%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP115^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n017.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 08:55:40 AM UTC 2026
% Result : Theorem 2.65s 1.57s
% Output : Refutation 2.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 9
% Syntax : Number of formulae : 55 ( 12 unt; 0 typ; 4 def)
% Number of atoms : 326 ( 123 equ; 0 cnn)
% Maximal formula atoms : 10 ( 5 avg)
% Number of connectives : 393 ( 70 ~; 70 |; 40 &; 202 @)
% ( 4 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 58 usr; 14 con; 0-4 aty)
% Number of variables : 29 ( 0 ^; 19 !; 10 ?; 29 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_a: $tType ).
thf(type_def_7,type,
set_b: $tType ).
thf(type_def_8,type,
b: $tType ).
thf(type_def_9,type,
a: $tType ).
thf(func_def_0,type,
set_b: $tType ).
thf(func_def_1,type,
set_a: $tType ).
thf(func_def_2,type,
b: $tType ).
thf(func_def_3,type,
a: $tType ).
thf(func_def_4,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_5,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(func_def_6,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(func_def_7,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(func_def_8,type,
inf_inf_b: b > b > b ).
thf(func_def_9,type,
lower_464587817at_a_b: a > ( a > b ) > $o ).
thf(func_def_10,type,
bot_bot_a_o: a > $o ).
thf(func_def_11,type,
bot_bot_b_o: b > $o ).
thf(func_def_12,type,
bot_bot_set_a: set_a ).
thf(func_def_13,type,
bot_bot_set_b: set_b ).
thf(func_def_14,type,
bot_bot_b: b ).
thf(func_def_15,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_16,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(func_def_17,type,
ord_less_eq_b: b > b > $o ).
thf(func_def_18,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_19,type,
collect_b: ( b > $o ) > set_b ).
thf(func_def_20,type,
insert_a: a > set_a > set_a ).
thf(func_def_21,type,
insert_b: b > set_b > set_b ).
thf(func_def_22,type,
is_empty_a: set_a > $o ).
thf(func_def_23,type,
is_empty_b: set_b > $o ).
thf(func_def_24,type,
is_singleton_a: set_a > $o ).
thf(func_def_25,type,
is_singleton_b: set_b > $o ).
thf(func_def_26,type,
the_elem_a: set_a > a ).
thf(func_def_27,type,
the_elem_b: set_b > b ).
thf(func_def_28,type,
topolo1276428101open_a: set_a > $o ).
thf(func_def_29,type,
topolo1276428102open_b: set_b > $o ).
thf(func_def_30,type,
member_a: a > set_a > $o ).
thf(func_def_31,type,
member_b: b > set_b > $o ).
thf(func_def_32,type,
a2: b ).
thf(func_def_33,type,
f: a > b ).
thf(func_def_35,type,
x0: a ).
thf(func_def_45,type,
sP0: set_b > set_b > b > b > $o ).
thf(func_def_46,type,
sP1: b > set_b > b > set_b > $o ).
thf(func_def_47,type,
sK2: b > set_b > set_b ).
thf(func_def_48,type,
sK3: set_b ).
thf(func_def_49,type,
sK4: set_b ).
thf(func_def_50,type,
sK5: set_b > set_b > b ).
thf(func_def_51,type,
sK6: set_b > set_b > b ).
thf(func_def_52,type,
sK7: b > b > set_b ).
thf(func_def_53,type,
sK8: b > b > set_b ).
thf(func_def_54,type,
sK9: b > b > set_b ).
thf(func_def_55,type,
sK10: set_b > set_b > b ).
thf(func_def_56,type,
sK11: set_b > set_b > b ).
thf(func_def_57,type,
sK12: set_b > set_b > b > b > set_b ).
thf(func_def_58,type,
sK13: b > set_b > set_b ).
thf(func_def_59,type,
sK14: set_b > b ).
thf(func_def_60,type,
sK15: set_b > b ).
thf(func_def_61,type,
sK16: b > b > set_b ).
thf(func_def_62,type,
sK17: b > b > set_b ).
thf(func_def_63,type,
sK18: b > b > set_b ).
thf(func_def_64,type,
sK19: b > b > set_b ).
thf(func_def_65,type,
sK20: set_b > b ).
thf(func_def_66,type,
sK21: b > b > set_b ).
thf(f1496,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1346,f1355,f1417,f1423,f1495]) ).
thf(f1495,plain,
spl22_7,
inference(avatar_contradiction_clause,[status(thm)],[f1494]) ).
thf(f1494,plain,
( $false
| spl22_7 ),
inference(subsumption_resolution,[status(thm)],[f1493,f1261]) ).
thf(f1261,plain,
( a2
!= ( f @ x0 ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( a2
!= ( f @ x0 ) ),
file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062) ).
thf(f1493,plain,
( ( a2
= ( f @ x0 ) )
| spl22_7 ),
inference(trivial_inequality_removal,[status(thm)],[f1492]) ).
thf(f1492,plain,
( ( bot_bot_set_b != bot_bot_set_b )
| ( a2
= ( f @ x0 ) )
| spl22_7 ),
inference(superposition,[status(thm)],[f1412,f1104]) ).
thf(f1104,plain,
( ( bot_bot_set_b
= ( inf_inf_set_b @ sK3 @ sK4 ) )
| ( a2
= ( f @ x0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1026]) ).
thf(f1026,plain,
( ( a2
= ( f @ x0 ) )
| ( ( $true
= ( member_b @ ( f @ x0 ) @ sK3 ) )
& ( ( member_b @ a2 @ sK4 )
= $true )
& ( $true
= ( topolo1276428102open_b @ sK4 ) )
& ( $true
= ( topolo1276428102open_b @ sK3 ) )
& ( bot_bot_set_b
= ( inf_inf_set_b @ sK3 @ sK4 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4])],[f1009,f1025]) ).
thf(f1025,plain,
( ? [X0: set_b,X1: set_b] :
( ( ( member_b @ ( f @ x0 ) @ X0 )
= $true )
& ( ( member_b @ a2 @ X1 )
= $true )
& ( ( topolo1276428102open_b @ X1 )
= $true )
& ( ( topolo1276428102open_b @ X0 )
= $true )
& ( ( inf_inf_set_b @ X0 @ X1 )
= bot_bot_set_b ) )
=> ( ( $true
= ( member_b @ ( f @ x0 ) @ sK3 ) )
& ( ( member_b @ a2 @ sK4 )
= $true )
& ( $true
= ( topolo1276428102open_b @ sK4 ) )
& ( $true
= ( topolo1276428102open_b @ sK3 ) )
& ( bot_bot_set_b
= ( inf_inf_set_b @ sK3 @ sK4 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1009,plain,
( ( a2
= ( f @ x0 ) )
| ? [X0: set_b,X1: set_b] :
( ( ( member_b @ ( f @ x0 ) @ X0 )
= $true )
& ( ( member_b @ a2 @ X1 )
= $true )
& ( ( topolo1276428102open_b @ X1 )
= $true )
& ( ( topolo1276428102open_b @ X0 )
= $true )
& ( ( inf_inf_set_b @ X0 @ X1 )
= bot_bot_set_b ) ) ),
inference(ennf_transformation,[status(thm)],[f791]) ).
thf(f791,plain,
( ( a2
!= ( f @ x0 ) )
=> ? [X0: set_b,X1: set_b] :
( ( ( member_b @ ( f @ x0 ) @ X0 )
= $true )
& ( ( member_b @ a2 @ X1 )
= $true )
& ( ( topolo1276428102open_b @ X1 )
= $true )
& ( ( topolo1276428102open_b @ X0 )
= $true )
& ( ( inf_inf_set_b @ X0 @ X1 )
= bot_bot_set_b ) ) ),
inference(fool_elimination,[status(thm)],[f790]) ).
thf(f790,plain,
( ( a2
!= ( f @ x0 ) )
=> ? [X0: set_b,X1: set_b] :
( ( member_b @ a2 @ X1 )
& ( member_b @ ( f @ x0 ) @ X0 )
& ( topolo1276428102open_b @ X1 )
& ( topolo1276428102open_b @ X0 )
& ( ( inf_inf_set_b @ X0 @ X1 )
= bot_bot_set_b ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( a2
!= ( f @ x0 ) )
=> ? [X0: set_b,X1: set_b] :
( ( member_b @ a2 @ X1 )
& ( member_b @ ( f @ x0 ) @ X0 )
& ( topolo1276428102open_b @ X1 )
& ( topolo1276428102open_b @ X0 )
& ( ( inf_inf_set_b @ X0 @ X1 )
= bot_bot_set_b ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062) ).
thf(f1412,plain,
( ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ sK4 ) )
| spl22_7 ),
inference(avatar_component_clause,[status(thm)],[f1410]) ).
thf(f1410,definition,
( spl22_7
<=> ( bot_bot_set_b
= ( inf_inf_set_b @ sK3 @ sK4 ) ) ),
introduced(definition,[new_symbols(naming,[spl22_7])],[avatar_definition]) ).
thf(f1423,plain,
spl22_8,
inference(avatar_contradiction_clause,[status(thm)],[f1422]) ).
thf(f1422,plain,
( $false
| spl22_8 ),
inference(subsumption_resolution,[status(thm)],[f1421,f1261]) ).
thf(f1421,plain,
( ( a2
= ( f @ x0 ) )
| spl22_8 ),
inference(trivial_inequality_removal,[status(thm)],[f1420]) ).
thf(f1420,plain,
( ( $true != $true )
| ( a2
= ( f @ x0 ) )
| spl22_8 ),
inference(superposition,[status(thm)],[f1416,f1106]) ).
thf(f1106,plain,
( ( $true
= ( topolo1276428102open_b @ sK4 ) )
| ( a2
= ( f @ x0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1026]) ).
thf(f1416,plain,
( ( $true
!= ( topolo1276428102open_b @ sK4 ) )
| spl22_8 ),
inference(avatar_component_clause,[status(thm)],[f1414]) ).
thf(f1414,definition,
( spl22_8
<=> ( $true
= ( topolo1276428102open_b @ sK4 ) ) ),
introduced(definition,[new_symbols(naming,[spl22_8])],[avatar_definition]) ).
thf(f1417,plain,
( ~ spl22_7
| ~ spl22_8
| ~ spl22_4 ),
inference(avatar_split_clause,[status(thm)],[f1408,f1344,f1414,f1410]) ).
thf(f1344,definition,
( spl22_4
<=> ! [X0: set_b] :
( ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ X0 ) )
| ( ( member_b @ a2 @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X0 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl22_4])],[avatar_definition]) ).
thf(f1408,plain,
( ( $true
!= ( topolo1276428102open_b @ sK4 ) )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ sK4 ) )
| ~ spl22_4 ),
inference(subsumption_resolution,[status(thm)],[f1384,f1261]) ).
thf(f1384,plain,
( ( $true
!= ( topolo1276428102open_b @ sK4 ) )
| ( a2
= ( f @ x0 ) )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ sK4 ) )
| ~ spl22_4 ),
inference(trivial_inequality_removal,[status(thm)],[f1375]) ).
thf(f1375,plain,
( ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ sK4 ) )
| ( $true != $true )
| ( a2
= ( f @ x0 ) )
| ( $true
!= ( topolo1276428102open_b @ sK4 ) )
| ~ spl22_4 ),
inference(superposition,[status(thm)],[f1345,f1107]) ).
thf(f1107,plain,
( ( ( member_b @ a2 @ sK4 )
= $true )
| ( a2
= ( f @ x0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1026]) ).
thf(f1345,plain,
( ! [X0: set_b] :
( ( ( member_b @ a2 @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X0 )
!= $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ X0 ) ) )
| ~ spl22_4 ),
inference(avatar_component_clause,[status(thm)],[f1344]) ).
thf(f1355,plain,
spl22_3,
inference(avatar_contradiction_clause,[status(thm)],[f1354]) ).
thf(f1354,plain,
( $false
| spl22_3 ),
inference(subsumption_resolution,[status(thm)],[f1353,f1261]) ).
thf(f1353,plain,
( ( a2
= ( f @ x0 ) )
| spl22_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1352]) ).
thf(f1352,plain,
( ( a2
= ( f @ x0 ) )
| ( $true != $true )
| spl22_3 ),
inference(superposition,[status(thm)],[f1342,f1105]) ).
thf(f1105,plain,
( ( $true
= ( topolo1276428102open_b @ sK3 ) )
| ( a2
= ( f @ x0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1026]) ).
thf(f1342,plain,
( ( $true
!= ( topolo1276428102open_b @ sK3 ) )
| spl22_3 ),
inference(avatar_component_clause,[status(thm)],[f1340]) ).
thf(f1340,definition,
( spl22_3
<=> ( $true
= ( topolo1276428102open_b @ sK3 ) ) ),
introduced(definition,[new_symbols(naming,[spl22_3])],[avatar_definition]) ).
thf(f1346,plain,
( ~ spl22_3
| spl22_4 ),
inference(avatar_split_clause,[status(thm)],[f1338,f1344,f1340]) ).
thf(f1338,plain,
! [X0: set_b] :
( ( $true
!= ( topolo1276428102open_b @ sK3 ) )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ X0 ) )
| ( ( topolo1276428102open_b @ X0 )
!= $true )
| ( ( member_b @ a2 @ X0 )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f1322,f1261]) ).
thf(f1322,plain,
! [X0: set_b] :
( ( ( topolo1276428102open_b @ X0 )
!= $true )
| ( ( member_b @ a2 @ X0 )
!= $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ X0 ) )
| ( $true
!= ( topolo1276428102open_b @ sK3 ) )
| ( a2
= ( f @ x0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1308]) ).
thf(f1308,plain,
! [X0: set_b] :
( ( ( topolo1276428102open_b @ X0 )
!= $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ sK3 @ X0 ) )
| ( a2
= ( f @ x0 ) )
| ( ( member_b @ a2 @ X0 )
!= $true )
| ( $true != $true )
| ( $true
!= ( topolo1276428102open_b @ sK3 ) ) ),
inference(superposition,[status(thm)],[f1289,f1108]) ).
thf(f1108,plain,
( ( $true
= ( member_b @ ( f @ x0 ) @ sK3 ) )
| ( a2
= ( f @ x0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1026]) ).
thf(f1289,plain,
! [X0: set_b,X1: set_b] :
( ( $true
!= ( member_b @ ( f @ x0 ) @ X1 ) )
| ( ( member_b @ a2 @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X1 )
!= $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ X1 @ X0 ) ) ),
inference(subsumption_resolution,[status(thm)],[f1116,f1121]) ).
thf(f1121,plain,
thesis != $true,
inference(cnf_transformation,[status(thm)],[f900]) ).
thf(f900,plain,
thesis != $true,
inference(flattening,[status(thm)],[f835]) ).
thf(f835,plain,
thesis != $true,
inference(fool_elimination,[status(thm)],[f834]) ).
thf(f834,plain,
~ thesis,
inference(rectify,[status(thm)],[f358]) ).
thf(f358,negated_conjecture,
~ thesis,
inference(negated_conjecture,[status(cth)],[f357]) ).
thf(f357,conjecture,
thesis,
file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',conj_1) ).
thf(f1116,plain,
! [X0: set_b,X1: set_b] :
( ( ( topolo1276428102open_b @ X1 )
!= $true )
| ( $true
!= ( member_b @ ( f @ x0 ) @ X1 ) )
| ( thesis = $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ X1 @ X0 ) )
| ( ( member_b @ a2 @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1015]) ).
thf(f1015,plain,
! [X0: set_b,X1: set_b] :
( ( ( member_b @ a2 @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X1 )
!= $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ X1 @ X0 ) )
| ( thesis = $true )
| ( $true
!= ( member_b @ ( f @ x0 ) @ X1 ) )
| ( ( topolo1276428102open_b @ X0 )
!= $true ) ),
inference(flattening,[status(thm)],[f1014]) ).
thf(f1014,plain,
! [X0: set_b,X1: set_b] :
( ( thesis = $true )
| ( ( member_b @ a2 @ X0 )
!= $true )
| ( ( topolo1276428102open_b @ X1 )
!= $true )
| ( bot_bot_set_b
!= ( inf_inf_set_b @ X1 @ X0 ) )
| ( $true
!= ( member_b @ ( f @ x0 ) @ X1 ) )
| ( ( topolo1276428102open_b @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f423]) ).
thf(f423,plain,
! [X0: set_b,X1: set_b] :
( ( ( ( member_b @ a2 @ X0 )
= $true )
& ( ( topolo1276428102open_b @ X1 )
= $true )
& ( bot_bot_set_b
= ( inf_inf_set_b @ X1 @ X0 ) )
& ( $true
= ( member_b @ ( f @ x0 ) @ X1 ) )
& ( ( topolo1276428102open_b @ X0 )
= $true ) )
=> ( thesis = $true ) ),
inference(fool_elimination,[status(thm)],[f422]) ).
thf(f422,plain,
! [X0: set_b,X1: set_b] :
( ( ( topolo1276428102open_b @ X1 )
& ( topolo1276428102open_b @ X0 )
& ( member_b @ a2 @ X0 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ X1 @ X0 ) )
& ( member_b @ ( f @ x0 ) @ X1 ) )
=> thesis ),
inference(rectify,[status(thm)],[f356]) ).
thf(f356,axiom,
! [X38: set_b,X15: set_b] :
( ( ( topolo1276428102open_b @ X15 )
& ( topolo1276428102open_b @ X38 )
& ( member_b @ a2 @ X38 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ X15 @ X38 ) )
& ( member_b @ ( f @ x0 ) @ X15 ) )
=> thesis ),
file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.13 % Problem : ITP115^1 : TPTP v9.2.1. Released v7.5.0.
% 0.10/0.14 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n017.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 08:25:11 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP115^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer Lower_Semicontinuous problem prob_96__6247558_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : Lower_Semicontinuous/prob_96__6247558_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 393 ( 202 unt; 36 typ; 0 def)
% 0.29/0.45 % Number of atoms : 865 ( 374 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.29/0.45 % Number of connectives : 2540 ( 120 ~; 5 |; 74 &;2074 @)
% 0.29/0.45 % ( 0 <=>; 267 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 15 ( 5 avg)
% 0.29/0.45 % Number of types : 5 ( 4 usr)
% 0.29/0.45 % Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 33 ( 32 usr; 6 con; 0-2 aty)
% 0.29/0.45 % Number of variables : 878 ( 62 ^; 796 !; 20 ?; 878 :)
% 0.29/0.45 % SPC : TH0_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:40:18.025
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (4)
% 0.29/0.45 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 0.29/0.45 set_b: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/0.45 set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_tf__b,type,
% 0.29/0.45 b: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_tf__a,type,
% 0.29/0.45 a: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (32)
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.45 minus_minus_set_a: set_a > set_a > set_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
% 0.29/0.45 minus_minus_set_b: set_b > set_b > set_b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.45 inf_inf_set_a: set_a > set_a > set_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
% 0.29/0.45 inf_inf_set_b: set_b > set_b > set_b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001tf__b,type,
% 0.29/0.45 inf_inf_b: b > b > b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001tf__b,type,
% 0.29/0.45 lower_464587817at_a_b: a > ( a > b ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.45 bot_bot_a_o: a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
% 0.29/0.45 bot_bot_b_o: b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.45 bot_bot_set_a: set_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
% 0.29/0.45 bot_bot_set_b: set_b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_001tf__b,type,
% 0.29/0.45 bot_bot_b: b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.45 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
% 0.29/0.45 ord_less_eq_set_b: set_b > set_b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
% 0.29/0.45 ord_less_eq_b: b > b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.29/0.45 collect_a: ( a > $o ) > set_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect_001tf__b,type,
% 0.29/0.45 collect_b: ( b > $o ) > set_b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_001tf__a,type,
% 0.29/0.45 insert_a: a > set_a > set_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_001tf__b,type,
% 0.29/0.45 insert_b: b > set_b > set_b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 0.29/0.45 is_empty_a: set_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ois__empty_001tf__b,type,
% 0.29/0.45 is_empty_b: set_b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ois__singleton_001tf__a,type,
% 0.29/0.45 is_singleton_a: set_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ois__singleton_001tf__b,type,
% 0.29/0.45 is_singleton_b: set_b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 0.29/0.45 the_elem_a: set_a > a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Othe__elem_001tf__b,type,
% 0.29/0.45 the_elem_b: set_b > b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
% 0.29/0.45 topolo1276428101open_a: set_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__b,type,
% 0.29/0.45 topolo1276428102open_b: set_b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_001tf__a,type,
% 0.29/0.45 member_a: a > set_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_001tf__b,type,
% 0.29/0.45 member_b: b > set_b > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_A____,type,
% 0.29/0.45 a2: b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_f,type,
% 0.29/0.45 f: a > b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_thesis____,type,
% 0.29/0.45 thesis: $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_x0,type,
% 0.29/0.45 x0: a ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (355)
% 0.29/0.45 thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062,axiom,
% 0.29/0.45 ( a2
% 0.29/0.45 != ( f @ x0 ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>A \<noteq> f x0\<close>
% 0.29/0.45 thf(fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062,axiom,
% 0.29/0.45 ( ( ( f @ x0 )
% 0.29/0.45 != a2 )
% 0.29/0.45 => ? [U: set_b,V: set_b] :
% 0.29/0.45 ( ( topolo1276428102open_b @ U )
% 0.29/0.45 & ( topolo1276428102open_b @ V )
% 0.29/0.45 & ( member_b @ ( f @ x0 ) @ U )
% 0.29/0.45 & ( member_b @ a2 @ V )
% 0.29/0.45 & ( ( inf_inf_set_b @ U @ V )
% 0.29/0.45 = bot_bot_set_b ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>f x0 \<noteq> A \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> f x0 \<in> U \<and> A \<in> V \<and> U \<inter> V = {}\<close>
% 0.29/0.45 thf(fact_2_open__Int,axiom,
% 0.29/0.45 ! [S: set_b,T: set_b] :
% 0.29/0.45 ( ( topolo1276428102open_b @ S )
% 0.29/0.45 => ( ( topolo1276428102open_b @ T )
% 0.29/0.45 => ( topolo1276428102open_b @ ( inf_inf_set_b @ S @ T ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % open_Int
% 0.29/0.45 thf(fact_3_open__Int,axiom,
% 0.29/0.45 ! [S: set_a,T: set_a] :
% 0.29/0.45 ( ( topolo1276428101open_a @ S )
% 0.29/0.45 => ( ( topolo1276428101open_a @ T )
% 0.29/0.45 => ( topolo1276428101open_a @ ( inf_inf_set_a @ S @ T ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % open_Int
% 0.29/0.45 thf(fact_4_open__empty,axiom,
% 0.29/0.45 topolo1276428102open_b @ bot_bot_set_b ).
