( ( binary339557810e_rm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( binary339557810e_rm_a @ H @ T2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_11_minf_I7_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ~ ( ord_less_int @ T @ X2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_12_minf_I5_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ( ord_less_int @ X2 @ T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_13_minf_I4_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ( X2 != T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_14_minf_I3_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ( X2 != T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_15_minf_I2_J,axiom,
% 2.78/1.80 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 2.78/1.80 ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ X3 @ Z2 )
% 2.78/1.80 => ( ( P @ X3 )
% 2.78/1.80 = ( P2 @ X3 ) ) )
% 2.78/1.80 => ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ X3 @ Z2 )
% 2.78/1.80 => ( ( Q @ X3 )
% 2.78/1.80 = ( Q2 @ X3 ) ) )
% 2.78/1.80 => ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ( ( ( P @ X2 )
% 2.78/1.80 | ( Q @ X2 ) )
% 2.78/1.80 = ( ( P2 @ X2 )
% 2.78/1.80 | ( Q2 @ X2 ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_16_sortedTree_Osimps_I1_J,axiom,
% 2.78/1.80 ! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_17_wrm__sort,axiom,
% 2.78/1.80 ! [T: binary1439146945Tree_a,H: a > int] :
% 2.78/1.80 ( ( ( T != binary476621312_Tip_a )
% 2.78/1.80 & ( binary1721989714Tree_a @ H @ T ) )
% 2.78/1.80 => ( binary1721989714Tree_a @ H @ ( binary1217730267_wrm_a @ H @ T ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_18_binsert__sorted,axiom,
% 2.78/1.80 ! [H: a > int,T: binary1439146945Tree_a,X: a] :
% 2.78/1.80 ( ( binary1721989714Tree_a @ H @ T )
% 2.78/1.80 => ( binary1721989714Tree_a @ H @ ( binary1226383794sert_a @ H @ X @ T ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_19_rm__set,axiom,
% 2.78/1.80 ! [T: binary1439146945Tree_a,H: a > int] :
% 2.78/1.80 ( ( ( T != binary476621312_Tip_a )
% 2.78/1.80 & ( binary1721989714Tree_a @ H @ T ) )
% 2.78/1.80 => ( member_a @ ( binary339557810e_rm_a @ H @ T ) @ ( binary945792244etOf_a @ T ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_20_binsert_Osimps_I1_J,axiom,
% 2.78/1.80 ! [H: a > int,E: a] :
% 2.78/1.80 ( ( binary1226383794sert_a @ H @ E @ binary476621312_Tip_a )
% 2.78/1.80 = ( binary717961607le_T_a @ binary476621312_Tip_a @ E @ binary476621312_Tip_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_21_wrm_Osimps,axiom,
% 2.78/1.80 ! [T2: binary1439146945Tree_a,H: a > int,T1: binary1439146945Tree_a,X: a] :
% 2.78/1.80 ( ( ( T2 = binary476621312_Tip_a )
% 2.78/1.80 => ( ( binary1217730267_wrm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = T1 ) )
% 2.78/1.80 & ( ( T2 != binary476621312_Tip_a )
% 2.78/1.80 => ( ( binary1217730267_wrm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( binary717961607le_T_a @ T1 @ X @ ( binary1217730267_wrm_a @ H @ T2 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_22_sortedTree_Osimps_I2_J,axiom,
% 2.78/1.80 ! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 2.78/1.80 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( ( binary1721989714Tree_a @ H @ T1 )
% 2.78/1.80 & ! [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ ( binary945792244etOf_a @ T1 ) )
% 2.78/1.80 => ( ord_less_int @ ( H @ X4 ) @ ( H @ X ) ) )
% 2.78/1.80 & ! [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ ( binary945792244etOf_a @ T2 ) )
% 2.78/1.80 => ( ord_less_int @ ( H @ X ) @ ( H @ X4 ) ) )
% 2.78/1.80 & ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_23_Tree_Oexhaust,axiom,
% 2.78/1.80 ! [Y: binary1439146945Tree_a] :
% 2.78/1.80 ( ( Y != binary476621312_Tip_a )
% 2.78/1.80 => ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
% 2.78/1.80 ( Y
% 2.78/1.80 != ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_24_Tree_Oinduct,axiom,
% 2.78/1.80 ! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
% 2.78/1.80 ( ( P @ binary476621312_Tip_a )
% 2.78/1.80 => ( ! [X1: binary1439146945Tree_a,X24: a,X32: binary1439146945Tree_a] :
% 2.78/1.80 ( ( P @ X1 )
% 2.78/1.80 => ( ( P @ X32 )
% 2.78/1.80 => ( P @ ( binary717961607le_T_a @ X1 @ X24 @ X32 ) ) ) )
% 2.78/1.80 => ( P @ Tree ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_25_Tree_Odistinct_I1_J,axiom,
% 2.78/1.80 ! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
% 2.78/1.80 ( binary476621312_Tip_a
% 2.78/1.80 != ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_26_sortLemmaR,axiom,
% 2.78/1.80 ! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 2.78/1.80 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 => ( binary1721989714Tree_a @ H @ T2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_27_sortLemmaL,axiom,
% 2.78/1.80 ! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 2.78/1.80 ( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 => ( binary1721989714Tree_a @ H @ T1 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_28_pinf_I1_J,axiom,
% 2.78/1.80 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 2.78/1.80 ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ Z2 @ X3 )
% 2.78/1.80 => ( ( P @ X3 )
% 2.78/1.80 = ( P2 @ X3 ) ) )
% 2.78/1.80 => ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ Z2 @ X3 )
% 2.78/1.80 => ( ( Q @ X3 )
% 2.78/1.80 = ( Q2 @ X3 ) ) )
% 2.78/1.80 => ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ( ( ( P @ X2 )
% 2.78/1.80 & ( Q @ X2 ) )
% 2.78/1.80 = ( ( P2 @ X2 )
% 2.78/1.80 & ( Q2 @ X2 ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_29_pinf_I2_J,axiom,
% 2.78/1.80 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 2.78/1.80 ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ Z2 @ X3 )
% 2.78/1.80 => ( ( P @ X3 )
% 2.78/1.80 = ( P2 @ X3 ) ) )
% 2.78/1.80 => ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ Z2 @ X3 )
% 2.78/1.80 => ( ( Q @ X3 )
% 2.78/1.80 = ( Q2 @ X3 ) ) )
% 2.78/1.80 => ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ( ( ( P @ X2 )
% 2.78/1.80 | ( Q @ X2 ) )
% 2.78/1.80 = ( ( P2 @ X2 )
% 2.78/1.80 | ( Q2 @ X2 ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_30_pinf_I3_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ( X2 != T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_31_pinf_I4_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ( X2 != T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_32_pinf_I5_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ~ ( ord_less_int @ X2 @ T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_33_pinf_I7_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ( ord_less_int @ T @ X2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_34_minf_I1_J,axiom,
% 2.78/1.80 ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
% 2.78/1.80 ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ X3 @ Z2 )
% 2.78/1.80 => ( ( P @ X3 )
% 2.78/1.80 = ( P2 @ X3 ) ) )
% 2.78/1.80 => ( ? [Z2: int] :
% 2.78/1.80 ! [X3: int] :
% 2.78/1.80 ( ( ord_less_int @ X3 @ Z2 )
% 2.78/1.80 => ( ( Q @ X3 )
% 2.78/1.80 = ( Q2 @ X3 ) ) )
% 2.78/1.80 => ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ( ( ( P @ X2 )
% 2.78/1.80 & ( Q @ X2 ) )
% 2.78/1.80 = ( ( P2 @ X2 )
% 2.78/1.80 & ( Q2 @ X2 ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_35_h1,axiom,
% 2.78/1.80 ( ( ( t1 != binary476621312_Tip_a )
% 2.78/1.80 & ( binary1721989714Tree_a @ h @ t1 ) )
% 2.78/1.80 => ! [X2: a] :
% 2.78/1.80 ( ( member_a @ X2 @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t1 ) ) )
% 2.78/1.80 => ( ord_less_int @ ( h @ X2 ) @ ( h @ ( binary339557810e_rm_a @ h @ t1 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_36_h2,axiom,
% 2.78/1.80 ( ( ( t2 != binary476621312_Tip_a )
% 2.78/1.80 & ( binary1721989714Tree_a @ h @ t2 ) )
% 2.78/1.80 => ! [X2: a] :
% 2.78/1.80 ( ( member_a @ X2 @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t2 ) ) )
% 2.78/1.80 => ( ord_less_int @ ( h @ X2 ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_37_memb__spec,axiom,
% 2.78/1.80 ! [H: a > int,T: binary1439146945Tree_a,X: a] :
% 2.78/1.80 ( ( binary1721989714Tree_a @ H @ T )
% 2.78/1.80 => ( ( binary2053421120memb_a @ H @ X @ T )
% 2.78/1.80 = ( member_a @ X @ ( binary945792244etOf_a @ T ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_38_sorted__distinct__pred__def,axiom,
% 2.78/1.80 ( binary670562003pred_a
% 2.78/1.80 = ( ^ [H2: a > int,A: a,B: a,T3: binary1439146945Tree_a] :
% 2.78/1.80 ( ( ( binary1721989714Tree_a @ H2 @ T3 )
% 2.78/1.80 & ( member_a @ A @ ( binary945792244etOf_a @ T3 ) )
% 2.78/1.80 & ( member_a @ B @ ( binary945792244etOf_a @ T3 ) )
% 2.78/1.80 & ( ( H2 @ A )
% 2.78/1.80 = ( H2 @ B ) ) )
% 2.78/1.80 => ( A = B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_39_l__scope,axiom,
% 2.78/1.80 member_a @ l @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ x @ bot_bot_set_a ) @ ( binary945792244etOf_a @ t1 ) ) @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_40_wrm__set1,axiom,
% 2.78/1.80 ! [T: binary1439146945Tree_a,H: a > int] :
% 2.78/1.80 ( ( ( T != binary476621312_Tip_a )
% 2.78/1.80 & ( binary1721989714Tree_a @ H @ T ) )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ H @ T ) ) @ ( binary945792244etOf_a @ T ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_41_remove_Osimps_I1_J,axiom,
% 2.78/1.80 ! [H: a > int,E: a] :
% 2.78/1.80 ( ( binary1804682569move_a @ H @ E @ binary476621312_Tip_a )
% 2.78/1.80 = binary476621312_Tip_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_42_mem__Collect__eq,axiom,
% 2.78/1.80 ! [A2: a,P: a > $o] :
% 2.78/1.80 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 2.78/1.80 = ( P @ A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_43_Collect__mem__eq,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
% 2.78/1.80 = A3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_44_Collect__cong,axiom,
% 2.78/1.80 ! [P: a > $o,Q: a > $o] :
% 2.78/1.80 ( ! [X3: a] :
% 2.78/1.80 ( ( P @ X3 )
% 2.78/1.80 = ( Q @ X3 ) )
% 2.78/1.80 => ( ( collect_a @ P )
% 2.78/1.80 = ( collect_a @ Q ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_45_linorder__neqE__linordered__idom,axiom,
% 2.78/1.80 ! [X: int,Y: int] :
% 2.78/1.80 ( ( X != Y )
% 2.78/1.80 => ( ~ ( ord_less_int @ X @ Y )
% 2.78/1.80 => ( ord_less_int @ Y @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_46_complete__interval,axiom,
% 2.78/1.80 ! [A2: int,B2: int,P: int > $o] :
% 2.78/1.80 ( ( ord_less_int @ A2 @ B2 )
% 2.78/1.80 => ( ( P @ A2 )
% 2.78/1.80 => ( ~ ( P @ B2 )
% 2.78/1.80 => ? [C: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ A2 @ C )
% 2.78/1.80 & ( ord_less_eq_int @ C @ B2 )
% 2.78/1.80 & ! [X2: int] :
% 2.78/1.80 ( ( ( ord_less_eq_int @ A2 @ X2 )
% 2.78/1.80 & ( ord_less_int @ X2 @ C ) )
% 2.78/1.80 => ( P @ X2 ) )
% 2.78/1.80 & ! [D: int] :
% 2.78/1.80 ( ! [X3: int] :
% 2.78/1.80 ( ( ( ord_less_eq_int @ A2 @ X3 )
% 2.78/1.80 & ( ord_less_int @ X3 @ D ) )
% 2.78/1.80 => ( P @ X3 ) )
% 2.78/1.80 => ( ord_less_eq_int @ D @ C ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_47_setOf_Osimps_I2_J,axiom,
% 2.78/1.80 ! [T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
% 2.78/1.80 ( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
% 2.78/1.80 = ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_48_minf_I8_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_49_minf_I6_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ X2 @ Z )
% 2.78/1.80 => ( ord_less_eq_int @ X2 @ T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_50_pinf_I8_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ( ord_less_eq_int @ T @ X2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_51_pinf_I6_J,axiom,
% 2.78/1.80 ! [T: int] :
% 2.78/1.80 ? [Z: int] :
% 2.78/1.80 ! [X2: int] :
% 2.78/1.80 ( ( ord_less_int @ Z @ X2 )
% 2.78/1.80 => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_52_sorted__distinct,axiom,
% 2.78/1.80 ! [H: a > int,A2: a,B2: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A2 @ B2 @ T ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_53_setOf_Osimps_I1_J,axiom,
% 2.78/1.80 ( ( binary945792244etOf_a @ binary476621312_Tip_a )