% 0.29/0.45
% 0.29/0.45 % open_empty
% 0.29/0.45 thf(fact_5_open__empty,axiom,
% 0.29/0.45 topolo1276428101open_a @ bot_bot_set_a ).
% 0.29/0.45
% 0.29/0.45 % open_empty
% 0.29/0.45 thf(fact_6_inf__bot__left,axiom,
% 0.29/0.45 ! [X: set_a] :
% 0.29/0.45 ( ( inf_inf_set_a @ bot_bot_set_a @ X )
% 0.29/0.45 = bot_bot_set_a ) ).
% 0.29/0.45
% 0.29/0.45 % inf_bot_left
% 0.29/0.45 thf(fact_7_inf__bot__left,axiom,
% 0.29/0.45 ! [X: set_b] :
% 0.29/0.45 ( ( inf_inf_set_b @ bot_bot_set_b @ X )
% 0.29/0.45 = bot_bot_set_b ) ).
% 0.29/0.45
% 0.29/0.45 % inf_bot_left
% 0.29/0.45 thf(fact_8_inf__bot__right,axiom,
% 0.29/0.45 ! [X: set_a] :
% 0.29/0.45 ( ( inf_inf_set_a @ X @ bot_bot_set_a )
% 0.29/0.45 = bot_bot_set_a ) ).
% 0.29/0.45
% 0.29/0.45 % inf_bot_right
% 0.29/0.45 thf(fact_9_inf__bot__right,axiom,
% 0.29/0.45 ! [X: set_b] :
% 0.29/0.45 ( ( inf_inf_set_b @ X @ bot_bot_set_b )
% 0.29/0.45 = bot_bot_set_b ) ).
% 0.29/0.45
% 0.29/0.45 % inf_bot_right
% 0.29/0.45 thf(fact_10_hausdorff,axiom,
% 0.29/0.45 ! [X: b,Y: b] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ? [U: set_b,V: set_b] :
% 0.29/0.45 ( ( topolo1276428102open_b @ U )
% 0.29/0.45 & ( topolo1276428102open_b @ V )
% 0.29/0.45 & ( member_b @ X @ U )
% 0.29/0.45 & ( member_b @ Y @ V )
% 0.29/0.45 & ( ( inf_inf_set_b @ U @ V )
% 0.29/0.45 = bot_bot_set_b ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hausdorff
% 0.29/0.45 thf(fact_11_separation__t2,axiom,
% 0.29/0.45 ! [X: b,Y: b] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 = ( ? [U2: set_b,V2: set_b] :
% 0.29/0.45 ( ( topolo1276428102open_b @ U2 )
% 0.29/0.45 & ( topolo1276428102open_b @ V2 )
% 0.29/0.45 & ( member_b @ X @ U2 )
% 0.29/0.45 & ( member_b @ Y @ V2 )
% 0.29/0.45 & ( ( inf_inf_set_b @ U2 @ V2 )
% 0.29/0.45 = bot_bot_set_b ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % separation_t2
% 0.29/0.45 thf(fact_12_IntI,axiom,
% 0.29/0.45 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.45 ( ( member_a @ C @ A )
% 0.29/0.45 => ( ( member_a @ C @ B )
% 0.29/0.45 => ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntI
% 0.29/0.45 thf(fact_13_IntI,axiom,
% 0.29/0.45 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.45 ( ( member_b @ C @ A )
% 0.29/0.45 => ( ( member_b @ C @ B )
% 0.29/0.45 => ( member_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntI
% 0.29/0.45 thf(fact_14_Int__iff,axiom,
% 0.29/0.45 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.45 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 0.29/0.45 = ( ( member_a @ C @ A )
% 0.29/0.45 & ( member_a @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_iff
% 0.29/0.45 thf(fact_15_Int__iff,axiom,
% 0.29/0.45 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.45 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 0.29/0.45 = ( ( member_b @ C @ A )
% 0.29/0.45 & ( member_b @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_iff
% 0.29/0.45 thf(fact_16_inf_Oidem,axiom,
% 0.29/0.45 ! [A2: set_b] :
% 0.29/0.45 ( ( inf_inf_set_b @ A2 @ A2 )
% 0.29/0.45 = A2 ) ).
% 0.29/0.45
% 0.29/0.45 % inf.idem
% 0.29/0.45 thf(fact_17_inf_Oidem,axiom,
% 0.29/0.45 ! [A2: set_a] :
% 0.29/0.45 ( ( inf_inf_set_a @ A2 @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf.idem
% 0.29/0.46 thf(fact_18_inf__idem,axiom,
% 0.29/0.46 ! [X: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ X @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_idem
% 0.29/0.46 thf(fact_19_inf__idem,axiom,
% 0.29/0.46 ! [X: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ X @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_idem
% 0.29/0.46 thf(fact_20_inf_Oleft__idem,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B2 ) )
% 0.29/0.46 = ( inf_inf_set_b @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_idem
% 0.29/0.46 thf(fact_21_inf_Oleft__idem,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
% 0.29/0.46 = ( inf_inf_set_a @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_idem
% 0.29/0.46 thf(fact_22_empty__Collect__eq,axiom,
% 0.29/0.46 ! [P: b > $o] :
% 0.29/0.46 ( ( bot_bot_set_b
% 0.29/0.46 = ( collect_b @ P ) )
% 0.29/0.46 = ( ! [X2: b] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Collect_eq
% 0.29/0.46 thf(fact_23_empty__Collect__eq,axiom,
% 0.29/0.46 ! [P: a > $o] :
% 0.29/0.46 ( ( bot_bot_set_a
% 0.29/0.46 = ( collect_a @ P ) )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Collect_eq
% 0.29/0.46 thf(fact_24_Collect__empty__eq,axiom,
% 0.29/0.46 ! [P: b > $o] :
% 0.29/0.46 ( ( ( collect_b @ P )
% 0.29/0.46 = bot_bot_set_b )
% 0.29/0.46 = ( ! [X2: b] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq
% 0.29/0.46 thf(fact_25_Collect__empty__eq,axiom,
% 0.29/0.46 ! [P: a > $o] :
% 0.29/0.46 ( ( ( collect_a @ P )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq
% 0.29/0.46 thf(fact_26_all__not__in__conv,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( ! [X2: b] :
% 0.29/0.46 ~ ( member_b @ X2 @ A ) )
% 0.29/0.46 = ( A = bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % all_not_in_conv
% 0.29/0.46 thf(fact_27_all__not__in__conv,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( ! [X2: a] :
% 0.29/0.46 ~ ( member_a @ X2 @ A ) )
% 0.29/0.46 = ( A = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % all_not_in_conv
% 0.29/0.46 thf(fact_28_empty__iff,axiom,
% 0.29/0.46 ! [C: b] :
% 0.29/0.46 ~ ( member_b @ C @ bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % empty_iff
% 0.29/0.46 thf(fact_29_empty__iff,axiom,
% 0.29/0.46 ! [C: a] :
% 0.29/0.46 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % empty_iff
% 0.29/0.46 thf(fact_30_inf__right__idem,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y )
% 0.29/0.46 = ( inf_inf_set_b @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_right_idem
% 0.29/0.46 thf(fact_31_inf__right__idem,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
% 0.29/0.46 = ( inf_inf_set_a @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_right_idem
% 0.29/0.46 thf(fact_32_inf_Oright__idem,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 )
% 0.29/0.46 = ( inf_inf_set_b @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.right_idem
% 0.29/0.46 thf(fact_33_inf_Oright__idem,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
% 0.29/0.46 = ( inf_inf_set_a @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.right_idem
% 0.29/0.46 thf(fact_34_inf__left__idem,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
% 0.29/0.46 = ( inf_inf_set_b @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_idem
% 0.29/0.46 thf(fact_35_inf__left__idem,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
% 0.29/0.46 = ( inf_inf_set_a @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_idem
% 0.29/0.46 thf(fact_36__092_060open_062_092_060forall_062S_O_Aopen_AS_A_092_060and_062_Af_Ax0_A_092_060in_062_AS_A_092_060longrightarrow_062_A_I_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062x_H_092_060in_062T_O_Af_Ax_H_A_092_060le_062_Af_Ax0_A_092_060longrightarrow_062_Af_Ax_H_A_092_060in_062_AS_J_J_092_060close_062,axiom,
% 0.29/0.46 ! [S2: set_b] :
% 0.29/0.46 ( ( ( topolo1276428102open_b @ S2 )
% 0.29/0.46 & ( member_b @ ( f @ x0 ) @ S2 ) )
% 0.29/0.46 => ? [T2: set_a] :
% 0.29/0.46 ( ( topolo1276428101open_a @ T2 )
% 0.29/0.46 & ( member_a @ x0 @ T2 )
% 0.29/0.46 & ! [X3: a] :
% 0.29/0.46 ( ( member_a @ X3 @ T2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ ( f @ X3 ) @ ( f @ x0 ) )
% 0.29/0.46 => ( member_b @ ( f @ X3 ) @ S2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>\<forall>S. open S \<and> f x0 \<in> S \<longrightarrow> (\<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>x'\<in>T. f x' \<le> f x0 \<longrightarrow> f x' \<in> S))\<close>
% 0.29/0.46 thf(fact_37_bot__set__def,axiom,
% 0.29/0.46 ( bot_bot_set_b
% 0.29/0.46 = ( collect_b @ bot_bot_b_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_set_def
% 0.29/0.46 thf(fact_38_bot__set__def,axiom,
% 0.29/0.46 ( bot_bot_set_a
% 0.29/0.46 = ( collect_a @ bot_bot_a_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_set_def
% 0.29/0.46 thf(fact_39_ex__in__conv,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( ? [X2: b] : ( member_b @ X2 @ A ) )
% 0.29/0.46 = ( A != bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_in_conv
% 0.29/0.46 thf(fact_40_ex__in__conv,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( ? [X2: a] : ( member_a @ X2 @ A ) )
% 0.29/0.46 = ( A != bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_in_conv
% 0.29/0.46 thf(fact_41_equals0I,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ! [Y2: b] :
% 0.29/0.46 ~ ( member_b @ Y2 @ A )
% 0.29/0.46 => ( A = bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0I
% 0.29/0.46 thf(fact_42_equals0I,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ! [Y2: a] :
% 0.29/0.46 ~ ( member_a @ Y2 @ A )
% 0.29/0.46 => ( A = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0I
% 0.29/0.46 thf(fact_43_equals0D,axiom,
% 0.29/0.46 ! [A: set_b,A2: b] :
% 0.29/0.46 ( ( A = bot_bot_set_b )
% 0.29/0.46 => ~ ( member_b @ A2 @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0D
% 0.29/0.46 thf(fact_44_equals0D,axiom,
% 0.29/0.46 ! [A: set_a,A2: a] :
% 0.29/0.46 ( ( A = bot_bot_set_a )
% 0.29/0.46 => ~ ( member_a @ A2 @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0D
% 0.29/0.46 thf(fact_45_emptyE,axiom,
% 0.29/0.46 ! [A2: b] :
% 0.29/0.46 ~ ( member_b @ A2 @ bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % emptyE
% 0.29/0.46 thf(fact_46_emptyE,axiom,
% 0.29/0.46 ! [A2: a] :
% 0.29/0.46 ~ ( member_a @ A2 @ bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % emptyE
% 0.29/0.46 thf(fact_47_inf__left__commute,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b,Z: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
% 0.29/0.46 = ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_commute
% 0.29/0.46 thf(fact_48_inf__left__commute,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
% 0.29/0.46 = ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_commute
% 0.29/0.46 thf(fact_49_inf_Oleft__commute,axiom,
% 0.29/0.46 ! [B2: set_b,A2: set_b,C: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ B2 @ ( inf_inf_set_b @ A2 @ C ) )
% 0.29/0.46 = ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_commute
% 0.29/0.46 thf(fact_50_inf_Oleft__commute,axiom,
% 0.29/0.46 ! [B2: set_a,A2: set_a,C: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
% 0.29/0.46 = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_commute
% 0.29/0.46 thf(fact_51_inf__commute,axiom,
% 0.29/0.46 ( inf_inf_set_b
% 0.29/0.46 = ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_commute
% 0.29/0.46 thf(fact_52_inf__commute,axiom,
% 0.29/0.46 ( inf_inf_set_a
% 0.29/0.46 = ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_commute
% 0.29/0.46 thf(fact_53_inf_Ocommute,axiom,
% 0.29/0.46 ( inf_inf_set_b
% 0.29/0.46 = ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.commute
% 0.29/0.46 thf(fact_54_inf_Ocommute,axiom,
% 0.29/0.46 ( inf_inf_set_a
% 0.29/0.46 = ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.commute
% 0.29/0.46 thf(fact_55_inf__assoc,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b,Z: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
% 0.29/0.46 = ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_assoc
% 0.29/0.46 thf(fact_56_inf__assoc,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
% 0.29/0.46 = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_assoc
% 0.29/0.46 thf(fact_57_inf_Oassoc,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b,C: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.assoc
% 0.29/0.46 thf(fact_58_inf_Oassoc,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a,C: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.assoc
% 0.29/0.46 thf(fact_59_boolean__algebra__cancel_Oinf2,axiom,
% 0.29/0.46 ! [B: set_b,K: set_b,B2: set_b,A2: set_b] :
% 0.29/0.46 ( ( B
% 0.29/0.46 = ( inf_inf_set_b @ K @ B2 ) )
% 0.29/0.46 => ( ( inf_inf_set_b @ A2 @ B )
% 0.29/0.46 = ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf2
% 0.29/0.46 thf(fact_60_boolean__algebra__cancel_Oinf2,axiom,
% 0.29/0.46 ! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
% 0.29/0.46 ( ( B
% 0.29/0.46 = ( inf_inf_set_a @ K @ B2 ) )
% 0.29/0.46 => ( ( inf_inf_set_a @ A2 @ B )
% 0.29/0.46 = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf2
% 0.29/0.46 thf(fact_61_boolean__algebra__cancel_Oinf1,axiom,
% 0.29/0.46 ! [A: set_b,K: set_b,A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( inf_inf_set_b @ K @ A2 ) )
% 0.29/0.46 => ( ( inf_inf_set_b @ A @ B2 )
% 0.29/0.46 = ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf1
% 0.29/0.46 thf(fact_62_boolean__algebra__cancel_Oinf1,axiom,
% 0.29/0.46 ! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( inf_inf_set_a @ K @ A2 ) )
% 0.29/0.46 => ( ( inf_inf_set_a @ A @ B2 )
% 0.29/0.46 = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf1
% 0.29/0.46 thf(fact_63_inf__sup__aci_I1_J,axiom,
% 0.29/0.46 ( inf_inf_set_b
% 0.29/0.46 = ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(1)
% 0.29/0.46 thf(fact_64_inf__sup__aci_I1_J,axiom,
% 0.29/0.46 ( inf_inf_set_a
% 0.29/0.46 = ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(1)
% 0.29/0.46 thf(fact_65_inf__sup__aci_I2_J,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b,Z: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
% 0.29/0.46 = ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(2)
% 0.29/0.46 thf(fact_66_inf__sup__aci_I2_J,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
% 0.29/0.46 = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(2)
% 0.29/0.46 thf(fact_67_inf__sup__aci_I3_J,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b,Z: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
% 0.29/0.46 = ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(3)
% 0.29/0.46 thf(fact_68_inf__sup__aci_I3_J,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
% 0.29/0.46 = ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(3)
% 0.29/0.46 thf(fact_69_inf__sup__aci_I4_J,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
% 0.29/0.46 = ( inf_inf_set_b @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(4)
% 0.29/0.46 thf(fact_70_inf__sup__aci_I4_J,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
% 0.29/0.46 = ( inf_inf_set_a @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(4)
% 0.29/0.46 thf(fact_71_Int__left__commute,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b,C2: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
% 0.29/0.46 = ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_left_commute
% 0.29/0.46 thf(fact_72_Int__left__commute,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
% 0.29/0.46 = ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_left_commute
% 0.29/0.46 thf(fact_73_Int__left__absorb,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B ) )
% 0.29/0.46 = ( inf_inf_set_b @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_left_absorb
% 0.29/0.46 thf(fact_74_Int__left__absorb,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
% 0.29/0.46 = ( inf_inf_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_left_absorb
% 0.29/0.46 thf(fact_75_Int__commute,axiom,
% 0.29/0.46 ( inf_inf_set_b
% 0.29/0.46 = ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_commute
% 0.29/0.46 thf(fact_76_Int__commute,axiom,
% 0.29/0.46 ( inf_inf_set_a
% 0.29/0.46 = ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_commute
% 0.29/0.46 thf(fact_77_Int__absorb,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ A @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Int_absorb
% 0.29/0.46 thf(fact_78_Int__absorb,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ A @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Int_absorb
% 0.29/0.46 thf(fact_79_Int__assoc,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b,C2: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_assoc
% 0.29/0.46 thf(fact_80_Int__assoc,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_assoc
% 0.29/0.46 thf(fact_81_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A2: b,P: b > $o] :
% 0.29/0.46 ( ( member_b @ A2 @ ( collect_b @ P ) )
% 0.29/0.46 = ( P @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_82_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A2: a,P: a > $o] :
% 0.29/0.46 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 0.29/0.46 = ( P @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_83_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( collect_b
% 0.29/0.46 @ ^ [X2: b] : ( member_b @ X2 @ A ) )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_84_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( collect_a
% 0.29/0.46 @ ^ [X2: a] : ( member_a @ X2 @ A ) )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_85_IntD2,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 0.29/0.46 => ( member_b @ C @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD2
% 0.29/0.46 thf(fact_86_IntD2,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 0.29/0.46 => ( member_a @ C @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD2
% 0.29/0.46 thf(fact_87_IntD1,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 0.29/0.46 => ( member_b @ C @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD1
% 0.29/0.46 thf(fact_88_IntD1,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 0.29/0.46 => ( member_a @ C @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD1
% 0.29/0.46 thf(fact_89_IntE,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 0.29/0.46 => ~ ( ( member_b @ C @ A )
% 0.29/0.46 => ~ ( member_b @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntE
% 0.29/0.46 thf(fact_90_IntE,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 0.29/0.46 => ~ ( ( member_a @ C @ A )
% 0.29/0.46 => ~ ( member_a @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntE
% 0.29/0.46 thf(fact_91_separation__t1,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 = ( ? [U2: set_b] :
% 0.29/0.46 ( ( topolo1276428102open_b @ U2 )
% 0.29/0.46 & ( member_b @ X @ U2 )
% 0.29/0.46 & ~ ( member_b @ Y @ U2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % separation_t1
% 0.29/0.46 thf(fact_92_separation__t0,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 = ( ? [U2: set_b] :
% 0.29/0.46 ( ( topolo1276428102open_b @ U2 )
% 0.29/0.46 & ( ( member_b @ X @ U2 )
% 0.29/0.46 != ( member_b @ Y @ U2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % separation_t0
% 0.29/0.46 thf(fact_93_t1__space,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ? [U: set_b] :
% 0.29/0.46 ( ( topolo1276428102open_b @ U )
% 0.29/0.46 & ( member_b @ X @ U )
% 0.29/0.46 & ~ ( member_b @ Y @ U ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % t1_space
% 0.29/0.46 thf(fact_94_t0__space,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ? [U: set_b] :
% 0.29/0.46 ( ( topolo1276428102open_b @ U )
% 0.29/0.46 & ( ( member_b @ X @ U )
% 0.29/0.46 != ( member_b @ Y @ U ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % t0_space
% 0.29/0.46 thf(fact_95_disjoint__iff__not__equal,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b] :
% 0.29/0.46 ( ( ( inf_inf_set_b @ A @ B )
% 0.29/0.46 = bot_bot_set_b )
% 0.29/0.46 = ( ! [X2: b] :
% 0.29/0.46 ( ( member_b @ X2 @ A )
% 0.29/0.46 => ! [Y3: b] :
% 0.29/0.46 ( ( member_b @ Y3 @ B )
% 0.29/0.46 => ( X2 != Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_iff_not_equal
% 0.29/0.46 thf(fact_96_disjoint__iff__not__equal,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ( inf_inf_set_a @ A @ B )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A )
% 0.29/0.46 => ! [Y3: a] :
% 0.29/0.46 ( ( member_a @ Y3 @ B )
% 0.29/0.46 => ( X2 != Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_iff_not_equal
% 0.29/0.46 thf(fact_97_Int__empty__right,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ A @ bot_bot_set_b )
% 0.29/0.46 = bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % Int_empty_right
% 0.29/0.46 thf(fact_98_Int__empty__right,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ A @ bot_bot_set_a )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % Int_empty_right
% 0.29/0.46 thf(fact_99_Int__empty__left,axiom,
% 0.29/0.46 ! [B: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ bot_bot_set_b @ B )
% 0.29/0.46 = bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % Int_empty_left
% 0.29/0.46 thf(fact_100_Int__empty__left,axiom,
% 0.29/0.46 ! [B: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ bot_bot_set_a @ B )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % Int_empty_left
% 0.29/0.46 thf(fact_101_disjoint__iff,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b] :
% 0.29/0.46 ( ( ( inf_inf_set_b @ A @ B )
% 0.29/0.46 = bot_bot_set_b )
% 0.29/0.46 = ( ! [X2: b] :
% 0.29/0.46 ( ( member_b @ X2 @ A )
% 0.29/0.46 => ~ ( member_b @ X2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_iff
% 0.29/0.46 thf(fact_102_disjoint__iff,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( ( inf_inf_set_a @ A @ B )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ! [X2: a] :
% 0.29/0.46 ( ( member_a @ X2 @ A )
% 0.29/0.46 => ~ ( member_a @ X2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_iff
% 0.29/0.46 thf(fact_103_Int__emptyI,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b] :
% 0.29/0.46 ( ! [X4: b] :
% 0.29/0.46 ( ( member_b @ X4 @ A )
% 0.29/0.46 => ~ ( member_b @ X4 @ B ) )
% 0.29/0.46 => ( ( inf_inf_set_b @ A @ B )
% 0.29/0.46 = bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_emptyI
% 0.29/0.46 thf(fact_104_Int__emptyI,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ! [X4: a] :
% 0.29/0.46 ( ( member_a @ X4 @ A )
% 0.29/0.46 => ~ ( member_a @ X4 @ B ) )
% 0.29/0.46 => ( ( inf_inf_set_a @ A @ B )
% 0.29/0.46 = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_emptyI
% 0.29/0.46 thf(fact_105_calculation,axiom,
% 0.29/0.46 ( ~ ! [S2: set_b] :
% 0.29/0.46 ( ( ( topolo1276428102open_b @ S2 )
% 0.29/0.46 & ( member_b @ ( f @ x0 ) @ S2 ) )
% 0.29/0.46 => ? [T2: set_a] :
% 0.29/0.46 ( ( topolo1276428101open_a @ T2 )
% 0.29/0.46 & ( member_a @ x0 @ T2 )
% 0.29/0.46 & ! [X3: a] :
% 0.29/0.46 ( ( member_a @ X3 @ T2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ ( f @ X3 ) @ ( f @ x0 ) )
% 0.29/0.46 => ( member_b @ ( f @ X3 ) @ S2 ) ) ) ) )
% 0.29/0.46 => ~ ( lower_464587817at_a_b @ x0 @ f ) ) ).