% 2.78/1.80 = bot_bot_set_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_54_singleton__insert__inj__eq_H,axiom,
% 2.78/1.80 ! [A2: a,A3: set_a,B2: a] :
% 2.78/1.80 ( ( ( insert_a @ A2 @ A3 )
% 2.78/1.80 = ( insert_a @ B2 @ bot_bot_set_a ) )
% 2.78/1.80 = ( ( A2 = B2 )
% 2.78/1.80 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_55_singleton__insert__inj__eq,axiom,
% 2.78/1.80 ! [B2: a,A2: a,A3: set_a] :
% 2.78/1.80 ( ( ( insert_a @ B2 @ bot_bot_set_a )
% 2.78/1.80 = ( insert_a @ A2 @ A3 ) )
% 2.78/1.80 = ( ( A2 = B2 )
% 2.78/1.80 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_56_singleton__conv2,axiom,
% 2.78/1.80 ! [A2: a] :
% 2.78/1.80 ( ( collect_a
% 2.78/1.80 @ ( ^ [Y2: a,Z3: a] : ( Y2 = Z3 )
% 2.78/1.80 @ A2 ) )
% 2.78/1.80 = ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_57_singleton__conv,axiom,
% 2.78/1.80 ! [A2: a] :
% 2.78/1.80 ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] : ( X4 = A2 ) )
% 2.78/1.80 = ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_58_Un__insert__right,axiom,
% 2.78/1.80 ! [A3: set_a,A2: a,B3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A3 @ ( insert_a @ A2 @ B3 ) )
% 2.78/1.80 = ( insert_a @ A2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_59_Un__insert__left,axiom,
% 2.78/1.80 ! [A2: a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
% 2.78/1.80 = ( insert_a @ A2 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_60_Un__subset__iff,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
% 2.78/1.80 = ( ( ord_less_eq_set_a @ A3 @ C2 )
% 2.78/1.80 & ( ord_less_eq_set_a @ B3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_61_Un__empty,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 2.78/1.80 = bot_bot_set_a )
% 2.78/1.80 = ( ( A3 = bot_bot_set_a )
% 2.78/1.80 & ( B3 = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_62_empty__iff,axiom,
% 2.78/1.80 ! [C3: a] :
% 2.78/1.80 ~ ( member_a @ C3 @ bot_bot_set_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_63_all__not__in__conv,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ( ! [X4: a] :
% 2.78/1.80 ~ ( member_a @ X4 @ A3 ) )
% 2.78/1.80 = ( A3 = bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_64_Collect__empty__eq,axiom,
% 2.78/1.80 ! [P: a > $o] :
% 2.78/1.80 ( ( ( collect_a @ P )
% 2.78/1.80 = bot_bot_set_a )
% 2.78/1.80 = ( ! [X4: a] :
% 2.78/1.80 ~ ( P @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_65_empty__Collect__eq,axiom,
% 2.78/1.80 ! [P: a > $o] :
% 2.78/1.80 ( ( bot_bot_set_a
% 2.78/1.80 = ( collect_a @ P ) )
% 2.78/1.80 = ( ! [X4: a] :
% 2.78/1.80 ~ ( P @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_66_subsetI,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ! [X3: a] :
% 2.78/1.80 ( ( member_a @ X3 @ A3 )
% 2.78/1.80 => ( member_a @ X3 @ B3 ) )
% 2.78/1.80 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_67_subset__antisym,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B3 @ A3 )
% 2.78/1.80 => ( A3 = B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_68_psubsetI,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( A3 != B3 )
% 2.78/1.80 => ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_69_insertCI,axiom,
% 2.78/1.80 ! [A2: a,B3: set_a,B2: a] :
% 2.78/1.80 ( ( ~ ( member_a @ A2 @ B3 )
% 2.78/1.80 => ( A2 = B2 ) )
% 2.78/1.80 => ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_70_insert__iff,axiom,
% 2.78/1.80 ! [A2: a,B2: a,A3: set_a] :
% 2.78/1.80 ( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
% 2.78/1.80 = ( ( A2 = B2 )
% 2.78/1.80 | ( member_a @ A2 @ A3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_71_insert__absorb2,axiom,
% 2.78/1.80 ! [X: a,A3: set_a] :
% 2.78/1.80 ( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
% 2.78/1.80 = ( insert_a @ X @ A3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_72_UnCI,axiom,
% 2.78/1.80 ! [C3: a,B3: set_a,A3: set_a] :
% 2.78/1.80 ( ( ~ ( member_a @ C3 @ B3 )
% 2.78/1.80 => ( member_a @ C3 @ A3 ) )
% 2.78/1.80 => ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_73_Un__iff,axiom,
% 2.78/1.80 ! [C3: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 = ( ( member_a @ C3 @ A3 )
% 2.78/1.80 | ( member_a @ C3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_74_subset__empty,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 2.78/1.80 = ( A3 = bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_75_empty__subsetI,axiom,
% 2.78/1.80 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_76_singletonI,axiom,
% 2.78/1.80 ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_77_insert__subset,axiom,
% 2.78/1.80 ! [X: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B3 )
% 2.78/1.80 = ( ( member_a @ X @ B3 )
% 2.78/1.80 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_78_less__eq__set__def,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ord_less_eq_a_o
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ A4 )
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_79_emptyE,axiom,
% 2.78/1.80 ! [A2: a] :
% 2.78/1.80 ~ ( member_a @ A2 @ bot_bot_set_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_80_equals0D,axiom,
% 2.78/1.80 ! [A3: set_a,A2: a] :
% 2.78/1.80 ( ( A3 = bot_bot_set_a )
% 2.78/1.80 => ~ ( member_a @ A2 @ A3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_81_equals0I,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ! [Y3: a] :
% 2.78/1.80 ~ ( member_a @ Y3 @ A3 )
% 2.78/1.80 => ( A3 = bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_82_ex__in__conv,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
% 2.78/1.80 = ( A3 != bot_bot_set_a ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_83_not__psubset__empty,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_84_in__mono,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,X: a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( member_a @ X @ A3 )
% 2.78/1.80 => ( member_a @ X @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_85_subsetD,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C3: a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( member_a @ C3 @ A3 )
% 2.78/1.80 => ( member_a @ C3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_86_equalityE,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( A3 = B3 )
% 2.78/1.80 => ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_87_subset__eq,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ! [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ A4 )
% 2.78/1.80 => ( member_a @ X4 @ B4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_88_equalityD1,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( A3 = B3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_89_equalityD2,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( A3 = B3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_90_subset__iff,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ! [T3: a] :
% 2.78/1.80 ( ( member_a @ T3 @ A4 )
% 2.78/1.80 => ( member_a @ T3 @ B4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_91_subset__refl,axiom,
% 2.78/1.80 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_92_Collect__mono,axiom,
% 2.78/1.80 ! [P: a > $o,Q: a > $o] :
% 2.78/1.80 ( ! [X3: a] :
% 2.78/1.80 ( ( P @ X3 )
% 2.78/1.80 => ( Q @ X3 ) )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_93_subset__trans,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_94_set__eq__subset,axiom,
% 2.78/1.80 ( ( ^ [Y2: set_a,Z3: set_a] : ( Y2 = Z3 ) )
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 2.78/1.80 & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_95_Collect__mono__iff,axiom,
% 2.78/1.80 ! [P: a > $o,Q: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 2.78/1.80 = ( ! [X4: a] :
% 2.78/1.80 ( ( P @ X4 )
% 2.78/1.80 => ( Q @ X4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_96_subset__iff__psubset__eq,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ A4 @ B4 )
% 2.78/1.80 | ( A4 = B4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_97_subset__psubset__trans,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( ord_less_set_a @ B3 @ C2 )
% 2.78/1.80 => ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_98_subset__not__subset__eq,axiom,
% 2.78/1.80 ( ord_less_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 2.78/1.80 & ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_99_psubset__subset__trans,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 2.78/1.80 => ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_100_psubset__imp__subset,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_101_psubset__eq,axiom,
% 2.78/1.80 ( ord_less_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A4 @ B4 )
% 2.78/1.80 & ( A4 != B4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_102_psubsetE,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ A3 @ B3 )
% 2.78/1.80 => ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_103_insertE,axiom,
% 2.78/1.80 ! [A2: a,B2: a,A3: set_a] :
% 2.78/1.80 ( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
% 2.78/1.80 => ( ( A2 != B2 )
% 2.78/1.80 => ( member_a @ A2 @ A3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_104_insertI1,axiom,
% 2.78/1.80 ! [A2: a,B3: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_105_insertI2,axiom,
% 2.78/1.80 ! [A2: a,B3: set_a,B2: a] :
% 2.78/1.80 ( ( member_a @ A2 @ B3 )
% 2.78/1.80 => ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_106_Set_Oset__insert,axiom,
% 2.78/1.80 ! [X: a,A3: set_a] :
% 2.78/1.80 ( ( member_a @ X @ A3 )
% 2.78/1.80 => ~ ! [B5: set_a] :
% 2.78/1.80 ( ( A3
% 2.78/1.80 = ( insert_a @ X @ B5 ) )
% 2.78/1.80 => ( member_a @ X @ B5 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_107_insert__ident,axiom,
% 2.78/1.80 ! [X: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ~ ( member_a @ X @ A3 )
% 2.78/1.80 => ( ~ ( member_a @ X @ B3 )
% 2.78/1.80 => ( ( ( insert_a @ X @ A3 )
% 2.78/1.80 = ( insert_a @ X @ B3 ) )
% 2.78/1.80 = ( A3 = B3 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_108_insert__absorb,axiom,
% 2.78/1.80 ! [A2: a,A3: set_a] :
% 2.78/1.80 ( ( member_a @ A2 @ A3 )
% 2.78/1.80 => ( ( insert_a @ A2 @ A3 )
% 2.78/1.80 = A3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_109_insert__eq__iff,axiom,
% 2.78/1.80 ! [A2: a,A3: set_a,B2: a,B3: set_a] :
% 2.78/1.80 ( ~ ( member_a @ A2 @ A3 )
% 2.78/1.80 => ( ~ ( member_a @ B2 @ B3 )
% 2.78/1.80 => ( ( ( insert_a @ A2 @ A3 )
% 2.78/1.80 = ( insert_a @ B2 @ B3 ) )
% 2.78/1.80 = ( ( ( A2 = B2 )
% 2.78/1.80 => ( A3 = B3 ) )
% 2.78/1.80 & ( ( A2 != B2 )
% 2.78/1.80 => ? [C4: set_a] :
% 2.78/1.80 ( ( A3
% 2.78/1.80 = ( insert_a @ B2 @ C4 ) )
% 2.78/1.80 & ~ ( member_a @ B2 @ C4 )
% 2.78/1.80 & ( B3
% 2.78/1.80 = ( insert_a @ A2 @ C4 ) )
% 2.78/1.80 & ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_110_insert__commute,axiom,
% 2.78/1.80 ! [X: a,Y: a,A3: set_a] :
% 2.78/1.80 ( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
% 2.78/1.80 = ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_111_mk__disjoint__insert,axiom,
% 2.78/1.80 ! [A2: a,A3: set_a] :
% 2.78/1.80 ( ( member_a @ A2 @ A3 )
% 2.78/1.80 => ? [B5: set_a] :
% 2.78/1.80 ( ( A3
% 2.78/1.80 = ( insert_a @ A2 @ B5 ) )
% 2.78/1.80 & ~ ( member_a @ A2 @ B5 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_112_UnE,axiom,
% 2.78/1.80 ! [C3: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 => ( ~ ( member_a @ C3 @ A3 )
% 2.78/1.80 => ( member_a @ C3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_113_UnI1,axiom,
% 2.78/1.80 ! [C3: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( member_a @ C3 @ A3 )
% 2.78/1.80 => ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_114_UnI2,axiom,
% 2.78/1.80 ! [C3: a,B3: set_a,A3: set_a] :
% 2.78/1.80 ( ( member_a @ C3 @ B3 )
% 2.78/1.80 => ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_115_bex__Un,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,P: a > $o] :
% 2.78/1.80 ( ( ? [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 & ( P @ X4 ) ) )
% 2.78/1.80 = ( ? [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ A3 )
% 2.78/1.80 & ( P @ X4 ) )
% 2.78/1.80 | ? [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ B3 )