% 0.29/0.46
% 0.29/0.46 % calculation
% 0.29/0.46 thf(fact_106_Set_Ois__empty__def,axiom,
% 0.29/0.46 ( is_empty_b
% 0.29/0.46 = ( ^ [A4: set_b] : ( A4 = bot_bot_set_b ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.is_empty_def
% 0.29/0.46 thf(fact_107_Set_Ois__empty__def,axiom,
% 0.29/0.46 ( is_empty_a
% 0.29/0.46 = ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.is_empty_def
% 0.29/0.46 thf(fact_108_Collect__empty__eq__bot,axiom,
% 0.29/0.46 ! [P: b > $o] :
% 0.29/0.46 ( ( ( collect_b @ P )
% 0.29/0.46 = bot_bot_set_b )
% 0.29/0.46 = ( P = bot_bot_b_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq_bot
% 0.29/0.46 thf(fact_109_Collect__empty__eq__bot,axiom,
% 0.29/0.46 ! [P: a > $o] :
% 0.29/0.46 ( ( ( collect_a @ P )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( P = bot_bot_a_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq_bot
% 0.29/0.46 thf(fact_110_bot__empty__eq,axiom,
% 0.29/0.46 ( bot_bot_b_o
% 0.29/0.46 = ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq
% 0.29/0.46 thf(fact_111_bot__empty__eq,axiom,
% 0.29/0.46 ( bot_bot_a_o
% 0.29/0.46 = ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq
% 0.29/0.46 thf(fact_112_is__singletonI_H,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( A != bot_bot_set_b )
% 0.29/0.46 => ( ! [X4: b,Y2: b] :
% 0.29/0.46 ( ( member_b @ X4 @ A )
% 0.29/0.46 => ( ( member_b @ Y2 @ A )
% 0.29/0.46 => ( X4 = Y2 ) ) )
% 0.29/0.46 => ( is_singleton_b @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI'
% 0.29/0.46 thf(fact_113_is__singletonI_H,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( A != bot_bot_set_a )
% 0.29/0.46 => ( ! [X4: a,Y2: a] :
% 0.29/0.46 ( ( member_a @ X4 @ A )
% 0.29/0.46 => ( ( member_a @ Y2 @ A )
% 0.29/0.46 => ( X4 = Y2 ) ) )
% 0.29/0.46 => ( is_singleton_a @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI'
% 0.29/0.46 thf(fact_114_Diff__disjoint,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ A ) )
% 0.29/0.46 = bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_disjoint
% 0.29/0.46 thf(fact_115_Diff__disjoint,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_disjoint
% 0.29/0.46 thf(fact_116_insert__disjoint_I1_J,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B )
% 0.29/0.46 = bot_bot_set_b )
% 0.29/0.46 = ( ~ ( member_b @ A2 @ B )
% 0.29/0.46 & ( ( inf_inf_set_b @ A @ B )
% 0.29/0.46 = bot_bot_set_b ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_disjoint(1)
% 0.29/0.46 thf(fact_117_insert__disjoint_I1_J,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ~ ( member_a @ A2 @ B )
% 0.29/0.46 & ( ( inf_inf_set_a @ A @ B )
% 0.29/0.46 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_disjoint(1)
% 0.29/0.46 thf(fact_118_insert__disjoint_I2_J,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( bot_bot_set_b
% 0.29/0.46 = ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B ) )
% 0.29/0.46 = ( ~ ( member_b @ A2 @ B )
% 0.29/0.46 & ( bot_bot_set_b
% 0.29/0.46 = ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_disjoint(2)
% 0.29/0.46 thf(fact_119_insert__disjoint_I2_J,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( bot_bot_set_a
% 0.29/0.46 = ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
% 0.29/0.46 = ( ~ ( member_a @ A2 @ B )
% 0.29/0.46 & ( bot_bot_set_a
% 0.29/0.46 = ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_disjoint(2)
% 0.29/0.46 thf(fact_120_order__refl,axiom,
% 0.29/0.46 ! [X: b] : ( ord_less_eq_b @ X @ X ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_121_insertCI,axiom,
% 0.29/0.46 ! [A2: b,B: set_b,B2: b] :
% 0.29/0.46 ( ( ~ ( member_b @ A2 @ B )
% 0.29/0.46 => ( A2 = B2 ) )
% 0.29/0.46 => ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_122_insertCI,axiom,
% 0.29/0.46 ! [A2: a,B: set_a,B2: a] :
% 0.29/0.46 ( ( ~ ( member_a @ A2 @ B )
% 0.29/0.46 => ( A2 = B2 ) )
% 0.29/0.46 => ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_123_insert__iff,axiom,
% 0.29/0.46 ! [A2: b,B2: b,A: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
% 0.29/0.46 = ( ( A2 = B2 )
% 0.29/0.46 | ( member_b @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_124_insert__iff,axiom,
% 0.29/0.46 ! [A2: a,B2: a,A: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
% 0.29/0.46 = ( ( A2 = B2 )
% 0.29/0.46 | ( member_a @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_125_DiffI,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ A )
% 0.29/0.46 => ( ~ ( member_b @ C @ B )
% 0.29/0.46 => ( member_b @ C @ ( minus_minus_set_b @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffI
% 0.29/0.46 thf(fact_126_DiffI,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ A )
% 0.29/0.46 => ( ~ ( member_a @ C @ B )
% 0.29/0.46 => ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffI
% 0.29/0.46 thf(fact_127_Diff__iff,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.46 = ( ( member_b @ C @ A )
% 0.29/0.46 & ~ ( member_b @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_iff
% 0.29/0.46 thf(fact_128_Diff__iff,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.46 = ( ( member_a @ C @ A )
% 0.29/0.46 & ~ ( member_a @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_iff
% 0.29/0.46 thf(fact_129_le__inf__iff,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b,Z: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
% 0.29/0.46 = ( ( ord_less_eq_set_b @ X @ Y )
% 0.29/0.46 & ( ord_less_eq_set_b @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_inf_iff
% 0.29/0.46 thf(fact_130_le__inf__iff,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
% 0.29/0.46 = ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/0.46 & ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_inf_iff
% 0.29/0.46 thf(fact_131_le__inf__iff,axiom,
% 0.29/0.46 ! [X: b,Y: b,Z: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ ( inf_inf_b @ Y @ Z ) )
% 0.29/0.46 = ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 & ( ord_less_eq_b @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_inf_iff
% 0.29/0.46 thf(fact_132_inf_Obounded__iff,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b,C: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
% 0.29/0.46 = ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.29/0.46 & ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.bounded_iff
% 0.29/0.46 thf(fact_133_inf_Obounded__iff,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a,C: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
% 0.29/0.46 = ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.46 & ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.bounded_iff
% 0.29/0.46 thf(fact_134_inf_Obounded__iff,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) )
% 0.29/0.46 = ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 & ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.bounded_iff
% 0.29/0.46 thf(fact_135_singletonI,axiom,
% 0.29/0.46 ! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonI
% 0.29/0.46 thf(fact_136_singletonI,axiom,
% 0.29/0.46 ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonI
% 0.29/0.46 thf(fact_137_Int__insert__left__if0,axiom,
% 0.29/0.46 ! [A2: b,C2: set_b,B: set_b] :
% 0.29/0.46 ( ~ ( member_b @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_b @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if0
% 0.29/0.46 thf(fact_138_Int__insert__left__if0,axiom,
% 0.29/0.46 ! [A2: a,C2: set_a,B: set_a] :
% 0.29/0.46 ( ~ ( member_a @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_a @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if0
% 0.29/0.46 thf(fact_139_Int__insert__left__if1,axiom,
% 0.29/0.46 ! [A2: b,C2: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if1
% 0.29/0.46 thf(fact_140_Int__insert__left__if1,axiom,
% 0.29/0.46 ! [A2: a,C2: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if1
% 0.29/0.46 thf(fact_141_insert__inter__insert,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_inter_insert
% 0.29/0.46 thf(fact_142_insert__inter__insert,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_inter_insert
% 0.29/0.46 thf(fact_143_Int__insert__right__if0,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ~ ( member_b @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( inf_inf_set_b @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if0
% 0.29/0.46 thf(fact_144_Int__insert__right__if0,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ~ ( member_a @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( inf_inf_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if0
% 0.29/0.46 thf(fact_145_Int__insert__right__if1,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if1
% 0.29/0.46 thf(fact_146_Int__insert__right__if1,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if1
% 0.29/0.46 thf(fact_147_Diff__empty,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ A @ bot_bot_set_b )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_empty
% 0.29/0.46 thf(fact_148_Diff__empty,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A @ bot_bot_set_a )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_empty
% 0.29/0.46 thf(fact_149_empty__Diff,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ bot_bot_set_b @ A )
% 0.29/0.46 = bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Diff
% 0.29/0.46 thf(fact_150_empty__Diff,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ bot_bot_set_a @ A )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Diff
% 0.29/0.46 thf(fact_151_Diff__cancel,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ A @ A )
% 0.29/0.46 = bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_cancel
% 0.29/0.46 thf(fact_152_Diff__cancel,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A @ A )
% 0.29/0.46 = bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_cancel
% 0.29/0.46 thf(fact_153_Diff__insert0,axiom,
% 0.29/0.46 ! [X: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ~ ( member_b @ X @ A )
% 0.29/0.46 => ( ( minus_minus_set_b @ A @ ( insert_b @ X @ B ) )
% 0.29/0.46 = ( minus_minus_set_b @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert0
% 0.29/0.46 thf(fact_154_Diff__insert0,axiom,
% 0.29/0.46 ! [X: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ~ ( member_a @ X @ A )
% 0.29/0.46 => ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
% 0.29/0.46 = ( minus_minus_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert0
% 0.29/0.46 thf(fact_155_insert__Diff1,axiom,
% 0.29/0.46 ! [X: b,B: set_b,A: set_b] :
% 0.29/0.46 ( ( member_b @ X @ B )
% 0.29/0.46 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
% 0.29/0.46 = ( minus_minus_set_b @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff1
% 0.29/0.46 thf(fact_156_insert__Diff1,axiom,
% 0.29/0.46 ! [X: a,B: set_a,A: set_a] :
% 0.29/0.46 ( ( member_a @ X @ B )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
% 0.29/0.46 = ( minus_minus_set_a @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff1
% 0.29/0.46 thf(fact_157_disjoint__insert_I2_J,axiom,
% 0.29/0.46 ! [A: set_b,B2: b,B: set_b] :
% 0.29/0.46 ( ( bot_bot_set_b
% 0.29/0.46 = ( inf_inf_set_b @ A @ ( insert_b @ B2 @ B ) ) )
% 0.29/0.46 = ( ~ ( member_b @ B2 @ A )
% 0.29/0.46 & ( bot_bot_set_b
% 0.29/0.46 = ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_insert(2)
% 0.29/0.46 thf(fact_158_disjoint__insert_I2_J,axiom,
% 0.29/0.46 ! [A: set_a,B2: a,B: set_a] :
% 0.29/0.46 ( ( bot_bot_set_a
% 0.29/0.46 = ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
% 0.29/0.46 = ( ~ ( member_a @ B2 @ A )
% 0.29/0.46 & ( bot_bot_set_a
% 0.29/0.46 = ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_insert(2)
% 0.29/0.46 thf(fact_159_disjoint__insert_I1_J,axiom,
% 0.29/0.46 ! [B: set_b,A2: b,A: set_b] :
% 0.29/0.46 ( ( ( inf_inf_set_b @ B @ ( insert_b @ A2 @ A ) )
% 0.29/0.46 = bot_bot_set_b )
% 0.29/0.46 = ( ~ ( member_b @ A2 @ B )
% 0.29/0.46 & ( ( inf_inf_set_b @ B @ A )
% 0.29/0.46 = bot_bot_set_b ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_insert(1)
% 0.29/0.46 thf(fact_160_disjoint__insert_I1_J,axiom,
% 0.29/0.46 ! [B: set_a,A2: a,A: set_a] :
% 0.29/0.46 ( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
% 0.29/0.46 = bot_bot_set_a )
% 0.29/0.46 = ( ~ ( member_a @ A2 @ B )
% 0.29/0.46 & ( ( inf_inf_set_a @ B @ A )
% 0.29/0.46 = bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_insert(1)
% 0.29/0.46 thf(fact_161_insert__Diff__single,axiom,
% 0.29/0.46 ! [A2: b,A: set_b] :
% 0.29/0.46 ( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
% 0.29/0.46 = ( insert_b @ A2 @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_single
% 0.29/0.46 thf(fact_162_insert__Diff__single,axiom,
% 0.29/0.46 ! [A2: a,A: set_a] :
% 0.29/0.46 ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 0.29/0.46 = ( insert_a @ A2 @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_single
% 0.29/0.46 thf(fact_163_is__singletonI,axiom,
% 0.29/0.46 ! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI
% 0.29/0.46 thf(fact_164_is__singletonI,axiom,
% 0.29/0.46 ! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI
% 0.29/0.46 thf(fact_165_DiffE,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.46 => ~ ( ( member_b @ C @ A )
% 0.29/0.46 => ( member_b @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffE
% 0.29/0.46 thf(fact_166_DiffE,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.46 => ~ ( ( member_a @ C @ A )
% 0.29/0.46 => ( member_a @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffE
% 0.29/0.46 thf(fact_167_DiffD1,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.46 => ( member_b @ C @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD1
% 0.29/0.46 thf(fact_168_DiffD1,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.46 => ( member_a @ C @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD1
% 0.29/0.46 thf(fact_169_DiffD2,axiom,
% 0.29/0.46 ! [C: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.46 => ~ ( member_b @ C @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD2
% 0.29/0.46 thf(fact_170_DiffD2,axiom,
% 0.29/0.46 ! [C: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.46 => ~ ( member_a @ C @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD2
% 0.29/0.46 thf(fact_171_insertE,axiom,
% 0.29/0.46 ! [A2: b,B2: b,A: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
% 0.29/0.46 => ( ( A2 != B2 )
% 0.29/0.46 => ( member_b @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertE
% 0.29/0.46 thf(fact_172_insertE,axiom,
% 0.29/0.46 ! [A2: a,B2: a,A: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
% 0.29/0.46 => ( ( A2 != B2 )
% 0.29/0.46 => ( member_a @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertE
% 0.29/0.46 thf(fact_173_insertI1,axiom,
% 0.29/0.46 ! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI1
% 0.29/0.46 thf(fact_174_insertI1,axiom,
% 0.29/0.46 ! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI1
% 0.29/0.46 thf(fact_175_insertI2,axiom,
% 0.29/0.46 ! [A2: b,B: set_b,B2: b] :
% 0.29/0.46 ( ( member_b @ A2 @ B )
% 0.29/0.46 => ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI2
% 0.29/0.46 thf(fact_176_insertI2,axiom,
% 0.29/0.46 ! [A2: a,B: set_a,B2: a] :
% 0.29/0.46 ( ( member_a @ A2 @ B )
% 0.29/0.46 => ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI2
% 0.29/0.46 thf(fact_177_Set_Oset__insert,axiom,
% 0.29/0.46 ! [X: b,A: set_b] :
% 0.29/0.46 ( ( member_b @ X @ A )
% 0.29/0.46 => ~ ! [B5: set_b] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_b @ X @ B5 ) )
% 0.29/0.46 => ( member_b @ X @ B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.set_insert
% 0.29/0.46 thf(fact_178_Set_Oset__insert,axiom,
% 0.29/0.46 ! [X: a,A: set_a] :
% 0.29/0.46 ( ( member_a @ X @ A )
% 0.29/0.46 => ~ ! [B5: set_a] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_a @ X @ B5 ) )
% 0.29/0.46 => ( member_a @ X @ B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.set_insert
% 0.29/0.46 thf(fact_179_insert__ident,axiom,
% 0.29/0.46 ! [X: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ~ ( member_b @ X @ A )
% 0.29/0.46 => ( ~ ( member_b @ X @ B )
% 0.29/0.46 => ( ( ( insert_b @ X @ A )
% 0.29/0.46 = ( insert_b @ X @ B ) )
% 0.29/0.46 = ( A = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_180_insert__ident,axiom,
% 0.29/0.46 ! [X: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ~ ( member_a @ X @ A )
% 0.29/0.46 => ( ~ ( member_a @ X @ B )
% 0.29/0.46 => ( ( ( insert_a @ X @ A )
% 0.29/0.46 = ( insert_a @ X @ B ) )
% 0.29/0.46 = ( A = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_181_insert__absorb,axiom,
% 0.29/0.46 ! [A2: b,A: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ A )
% 0.29/0.46 => ( ( insert_b @ A2 @ A )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_182_insert__absorb,axiom,
% 0.29/0.46 ! [A2: a,A: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ A )
% 0.29/0.46 => ( ( insert_a @ A2 @ A )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_183_insert__eq__iff,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B2: b,B: set_b] :
% 0.29/0.46 ( ~ ( member_b @ A2 @ A )
% 0.29/0.46 => ( ~ ( member_b @ B2 @ B )
% 0.29/0.46 => ( ( ( insert_b @ A2 @ A )
% 0.29/0.46 = ( insert_b @ B2 @ B ) )
% 0.29/0.46 = ( ( ( A2 = B2 )
% 0.29/0.46 => ( A = B ) )
% 0.29/0.46 & ( ( A2 != B2 )
% 0.29/0.46 => ? [C3: set_b] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_b @ B2 @ C3 ) )
% 0.29/0.46 & ~ ( member_b @ B2 @ C3 )
% 0.29/0.46 & ( B
% 0.29/0.46 = ( insert_b @ A2 @ C3 ) )
% 0.29/0.46 & ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_184_insert__eq__iff,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B2: a,B: set_a] :
% 0.29/0.46 ( ~ ( member_a @ A2 @ A )
% 0.29/0.46 => ( ~ ( member_a @ B2 @ B )
% 0.29/0.46 => ( ( ( insert_a @ A2 @ A )
% 0.29/0.46 = ( insert_a @ B2 @ B ) )
% 0.29/0.46 = ( ( ( A2 = B2 )
% 0.29/0.46 => ( A = B ) )
% 0.29/0.46 & ( ( A2 != B2 )
% 0.29/0.46 => ? [C3: set_a] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_a @ B2 @ C3 ) )
% 0.29/0.46 & ~ ( member_a @ B2 @ C3 )
% 0.29/0.46 & ( B
% 0.29/0.46 = ( insert_a @ A2 @ C3 ) )
% 0.29/0.46 & ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_185_insert__Diff__if,axiom,
% 0.29/0.46 ! [X: b,B: set_b,A: set_b] :
% 0.29/0.46 ( ( ( member_b @ X @ B )
% 0.29/0.46 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
% 0.29/0.46 = ( minus_minus_set_b @ A @ B ) ) )
% 0.29/0.46 & ( ~ ( member_b @ X @ B )
% 0.29/0.46 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
% 0.29/0.46 = ( insert_b @ X @ ( minus_minus_set_b @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_if
% 0.29/0.46 thf(fact_186_insert__Diff__if,axiom,
% 0.29/0.46 ! [X: a,B: set_a,A: set_a] :
% 0.29/0.46 ( ( ( member_a @ X @ B )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
% 0.29/0.46 = ( minus_minus_set_a @ A @ B ) ) )
% 0.29/0.46 & ( ~ ( member_a @ X @ B )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
% 0.29/0.46 = ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_if
% 0.29/0.46 thf(fact_187_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A2: b,A: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ A )
% 0.29/0.46 => ? [B5: set_b] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_b @ A2 @ B5 ) )
% 0.29/0.46 & ~ ( member_b @ A2 @ B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_188_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A2: a,A: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ A )
% 0.29/0.46 => ? [B5: set_a] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_a @ A2 @ B5 ) )
% 0.29/0.46 & ~ ( member_a @ A2 @ B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_189_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B2: b,A2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ B2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( A2 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_190_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [A3: b,B3: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ B3 @ A3 )
% 0.29/0.46 & ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_191_dual__order_Otrans,axiom,
% 0.29/0.46 ! [B2: b,A2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ B2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ C @ B2 )
% 0.29/0.46 => ( ord_less_eq_b @ C @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_192_linorder__wlog,axiom,
% 0.29/0.46 ! [P: b > b > $o,A2: b,B2: b] :
% 0.29/0.46 ( ! [A5: b,B6: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A5 @ B6 )
% 0.29/0.46 => ( P @ A5 @ B6 ) )
% 0.29/0.46 => ( ! [A5: b,B6: b] :
% 0.29/0.46 ( ( P @ B6 @ A5 )
% 0.29/0.46 => ( P @ A5 @ B6 ) )
% 0.29/0.46 => ( P @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_193_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A2: b] : ( ord_less_eq_b @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_194_order__trans,axiom,
% 0.29/0.46 ! [X: b,Y: b,Z: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq_b @ Y @ Z )
% 0.29/0.46 => ( ord_less_eq_b @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_195_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A2: b,B2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ B2 @ A2 )
% 0.29/0.46 => ( A2 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_196_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( B2 = C )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_197_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b] :
% 0.29/0.46 ( ( A2 = B2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ B2 @ C )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_198_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [A3: b,B3: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A3 @ B3 )
% 0.29/0.46 & ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_199_antisym__conv,axiom,
% 0.29/0.46 ! [Y: b,X: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_200_le__cases3,axiom,
% 0.29/0.46 ! [X: b,Y: b,Z: b] :
% 0.29/0.46 ( ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 => ~ ( ord_less_eq_b @ Y @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq_b @ Y @ X )
% 0.29/0.46 => ~ ( ord_less_eq_b @ X @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq_b @ X @ Z )
% 0.29/0.46 => ~ ( ord_less_eq_b @ Z @ Y ) )
% 0.29/0.46 => ( ( ( ord_less_eq_b @ Z @ Y )
% 0.29/0.46 => ~ ( ord_less_eq_b @ Y @ X ) )
% 0.29/0.46 => ( ( ( ord_less_eq_b @ Y @ Z )
% 0.29/0.46 => ~ ( ord_less_eq_b @ Z @ X ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_b @ Z @ X )
% 0.29/0.46 => ~ ( ord_less_eq_b @ X @ Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_201_order_Otrans,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ B2 @ C )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_202_le__cases,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ~ ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 => ( ord_less_eq_b @ Y @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_203_eq__refl,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( X = Y )
% 0.29/0.46 => ( ord_less_eq_b @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_204_linear,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 | ( ord_less_eq_b @ Y @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_205_antisym,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq_b @ Y @ X )
% 0.29/0.46 => ( X = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_206_eq__iff,axiom,
% 0.29/0.46 ( ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [X2: b,Y3: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X2 @ Y3 )
% 0.29/0.46 & ( ord_less_eq_b @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_207_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A2: b,B2: b,F: b > b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( ( F @ B2 )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X4: b,Y2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq_b @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_208_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A2: b,F: b > b,B2: b,C: b] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( F @ B2 ) )
% 0.29/0.46 => ( ( ord_less_eq_b @ B2 @ C )
% 0.29/0.46 => ( ! [X4: b,Y2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_209_order__subst2,axiom,
% 0.29/0.46 ! [A2: b,B2: b,F: b > b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
% 0.29/0.46 => ( ! [X4: b,Y2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq_b @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_210_order__subst1,axiom,
% 0.29/0.46 ! [A2: b,F: b > b,B2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ ( F @ B2 ) )
% 0.29/0.46 => ( ( ord_less_eq_b @ B2 @ C )
% 0.29/0.46 => ( ! [X4: b,Y2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_211_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X: b,A: set_b] :
% 0.29/0.46 ( ~ ( member_b @ X @ A )
% 0.29/0.46 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert_absorb
% 0.29/0.46 thf(fact_212_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X: a,A: set_a] :
% 0.29/0.46 ( ~ ( member_a @ X @ A )
% 0.29/0.46 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert_absorb
% 0.29/0.46 thf(fact_213_Diff__insert2,axiom,
% 0.29/0.46 ! [A: set_b,A2: b,B: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ A @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( minus_minus_set_b @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert2
% 0.29/0.46 thf(fact_214_Diff__insert2,axiom,
% 0.29/0.46 ! [A: set_a,A2: a,B: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert2
% 0.29/0.46 thf(fact_215_insert__Diff,axiom,
% 0.29/0.46 ! [A2: b,A: set_b] :
% 0.29/0.46 ( ( member_b @ A2 @ A )
% 0.29/0.46 => ( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff
% 0.29/0.46 thf(fact_216_insert__Diff,axiom,
% 0.29/0.46 ! [A2: a,A: set_a] :
% 0.29/0.46 ( ( member_a @ A2 @ A )
% 0.29/0.46 => ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff
% 0.29/0.46 thf(fact_217_Diff__insert,axiom,
% 0.29/0.46 ! [A: set_b,A2: b,B: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ A @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( minus_minus_set_b @ ( minus_minus_set_b @ A @ B ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert
% 0.29/0.46 thf(fact_218_Diff__insert,axiom,
% 0.29/0.46 ! [A: set_a,A2: a,B: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert
% 0.29/0.46 thf(fact_219_bot_Oextremum,axiom,
% 0.29/0.46 ! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_220_bot_Oextremum,axiom,
% 0.29/0.46 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_221_bot_Oextremum,axiom,
% 0.29/0.46 ! [A2: b] : ( ord_less_eq_b @ bot_bot_b @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_222_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
% 0.29/0.46 = ( A2 = bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_223_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.29/0.46 = ( A2 = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_224_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ bot_bot_b )
% 0.29/0.46 = ( A2 = bot_bot_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_225_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
% 0.29/0.46 => ( A2 = bot_bot_set_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_226_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.29/0.46 => ( A2 = bot_bot_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_227_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ bot_bot_b )
% 0.29/0.46 => ( A2 = bot_bot_b ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_228_is__singleton__def,axiom,
% 0.29/0.46 ( is_singleton_b
% 0.29/0.46 = ( ^ [A4: set_b] :
% 0.29/0.46 ? [X2: b] :
% 0.29/0.46 ( A4
% 0.29/0.46 = ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singleton_def
% 0.29/0.46 thf(fact_229_is__singleton__def,axiom,
% 0.29/0.46 ( is_singleton_a
% 0.29/0.46 = ( ^ [A4: set_a] :
% 0.29/0.46 ? [X2: a] :
% 0.29/0.46 ( A4
% 0.29/0.46 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singleton_def
% 0.29/0.46 thf(fact_230_is__singletonE,axiom,
% 0.29/0.46 ! [A: set_b] :
% 0.29/0.46 ( ( is_singleton_b @ A )
% 0.29/0.46 => ~ ! [X4: b] :
% 0.29/0.46 ( A
% 0.29/0.46 != ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonE
% 0.29/0.46 thf(fact_231_is__singletonE,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( is_singleton_a @ A )
% 0.29/0.46 => ~ ! [X4: a] :
% 0.29/0.46 ( A
% 0.29/0.46 != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonE
% 0.29/0.46 thf(fact_232_singletonD,axiom,
% 0.29/0.46 ! [B2: b,A2: b] :
% 0.29/0.46 ( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
% 0.29/0.46 => ( B2 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonD
% 0.29/0.46 thf(fact_233_singletonD,axiom,
% 0.29/0.46 ! [B2: a,A2: a] :
% 0.29/0.46 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 0.29/0.46 => ( B2 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonD
% 0.29/0.46 thf(fact_234_singleton__iff,axiom,
% 0.29/0.46 ! [B2: b,A2: b] :
% 0.29/0.46 ( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
% 0.29/0.46 = ( B2 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_iff
% 0.29/0.46 thf(fact_235_singleton__iff,axiom,
% 0.29/0.46 ! [B2: a,A2: a] :
% 0.29/0.46 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 0.29/0.46 = ( B2 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_iff
% 0.29/0.46 thf(fact_236_doubleton__eq__iff,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b,D: b] :
% 0.29/0.46 ( ( ( insert_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) )
% 0.29/0.46 = ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
% 0.29/0.46 = ( ( ( A2 = C )
% 0.29/0.46 & ( B2 = D ) )
% 0.29/0.46 | ( ( A2 = D )
% 0.29/0.46 & ( B2 = C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % doubleton_eq_iff
% 0.29/0.46 thf(fact_237_doubleton__eq__iff,axiom,
% 0.29/0.46 ! [A2: a,B2: a,C: a,D: a] :
% 0.29/0.46 ( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
% 0.29/0.46 = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 0.29/0.46 = ( ( ( A2 = C )
% 0.29/0.46 & ( B2 = D ) )
% 0.29/0.46 | ( ( A2 = D )
% 0.29/0.46 & ( B2 = C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % doubleton_eq_iff
% 0.29/0.46 thf(fact_238_insert__not__empty,axiom,
% 0.29/0.46 ! [A2: b,A: set_b] :
% 0.29/0.46 ( ( insert_b @ A2 @ A )
% 0.29/0.46 != bot_bot_set_b ) ).
% 0.29/0.46
% 0.29/0.46 % insert_not_empty
% 0.29/0.46 thf(fact_239_insert__not__empty,axiom,
% 0.29/0.46 ! [A2: a,A: set_a] :
% 0.29/0.46 ( ( insert_a @ A2 @ A )
% 0.29/0.46 != bot_bot_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % insert_not_empty
% 0.29/0.46 thf(fact_240_singleton__inject,axiom,
% 0.29/0.46 ! [A2: b,B2: b] :
% 0.29/0.46 ( ( ( insert_b @ A2 @ bot_bot_set_b )
% 0.29/0.46 = ( insert_b @ B2 @ bot_bot_set_b ) )
% 0.29/0.46 => ( A2 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_inject
% 0.29/0.46 thf(fact_241_singleton__inject,axiom,
% 0.29/0.46 ! [A2: a,B2: a] :
% 0.29/0.46 ( ( ( insert_a @ A2 @ bot_bot_set_a )
% 0.29/0.46 = ( insert_a @ B2 @ bot_bot_set_a ) )
% 0.29/0.46 => ( A2 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_inject
% 0.29/0.46 thf(fact_242_Int__insert__left,axiom,
% 0.29/0.46 ! [A2: b,C2: set_b,B: set_b] :
% 0.29/0.46 ( ( ( member_b @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) )
% 0.29/0.46 & ( ~ ( member_b @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left
% 0.29/0.46 thf(fact_243_Int__insert__left,axiom,
% 0.29/0.46 ! [A2: a,C2: set_a,B: set_a] :
% 0.29/0.46 ( ( ( member_a @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
% 0.29/0.46 & ( ~ ( member_a @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left
% 0.29/0.46 thf(fact_244_Int__insert__right,axiom,
% 0.29/0.46 ! [A2: b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( ( member_b @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) )
% 0.29/0.46 & ( ~ ( member_b @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 0.29/0.46 = ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right
% 0.29/0.46 thf(fact_245_Int__insert__right,axiom,
% 0.29/0.46 ! [A2: a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( ( member_a @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
% 0.29/0.46 & ( ~ ( member_a @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 0.29/0.46 = ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right
% 0.29/0.46 thf(fact_246_Int__Diff,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b,C2: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_Diff
% 0.29/0.46 thf(fact_247_Int__Diff,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_Diff
% 0.29/0.46 thf(fact_248_Diff__Int2,axiom,
% 0.29/0.46 ! [A: set_b,C2: set_b,B: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) )
% 0.29/0.46 = ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int2
% 0.29/0.46 thf(fact_249_Diff__Int2,axiom,
% 0.29/0.46 ! [A: set_a,C2: set_a,B: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
% 0.29/0.46 = ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int2
% 0.29/0.46 thf(fact_250_Diff__Diff__Int,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b] :
% 0.29/0.46 ( ( minus_minus_set_b @ A @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.46 = ( inf_inf_set_b @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Diff_Int
% 0.29/0.46 thf(fact_251_Diff__Diff__Int,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a] :
% 0.29/0.46 ( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.46 = ( inf_inf_set_a @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Diff_Int
% 0.29/0.46 thf(fact_252_Diff__Int__distrib,axiom,
% 0.29/0.46 ! [C2: set_b,A: set_b,B: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ C2 @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.46 = ( minus_minus_set_b @ ( inf_inf_set_b @ C2 @ A ) @ ( inf_inf_set_b @ C2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int_distrib
% 0.29/0.46 thf(fact_253_Diff__Int__distrib,axiom,
% 0.29/0.46 ! [C2: set_a,A: set_a,B: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.46 = ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int_distrib
% 0.29/0.46 thf(fact_254_Diff__Int__distrib2,axiom,
% 0.29/0.46 ! [A: set_b,B: set_b,C2: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ ( minus_minus_set_b @ A @ B ) @ C2 )
% 0.29/0.46 = ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int_distrib2
% 0.29/0.46 thf(fact_255_Diff__Int__distrib2,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
% 0.29/0.46 = ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int_distrib2
% 0.29/0.46 thf(fact_256_inf__sup__ord_I2_J,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(2)
% 0.29/0.46 thf(fact_257_inf__sup__ord_I2_J,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(2)
% 0.29/0.46 thf(fact_258_inf__sup__ord_I2_J,axiom,
% 0.29/0.46 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ Y ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(2)
% 0.29/0.46 thf(fact_259_inf__sup__ord_I1_J,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(1)
% 0.29/0.46 thf(fact_260_inf__sup__ord_I1_J,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(1)
% 0.29/0.46 thf(fact_261_inf__sup__ord_I1_J,axiom,
% 0.29/0.46 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(1)
% 0.29/0.46 thf(fact_262_inf__le1,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le1
% 0.29/0.46 thf(fact_263_inf__le1,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le1
% 0.29/0.46 thf(fact_264_inf__le1,axiom,
% 0.29/0.46 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le1
% 0.29/0.46 thf(fact_265_inf__le2,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le2
% 0.29/0.46 thf(fact_266_inf__le2,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le2
% 0.29/0.46 thf(fact_267_inf__le2,axiom,
% 0.29/0.46 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ Y ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le2
% 0.29/0.46 thf(fact_268_le__infE,axiom,
% 0.29/0.46 ! [X: set_b,A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_set_b @ X @ A2 )
% 0.29/0.46 => ~ ( ord_less_eq_set_b @ X @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infE
% 0.29/0.46 thf(fact_269_le__infE,axiom,
% 0.29/0.46 ! [X: set_a,A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_set_a @ X @ A2 )
% 0.29/0.46 => ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infE
% 0.29/0.46 thf(fact_270_le__infE,axiom,
% 0.29/0.46 ! [X: b,A2: b,B2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ ( inf_inf_b @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_b @ X @ A2 )
% 0.29/0.46 => ~ ( ord_less_eq_b @ X @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infE
% 0.29/0.46 thf(fact_271_le__infI,axiom,
% 0.29/0.46 ! [X: set_b,A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_set_b @ X @ B2 )
% 0.29/0.46 => ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI
% 0.29/0.46 thf(fact_272_le__infI,axiom,
% 0.29/0.46 ! [X: set_a,A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ X @ B2 )
% 0.29/0.46 => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI
% 0.29/0.46 thf(fact_273_le__infI,axiom,
% 0.29/0.46 ! [X: b,A2: b,B2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ X @ B2 )
% 0.29/0.46 => ( ord_less_eq_b @ X @ ( inf_inf_b @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI
% 0.29/0.46 thf(fact_274_inf__mono,axiom,
% 0.29/0.46 ! [A2: set_b,C: set_b,B2: set_b,D: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ C )
% 0.29/0.46 => ( ( ord_less_eq_set_b @ B2 @ D )
% 0.29/0.46 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_mono
% 0.29/0.46 thf(fact_275_inf__mono,axiom,
% 0.29/0.46 ! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ C )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ B2 @ D )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_mono
% 0.29/0.46 thf(fact_276_inf__mono,axiom,
% 0.29/0.46 ! [A2: b,C: b,B2: b,D: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ C )
% 0.29/0.46 => ( ( ord_less_eq_b @ B2 @ D )
% 0.29/0.46 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ ( inf_inf_b @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_mono
% 0.29/0.46 thf(fact_277_le__infI1,axiom,
% 0.29/0.46 ! [A2: set_b,X: set_b,B2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ X )
% 0.29/0.46 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI1
% 0.29/0.46 thf(fact_278_le__infI1,axiom,
% 0.29/0.46 ! [A2: set_a,X: set_a,B2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ X )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI1
% 0.29/0.46 thf(fact_279_le__infI1,axiom,
% 0.29/0.46 ! [A2: b,X: b,B2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ X )
% 0.29/0.46 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI1
% 0.29/0.46 thf(fact_280_le__infI2,axiom,
% 0.29/0.46 ! [B2: set_b,X: set_b,A2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ B2 @ X )
% 0.29/0.46 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI2
% 0.29/0.46 thf(fact_281_le__infI2,axiom,
% 0.29/0.46 ! [B2: set_a,X: set_a,A2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ B2 @ X )
% 0.29/0.46 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI2
% 0.29/0.46 thf(fact_282_le__infI2,axiom,
% 0.29/0.46 ! [B2: b,X: b,A2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ B2 @ X )
% 0.29/0.46 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI2
% 0.29/0.46 thf(fact_283_inf_OorderE,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderE
% 0.29/0.46 thf(fact_284_inf_OorderE,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderE
% 0.29/0.46 thf(fact_285_inf_OorderE,axiom,
% 0.29/0.46 ! [A2: b,B2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( inf_inf_b @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderE
% 0.29/0.46 thf(fact_286_inf_OorderI,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( inf_inf_set_b @ A2 @ B2 ) )
% 0.29/0.46 => ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderI
% 0.29/0.46 thf(fact_287_inf_OorderI,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( inf_inf_set_a @ A2 @ B2 ) )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderI
% 0.29/0.46 thf(fact_288_inf_OorderI,axiom,
% 0.29/0.46 ! [A2: b,B2: b] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( inf_inf_b @ A2 @ B2 ) )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderI
% 0.29/0.46 thf(fact_289_inf__unique,axiom,
% 0.29/0.46 ! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
% 0.29/0.46 ( ! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y2 ) @ X4 )
% 0.29/0.46 => ( ! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y2 ) @ Y2 )
% 0.29/0.46 => ( ! [X4: set_b,Y2: set_b,Z3: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_set_b @ X4 @ Z3 )
% 0.29/0.46 => ( ord_less_eq_set_b @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
% 0.29/0.46 => ( ( inf_inf_set_b @ X @ Y )
% 0.29/0.46 = ( F @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_unique
% 0.29/0.46 thf(fact_290_inf__unique,axiom,
% 0.29/0.46 ! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
% 0.29/0.46 ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ X4 )
% 0.29/0.46 => ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
% 0.29/0.46 => ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ X4 @ Z3 )
% 0.29/0.46 => ( ord_less_eq_set_a @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
% 0.29/0.46 => ( ( inf_inf_set_a @ X @ Y )
% 0.29/0.46 = ( F @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_unique
% 0.29/0.46 thf(fact_291_inf__unique,axiom,
% 0.29/0.46 ! [F: b > b > b,X: b,Y: b] :
% 0.29/0.46 ( ! [X4: b,Y2: b] : ( ord_less_eq_b @ ( F @ X4 @ Y2 ) @ X4 )
% 0.29/0.46 => ( ! [X4: b,Y2: b] : ( ord_less_eq_b @ ( F @ X4 @ Y2 ) @ Y2 )
% 0.29/0.46 => ( ! [X4: b,Y2: b,Z3: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ X4 @ Z3 )
% 0.29/0.46 => ( ord_less_eq_b @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
% 0.29/0.46 => ( ( inf_inf_b @ X @ Y )
% 0.29/0.46 = ( F @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_unique
% 0.29/0.46 thf(fact_292_le__iff__inf,axiom,
% 0.29/0.46 ( ord_less_eq_set_b
% 0.29/0.46 = ( ^ [X2: set_b,Y3: set_b] :
% 0.29/0.46 ( ( inf_inf_set_b @ X2 @ Y3 )
% 0.29/0.46 = X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_inf
% 0.29/0.46 thf(fact_293_le__iff__inf,axiom,
% 0.29/0.46 ( ord_less_eq_set_a
% 0.29/0.46 = ( ^ [X2: set_a,Y3: set_a] :
% 0.29/0.46 ( ( inf_inf_set_a @ X2 @ Y3 )
% 0.29/0.46 = X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_inf
% 0.29/0.46 thf(fact_294_le__iff__inf,axiom,
% 0.29/0.46 ( ord_less_eq_b
% 0.29/0.46 = ( ^ [X2: b,Y3: b] :
% 0.29/0.46 ( ( inf_inf_b @ X2 @ Y3 )
% 0.29/0.46 = X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_inf
% 0.29/0.46 thf(fact_295_inf_Oabsorb1,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.29/0.46 => ( ( inf_inf_set_b @ A2 @ B2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb1
% 0.29/0.46 thf(fact_296_inf_Oabsorb1,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.46 => ( ( inf_inf_set_a @ A2 @ B2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb1
% 0.29/0.46 thf(fact_297_inf_Oabsorb1,axiom,
% 0.29/0.46 ! [A2: b,B2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( inf_inf_b @ A2 @ B2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb1
% 0.29/0.46 thf(fact_298_inf_Oabsorb2,axiom,
% 0.29/0.46 ! [B2: set_b,A2: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ B2 @ A2 )
% 0.29/0.46 => ( ( inf_inf_set_b @ A2 @ B2 )
% 0.29/0.46 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb2
% 0.29/0.46 thf(fact_299_inf_Oabsorb2,axiom,
% 0.29/0.46 ! [B2: set_a,A2: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.29/0.46 => ( ( inf_inf_set_a @ A2 @ B2 )
% 0.29/0.46 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb2
% 0.29/0.46 thf(fact_300_inf_Oabsorb2,axiom,
% 0.29/0.46 ! [B2: b,A2: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ B2 @ A2 )
% 0.29/0.46 => ( ( inf_inf_b @ A2 @ B2 )
% 0.29/0.46 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb2
% 0.29/0.46 thf(fact_301_inf__absorb1,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ X @ Y )
% 0.29/0.46 => ( ( inf_inf_set_b @ X @ Y )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb1
% 0.29/0.46 thf(fact_302_inf__absorb1,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/0.46 => ( ( inf_inf_set_a @ X @ Y )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb1
% 0.29/0.46 thf(fact_303_inf__absorb1,axiom,
% 0.29/0.46 ! [X: b,Y: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.46 => ( ( inf_inf_b @ X @ Y )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb1
% 0.29/0.46 thf(fact_304_inf__absorb2,axiom,
% 0.29/0.46 ! [Y: set_b,X: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ Y @ X )
% 0.29/0.46 => ( ( inf_inf_set_b @ X @ Y )
% 0.29/0.46 = Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb2
% 0.29/0.46 thf(fact_305_inf__absorb2,axiom,
% 0.29/0.46 ! [Y: set_a,X: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ Y @ X )
% 0.29/0.46 => ( ( inf_inf_set_a @ X @ Y )
% 0.29/0.46 = Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb2
% 0.29/0.46 thf(fact_306_inf__absorb2,axiom,
% 0.29/0.46 ! [Y: b,X: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ Y @ X )
% 0.29/0.46 => ( ( inf_inf_b @ X @ Y )
% 0.29/0.46 = Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb2
% 0.29/0.46 thf(fact_307_inf_OboundedE,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b,C: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.29/0.46 => ~ ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedE
% 0.29/0.46 thf(fact_308_inf_OboundedE,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a,C: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.46 => ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedE
% 0.29/0.46 thf(fact_309_inf_OboundedE,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) )
% 0.29/0.46 => ~ ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ~ ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedE
% 0.29/0.46 thf(fact_310_inf_OboundedI,axiom,
% 0.29/0.46 ! [A2: set_b,B2: set_b,C: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq_set_b @ A2 @ C )
% 0.29/0.46 => ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedI
% 0.29/0.46 thf(fact_311_inf_OboundedI,axiom,
% 0.29/0.46 ! [A2: set_a,B2: set_a,C: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ A2 @ C )
% 0.29/0.46 => ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedI
% 0.29/0.46 thf(fact_312_inf_OboundedI,axiom,
% 0.29/0.46 ! [A2: b,B2: b,C: b] :
% 0.29/0.46 ( ( ord_less_eq_b @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq_b @ A2 @ C )
% 0.29/0.46 => ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedI
% 0.29/0.46 thf(fact_313_inf__greatest,axiom,
% 0.29/0.46 ! [X: set_b,Y: set_b,Z: set_b] :
% 0.29/0.46 ( ( ord_less_eq_set_b @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq_set_b @ X @ Z )
% 0.29/0.46 => ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_greatest
% 0.29/0.46 thf(fact_314_inf__greatest,axiom,
% 0.29/0.46 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq_set_a @ X @ Z )
% 0.29/0.46 => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf_greatest
% 0.29/0.47 thf(fact_315_inf__greatest,axiom,
% 0.29/0.47 ! [X: b,Y: b,Z: b] :
% 0.29/0.47 ( ( ord_less_eq_b @ X @ Y )
% 0.29/0.47 => ( ( ord_less_eq_b @ X @ Z )
% 0.29/0.47 => ( ord_less_eq_b @ X @ ( inf_inf_b @ Y @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf_greatest
% 0.29/0.47 thf(fact_316_inf_Oorder__iff,axiom,
% 0.29/0.47 ( ord_less_eq_set_b
% 0.29/0.47 = ( ^ [A3: set_b,B3: set_b] :
% 0.29/0.47 ( A3
% 0.29/0.47 = ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.order_iff
% 0.29/0.47 thf(fact_317_inf_Oorder__iff,axiom,
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 = ( ^ [A3: set_a,B3: set_a] :
% 0.29/0.47 ( A3
% 0.29/0.47 = ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.order_iff
% 0.29/0.47 thf(fact_318_inf_Oorder__iff,axiom,
% 0.29/0.47 ( ord_less_eq_b
% 0.29/0.47 = ( ^ [A3: b,B3: b] :
% 0.29/0.47 ( A3
% 0.29/0.47 = ( inf_inf_b @ A3 @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.order_iff
% 0.29/0.47 thf(fact_319_inf_Ocobounded1,axiom,
% 0.29/0.47 ! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % inf.cobounded1
% 0.29/0.47 thf(fact_320_inf_Ocobounded1,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % inf.cobounded1
% 0.29/0.47 thf(fact_321_inf_Ocobounded1,axiom,
% 0.29/0.47 ! [A2: b,B2: b] : ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % inf.cobounded1
% 0.29/0.47 thf(fact_322_inf_Ocobounded2,axiom,
% 0.29/0.47 ! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).
% 0.29/0.47
% 0.29/0.47 % inf.cobounded2
% 0.29/0.47 thf(fact_323_inf_Ocobounded2,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% 0.29/0.47
% 0.29/0.47 % inf.cobounded2
% 0.29/0.47 thf(fact_324_inf_Ocobounded2,axiom,
% 0.29/0.47 ! [A2: b,B2: b] : ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ B2 ) ).
% 0.29/0.47
% 0.29/0.47 % inf.cobounded2
% 0.29/0.47 thf(fact_325_inf_Oabsorb__iff1,axiom,
% 0.29/0.47 ( ord_less_eq_set_b
% 0.29/0.47 = ( ^ [A3: set_b,B3: set_b] :
% 0.29/0.47 ( ( inf_inf_set_b @ A3 @ B3 )
% 0.29/0.47 = A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.absorb_iff1
% 0.29/0.47 thf(fact_326_inf_Oabsorb__iff1,axiom,
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 = ( ^ [A3: set_a,B3: set_a] :
% 0.29/0.47 ( ( inf_inf_set_a @ A3 @ B3 )
% 0.29/0.47 = A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.absorb_iff1
% 0.29/0.47 thf(fact_327_inf_Oabsorb__iff1,axiom,
% 0.29/0.47 ( ord_less_eq_b
% 0.29/0.47 = ( ^ [A3: b,B3: b] :
% 0.29/0.47 ( ( inf_inf_b @ A3 @ B3 )
% 0.29/0.47 = A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.absorb_iff1
% 0.29/0.47 thf(fact_328_inf_Oabsorb__iff2,axiom,
% 0.29/0.47 ( ord_less_eq_set_b
% 0.29/0.47 = ( ^ [B3: set_b,A3: set_b] :
% 0.29/0.47 ( ( inf_inf_set_b @ A3 @ B3 )
% 0.29/0.47 = B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.absorb_iff2
% 0.29/0.47 thf(fact_329_inf_Oabsorb__iff2,axiom,
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 = ( ^ [B3: set_a,A3: set_a] :
% 0.29/0.47 ( ( inf_inf_set_a @ A3 @ B3 )
% 0.29/0.47 = B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.absorb_iff2
% 0.29/0.47 thf(fact_330_inf_Oabsorb__iff2,axiom,
% 0.29/0.47 ( ord_less_eq_b
% 0.29/0.47 = ( ^ [B3: b,A3: b] :
% 0.29/0.47 ( ( inf_inf_b @ A3 @ B3 )
% 0.29/0.47 = B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.absorb_iff2
% 0.29/0.47 thf(fact_331_inf_OcoboundedI1,axiom,
% 0.29/0.47 ! [A2: set_b,C: set_b,B2: set_b] :
% 0.29/0.47 ( ( ord_less_eq_set_b @ A2 @ C )
% 0.29/0.47 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.coboundedI1
% 0.29/0.47 thf(fact_332_inf_OcoboundedI1,axiom,
% 0.29/0.47 ! [A2: set_a,C: set_a,B2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ C )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.coboundedI1
% 0.29/0.47 thf(fact_333_inf_OcoboundedI1,axiom,
% 0.29/0.47 ! [A2: b,C: b,B2: b] :
% 0.29/0.47 ( ( ord_less_eq_b @ A2 @ C )
% 0.29/0.47 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.coboundedI1
% 0.29/0.47 thf(fact_334_inf_OcoboundedI2,axiom,
% 0.29/0.47 ! [B2: set_b,C: set_b,A2: set_b] :
% 0.29/0.47 ( ( ord_less_eq_set_b @ B2 @ C )
% 0.29/0.47 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.coboundedI2
% 0.29/0.47 thf(fact_335_inf_OcoboundedI2,axiom,
% 0.29/0.47 ! [B2: set_a,C: set_a,A2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ B2 @ C )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.coboundedI2
% 0.29/0.47 thf(fact_336_inf_OcoboundedI2,axiom,
% 0.29/0.47 ! [B2: b,C: b,A2: b] :
% 0.29/0.47 ( ( ord_less_eq_b @ B2 @ C )
% 0.29/0.47 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 % inf.coboundedI2
% 0.29/0.47 thf(fact_337_Diff__triv,axiom,
% 0.29/0.47 ! [A: set_b,B: set_b] :
% 0.29/0.47 ( ( ( inf_inf_set_b @ A @ B )
% 0.29/0.47 = bot_bot_set_b )
% 0.29/0.47 => ( ( minus_minus_set_b @ A @ B )
% 0.29/0.47 = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % Diff_triv
% 0.29/0.47 thf(fact_338_Diff__triv,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a] :
% 0.29/0.47 ( ( ( inf_inf_set_a @ A @ B )
% 0.29/0.47 = bot_bot_set_a )
% 0.29/0.47 => ( ( minus_minus_set_a @ A @ B )
% 0.29/0.47 = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % Diff_triv
% 0.29/0.47 thf(fact_339_open__delete,axiom,
% 0.29/0.47 ! [S3: set_b,X: b] :
% 0.29/0.47 ( ( topolo1276428102open_b @ S3 )
% 0.29/0.47 => ( topolo1276428102open_b @ ( minus_minus_set_b @ S3 @ ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % open_delete
% 0.29/0.47 thf(fact_340_Int__Diff__disjoint,axiom,
% 0.29/0.47 ! [A: set_b,B: set_b] :
% 0.29/0.47 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ B ) )
% 0.29/0.47 = bot_bot_set_b ) ).
% 0.29/0.47
% 0.29/0.47 % Int_Diff_disjoint
% 0.29/0.47 thf(fact_341_Int__Diff__disjoint,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a] :
% 0.29/0.47 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
% 0.29/0.47 = bot_bot_set_a ) ).
% 0.29/0.47
% 0.29/0.47 % Int_Diff_disjoint
% 0.29/0.47 thf(fact_342_is__singleton__the__elem,axiom,
% 0.29/0.47 ( is_singleton_b
% 0.29/0.47 = ( ^ [A4: set_b] :
% 0.29/0.47 ( A4
% 0.29/0.47 = ( insert_b @ ( the_elem_b @ A4 ) @ bot_bot_set_b ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_singleton_the_elem
% 0.29/0.47 thf(fact_343_is__singleton__the__elem,axiom,
% 0.29/0.47 ( is_singleton_a
% 0.29/0.47 = ( ^ [A4: set_a] :
% 0.29/0.47 ( A4
% 0.29/0.47 = ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % is_singleton_the_elem
% 0.29/0.47 thf(fact_344_the__elem__eq,axiom,
% 0.29/0.47 ! [X: b] :
% 0.29/0.47 ( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 % the_elem_eq
% 0.29/0.47 thf(fact_345_the__elem__eq,axiom,
% 0.29/0.47 ! [X: a] :
% 0.29/0.47 ( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 % the_elem_eq
% 0.29/0.47 thf(fact_346_empty__subsetI,axiom,
% 0.29/0.47 ! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_347_empty__subsetI,axiom,
% 0.29/0.47 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_348_subset__empty,axiom,
% 0.29/0.47 ! [A: set_b] :
% 0.29/0.47 ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
% 0.29/0.47 = ( A = bot_bot_set_b ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_empty
% 0.29/0.47 thf(fact_349_subset__empty,axiom,
% 0.29/0.47 ! [A: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.29/0.47 = ( A = bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_empty
% 0.29/0.47 thf(fact_350_insert__subset,axiom,
% 0.29/0.47 ! [X: b,A: set_b,B: set_b] :
% 0.29/0.47 ( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B )
% 0.29/0.47 = ( ( member_b @ X @ B )
% 0.29/0.47 & ( ord_less_eq_set_b @ A @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % insert_subset
% 0.29/0.47 thf(fact_351_insert__subset,axiom,
% 0.29/0.47 ! [X: a,A: set_a,B: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
% 0.29/0.47 = ( ( member_a @ X @ B )
% 0.29/0.47 & ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % insert_subset
% 0.29/0.47 thf(fact_352_Int__subset__iff,axiom,
% 0.29/0.47 ! [C2: set_b,A: set_b,B: set_b] :
% 0.29/0.47 ( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
% 0.29/0.47 = ( ( ord_less_eq_set_b @ C2 @ A )
% 0.29/0.47 & ( ord_less_eq_set_b @ C2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Int_subset_iff
% 0.29/0.47 thf(fact_353_Int__subset__iff,axiom,
% 0.29/0.47 ! [C2: set_a,A: set_a,B: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ C2 @ A )
% 0.29/0.47 & ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Int_subset_iff
% 0.29/0.47 thf(fact_354_singleton__insert__inj__eq,axiom,
% 0.29/0.47 ! [B2: a,A2: a,A: set_a] :
% 0.29/0.47 ( ( ( insert_a @ B2 @ bot_bot_set_a )
% 0.29/0.47 = ( insert_a @ A2 @ A ) )
% 0.29/0.47 = ( ( A2 = B2 )
% 0.29/0.47 & ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % singleton_insert_inj_eq
% 0.29/0.47
% 0.29/0.47 % Conjectures (2)
% 0.29/0.47 thf(conj_0,hypothesis,
% 0.29/0.47 ! [S2: set_b,V3: set_b] :
% 0.29/0.47 ( ( ( topolo1276428102open_b @ S2 )
% 0.29/0.47 & ( topolo1276428102open_b @ V3 )
% 0.29/0.47 & ( member_b @ ( f @ x0 ) @ S2 )
% 0.29/0.47 & ( member_b @ a2 @ V3 )
% 0.29/0.47 & ( ( inf_inf_set_b @ S2 @ V3 )
% 0.29/0.47 = bot_bot_set_b ) )
% 0.29/0.47 => thesis ) ).
% 0.29/0.47
% 0.29/0.47 thf(conj_1,conjecture,
% 0.29/0.47 thesis ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.48/1.45 ---- Embedded in THF ---
% 2.48/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.48/1.45 %%% Generated on Tue Feb 24 08:25:12 EST 2026
% 2.48/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.48/1.45 %%% Logic specification used:
% 2.48/1.45 %%% thf(spec, logic, $$dhol).
% 2.48/1.45
% 2.48/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2DTF12126.p
% See solution above
% 2.48/1.46 ------------------------
% 2.48/1.48 ---- Cleaned THF ---
% 2.48/1.48 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 2.48/1.48 set_b: $tType ).
% 2.48/1.48
% 2.48/1.48 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.48/1.48 set_a: $tType ).
% 2.48/1.48
% 2.48/1.48 thf(ty_n_tf__b,type,
% 2.48/1.48 b: $tType ).
% 2.48/1.48
% 2.48/1.48 thf(ty_n_tf__a,type,
% 2.48/1.48 a: $tType ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 2.48/1.48 minus_minus_set_a: set_a > set_a > set_a ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
% 2.48/1.48 minus_minus_set_b: set_b > set_b > set_b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
% 2.48/1.48 inf_inf_set_a: set_a > set_a > set_a ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
% 2.48/1.48 inf_inf_set_b: set_b > set_b > set_b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Lattices_Oinf__class_Oinf_001tf__b,type,
% 2.48/1.48 inf_inf_b: b > b > b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001tf__b,type,
% 2.48/1.48 lower_464587817at_a_b: a > ( a > b ) > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 2.48/1.48 bot_bot_a_o: a > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
% 2.48/1.48 bot_bot_b_o: b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 2.48/1.48 bot_bot_set_a: set_a ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
% 2.48/1.48 bot_bot_set_b: set_b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Obot__class_Obot_001tf__b,type,
% 2.48/1.48 bot_bot_b: b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.48/1.48 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
% 2.48/1.48 ord_less_eq_set_b: set_b > set_b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
% 2.48/1.48 ord_less_eq_b: b > b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.48/1.48 collect_a: ( a > $o ) > set_a ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_OCollect_001tf__b,type,
% 2.48/1.48 collect_b: ( b > $o ) > set_b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Oinsert_001tf__a,type,
% 2.48/1.48 insert_a: a > set_a > set_a ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Oinsert_001tf__b,type,
% 2.48/1.48 insert_b: b > set_b > set_b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 2.48/1.48 is_empty_a: set_a > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Ois__empty_001tf__b,type,
% 2.48/1.48 is_empty_b: set_b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Ois__singleton_001tf__a,type,
% 2.48/1.48 is_singleton_a: set_a > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Ois__singleton_001tf__b,type,
% 2.48/1.48 is_singleton_b: set_b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 2.48/1.48 the_elem_a: set_a > a ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Set_Othe__elem_001tf__b,type,
% 2.48/1.48 the_elem_b: set_b > b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
% 2.48/1.48 topolo1276428101open_a: set_a > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__b,type,
% 2.48/1.48 topolo1276428102open_b: set_b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_member_001tf__a,type,
% 2.48/1.48 member_a: a > set_a > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_c_member_001tf__b,type,
% 2.48/1.48 member_b: b > set_b > $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_v_A____,type,
% 2.48/1.48 a2: b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_v_f,type,
% 2.48/1.48 f: a > b ).
% 2.48/1.48
% 2.48/1.48 thf(sy_v_thesis____,type,
% 2.48/1.48 thesis: $o ).
% 2.48/1.48
% 2.48/1.48 thf(sy_v_x0,type,
% 2.48/1.48 x0: a ).