% 2.78/1.80 & ( P @ X4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_116_ball__Un,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,P: a > $o] :
% 2.78/1.80 ( ( ! [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 => ( P @ X4 ) ) )
% 2.78/1.80 = ( ! [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ A3 )
% 2.78/1.80 => ( P @ X4 ) )
% 2.78/1.80 & ! [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ B3 )
% 2.78/1.80 => ( P @ X4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_117_Un__assoc,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
% 2.78/1.80 = ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_118_Un__absorb,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A3 @ A3 )
% 2.78/1.80 = A3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_119_Un__commute,axiom,
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_120_Un__left__absorb,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 = ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_121_Un__left__commute,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) )
% 2.78/1.80 = ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_122_empty__def,axiom,
% 2.78/1.80 ( bot_bot_set_a
% 2.78/1.80 = ( collect_a
% 2.78/1.80 @ ^ [X4: a] : $false ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_123_Collect__subset,axiom,
% 2.78/1.80 ! [A3: set_a,P: a > $o] :
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 @ ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ A3 )
% 2.78/1.80 & ( P @ X4 ) ) )
% 2.78/1.80 @ A3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_124_insert__compr,axiom,
% 2.78/1.80 ( insert_a
% 2.78/1.80 = ( ^ [A: a,B4: set_a] :
% 2.78/1.80 ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( X4 = A )
% 2.78/1.80 | ( member_a @ X4 @ B4 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_125_insert__Collect,axiom,
% 2.78/1.80 ! [A2: a,P: a > $o] :
% 2.78/1.80 ( ( insert_a @ A2 @ ( collect_a @ P ) )
% 2.78/1.80 = ( collect_a
% 2.78/1.80 @ ^ [U: a] :
% 2.78/1.80 ( ( U != A2 )
% 2.78/1.80 => ( P @ U ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_126_Un__def,axiom,
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( member_a @ X4 @ A4 )
% 2.78/1.80 | ( member_a @ X4 @ B4 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_127_Collect__disj__eq,axiom,
% 2.78/1.80 ! [P: a > $o,Q: a > $o] :
% 2.78/1.80 ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( P @ X4 )
% 2.78/1.80 | ( Q @ X4 ) ) )
% 2.78/1.80 = ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_128_singletonD,axiom,
% 2.78/1.80 ! [B2: a,A2: a] :
% 2.78/1.80 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 2.78/1.80 => ( B2 = A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_129_singleton__iff,axiom,
% 2.78/1.80 ! [B2: a,A2: a] :
% 2.78/1.80 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 2.78/1.80 = ( B2 = A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_130_doubleton__eq__iff,axiom,
% 2.78/1.80 ! [A2: a,B2: a,C3: a,D2: a] :
% 2.78/1.80 ( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
% 2.78/1.80 = ( insert_a @ C3 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
% 2.78/1.80 = ( ( ( A2 = C3 )
% 2.78/1.80 & ( B2 = D2 ) )
% 2.78/1.80 | ( ( A2 = D2 )
% 2.78/1.80 & ( B2 = C3 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_131_insert__not__empty,axiom,
% 2.78/1.80 ! [A2: a,A3: set_a] :
% 2.78/1.80 ( ( insert_a @ A2 @ A3 )
% 2.78/1.80 != bot_bot_set_a ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_132_singleton__inject,axiom,
% 2.78/1.80 ! [A2: a,B2: a] :
% 2.78/1.80 ( ( ( insert_a @ A2 @ bot_bot_set_a )
% 2.78/1.80 = ( insert_a @ B2 @ bot_bot_set_a ) )
% 2.78/1.80 => ( A2 = B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_133_insert__mono,axiom,
% 2.78/1.80 ! [C2: set_a,D3: set_a,A2: a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ C2 @ D3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_134_subset__insert,axiom,
% 2.78/1.80 ! [X: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ~ ( member_a @ X @ A3 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B3 ) )
% 2.78/1.80 = ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_135_subset__insertI,axiom,
% 2.78/1.80 ! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A2 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_136_subset__insertI2,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,B2: a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_137_Un__empty__left,axiom,
% 2.78/1.80 ! [B3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
% 2.78/1.80 = B3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_138_Un__empty__right,axiom,
% 2.78/1.80 ! [A3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
% 2.78/1.80 = A3 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_139_Un__mono,axiom,
% 2.78/1.80 ! [A3: set_a,C2: set_a,B3: set_a,D3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ C2 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B3 @ D3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ C2 @ D3 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_140_Un__least,axiom,
% 2.78/1.80 ! [A3: set_a,C2: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ C2 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B3 @ C2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_141_Un__upper1,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_142_Un__upper2,axiom,
% 2.78/1.80 ! [B3: set_a,A3: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_143_Un__absorb1,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( sup_sup_set_a @ A3 @ B3 )
% 2.78/1.80 = B3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_144_Un__absorb2,axiom,
% 2.78/1.80 ! [B3: set_a,A3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ B3 @ A3 )
% 2.78/1.80 => ( ( sup_sup_set_a @ A3 @ B3 )
% 2.78/1.80 = A3 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_145_subset__UnE,axiom,
% 2.78/1.80 ! [C2: set_a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 => ~ ! [A5: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A5 @ A3 )
% 2.78/1.80 => ! [B6: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ B6 @ B3 )
% 2.78/1.80 => ( C2
% 2.78/1.80 != ( sup_sup_set_a @ A5 @ B6 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_146_subset__Un__eq,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A4 @ B4 )
% 2.78/1.80 = B4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_147_Collect__conv__if,axiom,
% 2.78/1.80 ! [P: a > $o,A2: a] :
% 2.78/1.80 ( ( ( P @ A2 )
% 2.78/1.80 => ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( X4 = A2 )
% 2.78/1.80 & ( P @ X4 ) ) )
% 2.78/1.80 = ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 2.78/1.80 & ( ~ ( P @ A2 )
% 2.78/1.80 => ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( X4 = A2 )
% 2.78/1.80 & ( P @ X4 ) ) )
% 2.78/1.80 = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_148_Collect__conv__if2,axiom,
% 2.78/1.80 ! [P: a > $o,A2: a] :
% 2.78/1.80 ( ( ( P @ A2 )
% 2.78/1.80 => ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( A2 = X4 )
% 2.78/1.80 & ( P @ X4 ) ) )
% 2.78/1.80 = ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 2.78/1.80 & ( ~ ( P @ A2 )
% 2.78/1.80 => ( ( collect_a
% 2.78/1.80 @ ^ [X4: a] :
% 2.78/1.80 ( ( A2 = X4 )
% 2.78/1.80 & ( P @ X4 ) ) )
% 2.78/1.80 = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_149_insert__def,axiom,
% 2.78/1.80 ( insert_a
% 2.78/1.80 = ( ^ [A: a] :
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 @ ( collect_a
% 2.78/1.80 @ ^ [X4: a] : ( X4 = A ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_150_subset__singletonD,axiom,
% 2.78/1.80 ! [A3: set_a,X: a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 => ( ( A3 = bot_bot_set_a )
% 2.78/1.80 | ( A3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_151_subset__singleton__iff,axiom,
% 2.78/1.80 ! [X5: set_a,A2: a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 2.78/1.80 = ( ( X5 = bot_bot_set_a )
% 2.78/1.80 | ( X5
% 2.78/1.80 = ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_152_insert__is__Un,axiom,
% 2.78/1.80 ( insert_a
% 2.78/1.80 = ( ^ [A: a] : ( sup_sup_set_a @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_153_Un__singleton__iff,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,X: a] :
% 2.78/1.80 ( ( ( sup_sup_set_a @ A3 @ B3 )
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 = ( ( ( A3 = bot_bot_set_a )
% 2.78/1.80 & ( B3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) ) )
% 2.78/1.80 | ( ( A3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 & ( B3 = bot_bot_set_a ) )
% 2.78/1.80 | ( ( A3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 & ( B3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_154_singleton__Un__iff,axiom,
% 2.78/1.80 ! [X: a,A3: set_a,B3: set_a] :
% 2.78/1.80 ( ( ( insert_a @ X @ bot_bot_set_a )
% 2.78/1.80 = ( sup_sup_set_a @ A3 @ B3 ) )
% 2.78/1.80 = ( ( ( A3 = bot_bot_set_a )
% 2.78/1.80 & ( B3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) ) )
% 2.78/1.80 | ( ( A3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 & ( B3 = bot_bot_set_a ) )
% 2.78/1.80 | ( ( A3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 & ( B3
% 2.78/1.80 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_155_sup__bot__left,axiom,
% 2.78/1.80 ! [X: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ bot_bot_a_o @ X )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_156_sup__bot__left,axiom,
% 2.78/1.80 ! [X: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ bot_bot_set_a @ X )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_157_sup__bot__right,axiom,
% 2.78/1.80 ! [X: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X @ bot_bot_a_o )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_158_sup__bot__right,axiom,
% 2.78/1.80 ! [X: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X @ bot_bot_set_a )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_159_bot__eq__sup__iff,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] :
% 2.78/1.80 ( ( bot_bot_a_o
% 2.78/1.80 = ( sup_sup_a_o @ X @ Y ) )
% 2.78/1.80 = ( ( X = bot_bot_a_o )
% 2.78/1.80 & ( Y = bot_bot_a_o ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_160_bot__eq__sup__iff,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] :
% 2.78/1.80 ( ( bot_bot_set_a
% 2.78/1.80 = ( sup_sup_set_a @ X @ Y ) )
% 2.78/1.80 = ( ( X = bot_bot_set_a )
% 2.78/1.80 & ( Y = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_161_sup__eq__bot__iff,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] :
% 2.78/1.80 ( ( ( sup_sup_a_o @ X @ Y )
% 2.78/1.80 = bot_bot_a_o )
% 2.78/1.80 = ( ( X = bot_bot_a_o )
% 2.78/1.80 & ( Y = bot_bot_a_o ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_162_sup__eq__bot__iff,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] :
% 2.78/1.80 ( ( ( sup_sup_set_a @ X @ Y )
% 2.78/1.80 = bot_bot_set_a )
% 2.78/1.80 = ( ( X = bot_bot_set_a )
% 2.78/1.80 & ( Y = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_163_sup__bot_Oeq__neutr__iff,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ( sup_sup_a_o @ A2 @ B2 )
% 2.78/1.80 = bot_bot_a_o )
% 2.78/1.80 = ( ( A2 = bot_bot_a_o )
% 2.78/1.80 & ( B2 = bot_bot_a_o ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_164_sup__bot_Oeq__neutr__iff,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ( sup_sup_set_a @ A2 @ B2 )
% 2.78/1.80 = bot_bot_set_a )
% 2.78/1.80 = ( ( A2 = bot_bot_set_a )
% 2.78/1.80 & ( B2 = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_165_sup__bot_Oleft__neutral,axiom,
% 2.78/1.80 ! [A2: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_166_sup__bot_Oleft__neutral,axiom,
% 2.78/1.80 ! [A2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_167_sup__bot_Oneutr__eq__iff,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( bot_bot_a_o
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ B2 ) )
% 2.78/1.80 = ( ( A2 = bot_bot_a_o )
% 2.78/1.80 & ( B2 = bot_bot_a_o ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_168_sup__bot_Oneutr__eq__iff,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( bot_bot_set_a
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ B2 ) )
% 2.78/1.80 = ( ( A2 = bot_bot_set_a )