% 2.48/1.48
% 2.48/1.48 thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062,axiom,
% 2.48/1.48 ( a2
% 2.48/1.48 != ( f @ x0 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062,axiom,
% 2.48/1.48 ( ( ( f @ x0 )
% 2.48/1.48 != a2 )
% 2.48/1.48 => ? [U: set_b,V: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U )
% 2.48/1.48 & ( topolo1276428102open_b @ V )
% 2.48/1.48 & ( member_b @ ( f @ x0 ) @ U )
% 2.48/1.48 & ( member_b @ a2 @ V )
% 2.48/1.48 & ( ( inf_inf_set_b @ U @ V )
% 2.48/1.48 = bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_2_open__Int,axiom,
% 2.48/1.48 ! [S: set_b,T: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ S )
% 2.48/1.48 => ( ( topolo1276428102open_b @ T )
% 2.48/1.48 => ( topolo1276428102open_b @ ( inf_inf_set_b @ S @ T ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_3_open__Int,axiom,
% 2.48/1.48 ! [S: set_a,T: set_a] :
% 2.48/1.48 ( ( topolo1276428101open_a @ S )
% 2.48/1.48 => ( ( topolo1276428101open_a @ T )
% 2.48/1.48 => ( topolo1276428101open_a @ ( inf_inf_set_a @ S @ T ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_4_open__empty,axiom,
% 2.48/1.48 topolo1276428102open_b @ bot_bot_set_b ).
% 2.48/1.48
% 2.48/1.48 thf(fact_5_open__empty,axiom,
% 2.48/1.48 topolo1276428101open_a @ bot_bot_set_a ).
% 2.48/1.48
% 2.48/1.48 thf(fact_6_inf__bot__left,axiom,
% 2.48/1.48 ! [X: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ bot_bot_set_a @ X )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_7_inf__bot__left,axiom,
% 2.48/1.48 ! [X: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ bot_bot_set_b @ X )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_8_inf__bot__right,axiom,
% 2.48/1.48 ! [X: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X @ bot_bot_set_a )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_9_inf__bot__right,axiom,
% 2.48/1.48 ! [X: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X @ bot_bot_set_b )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_10_hausdorff,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X != Y )
% 2.48/1.48 => ? [U: set_b,V: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U )
% 2.48/1.48 & ( topolo1276428102open_b @ V )
% 2.48/1.48 & ( member_b @ X @ U )
% 2.48/1.48 & ( member_b @ Y @ V )
% 2.48/1.48 & ( ( inf_inf_set_b @ U @ V )
% 2.48/1.48 = bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_11_separation__t2,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X != Y )
% 2.48/1.48 = ( ? [U2: set_b,V2: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U2 )
% 2.48/1.48 & ( topolo1276428102open_b @ V2 )
% 2.48/1.48 & ( member_b @ X @ U2 )
% 2.48/1.48 & ( member_b @ Y @ V2 )
% 2.48/1.48 & ( ( inf_inf_set_b @ U2 @ V2 )
% 2.48/1.48 = bot_bot_set_b ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_12_IntI,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ A )
% 2.48/1.48 => ( ( member_a @ C @ B )
% 2.48/1.48 => ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_13_IntI,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ A )
% 2.48/1.48 => ( ( member_b @ C @ B )
% 2.48/1.48 => ( member_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_14_Int__iff,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 2.48/1.48 = ( ( member_a @ C @ A )
% 2.48/1.48 & ( member_a @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_15_Int__iff,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 2.48/1.48 = ( ( member_b @ C @ A )
% 2.48/1.48 & ( member_b @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_16_inf_Oidem,axiom,
% 2.48/1.48 ! [A2: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A2 @ A2 )
% 2.48/1.48 = A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_17_inf_Oidem,axiom,
% 2.48/1.48 ! [A2: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A2 @ A2 )
% 2.48/1.48 = A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_18_inf__idem,axiom,
% 2.48/1.48 ! [X: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X @ X )
% 2.48/1.48 = X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_19_inf__idem,axiom,
% 2.48/1.48 ! [X: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X @ X )
% 2.48/1.48 = X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_20_inf_Oleft__idem,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B2 ) )
% 2.48/1.48 = ( inf_inf_set_b @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_21_inf_Oleft__idem,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
% 2.48/1.48 = ( inf_inf_set_a @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_22_empty__Collect__eq,axiom,
% 2.48/1.48 ! [P: b > $o] :
% 2.48/1.48 ( ( bot_bot_set_b
% 2.48/1.48 = ( collect_b @ P ) )
% 2.48/1.48 = ( ! [X2: b] :
% 2.48/1.48 ~ ( P @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_23_empty__Collect__eq,axiom,
% 2.48/1.48 ! [P: a > $o] :
% 2.48/1.48 ( ( bot_bot_set_a
% 2.48/1.48 = ( collect_a @ P ) )
% 2.48/1.48 = ( ! [X2: a] :
% 2.48/1.48 ~ ( P @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_24_Collect__empty__eq,axiom,
% 2.48/1.48 ! [P: b > $o] :
% 2.48/1.48 ( ( ( collect_b @ P )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 = ( ! [X2: b] :
% 2.48/1.48 ~ ( P @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_25_Collect__empty__eq,axiom,
% 2.48/1.48 ! [P: a > $o] :
% 2.48/1.48 ( ( ( collect_a @ P )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 = ( ! [X2: a] :
% 2.48/1.48 ~ ( P @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_26_all__not__in__conv,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( ! [X2: b] :
% 2.48/1.48 ~ ( member_b @ X2 @ A ) )
% 2.48/1.48 = ( A = bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_27_all__not__in__conv,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( ! [X2: a] :
% 2.48/1.48 ~ ( member_a @ X2 @ A ) )
% 2.48/1.48 = ( A = bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_28_empty__iff,axiom,
% 2.48/1.48 ! [C: b] :
% 2.48/1.48 ~ ( member_b @ C @ bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_29_empty__iff,axiom,
% 2.48/1.48 ! [C: a] :
% 2.48/1.48 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_30_inf__right__idem,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y )
% 2.48/1.48 = ( inf_inf_set_b @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_31_inf__right__idem,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
% 2.48/1.48 = ( inf_inf_set_a @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_32_inf_Oright__idem,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 )
% 2.48/1.48 = ( inf_inf_set_b @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_33_inf_Oright__idem,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
% 2.48/1.48 = ( inf_inf_set_a @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_34_inf__left__idem,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
% 2.48/1.48 = ( inf_inf_set_b @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_35_inf__left__idem,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
% 2.48/1.48 = ( inf_inf_set_a @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_36__092_060open_062_092_060forall_062S_O_Aopen_AS_A_092_060and_062_Af_Ax0_A_092_060in_062_AS_A_092_060longrightarrow_062_A_I_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062x_H_092_060in_062T_O_Af_Ax_H_A_092_060le_062_Af_Ax0_A_092_060longrightarrow_062_Af_Ax_H_A_092_060in_062_AS_J_J_092_060close_062,axiom,
% 2.48/1.48 ! [S2: set_b] :
% 2.48/1.48 ( ( ( topolo1276428102open_b @ S2 )
% 2.48/1.48 & ( member_b @ ( f @ x0 ) @ S2 ) )
% 2.48/1.48 => ? [T2: set_a] :
% 2.48/1.48 ( ( topolo1276428101open_a @ T2 )
% 2.48/1.48 & ( member_a @ x0 @ T2 )
% 2.48/1.48 & ! [X3: a] :
% 2.48/1.48 ( ( member_a @ X3 @ T2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ ( f @ X3 ) @ ( f @ x0 ) )
% 2.48/1.48 => ( member_b @ ( f @ X3 ) @ S2 ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_37_bot__set__def,axiom,
% 2.48/1.48 ( bot_bot_set_b
% 2.48/1.48 = ( collect_b @ bot_bot_b_o ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_38_bot__set__def,axiom,
% 2.48/1.48 ( bot_bot_set_a
% 2.48/1.48 = ( collect_a @ bot_bot_a_o ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_39_ex__in__conv,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( ? [X2: b] : ( member_b @ X2 @ A ) )
% 2.48/1.48 = ( A != bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_40_ex__in__conv,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( ? [X2: a] : ( member_a @ X2 @ A ) )
% 2.48/1.48 = ( A != bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_41_equals0I,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ! [Y2: b] :
% 2.48/1.48 ~ ( member_b @ Y2 @ A )
% 2.48/1.48 => ( A = bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_42_equals0I,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ! [Y2: a] :
% 2.48/1.48 ~ ( member_a @ Y2 @ A )
% 2.48/1.48 => ( A = bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_43_equals0D,axiom,
% 2.48/1.48 ! [A: set_b,A2: b] :
% 2.48/1.48 ( ( A = bot_bot_set_b )
% 2.48/1.48 => ~ ( member_b @ A2 @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_44_equals0D,axiom,
% 2.48/1.48 ! [A: set_a,A2: a] :
% 2.48/1.48 ( ( A = bot_bot_set_a )
% 2.48/1.48 => ~ ( member_a @ A2 @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_45_emptyE,axiom,
% 2.48/1.48 ! [A2: b] :
% 2.48/1.48 ~ ( member_b @ A2 @ bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_46_emptyE,axiom,
% 2.48/1.48 ! [A2: a] :
% 2.48/1.48 ~ ( member_a @ A2 @ bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_47_inf__left__commute,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b,Z: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
% 2.48/1.48 = ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_48_inf__left__commute,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
% 2.48/1.48 = ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_49_inf_Oleft__commute,axiom,
% 2.48/1.48 ! [B2: set_b,A2: set_b,C: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ B2 @ ( inf_inf_set_b @ A2 @ C ) )
% 2.48/1.48 = ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_50_inf_Oleft__commute,axiom,
% 2.48/1.48 ! [B2: set_a,A2: set_a,C: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
% 2.48/1.48 = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_51_inf__commute,axiom,
% 2.48/1.48 ( inf_inf_set_b
% 2.48/1.48 = ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_52_inf__commute,axiom,
% 2.48/1.48 ( inf_inf_set_a
% 2.48/1.48 = ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_53_inf_Ocommute,axiom,
% 2.48/1.48 ( inf_inf_set_b
% 2.48/1.48 = ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_54_inf_Ocommute,axiom,
% 2.48/1.48 ( inf_inf_set_a
% 2.48/1.48 = ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_55_inf__assoc,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b,Z: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
% 2.48/1.48 = ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_56_inf__assoc,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
% 2.48/1.48 = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_57_inf_Oassoc,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b,C: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C )
% 2.48/1.48 = ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_58_inf_Oassoc,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a,C: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
% 2.48/1.48 = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_59_boolean__algebra__cancel_Oinf2,axiom,
% 2.48/1.48 ! [B: set_b,K: set_b,B2: set_b,A2: set_b] :
% 2.48/1.48 ( ( B
% 2.48/1.48 = ( inf_inf_set_b @ K @ B2 ) )
% 2.48/1.48 => ( ( inf_inf_set_b @ A2 @ B )
% 2.48/1.48 = ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_60_boolean__algebra__cancel_Oinf2,axiom,
% 2.48/1.48 ! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
% 2.48/1.48 ( ( B
% 2.48/1.48 = ( inf_inf_set_a @ K @ B2 ) )
% 2.48/1.48 => ( ( inf_inf_set_a @ A2 @ B )
% 2.48/1.48 = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_61_boolean__algebra__cancel_Oinf1,axiom,
% 2.48/1.48 ! [A: set_b,K: set_b,A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( inf_inf_set_b @ K @ A2 ) )
% 2.48/1.48 => ( ( inf_inf_set_b @ A @ B2 )
% 2.48/1.48 = ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_62_boolean__algebra__cancel_Oinf1,axiom,
% 2.48/1.48 ! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( inf_inf_set_a @ K @ A2 ) )
% 2.48/1.48 => ( ( inf_inf_set_a @ A @ B2 )
% 2.48/1.48 = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_63_inf__sup__aci_I1_J,axiom,
% 2.48/1.48 ( inf_inf_set_b
% 2.48/1.48 = ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_64_inf__sup__aci_I1_J,axiom,
% 2.48/1.48 ( inf_inf_set_a
% 2.48/1.48 = ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_65_inf__sup__aci_I2_J,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b,Z: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
% 2.48/1.48 = ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_66_inf__sup__aci_I2_J,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
% 2.48/1.48 = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_67_inf__sup__aci_I3_J,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b,Z: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
% 2.48/1.48 = ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_68_inf__sup__aci_I3_J,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
% 2.48/1.48 = ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_69_inf__sup__aci_I4_J,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
% 2.48/1.48 = ( inf_inf_set_b @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_70_inf__sup__aci_I4_J,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
% 2.48/1.48 = ( inf_inf_set_a @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_71_Int__left__commute,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b,C2: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
% 2.48/1.48 = ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_72_Int__left__commute,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a,C2: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
% 2.48/1.48 = ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_73_Int__left__absorb,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B ) )
% 2.48/1.48 = ( inf_inf_set_b @ A @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_74_Int__left__absorb,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
% 2.48/1.48 = ( inf_inf_set_a @ A @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_75_Int__commute,axiom,
% 2.48/1.48 ( inf_inf_set_b
% 2.48/1.48 = ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A4 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_76_Int__commute,axiom,
% 2.48/1.48 ( inf_inf_set_a
% 2.48/1.48 = ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_77_Int__absorb,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A @ A )
% 2.48/1.48 = A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_78_Int__absorb,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A @ A )
% 2.48/1.48 = A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_79_Int__assoc,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b,C2: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_80_Int__assoc,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a,C2: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_81_mem__Collect__eq,axiom,
% 2.48/1.48 ! [A2: b,P: b > $o] :
% 2.48/1.48 ( ( member_b @ A2 @ ( collect_b @ P ) )
% 2.48/1.48 = ( P @ A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_82_mem__Collect__eq,axiom,
% 2.48/1.48 ! [A2: a,P: a > $o] :
% 2.48/1.48 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 2.48/1.48 = ( P @ A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_83_Collect__mem__eq,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( collect_b
% 2.48/1.48 @ ^ [X2: b] : ( member_b @ X2 @ A ) )
% 2.48/1.48 = A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_84_Collect__mem__eq,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( collect_a
% 2.48/1.48 @ ^ [X2: a] : ( member_a @ X2 @ A ) )
% 2.48/1.48 = A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_85_IntD2,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 2.48/1.48 => ( member_b @ C @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_86_IntD2,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 2.48/1.48 => ( member_a @ C @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_87_IntD1,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 2.48/1.48 => ( member_b @ C @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_88_IntD1,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 2.48/1.48 => ( member_a @ C @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_89_IntE,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
% 2.48/1.48 => ~ ( ( member_b @ C @ A )
% 2.48/1.48 => ~ ( member_b @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_90_IntE,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
% 2.48/1.48 => ~ ( ( member_a @ C @ A )
% 2.48/1.48 => ~ ( member_a @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_91_separation__t1,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X != Y )
% 2.48/1.48 = ( ? [U2: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U2 )
% 2.48/1.48 & ( member_b @ X @ U2 )
% 2.48/1.48 & ~ ( member_b @ Y @ U2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_92_separation__t0,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X != Y )
% 2.48/1.48 = ( ? [U2: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U2 )
% 2.48/1.48 & ( ( member_b @ X @ U2 )
% 2.48/1.48 != ( member_b @ Y @ U2 ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_93_t1__space,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X != Y )
% 2.48/1.48 => ? [U: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U )
% 2.48/1.48 & ( member_b @ X @ U )
% 2.48/1.48 & ~ ( member_b @ Y @ U ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_94_t0__space,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X != Y )
% 2.48/1.48 => ? [U: set_b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ U )
% 2.48/1.48 & ( ( member_b @ X @ U )
% 2.48/1.48 != ( member_b @ Y @ U ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_95_disjoint__iff__not__equal,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( ( inf_inf_set_b @ A @ B )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 = ( ! [X2: b] :
% 2.48/1.48 ( ( member_b @ X2 @ A )
% 2.48/1.48 => ! [Y3: b] :
% 2.48/1.48 ( ( member_b @ Y3 @ B )
% 2.48/1.48 => ( X2 != Y3 ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_96_disjoint__iff__not__equal,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( ( inf_inf_set_a @ A @ B )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 = ( ! [X2: a] :
% 2.48/1.48 ( ( member_a @ X2 @ A )
% 2.48/1.48 => ! [Y3: a] :
% 2.48/1.48 ( ( member_a @ Y3 @ B )
% 2.48/1.48 => ( X2 != Y3 ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_97_Int__empty__right,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A @ bot_bot_set_b )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_98_Int__empty__right,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A @ bot_bot_set_a )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_99_Int__empty__left,axiom,
% 2.48/1.48 ! [B: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ bot_bot_set_b @ B )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_100_Int__empty__left,axiom,
% 2.48/1.48 ! [B: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ bot_bot_set_a @ B )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_101_disjoint__iff,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( ( inf_inf_set_b @ A @ B )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 = ( ! [X2: b] :
% 2.48/1.48 ( ( member_b @ X2 @ A )
% 2.48/1.48 => ~ ( member_b @ X2 @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_102_disjoint__iff,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( ( inf_inf_set_a @ A @ B )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 = ( ! [X2: a] :
% 2.48/1.48 ( ( member_a @ X2 @ A )
% 2.48/1.48 => ~ ( member_a @ X2 @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_103_Int__emptyI,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ! [X4: b] :
% 2.48/1.48 ( ( member_b @ X4 @ A )
% 2.48/1.48 => ~ ( member_b @ X4 @ B ) )
% 2.48/1.48 => ( ( inf_inf_set_b @ A @ B )
% 2.48/1.48 = bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_104_Int__emptyI,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ! [X4: a] :
% 2.48/1.48 ( ( member_a @ X4 @ A )
% 2.48/1.48 => ~ ( member_a @ X4 @ B ) )
% 2.48/1.48 => ( ( inf_inf_set_a @ A @ B )
% 2.48/1.48 = bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_105_calculation,axiom,
% 2.48/1.48 ( ~ ! [S2: set_b] :
% 2.48/1.48 ( ( ( topolo1276428102open_b @ S2 )
% 2.48/1.48 & ( member_b @ ( f @ x0 ) @ S2 ) )
% 2.48/1.48 => ? [T2: set_a] :
% 2.48/1.48 ( ( topolo1276428101open_a @ T2 )
% 2.48/1.48 & ( member_a @ x0 @ T2 )
% 2.48/1.48 & ! [X3: a] :
% 2.48/1.48 ( ( member_a @ X3 @ T2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ ( f @ X3 ) @ ( f @ x0 ) )
% 2.48/1.48 => ( member_b @ ( f @ X3 ) @ S2 ) ) ) ) )
% 2.48/1.48 => ~ ( lower_464587817at_a_b @ x0 @ f ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_106_Set_Ois__empty__def,axiom,
% 2.48/1.48 ( is_empty_b
% 2.48/1.48 = ( ^ [A4: set_b] : ( A4 = bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_107_Set_Ois__empty__def,axiom,
% 2.48/1.48 ( is_empty_a
% 2.48/1.48 = ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_108_Collect__empty__eq__bot,axiom,
% 2.48/1.48 ! [P: b > $o] :
% 2.48/1.48 ( ( ( collect_b @ P )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 = ( P = bot_bot_b_o ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_109_Collect__empty__eq__bot,axiom,
% 2.48/1.48 ! [P: a > $o] :
% 2.48/1.48 ( ( ( collect_a @ P )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 = ( P = bot_bot_a_o ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_110_bot__empty__eq,axiom,
% 2.48/1.48 ( bot_bot_b_o
% 2.48/1.48 = ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_111_bot__empty__eq,axiom,
% 2.48/1.48 ( bot_bot_a_o
% 2.48/1.48 = ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_112_is__singletonI_H,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( A != bot_bot_set_b )
% 2.48/1.48 => ( ! [X4: b,Y2: b] :
% 2.48/1.48 ( ( member_b @ X4 @ A )
% 2.48/1.48 => ( ( member_b @ Y2 @ A )
% 2.48/1.48 => ( X4 = Y2 ) ) )
% 2.48/1.48 => ( is_singleton_b @ A ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_113_is__singletonI_H,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( A != bot_bot_set_a )
% 2.48/1.48 => ( ! [X4: a,Y2: a] :
% 2.48/1.48 ( ( member_a @ X4 @ A )
% 2.48/1.48 => ( ( member_a @ Y2 @ A )
% 2.48/1.48 => ( X4 = Y2 ) ) )
% 2.48/1.48 => ( is_singleton_a @ A ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_114_Diff__disjoint,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ A ) )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_115_Diff__disjoint,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_116_insert__disjoint_I1_J,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 = ( ~ ( member_b @ A2 @ B )
% 2.48/1.48 & ( ( inf_inf_set_b @ A @ B )
% 2.48/1.48 = bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_117_insert__disjoint_I1_J,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 = ( ~ ( member_a @ A2 @ B )
% 2.48/1.48 & ( ( inf_inf_set_a @ A @ B )
% 2.48/1.48 = bot_bot_set_a ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_118_insert__disjoint_I2_J,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( bot_bot_set_b
% 2.48/1.48 = ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B ) )
% 2.48/1.48 = ( ~ ( member_b @ A2 @ B )
% 2.48/1.48 & ( bot_bot_set_b
% 2.48/1.48 = ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_119_insert__disjoint_I2_J,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( bot_bot_set_a
% 2.48/1.48 = ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
% 2.48/1.48 = ( ~ ( member_a @ A2 @ B )
% 2.48/1.48 & ( bot_bot_set_a
% 2.48/1.48 = ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_120_order__refl,axiom,
% 2.48/1.48 ! [X: b] : ( ord_less_eq_b @ X @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_121_insertCI,axiom,
% 2.48/1.48 ! [A2: b,B: set_b,B2: b] :
% 2.48/1.48 ( ( ~ ( member_b @ A2 @ B )
% 2.48/1.48 => ( A2 = B2 ) )
% 2.48/1.48 => ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_122_insertCI,axiom,
% 2.48/1.48 ! [A2: a,B: set_a,B2: a] :
% 2.48/1.48 ( ( ~ ( member_a @ A2 @ B )
% 2.48/1.48 => ( A2 = B2 ) )
% 2.48/1.48 => ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_123_insert__iff,axiom,
% 2.48/1.48 ! [A2: b,B2: b,A: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
% 2.48/1.48 = ( ( A2 = B2 )
% 2.48/1.48 | ( member_b @ A2 @ A ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_124_insert__iff,axiom,
% 2.48/1.48 ! [A2: a,B2: a,A: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
% 2.48/1.48 = ( ( A2 = B2 )
% 2.48/1.48 | ( member_a @ A2 @ A ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_125_DiffI,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ A )
% 2.48/1.48 => ( ~ ( member_b @ C @ B )
% 2.48/1.48 => ( member_b @ C @ ( minus_minus_set_b @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_126_DiffI,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ A )
% 2.48/1.48 => ( ~ ( member_a @ C @ B )
% 2.48/1.48 => ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_127_Diff__iff,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 = ( ( member_b @ C @ A )
% 2.48/1.48 & ~ ( member_b @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_128_Diff__iff,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 = ( ( member_a @ C @ A )
% 2.48/1.48 & ~ ( member_a @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_129_le__inf__iff,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b,Z: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
% 2.48/1.48 = ( ( ord_less_eq_set_b @ X @ Y )
% 2.48/1.48 & ( ord_less_eq_set_b @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_130_le__inf__iff,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
% 2.48/1.48 = ( ( ord_less_eq_set_a @ X @ Y )
% 2.48/1.48 & ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_131_le__inf__iff,axiom,
% 2.48/1.48 ! [X: b,Y: b,Z: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ ( inf_inf_b @ Y @ Z ) )
% 2.48/1.48 = ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 & ( ord_less_eq_b @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_132_inf_Obounded__iff,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b,C: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
% 2.48/1.48 = ( ( ord_less_eq_set_b @ A2 @ B2 )
% 2.48/1.48 & ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_133_inf_Obounded__iff,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a,C: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
% 2.48/1.48 = ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.48/1.48 & ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_134_inf_Obounded__iff,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) )
% 2.48/1.48 = ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 & ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_135_singletonI,axiom,
% 2.48/1.48 ! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_136_singletonI,axiom,
% 2.48/1.48 ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_137_Int__insert__left__if0,axiom,
% 2.48/1.48 ! [A2: b,C2: set_b,B: set_b] :
% 2.48/1.48 ( ~ ( member_b @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_b @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_138_Int__insert__left__if0,axiom,
% 2.48/1.48 ! [A2: a,C2: set_a,B: set_a] :
% 2.48/1.48 ( ~ ( member_a @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_a @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_139_Int__insert__left__if1,axiom,
% 2.48/1.48 ! [A2: b,C2: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_140_Int__insert__left__if1,axiom,
% 2.48/1.48 ! [A2: a,C2: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_141_insert__inter__insert,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_142_insert__inter__insert,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_143_Int__insert__right__if0,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ~ ( member_b @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( inf_inf_set_b @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_144_Int__insert__right__if0,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ~ ( member_a @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( inf_inf_set_a @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_145_Int__insert__right__if1,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_146_Int__insert__right__if1,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_147_Diff__empty,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ A @ bot_bot_set_b )
% 2.48/1.48 = A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_148_Diff__empty,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ A @ bot_bot_set_a )
% 2.48/1.48 = A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_149_empty__Diff,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ bot_bot_set_b @ A )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_150_empty__Diff,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ bot_bot_set_a @ A )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_151_Diff__cancel,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ A @ A )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_152_Diff__cancel,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ A @ A )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_153_Diff__insert0,axiom,
% 2.48/1.48 ! [X: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ~ ( member_b @ X @ A )
% 2.48/1.48 => ( ( minus_minus_set_b @ A @ ( insert_b @ X @ B ) )
% 2.48/1.48 = ( minus_minus_set_b @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_154_Diff__insert0,axiom,
% 2.48/1.48 ! [X: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ~ ( member_a @ X @ A )
% 2.48/1.48 => ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
% 2.48/1.48 = ( minus_minus_set_a @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_155_insert__Diff1,axiom,
% 2.48/1.48 ! [X: b,B: set_b,A: set_b] :
% 2.48/1.48 ( ( member_b @ X @ B )
% 2.48/1.48 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
% 2.48/1.48 = ( minus_minus_set_b @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_156_insert__Diff1,axiom,
% 2.48/1.48 ! [X: a,B: set_a,A: set_a] :
% 2.48/1.48 ( ( member_a @ X @ B )
% 2.48/1.48 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
% 2.48/1.48 = ( minus_minus_set_a @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_157_disjoint__insert_I2_J,axiom,
% 2.48/1.48 ! [A: set_b,B2: b,B: set_b] :
% 2.48/1.48 ( ( bot_bot_set_b
% 2.48/1.48 = ( inf_inf_set_b @ A @ ( insert_b @ B2 @ B ) ) )
% 2.48/1.48 = ( ~ ( member_b @ B2 @ A )
% 2.48/1.48 & ( bot_bot_set_b
% 2.48/1.48 = ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_158_disjoint__insert_I2_J,axiom,
% 2.48/1.48 ! [A: set_a,B2: a,B: set_a] :
% 2.48/1.48 ( ( bot_bot_set_a
% 2.48/1.48 = ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
% 2.48/1.48 = ( ~ ( member_a @ B2 @ A )
% 2.48/1.48 & ( bot_bot_set_a
% 2.48/1.48 = ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_159_disjoint__insert_I1_J,axiom,
% 2.48/1.48 ! [B: set_b,A2: b,A: set_b] :
% 2.48/1.48 ( ( ( inf_inf_set_b @ B @ ( insert_b @ A2 @ A ) )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 = ( ~ ( member_b @ A2 @ B )
% 2.48/1.48 & ( ( inf_inf_set_b @ B @ A )
% 2.48/1.48 = bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_160_disjoint__insert_I1_J,axiom,
% 2.48/1.48 ! [B: set_a,A2: a,A: set_a] :
% 2.48/1.48 ( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 = ( ~ ( member_a @ A2 @ B )
% 2.48/1.48 & ( ( inf_inf_set_a @ B @ A )
% 2.48/1.48 = bot_bot_set_a ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_161_insert__Diff__single,axiom,
% 2.48/1.48 ! [A2: b,A: set_b] :
% 2.48/1.48 ( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
% 2.48/1.48 = ( insert_b @ A2 @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_162_insert__Diff__single,axiom,
% 2.48/1.48 ! [A2: a,A: set_a] :
% 2.48/1.48 ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 2.48/1.48 = ( insert_a @ A2 @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_163_is__singletonI,axiom,
% 2.48/1.48 ! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_164_is__singletonI,axiom,
% 2.48/1.48 ! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_165_DiffE,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 => ~ ( ( member_b @ C @ A )
% 2.48/1.48 => ( member_b @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_166_DiffE,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 => ~ ( ( member_a @ C @ A )
% 2.48/1.48 => ( member_a @ C @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_167_DiffD1,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 => ( member_b @ C @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_168_DiffD1,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 => ( member_a @ C @ A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_169_DiffD2,axiom,
% 2.48/1.48 ! [C: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 => ~ ( member_b @ C @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_170_DiffD2,axiom,
% 2.48/1.48 ! [C: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 => ~ ( member_a @ C @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_171_insertE,axiom,
% 2.48/1.48 ! [A2: b,B2: b,A: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
% 2.48/1.48 => ( ( A2 != B2 )
% 2.48/1.48 => ( member_b @ A2 @ A ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_172_insertE,axiom,
% 2.48/1.48 ! [A2: a,B2: a,A: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
% 2.48/1.48 => ( ( A2 != B2 )
% 2.48/1.48 => ( member_a @ A2 @ A ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_173_insertI1,axiom,
% 2.48/1.48 ! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_174_insertI1,axiom,
% 2.48/1.48 ! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_175_insertI2,axiom,
% 2.48/1.48 ! [A2: b,B: set_b,B2: b] :
% 2.48/1.48 ( ( member_b @ A2 @ B )
% 2.48/1.48 => ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_176_insertI2,axiom,
% 2.48/1.48 ! [A2: a,B: set_a,B2: a] :
% 2.48/1.48 ( ( member_a @ A2 @ B )
% 2.48/1.48 => ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_177_Set_Oset__insert,axiom,
% 2.48/1.48 ! [X: b,A: set_b] :
% 2.48/1.48 ( ( member_b @ X @ A )
% 2.48/1.48 => ~ ! [B5: set_b] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( insert_b @ X @ B5 ) )
% 2.48/1.48 => ( member_b @ X @ B5 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_178_Set_Oset__insert,axiom,
% 2.48/1.48 ! [X: a,A: set_a] :
% 2.48/1.48 ( ( member_a @ X @ A )
% 2.48/1.48 => ~ ! [B5: set_a] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( insert_a @ X @ B5 ) )
% 2.48/1.48 => ( member_a @ X @ B5 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_179_insert__ident,axiom,
% 2.48/1.48 ! [X: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ~ ( member_b @ X @ A )
% 2.48/1.48 => ( ~ ( member_b @ X @ B )
% 2.48/1.48 => ( ( ( insert_b @ X @ A )
% 2.48/1.48 = ( insert_b @ X @ B ) )
% 2.48/1.48 = ( A = B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_180_insert__ident,axiom,
% 2.48/1.48 ! [X: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ~ ( member_a @ X @ A )
% 2.48/1.48 => ( ~ ( member_a @ X @ B )
% 2.48/1.48 => ( ( ( insert_a @ X @ A )
% 2.48/1.48 = ( insert_a @ X @ B ) )
% 2.48/1.48 = ( A = B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_181_insert__absorb,axiom,
% 2.48/1.48 ! [A2: b,A: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ A )
% 2.48/1.48 => ( ( insert_b @ A2 @ A )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_182_insert__absorb,axiom,
% 2.48/1.48 ! [A2: a,A: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ A )
% 2.48/1.48 => ( ( insert_a @ A2 @ A )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_183_insert__eq__iff,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B2: b,B: set_b] :
% 2.48/1.48 ( ~ ( member_b @ A2 @ A )
% 2.48/1.48 => ( ~ ( member_b @ B2 @ B )
% 2.48/1.48 => ( ( ( insert_b @ A2 @ A )
% 2.48/1.48 = ( insert_b @ B2 @ B ) )
% 2.48/1.48 = ( ( ( A2 = B2 )
% 2.48/1.48 => ( A = B ) )
% 2.48/1.48 & ( ( A2 != B2 )
% 2.48/1.48 => ? [C3: set_b] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( insert_b @ B2 @ C3 ) )
% 2.48/1.48 & ~ ( member_b @ B2 @ C3 )
% 2.48/1.48 & ( B
% 2.48/1.48 = ( insert_b @ A2 @ C3 ) )
% 2.48/1.48 & ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_184_insert__eq__iff,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B2: a,B: set_a] :
% 2.48/1.48 ( ~ ( member_a @ A2 @ A )
% 2.48/1.48 => ( ~ ( member_a @ B2 @ B )
% 2.48/1.48 => ( ( ( insert_a @ A2 @ A )
% 2.48/1.48 = ( insert_a @ B2 @ B ) )
% 2.48/1.48 = ( ( ( A2 = B2 )
% 2.48/1.48 => ( A = B ) )
% 2.48/1.48 & ( ( A2 != B2 )
% 2.48/1.48 => ? [C3: set_a] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( insert_a @ B2 @ C3 ) )
% 2.48/1.48 & ~ ( member_a @ B2 @ C3 )
% 2.48/1.48 & ( B
% 2.48/1.48 = ( insert_a @ A2 @ C3 ) )
% 2.48/1.48 & ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_185_insert__Diff__if,axiom,
% 2.48/1.48 ! [X: b,B: set_b,A: set_b] :
% 2.48/1.48 ( ( ( member_b @ X @ B )
% 2.48/1.48 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
% 2.48/1.48 = ( minus_minus_set_b @ A @ B ) ) )
% 2.48/1.48 & ( ~ ( member_b @ X @ B )
% 2.48/1.48 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
% 2.48/1.48 = ( insert_b @ X @ ( minus_minus_set_b @ A @ B ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_186_insert__Diff__if,axiom,
% 2.48/1.48 ! [X: a,B: set_a,A: set_a] :
% 2.48/1.48 ( ( ( member_a @ X @ B )
% 2.48/1.48 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
% 2.48/1.48 = ( minus_minus_set_a @ A @ B ) ) )
% 2.48/1.48 & ( ~ ( member_a @ X @ B )
% 2.48/1.48 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
% 2.48/1.48 = ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_187_mk__disjoint__insert,axiom,
% 2.48/1.48 ! [A2: b,A: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ A )
% 2.48/1.48 => ? [B5: set_b] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( insert_b @ A2 @ B5 ) )
% 2.48/1.48 & ~ ( member_b @ A2 @ B5 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_188_mk__disjoint__insert,axiom,
% 2.48/1.48 ! [A2: a,A: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ A )
% 2.48/1.48 => ? [B5: set_a] :
% 2.48/1.48 ( ( A
% 2.48/1.48 = ( insert_a @ A2 @ B5 ) )
% 2.48/1.48 & ~ ( member_a @ A2 @ B5 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_189_dual__order_Oantisym,axiom,
% 2.48/1.48 ! [B2: b,A2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ B2 @ A2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( A2 = B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_190_dual__order_Oeq__iff,axiom,
% 2.48/1.48 ( ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 ) )
% 2.48/1.48 = ( ^ [A3: b,B3: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ B3 @ A3 )
% 2.48/1.48 & ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_191_dual__order_Otrans,axiom,
% 2.48/1.48 ! [B2: b,A2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ B2 @ A2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ C @ B2 )
% 2.48/1.48 => ( ord_less_eq_b @ C @ A2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_192_linorder__wlog,axiom,
% 2.48/1.48 ! [P: b > b > $o,A2: b,B2: b] :
% 2.48/1.48 ( ! [A5: b,B6: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A5 @ B6 )
% 2.48/1.48 => ( P @ A5 @ B6 ) )
% 2.48/1.48 => ( ! [A5: b,B6: b] :
% 2.48/1.48 ( ( P @ B6 @ A5 )
% 2.48/1.48 => ( P @ A5 @ B6 ) )
% 2.48/1.48 => ( P @ A2 @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_193_dual__order_Orefl,axiom,
% 2.48/1.48 ! [A2: b] : ( ord_less_eq_b @ A2 @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_194_order__trans,axiom,
% 2.48/1.48 ! [X: b,Y: b,Z: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 => ( ( ord_less_eq_b @ Y @ Z )
% 2.48/1.48 => ( ord_less_eq_b @ X @ Z ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_195_order__class_Oorder_Oantisym,axiom,
% 2.48/1.48 ! [A2: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ B2 @ A2 )
% 2.48/1.48 => ( A2 = B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_196_ord__le__eq__trans,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( B2 = C )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_197_ord__eq__le__trans,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b] :
% 2.48/1.48 ( ( A2 = B2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ B2 @ C )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_198_order__class_Oorder_Oeq__iff,axiom,
% 2.48/1.48 ( ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 ) )
% 2.48/1.48 = ( ^ [A3: b,B3: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A3 @ B3 )
% 2.48/1.48 & ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_199_antisym__conv,axiom,
% 2.48/1.48 ! [Y: b,X: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ Y @ X )
% 2.48/1.48 => ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 = ( X = Y ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_200_le__cases3,axiom,
% 2.48/1.48 ! [X: b,Y: b,Z: b] :
% 2.48/1.48 ( ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 => ~ ( ord_less_eq_b @ Y @ Z ) )
% 2.48/1.48 => ( ( ( ord_less_eq_b @ Y @ X )
% 2.48/1.48 => ~ ( ord_less_eq_b @ X @ Z ) )
% 2.48/1.48 => ( ( ( ord_less_eq_b @ X @ Z )
% 2.48/1.48 => ~ ( ord_less_eq_b @ Z @ Y ) )
% 2.48/1.48 => ( ( ( ord_less_eq_b @ Z @ Y )
% 2.48/1.48 => ~ ( ord_less_eq_b @ Y @ X ) )
% 2.48/1.48 => ( ( ( ord_less_eq_b @ Y @ Z )
% 2.48/1.48 => ~ ( ord_less_eq_b @ Z @ X ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_b @ Z @ X )
% 2.48/1.48 => ~ ( ord_less_eq_b @ X @ Y ) ) ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_201_order_Otrans,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ B2 @ C )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_202_le__cases,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ~ ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 => ( ord_less_eq_b @ Y @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_203_eq__refl,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( X = Y )
% 2.48/1.48 => ( ord_less_eq_b @ X @ Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_204_linear,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 | ( ord_less_eq_b @ Y @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_205_antisym,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 => ( ( ord_less_eq_b @ Y @ X )
% 2.48/1.48 => ( X = Y ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_206_eq__iff,axiom,
% 2.48/1.48 ( ( ^ [Y4: b,Z2: b] : ( Y4 = Z2 ) )
% 2.48/1.48 = ( ^ [X2: b,Y3: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X2 @ Y3 )
% 2.48/1.48 & ( ord_less_eq_b @ Y3 @ X2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_207_ord__le__eq__subst,axiom,
% 2.48/1.48 ! [A2: b,B2: b,F: b > b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( ( F @ B2 )
% 2.48/1.48 = C )
% 2.48/1.48 => ( ! [X4: b,Y2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X4 @ Y2 )
% 2.48/1.48 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.48 => ( ord_less_eq_b @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_208_ord__eq__le__subst,axiom,
% 2.48/1.48 ! [A2: b,F: b > b,B2: b,C: b] :
% 2.48/1.48 ( ( A2
% 2.48/1.48 = ( F @ B2 ) )
% 2.48/1.48 => ( ( ord_less_eq_b @ B2 @ C )
% 2.48/1.48 => ( ! [X4: b,Y2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X4 @ Y2 )
% 2.48/1.48 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_209_order__subst2,axiom,
% 2.48/1.48 ! [A2: b,B2: b,F: b > b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
% 2.48/1.48 => ( ! [X4: b,Y2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X4 @ Y2 )
% 2.48/1.48 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.48 => ( ord_less_eq_b @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_210_order__subst1,axiom,
% 2.48/1.48 ! [A2: b,F: b > b,B2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ ( F @ B2 ) )
% 2.48/1.48 => ( ( ord_less_eq_b @ B2 @ C )
% 2.48/1.48 => ( ! [X4: b,Y2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X4 @ Y2 )
% 2.48/1.48 => ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_211_Diff__insert__absorb,axiom,
% 2.48/1.48 ! [X: b,A: set_b] :
% 2.48/1.48 ( ~ ( member_b @ X @ A )
% 2.48/1.48 => ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ ( insert_b @ X @ bot_bot_set_b ) )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_212_Diff__insert__absorb,axiom,
% 2.48/1.48 ! [X: a,A: set_a] :
% 2.48/1.48 ( ~ ( member_a @ X @ A )
% 2.48/1.48 => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_213_Diff__insert2,axiom,
% 2.48/1.48 ! [A: set_b,A2: b,B: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ A @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( minus_minus_set_b @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_214_Diff__insert2,axiom,
% 2.48/1.48 ! [A: set_a,A2: a,B: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_215_insert__Diff,axiom,
% 2.48/1.48 ! [A2: b,A: set_b] :
% 2.48/1.48 ( ( member_b @ A2 @ A )
% 2.48/1.48 => ( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_216_insert__Diff,axiom,
% 2.48/1.48 ! [A2: a,A: set_a] :
% 2.48/1.48 ( ( member_a @ A2 @ A )
% 2.48/1.48 => ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_217_Diff__insert,axiom,
% 2.48/1.48 ! [A: set_b,A2: b,B: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ A @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( minus_minus_set_b @ ( minus_minus_set_b @ A @ B ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_218_Diff__insert,axiom,
% 2.48/1.48 ! [A: set_a,A2: a,B: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_219_bot_Oextremum,axiom,
% 2.48/1.48 ! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_220_bot_Oextremum,axiom,
% 2.48/1.48 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_221_bot_Oextremum,axiom,
% 2.48/1.48 ! [A2: b] : ( ord_less_eq_b @ bot_bot_b @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_222_bot_Oextremum__unique,axiom,
% 2.48/1.48 ! [A2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