% 2.78/1.80 & ( B2 = bot_bot_set_a ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_169_sup_Oright__idem,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ B2 )
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_170_sup_Oright__idem,axiom,
% 2.78/1.80 ! [A2: int,B2: int] :
% 2.78/1.80 ( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ B2 )
% 2.78/1.80 = ( sup_sup_int @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_171_sup_Oright__idem,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_172_sup__left__idem,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
% 2.78/1.80 = ( sup_sup_a_o @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_173_sup__left__idem,axiom,
% 2.78/1.80 ! [X: int,Y: int] :
% 2.78/1.80 ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
% 2.78/1.80 = ( sup_sup_int @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_174_sup__left__idem,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
% 2.78/1.80 = ( sup_sup_set_a @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_175_sup_Oleft__idem,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ A2 @ B2 ) )
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_176_sup_Oleft__idem,axiom,
% 2.78/1.80 ! [A2: int,B2: int] :
% 2.78/1.80 ( ( sup_sup_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) )
% 2.78/1.80 = ( sup_sup_int @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_177_sup_Oleft__idem,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_178_sup__idem,axiom,
% 2.78/1.80 ! [X: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X @ X )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_179_sup__idem,axiom,
% 2.78/1.80 ! [X: int] :
% 2.78/1.80 ( ( sup_sup_int @ X @ X )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_180_sup__idem,axiom,
% 2.78/1.80 ! [X: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X @ X )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_181_sup_Oidem,axiom,
% 2.78/1.80 ! [A2: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ A2 @ A2 )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_182_sup_Oidem,axiom,
% 2.78/1.80 ! [A2: int] :
% 2.78/1.80 ( ( sup_sup_int @ A2 @ A2 )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_183_sup_Oidem,axiom,
% 2.78/1.80 ! [A2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A2 @ A2 )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_184_sup__apply,axiom,
% 2.78/1.80 ( sup_sup_a_o
% 2.78/1.80 = ( ^ [F: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_185_sup_Obounded__iff,axiom,
% 2.78/1.80 ! [B2: int,C3: int,A2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
% 2.78/1.80 = ( ( ord_less_eq_int @ B2 @ A2 )
% 2.78/1.80 & ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_186_sup_Obounded__iff,axiom,
% 2.78/1.80 ! [B2: a > $o,C3: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
% 2.78/1.80 = ( ( ord_less_eq_a_o @ B2 @ A2 )
% 2.78/1.80 & ( ord_less_eq_a_o @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_187_sup_Obounded__iff,axiom,
% 2.78/1.80 ! [B2: set_a,C3: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
% 2.78/1.80 = ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.78/1.80 & ( ord_less_eq_set_a @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_188_le__sup__iff,axiom,
% 2.78/1.80 ! [X: int,Y: int,Z4: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( ( ord_less_eq_int @ X @ Z4 )
% 2.78/1.80 & ( ord_less_eq_int @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_189_le__sup__iff,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( ( ord_less_eq_a_o @ X @ Z4 )
% 2.78/1.80 & ( ord_less_eq_a_o @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_190_le__sup__iff,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a,Z4: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( ( ord_less_eq_set_a @ X @ Z4 )
% 2.78/1.80 & ( ord_less_eq_set_a @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_191_sup__bot_Oright__neutral,axiom,
% 2.78/1.80 ! [A2: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_192_sup__bot_Oright__neutral,axiom,
% 2.78/1.80 ! [A2: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
% 2.78/1.80 = A2 ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_193_less__set__def,axiom,
% 2.78/1.80 ( ord_less_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( ord_less_a_o
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ A4 )
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_194_psubset__trans,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( ord_less_set_a @ B3 @ C2 )
% 2.78/1.80 => ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_195_psubsetD,axiom,
% 2.78/1.80 ! [A3: set_a,B3: set_a,C3: a] :
% 2.78/1.80 ( ( ord_less_set_a @ A3 @ B3 )
% 2.78/1.80 => ( ( member_a @ C3 @ A3 )
% 2.78/1.80 => ( member_a @ C3 @ B3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_196_bot__set__def,axiom,
% 2.78/1.80 ( bot_bot_set_a
% 2.78/1.80 = ( collect_a @ bot_bot_a_o ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_197_sup__set__def,axiom,
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 = ( ^ [A4: set_a,B4: set_a] :
% 2.78/1.80 ( collect_a
% 2.78/1.80 @ ( sup_sup_a_o
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ A4 )
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_198_sup__left__commute,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) )
% 2.78/1.80 = ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_199_sup__left__commute,axiom,
% 2.78/1.80 ! [X: int,Y: int,Z4: int] :
% 2.78/1.80 ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) )
% 2.78/1.80 = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_200_sup__left__commute,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a,Z4: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
% 2.78/1.80 = ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_201_sup_Oleft__commute,axiom,
% 2.78/1.80 ! [B2: a > $o,A2: a > $o,C3: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ B2 @ ( sup_sup_a_o @ A2 @ C3 ) )
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_202_sup_Oleft__commute,axiom,
% 2.78/1.80 ! [B2: int,A2: int,C3: int] :
% 2.78/1.80 ( ( sup_sup_int @ B2 @ ( sup_sup_int @ A2 @ C3 ) )
% 2.78/1.80 = ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_203_sup_Oleft__commute,axiom,
% 2.78/1.80 ! [B2: set_a,A2: set_a,C3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C3 ) )
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_204_sup__commute,axiom,
% 2.78/1.80 ( sup_sup_a_o
% 2.78/1.80 = ( ^ [X4: a > $o,Y4: a > $o] : ( sup_sup_a_o @ Y4 @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_205_sup__commute,axiom,
% 2.78/1.80 ( sup_sup_int
% 2.78/1.80 = ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_206_sup__commute,axiom,
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 = ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_207_sup_Ocommute,axiom,
% 2.78/1.80 ( sup_sup_a_o
% 2.78/1.80 = ( ^ [A: a > $o,B: a > $o] : ( sup_sup_a_o @ B @ A ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_208_sup_Ocommute,axiom,
% 2.78/1.80 ( sup_sup_int
% 2.78/1.80 = ( ^ [A: int,B: int] : ( sup_sup_int @ B @ A ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_209_sup_Ocommute,axiom,
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 = ( ^ [A: set_a,B: set_a] : ( sup_sup_set_a @ B @ A ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_210_sup__assoc,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_211_sup__assoc,axiom,
% 2.78/1.80 ! [X: int,Y: int,Z4: int] :
% 2.78/1.80 ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_212_sup__assoc,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a,Z4: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_213_sup_Oassoc,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o,C3: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ C3 )
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_214_sup_Oassoc,axiom,
% 2.78/1.80 ! [A2: int,B2: int,C3: int] :
% 2.78/1.80 ( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ C3 )
% 2.78/1.80 = ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_215_sup_Oassoc,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a,C3: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C3 )
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_216_boolean__algebra__cancel_Osup2,axiom,
% 2.78/1.80 ! [B3: a > $o,K: a > $o,B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( B3
% 2.78/1.80 = ( sup_sup_a_o @ K @ B2 ) )
% 2.78/1.80 => ( ( sup_sup_a_o @ A2 @ B3 )
% 2.78/1.80 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_217_boolean__algebra__cancel_Osup2,axiom,
% 2.78/1.80 ! [B3: int,K: int,B2: int,A2: int] :
% 2.78/1.80 ( ( B3
% 2.78/1.80 = ( sup_sup_int @ K @ B2 ) )
% 2.78/1.80 => ( ( sup_sup_int @ A2 @ B3 )
% 2.78/1.80 = ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_218_boolean__algebra__cancel_Osup2,axiom,
% 2.78/1.80 ! [B3: set_a,K: set_a,B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( B3
% 2.78/1.80 = ( sup_sup_set_a @ K @ B2 ) )
% 2.78/1.80 => ( ( sup_sup_set_a @ A2 @ B3 )
% 2.78/1.80 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_219_boolean__algebra__cancel_Osup1,axiom,
% 2.78/1.80 ! [A3: a > $o,K: a > $o,A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( A3
% 2.78/1.80 = ( sup_sup_a_o @ K @ A2 ) )
% 2.78/1.80 => ( ( sup_sup_a_o @ A3 @ B2 )
% 2.78/1.80 = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_220_boolean__algebra__cancel_Osup1,axiom,
% 2.78/1.80 ! [A3: int,K: int,A2: int,B2: int] :
% 2.78/1.80 ( ( A3
% 2.78/1.80 = ( sup_sup_int @ K @ A2 ) )
% 2.78/1.80 => ( ( sup_sup_int @ A3 @ B2 )
% 2.78/1.80 = ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_221_boolean__algebra__cancel_Osup1,axiom,
% 2.78/1.80 ! [A3: set_a,K: set_a,A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( A3
% 2.78/1.80 = ( sup_sup_set_a @ K @ A2 ) )
% 2.78/1.80 => ( ( sup_sup_set_a @ A3 @ B2 )
% 2.78/1.80 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_222_sup__fun__def,axiom,
% 2.78/1.80 ( sup_sup_a_o
% 2.78/1.80 = ( ^ [F: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_223_inf__sup__aci_I5_J,axiom,
% 2.78/1.80 ( sup_sup_a_o
% 2.78/1.80 = ( ^ [X4: a > $o,Y4: a > $o] : ( sup_sup_a_o @ Y4 @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_224_inf__sup__aci_I5_J,axiom,
% 2.78/1.80 ( sup_sup_int
% 2.78/1.80 = ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_225_inf__sup__aci_I5_J,axiom,
% 2.78/1.80 ( sup_sup_set_a
% 2.78/1.80 = ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_226_inf__sup__aci_I6_J,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_227_inf__sup__aci_I6_J,axiom,
% 2.78/1.80 ! [X: int,Y: int,Z4: int] :
% 2.78/1.80 ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_228_inf__sup__aci_I6_J,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a,Z4: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
% 2.78/1.80 = ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_229_inf__sup__aci_I7_J,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o,Z4: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) )
% 2.78/1.80 = ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_230_inf__sup__aci_I7_J,axiom,
% 2.78/1.80 ! [X: int,Y: int,Z4: int] :
% 2.78/1.80 ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) )
% 2.78/1.80 = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_231_inf__sup__aci_I7_J,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a,Z4: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
% 2.78/1.80 = ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_232_inf__sup__aci_I8_J,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
% 2.78/1.80 = ( sup_sup_a_o @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_233_inf__sup__aci_I8_J,axiom,
% 2.78/1.80 ! [X: int,Y: int] :
% 2.78/1.80 ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
% 2.78/1.80 = ( sup_sup_int @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_234_inf__sup__aci_I8_J,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
% 2.78/1.80 = ( sup_sup_set_a @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_235_sup_OcoboundedI2,axiom,
% 2.78/1.80 ! [C3: int,B2: int,A2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ C3 @ B2 )
% 2.78/1.80 => ( ord_less_eq_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_236_sup_OcoboundedI2,axiom,
% 2.78/1.80 ! [C3: a > $o,B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ C3 @ B2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_237_sup_OcoboundedI2,axiom,
% 2.78/1.80 ! [C3: set_a,B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ C3 @ B2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_238_sup_OcoboundedI1,axiom,
% 2.78/1.80 ! [C3: int,A2: int,B2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_eq_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_239_sup_OcoboundedI1,axiom,
% 2.78/1.80 ! [C3: a > $o,A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_240_sup_OcoboundedI1,axiom,
% 2.78/1.80 ! [C3: set_a,A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_241_sup_Oabsorb__iff2,axiom,
% 2.78/1.80 ( ord_less_eq_int
% 2.78/1.80 = ( ^ [A: int,B: int] :
% 2.78/1.80 ( ( sup_sup_int @ A @ B )
% 2.78/1.80 = B ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_242_sup_Oabsorb__iff2,axiom,
% 2.78/1.80 ( ord_less_eq_a_o
% 2.78/1.80 = ( ^ [A: a > $o,B: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ A @ B )
% 2.78/1.80 = B ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_243_sup_Oabsorb__iff2,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [A: set_a,B: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A @ B )
% 2.78/1.80 = B ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_244_sup_Oabsorb__iff1,axiom,
% 2.78/1.80 ( ord_less_eq_int
% 2.78/1.80 = ( ^ [B: int,A: int] :
% 2.78/1.80 ( ( sup_sup_int @ A @ B )
% 2.78/1.80 = A ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_245_sup_Oabsorb__iff1,axiom,
% 2.78/1.80 ( ord_less_eq_a_o
% 2.78/1.80 = ( ^ [B: a > $o,A: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ A @ B )
% 2.78/1.80 = A ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_246_sup_Oabsorb__iff1,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [B: set_a,A: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ A @ B )
% 2.78/1.80 = A ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_247_sup_Ocobounded2,axiom,
% 2.78/1.80 ! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_248_sup_Ocobounded2,axiom,
% 2.78/1.80 ! [B2: a > $o,A2: a > $o] : ( ord_less_eq_a_o @ B2 @ ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_249_sup_Ocobounded2,axiom,
% 2.78/1.80 ! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_250_sup_Ocobounded1,axiom,
% 2.78/1.80 ! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_251_sup_Ocobounded1,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] : ( ord_less_eq_a_o @ A2 @ ( sup_sup_a_o @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_252_sup_Ocobounded1,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_253_sup_Oorder__iff,axiom,
% 2.78/1.80 ( ord_less_eq_int
% 2.78/1.80 = ( ^ [B: int,A: int] :
% 2.78/1.80 ( A
% 2.78/1.80 = ( sup_sup_int @ A @ B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_254_sup_Oorder__iff,axiom,
% 2.78/1.80 ( ord_less_eq_a_o
% 2.78/1.80 = ( ^ [B: a > $o,A: a > $o] :
% 2.78/1.80 ( A
% 2.78/1.80 = ( sup_sup_a_o @ A @ B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_255_sup_Oorder__iff,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [B: set_a,A: set_a] :
% 2.78/1.80 ( A
% 2.78/1.80 = ( sup_sup_set_a @ A @ B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_256_sup_OboundedI,axiom,
% 2.78/1.80 ! [B2: int,A2: int,C3: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ B2 @ A2 )
% 2.78/1.80 => ( ( ord_less_eq_int @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_257_sup_OboundedI,axiom,
% 2.78/1.80 ! [B2: a > $o,A2: a > $o,C3: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ B2 @ A2 )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_258_sup_OboundedI,axiom,
% 2.78/1.80 ! [B2: set_a,A2: set_a,C3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_259_sup_OboundedE,axiom,
% 2.78/1.80 ! [B2: int,C3: int,A2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
% 2.78/1.80 => ~ ( ( ord_less_eq_int @ B2 @ A2 )
% 2.78/1.80 => ~ ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_260_sup_OboundedE,axiom,
% 2.78/1.80 ! [B2: a > $o,C3: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
% 2.78/1.80 => ~ ( ( ord_less_eq_a_o @ B2 @ A2 )
% 2.78/1.80 => ~ ( ord_less_eq_a_o @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_261_sup_OboundedE,axiom,
% 2.78/1.80 ! [B2: set_a,C3: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
% 2.78/1.80 => ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.78/1.80 => ~ ( ord_less_eq_set_a @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_262_sup__absorb2,axiom,
% 2.78/1.80 ! [X: int,Y: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ X @ Y )
% 2.78/1.80 => ( ( sup_sup_int @ X @ Y )
% 2.78/1.80 = Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_263_sup__absorb2,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ X @ Y )
% 2.78/1.80 => ( ( sup_sup_a_o @ X @ Y )
% 2.78/1.80 = Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_264_sup__absorb2,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ X @ Y )
% 2.78/1.80 => ( ( sup_sup_set_a @ X @ Y )
% 2.78/1.80 = Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_265_sup__absorb1,axiom,
% 2.78/1.80 ! [Y: int,X: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ Y @ X )
% 2.78/1.80 => ( ( sup_sup_int @ X @ Y )
% 2.78/1.80 = X ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_266_sup__absorb1,axiom,
% 2.78/1.80 ! [Y: a > $o,X: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ Y @ X )
% 2.78/1.80 => ( ( sup_sup_a_o @ X @ Y )
% 2.78/1.80 = X ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_267_sup__absorb1,axiom,
% 2.78/1.80 ! [Y: set_a,X: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ Y @ X )
% 2.78/1.80 => ( ( sup_sup_set_a @ X @ Y )
% 2.78/1.80 = X ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_268_sup_Oabsorb2,axiom,
% 2.78/1.80 ! [A2: int,B2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ A2 @ B2 )
% 2.78/1.80 => ( ( sup_sup_int @ A2 @ B2 )
% 2.78/1.80 = B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_269_sup_Oabsorb2,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ A2 @ B2 )
% 2.78/1.80 => ( ( sup_sup_a_o @ A2 @ B2 )
% 2.78/1.80 = B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_270_sup_Oabsorb2,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.78/1.80 => ( ( sup_sup_set_a @ A2 @ B2 )
% 2.78/1.80 = B2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_271_sup_Oabsorb1,axiom,
% 2.78/1.80 ! [B2: int,A2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ B2 @ A2 )
% 2.78/1.80 => ( ( sup_sup_int @ A2 @ B2 )
% 2.78/1.80 = A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_272_sup_Oabsorb1,axiom,
% 2.78/1.80 ! [B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ B2 @ A2 )
% 2.78/1.80 => ( ( sup_sup_a_o @ A2 @ B2 )
% 2.78/1.80 = A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_273_sup_Oabsorb1,axiom,
% 2.78/1.80 ! [B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.78/1.80 => ( ( sup_sup_set_a @ A2 @ B2 )
% 2.78/1.80 = A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_274_sup__unique,axiom,
% 2.78/1.80 ! [F2: int > int > int,X: int,Y: int] :
% 2.78/1.80 ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
% 2.78/1.80 => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 2.78/1.80 => ( ! [X3: int,Y3: int,Z: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ Y3 @ X3 )
% 2.78/1.80 => ( ( ord_less_eq_int @ Z @ X3 )
% 2.78/1.80 => ( ord_less_eq_int @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 2.78/1.80 => ( ( sup_sup_int @ X @ Y )
% 2.78/1.80 = ( F2 @ X @ Y ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_275_sup__unique,axiom,
% 2.78/1.80 ! [F2: ( a > $o ) > ( a > $o ) > a > $o,X: a > $o,Y: a > $o] :
% 2.78/1.80 ( ! [X3: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ X3 @ ( F2 @ X3 @ Y3 ) )
% 2.78/1.80 => ( ! [X3: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 2.78/1.80 => ( ! [X3: a > $o,Y3: a > $o,Z: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ Y3 @ X3 )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ Z @ X3 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 2.78/1.80 => ( ( sup_sup_a_o @ X @ Y )
% 2.78/1.80 = ( F2 @ X @ Y ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_276_sup__unique,axiom,
% 2.78/1.80 ! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
% 2.78/1.80 ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F2 @ X3 @ Y3 ) )
% 2.78/1.80 => ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F2 @ X3 @ Y3 ) )
% 2.78/1.80 => ( ! [X3: set_a,Y3: set_a,Z: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ Y3 @ X3 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ Z @ X3 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
% 2.78/1.80 => ( ( sup_sup_set_a @ X @ Y )
% 2.78/1.80 = ( F2 @ X @ Y ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_277_sup_OorderI,axiom,
% 2.78/1.80 ! [A2: int,B2: int] :
% 2.78/1.80 ( ( A2
% 2.78/1.80 = ( sup_sup_int @ A2 @ B2 ) )
% 2.78/1.80 => ( ord_less_eq_int @ B2 @ A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_278_sup_OorderI,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( A2
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ B2 ) )
% 2.78/1.80 => ( ord_less_eq_a_o @ B2 @ A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_279_sup_OorderI,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( A2
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ B2 ) )
% 2.78/1.80 => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_280_sup_OorderE,axiom,
% 2.78/1.80 ! [B2: int,A2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ B2 @ A2 )
% 2.78/1.80 => ( A2
% 2.78/1.80 = ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_281_sup_OorderE,axiom,
% 2.78/1.80 ! [B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ B2 @ A2 )
% 2.78/1.80 => ( A2
% 2.78/1.80 = ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_282_sup_OorderE,axiom,
% 2.78/1.80 ! [B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.78/1.80 => ( A2
% 2.78/1.80 = ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_283_le__iff__sup,axiom,
% 2.78/1.80 ( ord_less_eq_int
% 2.78/1.80 = ( ^ [X4: int,Y4: int] :
% 2.78/1.80 ( ( sup_sup_int @ X4 @ Y4 )
% 2.78/1.80 = Y4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_284_le__iff__sup,axiom,
% 2.78/1.80 ( ord_less_eq_a_o
% 2.78/1.80 = ( ^ [X4: a > $o,Y4: a > $o] :
% 2.78/1.80 ( ( sup_sup_a_o @ X4 @ Y4 )
% 2.78/1.80 = Y4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_285_le__iff__sup,axiom,
% 2.78/1.80 ( ord_less_eq_set_a
% 2.78/1.80 = ( ^ [X4: set_a,Y4: set_a] :
% 2.78/1.80 ( ( sup_sup_set_a @ X4 @ Y4 )
% 2.78/1.80 = Y4 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_286_sup__least,axiom,
% 2.78/1.80 ! [Y: int,X: int,Z4: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ Y @ X )
% 2.78/1.80 => ( ( ord_less_eq_int @ Z4 @ X )
% 2.78/1.80 => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z4 ) @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_287_sup__least,axiom,
% 2.78/1.80 ! [Y: a > $o,X: a > $o,Z4: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ Y @ X )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ Z4 @ X )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ Y @ Z4 ) @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_288_sup__least,axiom,
% 2.78/1.80 ! [Y: set_a,X: set_a,Z4: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ Y @ X )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ Z4 @ X )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_289_sup__mono,axiom,
% 2.78/1.80 ! [A2: int,C3: int,B2: int,D2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ A2 @ C3 )
% 2.78/1.80 => ( ( ord_less_eq_int @ B2 @ D2 )
% 2.78/1.80 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C3 @ D2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_290_sup__mono,axiom,
% 2.78/1.80 ! [A2: a > $o,C3: a > $o,B2: a > $o,D2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ A2 @ C3 )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ B2 @ D2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ ( sup_sup_a_o @ C3 @ D2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_291_sup__mono,axiom,
% 2.78/1.80 ! [A2: set_a,C3: set_a,B2: set_a,D2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A2 @ C3 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B2 @ D2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_292_sup_Omono,axiom,
% 2.78/1.80 ! [C3: int,A2: int,D2: int,B2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ C3 @ A2 )
% 2.78/1.80 => ( ( ord_less_eq_int @ D2 @ B2 )
% 2.78/1.80 => ( ord_less_eq_int @ ( sup_sup_int @ C3 @ D2 ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_293_sup_Omono,axiom,
% 2.78/1.80 ! [C3: a > $o,A2: a > $o,D2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ C3 @ A2 )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ D2 @ B2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ C3 @ D2 ) @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_294_sup_Omono,axiom,
% 2.78/1.80 ! [C3: set_a,A2: set_a,D2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ C3 @ A2 )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ D2 @ B2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C3 @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_295_le__supI2,axiom,
% 2.78/1.80 ! [X: int,B2: int,A2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ X @ B2 )
% 2.78/1.80 => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_296_le__supI2,axiom,
% 2.78/1.80 ! [X: a > $o,B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ X @ B2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_297_le__supI2,axiom,
% 2.78/1.80 ! [X: set_a,B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ X @ B2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_298_le__supI1,axiom,
% 2.78/1.80 ! [X: int,A2: int,B2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ X @ A2 )
% 2.78/1.80 => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_299_le__supI1,axiom,
% 2.78/1.80 ! [X: a > $o,A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ X @ A2 )
% 2.78/1.80 => ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_300_le__supI1,axiom,
% 2.78/1.80 ! [X: set_a,A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ X @ A2 )
% 2.78/1.80 => ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_301_sup__ge2,axiom,
% 2.78/1.80 ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_302_sup__ge2,axiom,
% 2.78/1.80 ! [Y: a > $o,X: a > $o] : ( ord_less_eq_a_o @ Y @ ( sup_sup_a_o @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_303_sup__ge2,axiom,
% 2.78/1.80 ! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_304_sup__ge1,axiom,
% 2.78/1.80 ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_305_sup__ge1,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_306_sup__ge1,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_307_le__supI,axiom,
% 2.78/1.80 ! [A2: int,X: int,B2: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ A2 @ X )
% 2.78/1.80 => ( ( ord_less_eq_int @ B2 @ X )
% 2.78/1.80 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_308_le__supI,axiom,
% 2.78/1.80 ! [A2: a > $o,X: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ A2 @ X )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ B2 @ X )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_309_le__supI,axiom,
% 2.78/1.80 ! [A2: set_a,X: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ A2 @ X )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B2 @ X )
% 2.78/1.80 => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_310_le__supE,axiom,
% 2.78/1.80 ! [A2: int,B2: int,X: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X )
% 2.78/1.80 => ~ ( ( ord_less_eq_int @ A2 @ X )
% 2.78/1.80 => ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_311_le__supE,axiom,
% 2.78/1.80 ! [A2: a > $o,B2: a > $o,X: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ X )
% 2.78/1.80 => ~ ( ( ord_less_eq_a_o @ A2 @ X )
% 2.78/1.80 => ~ ( ord_less_eq_a_o @ B2 @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_312_le__supE,axiom,
% 2.78/1.80 ! [A2: set_a,B2: set_a,X: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
% 2.78/1.80 => ~ ( ( ord_less_eq_set_a @ A2 @ X )
% 2.78/1.80 => ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_313_inf__sup__ord_I3_J,axiom,
% 2.78/1.80 ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_314_inf__sup__ord_I3_J,axiom,
% 2.78/1.80 ! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_315_inf__sup__ord_I3_J,axiom,
% 2.78/1.80 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_316_inf__sup__ord_I4_J,axiom,
% 2.78/1.80 ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_317_inf__sup__ord_I4_J,axiom,
% 2.78/1.80 ! [Y: a > $o,X: a > $o] : ( ord_less_eq_a_o @ Y @ ( sup_sup_a_o @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_318_inf__sup__ord_I4_J,axiom,
% 2.78/1.80 ! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_319_less__supI1,axiom,
% 2.78/1.80 ! [X: a > $o,A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_a_o @ X @ A2 )
% 2.78/1.80 => ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_320_less__supI1,axiom,
% 2.78/1.80 ! [X: set_a,A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ X @ A2 )
% 2.78/1.80 => ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_321_less__supI1,axiom,
% 2.78/1.80 ! [X: int,A2: int,B2: int] :
% 2.78/1.80 ( ( ord_less_int @ X @ A2 )
% 2.78/1.80 => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_322_less__supI2,axiom,
% 2.78/1.80 ! [X: a > $o,B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_a_o @ X @ B2 )
% 2.78/1.80 => ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_323_less__supI2,axiom,
% 2.78/1.80 ! [X: set_a,B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ X @ B2 )
% 2.78/1.80 => ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_324_less__supI2,axiom,
% 2.78/1.80 ! [X: int,B2: int,A2: int] :
% 2.78/1.80 ( ( ord_less_int @ X @ B2 )
% 2.78/1.80 => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_325_sup_Ostrict__boundedE,axiom,
% 2.78/1.80 ! [B2: a > $o,C3: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
% 2.78/1.80 => ~ ( ( ord_less_a_o @ B2 @ A2 )
% 2.78/1.80 => ~ ( ord_less_a_o @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_326_sup_Ostrict__boundedE,axiom,
% 2.78/1.80 ! [B2: set_a,C3: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
% 2.78/1.80 => ~ ( ( ord_less_set_a @ B2 @ A2 )
% 2.78/1.80 => ~ ( ord_less_set_a @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_327_sup_Ostrict__boundedE,axiom,
% 2.78/1.80 ! [B2: int,C3: int,A2: int] :
% 2.78/1.80 ( ( ord_less_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
% 2.78/1.80 => ~ ( ( ord_less_int @ B2 @ A2 )
% 2.78/1.80 => ~ ( ord_less_int @ C3 @ A2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_328_sup_Ostrict__order__iff,axiom,
% 2.78/1.80 ( ord_less_a_o
% 2.78/1.80 = ( ^ [B: a > $o,A: a > $o] :
% 2.78/1.80 ( ( A
% 2.78/1.80 = ( sup_sup_a_o @ A @ B ) )
% 2.78/1.80 & ( A != B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_329_sup_Ostrict__order__iff,axiom,
% 2.78/1.80 ( ord_less_set_a
% 2.78/1.80 = ( ^ [B: set_a,A: set_a] :
% 2.78/1.80 ( ( A
% 2.78/1.80 = ( sup_sup_set_a @ A @ B ) )
% 2.78/1.80 & ( A != B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_330_sup_Ostrict__order__iff,axiom,
% 2.78/1.80 ( ord_less_int
% 2.78/1.80 = ( ^ [B: int,A: int] :
% 2.78/1.80 ( ( A
% 2.78/1.80 = ( sup_sup_int @ A @ B ) )
% 2.78/1.80 & ( A != B ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_331_sup_Ostrict__coboundedI1,axiom,
% 2.78/1.80 ! [C3: a > $o,A2: a > $o,B2: a > $o] :
% 2.78/1.80 ( ( ord_less_a_o @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_332_sup_Ostrict__coboundedI1,axiom,
% 2.78/1.80 ! [C3: set_a,A2: set_a,B2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_333_sup_Ostrict__coboundedI1,axiom,
% 2.78/1.80 ! [C3: int,A2: int,B2: int] :
% 2.78/1.80 ( ( ord_less_int @ C3 @ A2 )
% 2.78/1.80 => ( ord_less_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_334_sup_Ostrict__coboundedI2,axiom,
% 2.78/1.80 ! [C3: a > $o,B2: a > $o,A2: a > $o] :
% 2.78/1.80 ( ( ord_less_a_o @ C3 @ B2 )
% 2.78/1.80 => ( ord_less_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_335_sup_Ostrict__coboundedI2,axiom,
% 2.78/1.80 ! [C3: set_a,B2: set_a,A2: set_a] :
% 2.78/1.80 ( ( ord_less_set_a @ C3 @ B2 )
% 2.78/1.80 => ( ord_less_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_336_sup_Ostrict__coboundedI2,axiom,
% 2.78/1.80 ! [C3: int,B2: int,A2: int] :
% 2.78/1.80 ( ( ord_less_int @ C3 @ B2 )
% 2.78/1.80 => ( ord_less_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_337_predicate1I,axiom,
% 2.78/1.80 ! [P: a > $o,Q: a > $o] :
% 2.78/1.80 ( ! [X3: a] :
% 2.78/1.80 ( ( P @ X3 )
% 2.78/1.80 => ( Q @ X3 ) )
% 2.78/1.80 => ( ord_less_eq_a_o @ P @ Q ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_338_pred__subset__eq,axiom,
% 2.78/1.80 ! [R: set_a,S: set_a] :
% 2.78/1.80 ( ( ord_less_eq_a_o
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ R )
% 2.78/1.80 @ ^ [X4: a] : ( member_a @ X4 @ S ) )
% 2.78/1.80 = ( ord_less_eq_set_a @ R @ S ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_339_the__elem__eq,axiom,
% 2.78/1.80 ! [X: a] :
% 2.78/1.80 ( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
% 2.78/1.80 = X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_340_order__refl,axiom,
% 2.78/1.80 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_341_order__refl,axiom,
% 2.78/1.80 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_342_order__refl,axiom,
% 2.78/1.80 ! [X: a > $o] : ( ord_less_eq_a_o @ X @ X ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_343_sup1CI,axiom,
% 2.78/1.80 ! [B3: a > $o,X: a,A3: a > $o] :
% 2.78/1.80 ( ( ~ ( B3 @ X )
% 2.78/1.80 => ( A3 @ X ) )
% 2.78/1.80 => ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_344_bot__apply,axiom,
% 2.78/1.80 ( bot_bot_a_o
% 2.78/1.80 = ( ^ [X4: a] : bot_bot_o ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_345_sup1I2,axiom,
% 2.78/1.80 ! [B3: a > $o,X: a,A3: a > $o] :
% 2.78/1.80 ( ( B3 @ X )
% 2.78/1.80 => ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_346_sup1I1,axiom,
% 2.78/1.80 ! [A3: a > $o,X: a,B3: a > $o] :
% 2.78/1.80 ( ( A3 @ X )
% 2.78/1.80 => ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_347_sup1E,axiom,
% 2.78/1.80 ! [A3: a > $o,B3: a > $o,X: a] :
% 2.78/1.80 ( ( sup_sup_a_o @ A3 @ B3 @ X )
% 2.78/1.80 => ( ~ ( A3 @ X )
% 2.78/1.80 => ( B3 @ X ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_348_order__subst1,axiom,
% 2.78/1.80 ! [A2: int,F2: ( a > $o ) > int,B2: a > $o,C3: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_int @ A2 @ ( F2 @ B2 ) )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ B2 @ C3 )
% 2.78/1.80 => ( ! [X3: a > $o,Y3: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ X3 @ Y3 )
% 2.78/1.80 => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.78/1.80 => ( ord_less_eq_int @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_349_order__subst1,axiom,
% 2.78/1.80 ! [A2: a > $o,F2: set_a > a > $o,B2: set_a,C3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
% 2.78/1.80 => ( ( ord_less_eq_set_a @ B2 @ C3 )
% 2.78/1.80 => ( ! [X3: set_a,Y3: set_a] :
% 2.78/1.80 ( ( ord_less_eq_set_a @ X3 @ Y3 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.78/1.80 => ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_350_order__subst1,axiom,
% 2.78/1.80 ! [A2: a > $o,F2: int > a > $o,B2: int,C3: int] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
% 2.78/1.80 => ( ( ord_less_eq_int @ B2 @ C3 )
% 2.78/1.80 => ( ! [X3: int,Y3: int] :
% 2.78/1.80 ( ( ord_less_eq_int @ X3 @ Y3 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.78/1.80 => ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(fact_351_order__subst1,axiom,
% 2.78/1.80 ! [A2: a > $o,F2: ( a > $o ) > a > $o,B2: a > $o,C3: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
% 2.78/1.80 => ( ( ord_less_eq_a_o @ B2 @ C3 )
% 2.78/1.80 => ( ! [X3: a > $o,Y3: a > $o] :
% 2.78/1.80 ( ( ord_less_eq_a_o @ X3 @ Y3 )
% 2.78/1.80 => ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.78/1.80 => ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% 2.78/1.80
% 2.78/1.80 thf(conj_0,conjecture,
% 2.78/1.80 ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ).
% 2.78/1.80 ------------------------
% 2.78/1.82 % (16917)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16811 for (2999ds/183Mi)
% 2.78/1.82 % (16919)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_16811 for (2999ds/27Mi)
% 2.78/1.82 % (16920)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16811 for (2999ds/2Mi)
% 2.78/1.82 % (16921)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_16811 for (2999ds/2Mi)
% 2.78/1.82 % (16922)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_16811 for (2999ds/275Mi)
% 2.78/1.82 % (16920)Instruction limit reached!
% 2.78/1.82 % (16920)------------------------------
% 2.78/1.82 % (16920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.82 % (16920)Termination reason: Unknown
% 2.78/1.82 % (16920)Termination phase: shuffling
% 2.78/1.82
% 2.78/1.82 % (16920)Memory used [KB]: 1407
% 2.78/1.82 % (16921)Instruction limit reached!
% 2.78/1.82 % (16921)------------------------------
% 2.78/1.82 % (16921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.82 % (16921)Termination reason: Unknown
% 2.78/1.82 % (16921)Termination phase: shuffling
% 2.78/1.82
% 2.78/1.82 % (16921)Memory used [KB]: 1407
% 2.78/1.82 % (16921)Time elapsed: 0.002 s
% 2.78/1.82 % (16921)Instructions burned: 3 (million)
% 2.78/1.82 % (16921)------------------------------
% 2.78/1.82 % (16921)------------------------------
% 2.78/1.82 % (16920)Time elapsed: 0.002 s
% 2.78/1.82 % (16920)Instructions burned: 3 (million)
% 2.78/1.82 % (16920)------------------------------
% 2.78/1.82 % (16920)------------------------------
% 2.78/1.82 % (16918)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_16811 for (2999ds/4Mi)
% 2.78/1.83 % (16918)Instruction limit reached!
% 2.78/1.83 % (16918)------------------------------
% 2.78/1.83 % (16918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.83 % (16918)Termination reason: Unknown
% 2.78/1.83 % (16918)Termination phase: shuffling
% 2.78/1.83
% 2.78/1.83 % (16918)Memory used [KB]: 1407
% 2.78/1.83 % (16918)Time elapsed: 0.004 s
% 2.78/1.83 % (16918)Instructions burned: 7 (million)
% 2.78/1.83 % (16918)------------------------------
% 2.78/1.83 % (16918)------------------------------
% 2.78/1.83 % (16923)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16811 for (2999ds/18Mi)
% 2.78/1.83 % (16919)Instruction limit reached!
% 2.78/1.83 % (16919)------------------------------
% 2.78/1.83 % (16919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.83 % (16919)Termination reason: Unknown
% 2.78/1.83 % (16919)Termination phase: Property scanning
% 2.78/1.83
% 2.78/1.83 % (16919)Memory used [KB]: 1791
% 2.78/1.83 % (16919)Time elapsed: 0.010 s
% 2.78/1.83 % (16919)Instructions burned: 29 (million)
% 2.78/1.83 % (16919)------------------------------
% 2.78/1.83 % (16919)------------------------------
% 2.78/1.83 % (16925)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16811 for (2999ds/3Mi)
% 2.78/1.83 % (16925)Instruction limit reached!
% 2.78/1.83 % (16925)------------------------------
% 2.78/1.83 % (16925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.83 % (16925)Termination reason: Unknown
% 2.78/1.83 % (16925)Termination phase: shuffling
% 2.78/1.83
% 2.78/1.83 % (16925)Memory used [KB]: 1407
% 2.78/1.83 % (16925)Time elapsed: 0.002 s
% 2.78/1.83 % (16925)Instructions burned: 3 (million)
% 2.78/1.83 % (16925)------------------------------
% 2.78/1.83 % (16925)------------------------------
% 2.78/1.84 % (16927)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_16811 for (2999ds/15Mi)
% 2.78/1.84 % (16923)Instruction limit reached!
% 2.78/1.84 % (16923)------------------------------
% 2.78/1.84 % (16923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.84 % (16923)Termination reason: Unknown
% 2.78/1.84 % (16923)Termination phase: shuffling
% 2.78/1.84
% 2.78/1.84 % (16923)Memory used [KB]: 1663
% 2.78/1.84 % (16923)Time elapsed: 0.007 s
% 2.78/1.84 % (16923)Instructions burned: 19 (million)
% 2.78/1.84 % (16923)------------------------------
% 2.78/1.84 % (16923)------------------------------
% 2.78/1.84 % (16926)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_16811 for (2999ds/37Mi)
% 2.78/1.84 % (16927)Instruction limit reached!
% 2.78/1.84 % (16927)------------------------------
% 2.78/1.84 % (16927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.84 % (16927)Termination reason: Unknown
% 2.78/1.84 % (16927)Termination phase: shuffling
% 2.78/1.84
% 2.78/1.84 % (16927)Memory used [KB]: 1663
% 2.78/1.84 % (16927)Time elapsed: 0.006 s
% 2.78/1.84 % (16927)Instructions burned: 16 (million)
% 2.78/1.84 % (16928)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_16811 for (2999ds/3Mi)
% 2.78/1.84 % (16927)------------------------------
% 2.78/1.84 % (16927)------------------------------
% 2.78/1.84 % (16928)Instruction limit reached!
% 2.78/1.84 % (16928)------------------------------
% 2.78/1.84 % (16928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.84 % (16928)Termination reason: Unknown
% 2.78/1.84 % (16928)Termination phase: shuffling
% 2.78/1.84
% 2.78/1.84 % (16928)Memory used [KB]: 1407
% 2.78/1.84 % (16928)Time elapsed: 0.002 s
% 2.78/1.84 % (16928)Instructions burned: 3 (million)
% 2.78/1.84 % (16928)------------------------------
% 2.78/1.84 % (16928)------------------------------
% 2.78/1.84 % (16929)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_16811 for (2999ds/1041Mi)
% 2.78/1.85 % (16930)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16811 for (2999ds/7Mi)
% 2.78/1.85 % (16930)Instruction limit reached!
% 2.78/1.85 % (16930)------------------------------
% 2.78/1.85 % (16930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.85 % (16930)Termination reason: Unknown
% 2.78/1.85 % (16930)Termination phase: shuffling
% 2.78/1.85
% 2.78/1.85 % (16930)Memory used [KB]: 1535
% 2.78/1.85 % (16930)Time elapsed: 0.004 s
% 2.78/1.85 % (16930)Instructions burned: 9 (million)
% 2.78/1.85 % (16930)------------------------------
% 2.78/1.85 % (16930)------------------------------
% 2.78/1.85 % (16931)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16811 for (2999ds/16Mi)
% 2.78/1.85 % (16932)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_16811 for (2999ds/3Mi)
% 2.78/1.85 % (16922)First to succeed.
% 2.78/1.85 % (16932)Instruction limit reached!
% 2.78/1.85 % (16932)------------------------------
% 2.78/1.85 % (16932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.85 % (16932)Termination reason: Unknown
% 2.78/1.85 % (16932)Termination phase: shuffling
% 2.78/1.85
% 2.78/1.85 % (16932)Memory used [KB]: 1407
% 2.78/1.85 % (16932)Time elapsed: 0.002 s
% 2.78/1.85 % (16932)Instructions burned: 3 (million)
% 2.78/1.85 % (16932)------------------------------
% 2.78/1.85 % (16932)------------------------------
% 2.78/1.85 % (16922)Refutation found. Thanks to Tanya!
% 2.78/1.85 % SZS status Theorem for DTF2THF_16811
% 2.78/1.85 % SZS output start Proof for DTF2THF_16811
% 2.78/1.85 thf(type_def_5, type, binary1439146945Tree_a: $tType).
% 2.78/1.85 thf(type_def_6, type, a: $tType).
% 2.78/1.85 thf(type_def_8, type, int: $tType).
% 2.78/1.85 thf(type_def_9, type, set_a: $tType).
% 2.78/1.85 thf(func_def_0, type, binary1439146945Tree_a: $tType).
% 2.78/1.85 thf(func_def_1, type, set_a: $tType).
% 2.78/1.85 thf(func_def_2, type, int: $tType).
% 2.78/1.85 thf(func_def_3, type, a: $tType).
% 2.78/1.85 thf(func_def_4, type, binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_5, type, binary476621312_Tip_a: binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_6, type, binary1226383794sert_a: (a > int) > a > binary1439146945Tree_a > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_7, type, binary2053421120memb_a: (a > int) > a > binary1439146945Tree_a > $o).
% 2.78/1.85 thf(func_def_8, type, binary1804682569move_a: (a > int) > a > binary1439146945Tree_a > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_9, type, binary339557810e_rm_a: (a > int) > binary1439146945Tree_a > a).
% 2.78/1.85 thf(func_def_10, type, binary945792244etOf_a: binary1439146945Tree_a > set_a).
% 2.78/1.85 thf(func_def_11, type, binary1721989714Tree_a: (a > int) > binary1439146945Tree_a > $o).
% 2.78/1.85 thf(func_def_12, type, binary670562003pred_a: (a > int) > a > a > binary1439146945Tree_a > $o).
% 2.78/1.85 thf(func_def_13, type, binary1217730267_wrm_a: (a > int) > binary1439146945Tree_a > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_14, type, sup_sup_a_o: (a > $o) > (a > $o) > a > $o).
% 2.78/1.85 thf(func_def_15, type, sup_sup_o: $o > $o > $o).
% 2.78/1.85 thf(func_def_16, type, sup_sup_int: int > int > int).
% 2.78/1.85 thf(func_def_17, type, sup_sup_set_a: set_a > set_a > set_a).
% 2.78/1.85 thf(func_def_18, type, bot_bot_a_o: a > $o).
% 2.78/1.85 thf(func_def_20, type, bot_bot_set_a: set_a).
% 2.78/1.85 thf(func_def_21, type, ord_less_a_o: (a > $o) > (a > $o) > $o).
% 2.78/1.85 thf(func_def_22, type, ord_less_int: int > int > $o).
% 2.78/1.85 thf(func_def_23, type, ord_less_set_a: set_a > set_a > $o).
% 2.78/1.85 thf(func_def_24, type, ord_less_eq_a_o: (a > $o) > (a > $o) > $o).
% 2.78/1.85 thf(func_def_25, type, ord_less_eq_int: int > int > $o).
% 2.78/1.85 thf(func_def_26, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.78/1.85 thf(func_def_27, type, collect_a: (a > $o) > set_a).
% 2.78/1.85 thf(func_def_28, type, insert_a: a > set_a > set_a).
% 2.78/1.85 thf(func_def_29, type, the_elem_a: set_a > a).
% 2.78/1.85 thf(func_def_30, type, member_a: a > set_a > $o).
% 2.78/1.85 thf(func_def_31, type, h: a > int).
% 2.78/1.85 thf(func_def_32, type, l: a).
% 2.78/1.85 thf(func_def_33, type, t1: binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_34, type, t2: binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_35, type, x: a).
% 2.78/1.85 thf(func_def_50, type, sP0: set_a > set_a > a > a > $o).
% 2.78/1.85 thf(func_def_51, type, sP1: set_a > set_a > a > a > $o).
% 2.78/1.85 thf(func_def_52, type, sP2: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 2.78/1.85 thf(func_def_53, type, sP3: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 2.78/1.85 thf(func_def_54, type, sP4: a > set_a > set_a > $o).
% 2.78/1.85 thf(func_def_55, type, sP5: set_a > a > set_a > $o).
% 2.78/1.85 thf(func_def_56, type, sP6: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 2.78/1.85 thf(func_def_57, type, sP7: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 2.78/1.85 thf(func_def_58, type, sK8: int > int).
% 2.78/1.85 thf(func_def_59, type, sK9: set_a > a > set_a).
% 2.78/1.85 thf(func_def_60, type, sK10: int > (int > $o) > int > int).
% 2.78/1.85 thf(func_def_61, type, sK11: int > int > (int > $o) > int).
% 2.78/1.85 thf(func_def_62, type, sK12: int > int).
% 2.78/1.85 thf(func_def_63, type, sK13: int > int).
% 2.78/1.85 thf(func_def_64, type, sK14: set_a > a).
% 2.78/1.85 thf(func_def_65, type, sK15: int > int).
% 2.78/1.85 thf(func_def_66, type, sK16: (a > $o) > set_a > set_a > a).
% 2.78/1.85 thf(func_def_67, type, sK17: (a > $o) > set_a > a).
% 2.78/1.85 thf(func_def_68, type, sK18: (a > $o) > set_a > a).
% 2.78/1.85 thf(func_def_69, type, sK19: set_a > set_a > a > a > set_a).
% 2.78/1.85 thf(func_def_70, type, sK20: binary1439146945Tree_a > a).
% 2.78/1.85 thf(func_def_71, type, sK21: binary1439146945Tree_a > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_72, type, sK22: binary1439146945Tree_a > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_73, type, sK23: set_a > (a > $o) > a).
% 2.78/1.85 thf(func_def_74, type, sK24: (a > $o) > set_a > a).
% 2.78/1.85 thf(func_def_75, type, sK25: set_a > (a > $o) > set_a > a).
% 2.78/1.85 thf(func_def_76, type, sK26: set_a > set_a > a).
% 2.78/1.85 thf(func_def_77, type, sK27: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_78, type, sK28: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_79, type, sK29: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_80, type, sK30: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_81, type, sK31: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_82, type, sK32: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_83, type, sK33: int > int).
% 2.78/1.85 thf(func_def_84, type, sK34: a > set_a > set_a).
% 2.78/1.85 thf(func_def_85, type, sK35: int > int).
% 2.78/1.85 thf(func_def_86, type, sK36: (int > int > int) > int).
% 2.78/1.85 thf(func_def_87, type, sK37: (int > int > int) > int).
% 2.78/1.85 thf(func_def_88, type, sK38: (int > int > int) > int).
% 2.78/1.85 thf(func_def_89, type, sK39: (int > int > int) > int).
% 2.78/1.85 thf(func_def_90, type, sK40: (int > int > int) > int).
% 2.78/1.85 thf(func_def_91, type, sK41: (int > int > int) > int).
% 2.78/1.85 thf(func_def_92, type, sK42: (int > int > int) > int).
% 2.78/1.85 thf(func_def_93, type, sK43: int > int).
% 2.78/1.85 thf(func_def_94, type, sK44: (int > a > $o) > int).
% 2.78/1.85 thf(func_def_95, type, sK45: (int > a > $o) > int).
% 2.78/1.85 thf(func_def_96, type, sK46: (binary1439146945Tree_a > $o) > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_97, type, sK47: (binary1439146945Tree_a > $o) > binary1439146945Tree_a).
% 2.78/1.85 thf(func_def_98, type, sK48: (binary1439146945Tree_a > $o) > a).
% 2.78/1.85 thf(func_def_99, type, sK49: ((a > $o) > int) > a > $o).
% 2.78/1.85 thf(func_def_100, type, sK50: ((a > $o) > int) > a > $o).
% 2.78/1.85 thf(func_def_101, type, sK51: int > int).
% 2.78/1.85 thf(func_def_102, type, sK52: a > (a > int) > binary1439146945Tree_a > a).
% 2.78/1.85 thf(func_def_103, type, sK53: binary1439146945Tree_a > a > (a > int) > a).
% 2.78/1.85 thf(func_def_104, type, sK54: int > int).
% 2.78/1.85 thf(func_def_105, type, sK55: int > int).
% 2.78/1.85 thf(func_def_106, type, sK56: set_a > a).
% 2.78/1.85 thf(func_def_107, type, sK57: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_108, type, sK58: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_109, type, sK59: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_110, type, sK60: int > int).
% 2.78/1.85 thf(func_def_111, type, sK61: set_a > a).
% 2.78/1.85 thf(func_def_112, type, sK62: int > int).
% 2.78/1.85 thf(func_def_113, type, sK63: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_114, type, sK64: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_115, type, sK65: int > (int > $o) > (int > $o) > int).
% 2.78/1.85 thf(func_def_117, type, ph67: !>[X0: $tType]:(X0)).
% 2.78/1.85 thf(f1730,plain,(
% 2.78/1.85 $false),
% 2.78/1.85 inference(subsumption_resolution,[status(thm)],[f1729,f1569])).
% 2.78/1.85 thf(f1569,plain,(
% 2.78/1.85 ((ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ t2))) = $true)),
% 2.78/1.85 inference(cnf_transformation,[status(thm)],[f801])).
% 2.78/1.85 thf(f801,plain,(
% 2.78/1.85 ((ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ t2))) = $true)),
% 2.78/1.85 inference(fool_elimination,[status(thm)],[f800])).
% 2.78/1.85 thf(f800,plain,(
% 2.78/1.85 (ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ t2)))),
% 2.78/1.85 inference(rectify,[status(thm)],[f1])).
% 2.78/1.85 thf(f1,axiom,(
% 2.78/1.85 (ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ t2)))),
% 2.78/1.85 file('/export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2THF_16811.p',fact_0_hLess)).
% 2.78/1.85 thf(f1729,plain,(
% 2.78/1.85 ((ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ t2))) != $true)),
% 2.78/1.85 inference(forward_demodulation,[status(thm)],[f1370,f1548])).
% 2.78/1.85 thf(f1548,plain,(
% 2.78/1.85 ((binary339557810e_rm_a @ h @ t2) = (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2)))),
% 2.78/1.85 inference(cnf_transformation,[status(thm)],[f2])).
% 2.78/1.85 thf(f2,axiom,(
% 2.78/1.85 ((binary339557810e_rm_a @ h @ t2) = (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2)))),
% 2.78/1.85 file('/export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2THF_16811.p',fact_1_rm__res)).
% 2.78/1.85 thf(f1370,plain,(
% 2.78/1.85 ((ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2)))) != $true)),
% 2.78/1.85 inference(cnf_transformation,[status(thm)],[f908])).
% 2.78/1.85 thf(f908,plain,(
% 2.78/1.85 ((ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2)))) != $true)),
% 2.78/1.85 inference(flattening,[status(thm)],[f363])).
% 2.78/1.85 thf(f363,plain,(
% 2.78/1.85 ~((ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2)))) = $true)),
% 2.78/1.85 inference(fool_elimination,[status(thm)],[f362])).
% 2.78/1.85 thf(f362,plain,(
% 2.78/1.85 ~(ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2))))),
% 2.78/1.85 inference(rectify,[status(thm)],[f354])).
% 2.78/1.85 thf(f354,negated_conjecture,(
% 2.78/1.85 ~(ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2))))),
% 2.78/1.85 inference(negated_conjecture,[status(cth)],[f353])).
% 2.78/1.85 thf(f353,conjecture,(
% 2.78/1.85 (ord_less_int @ (h @ l) @ (h @ (binary339557810e_rm_a @ h @ (binary717961607le_T_a @ t1 @ x @ t2))))),
% 2.78/1.85 file('/export/starexec/sandbox/tmp/tmp.F15I5u2EBd/DTF2THF_16811.p',conj_0)).
% 2.78/1.85 % SZS output end Proof for DTF2THF_16811
% 2.78/1.85 % (16922)------------------------------
% 2.78/1.85 % (16922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.85 % (16922)Termination reason: Refutation
% 2.78/1.85
% 2.78/1.85 % (16922)Memory used [KB]: 7036
% 2.78/1.85 % (16922)Time elapsed: 0.032 s
% 2.78/1.85 % (16922)Instructions burned: 105 (million)
% 2.78/1.85 % (16922)------------------------------
% 2.78/1.85 % (16922)------------------------------
% 2.78/1.85 % (16916)Success in time 0.036 s
%------------------------------------------------------------------------------