% 2.48/1.48 = ( A2 = bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_223_bot_Oextremum__unique,axiom,
% 2.48/1.48 ! [A2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 2.48/1.48 = ( A2 = bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_224_bot_Oextremum__unique,axiom,
% 2.48/1.48 ! [A2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ bot_bot_b )
% 2.48/1.48 = ( A2 = bot_bot_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_225_bot_Oextremum__uniqueI,axiom,
% 2.48/1.48 ! [A2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
% 2.48/1.48 => ( A2 = bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_226_bot_Oextremum__uniqueI,axiom,
% 2.48/1.48 ! [A2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 2.48/1.48 => ( A2 = bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_227_bot_Oextremum__uniqueI,axiom,
% 2.48/1.48 ! [A2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ bot_bot_b )
% 2.48/1.48 => ( A2 = bot_bot_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_228_is__singleton__def,axiom,
% 2.48/1.48 ( is_singleton_b
% 2.48/1.48 = ( ^ [A4: set_b] :
% 2.48/1.48 ? [X2: b] :
% 2.48/1.48 ( A4
% 2.48/1.48 = ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_229_is__singleton__def,axiom,
% 2.48/1.48 ( is_singleton_a
% 2.48/1.48 = ( ^ [A4: set_a] :
% 2.48/1.48 ? [X2: a] :
% 2.48/1.48 ( A4
% 2.48/1.48 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_230_is__singletonE,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( is_singleton_b @ A )
% 2.48/1.48 => ~ ! [X4: b] :
% 2.48/1.48 ( A
% 2.48/1.48 != ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_231_is__singletonE,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( is_singleton_a @ A )
% 2.48/1.48 => ~ ! [X4: a] :
% 2.48/1.48 ( A
% 2.48/1.48 != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_232_singletonD,axiom,
% 2.48/1.48 ! [B2: b,A2: b] :
% 2.48/1.48 ( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
% 2.48/1.48 => ( B2 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_233_singletonD,axiom,
% 2.48/1.48 ! [B2: a,A2: a] :
% 2.48/1.48 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 2.48/1.48 => ( B2 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_234_singleton__iff,axiom,
% 2.48/1.48 ! [B2: b,A2: b] :
% 2.48/1.48 ( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
% 2.48/1.48 = ( B2 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_235_singleton__iff,axiom,
% 2.48/1.48 ! [B2: a,A2: a] :
% 2.48/1.48 ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
% 2.48/1.48 = ( B2 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_236_doubleton__eq__iff,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b,D: b] :
% 2.48/1.48 ( ( ( insert_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) )
% 2.48/1.48 = ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
% 2.48/1.48 = ( ( ( A2 = C )
% 2.48/1.48 & ( B2 = D ) )
% 2.48/1.48 | ( ( A2 = D )
% 2.48/1.48 & ( B2 = C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_237_doubleton__eq__iff,axiom,
% 2.48/1.48 ! [A2: a,B2: a,C: a,D: a] :
% 2.48/1.48 ( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
% 2.48/1.48 = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 2.48/1.48 = ( ( ( A2 = C )
% 2.48/1.48 & ( B2 = D ) )
% 2.48/1.48 | ( ( A2 = D )
% 2.48/1.48 & ( B2 = C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_238_insert__not__empty,axiom,
% 2.48/1.48 ! [A2: b,A: set_b] :
% 2.48/1.48 ( ( insert_b @ A2 @ A )
% 2.48/1.48 != bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_239_insert__not__empty,axiom,
% 2.48/1.48 ! [A2: a,A: set_a] :
% 2.48/1.48 ( ( insert_a @ A2 @ A )
% 2.48/1.48 != bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_240_singleton__inject,axiom,
% 2.48/1.48 ! [A2: b,B2: b] :
% 2.48/1.48 ( ( ( insert_b @ A2 @ bot_bot_set_b )
% 2.48/1.48 = ( insert_b @ B2 @ bot_bot_set_b ) )
% 2.48/1.48 => ( A2 = B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_241_singleton__inject,axiom,
% 2.48/1.48 ! [A2: a,B2: a] :
% 2.48/1.48 ( ( ( insert_a @ A2 @ bot_bot_set_a )
% 2.48/1.48 = ( insert_a @ B2 @ bot_bot_set_a ) )
% 2.48/1.48 => ( A2 = B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_242_Int__insert__left,axiom,
% 2.48/1.48 ! [A2: b,C2: set_b,B: set_b] :
% 2.48/1.48 ( ( ( member_b @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) )
% 2.48/1.48 & ( ~ ( member_b @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_243_Int__insert__left,axiom,
% 2.48/1.48 ! [A2: a,C2: set_a,B: set_a] :
% 2.48/1.48 ( ( ( member_a @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
% 2.48/1.48 & ( ~ ( member_a @ A2 @ C2 )
% 2.48/1.48 => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_244_Int__insert__right,axiom,
% 2.48/1.48 ! [A2: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( ( member_b @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) )
% 2.48/1.48 & ( ~ ( member_b @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
% 2.48/1.48 = ( inf_inf_set_b @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_245_Int__insert__right,axiom,
% 2.48/1.48 ! [A2: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( ( member_a @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
% 2.48/1.48 & ( ~ ( member_a @ A2 @ A )
% 2.48/1.48 => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
% 2.48/1.48 = ( inf_inf_set_a @ A @ B ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_246_Int__Diff,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b,C2: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_247_Int__Diff,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a,C2: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
% 2.48/1.48 = ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_248_Diff__Int2,axiom,
% 2.48/1.48 ! [A: set_b,C2: set_b,B: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) )
% 2.48/1.48 = ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_249_Diff__Int2,axiom,
% 2.48/1.48 ! [A: set_a,C2: set_a,B: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
% 2.48/1.48 = ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_250_Diff__Diff__Int,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( minus_minus_set_b @ A @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 = ( inf_inf_set_b @ A @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_251_Diff__Diff__Int,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 = ( inf_inf_set_a @ A @ B ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_252_Diff__Int__distrib,axiom,
% 2.48/1.48 ! [C2: set_b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ C2 @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 = ( minus_minus_set_b @ ( inf_inf_set_b @ C2 @ A ) @ ( inf_inf_set_b @ C2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_253_Diff__Int__distrib,axiom,
% 2.48/1.48 ! [C2: set_a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 = ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_254_Diff__Int__distrib2,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b,C2: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( minus_minus_set_b @ A @ B ) @ C2 )
% 2.48/1.48 = ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_255_Diff__Int__distrib2,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a,C2: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
% 2.48/1.48 = ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_256_inf__sup__ord_I2_J,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_257_inf__sup__ord_I2_J,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_258_inf__sup__ord_I2_J,axiom,
% 2.48/1.48 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ Y ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_259_inf__sup__ord_I1_J,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_260_inf__sup__ord_I1_J,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_261_inf__sup__ord_I1_J,axiom,
% 2.48/1.48 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_262_inf__le1,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_263_inf__le1,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_264_inf__le1,axiom,
% 2.48/1.48 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_265_inf__le2,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_266_inf__le2,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_267_inf__le2,axiom,
% 2.48/1.48 ! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ Y ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_268_le__infE,axiom,
% 2.48/1.48 ! [X: set_b,A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_set_b @ X @ A2 )
% 2.48/1.48 => ~ ( ord_less_eq_set_b @ X @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_269_le__infE,axiom,
% 2.48/1.48 ! [X: set_a,A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_set_a @ X @ A2 )
% 2.48/1.48 => ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_270_le__infE,axiom,
% 2.48/1.48 ! [X: b,A2: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ ( inf_inf_b @ A2 @ B2 ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_b @ X @ A2 )
% 2.48/1.48 => ~ ( ord_less_eq_b @ X @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_271_le__infI,axiom,
% 2.48/1.48 ! [X: set_b,A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ X @ A2 )
% 2.48/1.48 => ( ( ord_less_eq_set_b @ X @ B2 )
% 2.48/1.48 => ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_272_le__infI,axiom,
% 2.48/1.48 ! [X: set_a,A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ X @ A2 )
% 2.48/1.48 => ( ( ord_less_eq_set_a @ X @ B2 )
% 2.48/1.48 => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_273_le__infI,axiom,
% 2.48/1.48 ! [X: b,A2: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ A2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ X @ B2 )
% 2.48/1.48 => ( ord_less_eq_b @ X @ ( inf_inf_b @ A2 @ B2 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_274_inf__mono,axiom,
% 2.48/1.48 ! [A2: set_b,C: set_b,B2: set_b,D: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ C )
% 2.48/1.48 => ( ( ord_less_eq_set_b @ B2 @ D )
% 2.48/1.48 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_275_inf__mono,axiom,
% 2.48/1.48 ! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ C )
% 2.48/1.48 => ( ( ord_less_eq_set_a @ B2 @ D )
% 2.48/1.48 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_276_inf__mono,axiom,
% 2.48/1.48 ! [A2: b,C: b,B2: b,D: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ C )
% 2.48/1.48 => ( ( ord_less_eq_b @ B2 @ D )
% 2.48/1.48 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ ( inf_inf_b @ C @ D ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_277_le__infI1,axiom,
% 2.48/1.48 ! [A2: set_b,X: set_b,B2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ X )
% 2.48/1.48 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_278_le__infI1,axiom,
% 2.48/1.48 ! [A2: set_a,X: set_a,B2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ X )
% 2.48/1.48 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_279_le__infI1,axiom,
% 2.48/1.48 ! [A2: b,X: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ X )
% 2.48/1.48 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_280_le__infI2,axiom,
% 2.48/1.48 ! [B2: set_b,X: set_b,A2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ B2 @ X )
% 2.48/1.48 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_281_le__infI2,axiom,
% 2.48/1.48 ! [B2: set_a,X: set_a,A2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ B2 @ X )
% 2.48/1.48 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_282_le__infI2,axiom,
% 2.48/1.48 ! [B2: b,X: b,A2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ B2 @ X )
% 2.48/1.48 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_283_inf_OorderE,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 2.48/1.48 => ( A2
% 2.48/1.48 = ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_284_inf_OorderE,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.48/1.48 => ( A2
% 2.48/1.48 = ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_285_inf_OorderE,axiom,
% 2.48/1.48 ! [A2: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( A2
% 2.48/1.48 = ( inf_inf_b @ A2 @ B2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_286_inf_OorderI,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( A2
% 2.48/1.48 = ( inf_inf_set_b @ A2 @ B2 ) )
% 2.48/1.48 => ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_287_inf_OorderI,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( A2
% 2.48/1.48 = ( inf_inf_set_a @ A2 @ B2 ) )
% 2.48/1.48 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_288_inf_OorderI,axiom,
% 2.48/1.48 ! [A2: b,B2: b] :
% 2.48/1.48 ( ( A2
% 2.48/1.48 = ( inf_inf_b @ A2 @ B2 ) )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_289_inf__unique,axiom,
% 2.48/1.48 ! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
% 2.48/1.48 ( ! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y2 ) @ X4 )
% 2.48/1.48 => ( ! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y2 ) @ Y2 )
% 2.48/1.48 => ( ! [X4: set_b,Y2: set_b,Z3: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ X4 @ Y2 )
% 2.48/1.48 => ( ( ord_less_eq_set_b @ X4 @ Z3 )
% 2.48/1.48 => ( ord_less_eq_set_b @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
% 2.48/1.48 => ( ( inf_inf_set_b @ X @ Y )
% 2.48/1.48 = ( F @ X @ Y ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_290_inf__unique,axiom,
% 2.48/1.48 ! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
% 2.48/1.48 ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ X4 )
% 2.48/1.48 => ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
% 2.48/1.48 => ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ X4 @ Y2 )
% 2.48/1.48 => ( ( ord_less_eq_set_a @ X4 @ Z3 )
% 2.48/1.48 => ( ord_less_eq_set_a @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
% 2.48/1.48 => ( ( inf_inf_set_a @ X @ Y )
% 2.48/1.48 = ( F @ X @ Y ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_291_inf__unique,axiom,
% 2.48/1.48 ! [F: b > b > b,X: b,Y: b] :
% 2.48/1.48 ( ! [X4: b,Y2: b] : ( ord_less_eq_b @ ( F @ X4 @ Y2 ) @ X4 )
% 2.48/1.48 => ( ! [X4: b,Y2: b] : ( ord_less_eq_b @ ( F @ X4 @ Y2 ) @ Y2 )
% 2.48/1.48 => ( ! [X4: b,Y2: b,Z3: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X4 @ Y2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ X4 @ Z3 )
% 2.48/1.48 => ( ord_less_eq_b @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
% 2.48/1.48 => ( ( inf_inf_b @ X @ Y )
% 2.48/1.48 = ( F @ X @ Y ) ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_292_le__iff__inf,axiom,
% 2.48/1.48 ( ord_less_eq_set_b
% 2.48/1.48 = ( ^ [X2: set_b,Y3: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ X2 @ Y3 )
% 2.48/1.48 = X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_293_le__iff__inf,axiom,
% 2.48/1.48 ( ord_less_eq_set_a
% 2.48/1.48 = ( ^ [X2: set_a,Y3: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ X2 @ Y3 )
% 2.48/1.48 = X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_294_le__iff__inf,axiom,
% 2.48/1.48 ( ord_less_eq_b
% 2.48/1.48 = ( ^ [X2: b,Y3: b] :
% 2.48/1.48 ( ( inf_inf_b @ X2 @ Y3 )
% 2.48/1.48 = X2 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_295_inf_Oabsorb1,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 2.48/1.48 => ( ( inf_inf_set_b @ A2 @ B2 )
% 2.48/1.48 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_296_inf_Oabsorb1,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.48/1.48 => ( ( inf_inf_set_a @ A2 @ B2 )
% 2.48/1.48 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_297_inf_Oabsorb1,axiom,
% 2.48/1.48 ! [A2: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( inf_inf_b @ A2 @ B2 )
% 2.48/1.48 = A2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_298_inf_Oabsorb2,axiom,
% 2.48/1.48 ! [B2: set_b,A2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ B2 @ A2 )
% 2.48/1.48 => ( ( inf_inf_set_b @ A2 @ B2 )
% 2.48/1.48 = B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_299_inf_Oabsorb2,axiom,
% 2.48/1.48 ! [B2: set_a,A2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ B2 @ A2 )
% 2.48/1.48 => ( ( inf_inf_set_a @ A2 @ B2 )
% 2.48/1.48 = B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_300_inf_Oabsorb2,axiom,
% 2.48/1.48 ! [B2: b,A2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ B2 @ A2 )
% 2.48/1.48 => ( ( inf_inf_b @ A2 @ B2 )
% 2.48/1.48 = B2 ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_301_inf__absorb1,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ X @ Y )
% 2.48/1.48 => ( ( inf_inf_set_b @ X @ Y )
% 2.48/1.48 = X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_302_inf__absorb1,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ X @ Y )
% 2.48/1.48 => ( ( inf_inf_set_a @ X @ Y )
% 2.48/1.48 = X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_303_inf__absorb1,axiom,
% 2.48/1.48 ! [X: b,Y: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 => ( ( inf_inf_b @ X @ Y )
% 2.48/1.48 = X ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_304_inf__absorb2,axiom,
% 2.48/1.48 ! [Y: set_b,X: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ Y @ X )
% 2.48/1.48 => ( ( inf_inf_set_b @ X @ Y )
% 2.48/1.48 = Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_305_inf__absorb2,axiom,
% 2.48/1.48 ! [Y: set_a,X: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ Y @ X )
% 2.48/1.48 => ( ( inf_inf_set_a @ X @ Y )
% 2.48/1.48 = Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_306_inf__absorb2,axiom,
% 2.48/1.48 ! [Y: b,X: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ Y @ X )
% 2.48/1.48 => ( ( inf_inf_b @ X @ Y )
% 2.48/1.48 = Y ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_307_inf_OboundedE,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b,C: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
% 2.48/1.48 => ~ ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_308_inf_OboundedE,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a,C: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.48/1.48 => ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_309_inf_OboundedE,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) )
% 2.48/1.48 => ~ ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ~ ( ord_less_eq_b @ A2 @ C ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_310_inf_OboundedI,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b,C: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 2.48/1.48 => ( ( ord_less_eq_set_b @ A2 @ C )
% 2.48/1.48 => ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_311_inf_OboundedI,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a,C: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 2.48/1.48 => ( ( ord_less_eq_set_a @ A2 @ C )
% 2.48/1.48 => ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_312_inf_OboundedI,axiom,
% 2.48/1.48 ! [A2: b,B2: b,C: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ B2 )
% 2.48/1.48 => ( ( ord_less_eq_b @ A2 @ C )
% 2.48/1.48 => ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_313_inf__greatest,axiom,
% 2.48/1.48 ! [X: set_b,Y: set_b,Z: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ X @ Y )
% 2.48/1.48 => ( ( ord_less_eq_set_b @ X @ Z )
% 2.48/1.48 => ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_314_inf__greatest,axiom,
% 2.48/1.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ X @ Y )
% 2.48/1.48 => ( ( ord_less_eq_set_a @ X @ Z )
% 2.48/1.48 => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_315_inf__greatest,axiom,
% 2.48/1.48 ! [X: b,Y: b,Z: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ X @ Y )
% 2.48/1.48 => ( ( ord_less_eq_b @ X @ Z )
% 2.48/1.48 => ( ord_less_eq_b @ X @ ( inf_inf_b @ Y @ Z ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_316_inf_Oorder__iff,axiom,
% 2.48/1.48 ( ord_less_eq_set_b
% 2.48/1.48 = ( ^ [A3: set_b,B3: set_b] :
% 2.48/1.48 ( A3
% 2.48/1.48 = ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_317_inf_Oorder__iff,axiom,
% 2.48/1.48 ( ord_less_eq_set_a
% 2.48/1.48 = ( ^ [A3: set_a,B3: set_a] :
% 2.48/1.48 ( A3
% 2.48/1.48 = ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_318_inf_Oorder__iff,axiom,
% 2.48/1.48 ( ord_less_eq_b
% 2.48/1.48 = ( ^ [A3: b,B3: b] :
% 2.48/1.48 ( A3
% 2.48/1.48 = ( inf_inf_b @ A3 @ B3 ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_319_inf_Ocobounded1,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_320_inf_Ocobounded1,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_321_inf_Ocobounded1,axiom,
% 2.48/1.48 ! [A2: b,B2: b] : ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ A2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_322_inf_Ocobounded2,axiom,
% 2.48/1.48 ! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_323_inf_Ocobounded2,axiom,
% 2.48/1.48 ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_324_inf_Ocobounded2,axiom,
% 2.48/1.48 ! [A2: b,B2: b] : ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ B2 ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_325_inf_Oabsorb__iff1,axiom,
% 2.48/1.48 ( ord_less_eq_set_b
% 2.48/1.48 = ( ^ [A3: set_b,B3: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A3 @ B3 )
% 2.48/1.48 = A3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_326_inf_Oabsorb__iff1,axiom,
% 2.48/1.48 ( ord_less_eq_set_a
% 2.48/1.48 = ( ^ [A3: set_a,B3: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A3 @ B3 )
% 2.48/1.48 = A3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_327_inf_Oabsorb__iff1,axiom,
% 2.48/1.48 ( ord_less_eq_b
% 2.48/1.48 = ( ^ [A3: b,B3: b] :
% 2.48/1.48 ( ( inf_inf_b @ A3 @ B3 )
% 2.48/1.48 = A3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_328_inf_Oabsorb__iff2,axiom,
% 2.48/1.48 ( ord_less_eq_set_b
% 2.48/1.48 = ( ^ [B3: set_b,A3: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ A3 @ B3 )
% 2.48/1.48 = B3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_329_inf_Oabsorb__iff2,axiom,
% 2.48/1.48 ( ord_less_eq_set_a
% 2.48/1.48 = ( ^ [B3: set_a,A3: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ A3 @ B3 )
% 2.48/1.48 = B3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_330_inf_Oabsorb__iff2,axiom,
% 2.48/1.48 ( ord_less_eq_b
% 2.48/1.48 = ( ^ [B3: b,A3: b] :
% 2.48/1.48 ( ( inf_inf_b @ A3 @ B3 )
% 2.48/1.48 = B3 ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_331_inf_OcoboundedI1,axiom,
% 2.48/1.48 ! [A2: set_b,C: set_b,B2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A2 @ C )
% 2.48/1.48 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_332_inf_OcoboundedI1,axiom,
% 2.48/1.48 ! [A2: set_a,C: set_a,B2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A2 @ C )
% 2.48/1.48 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_333_inf_OcoboundedI1,axiom,
% 2.48/1.48 ! [A2: b,C: b,B2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ A2 @ C )
% 2.48/1.48 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ C ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_334_inf_OcoboundedI2,axiom,
% 2.48/1.48 ! [B2: set_b,C: set_b,A2: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ B2 @ C )
% 2.48/1.48 => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_335_inf_OcoboundedI2,axiom,
% 2.48/1.48 ! [B2: set_a,C: set_a,A2: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ B2 @ C )
% 2.48/1.48 => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_336_inf_OcoboundedI2,axiom,
% 2.48/1.48 ! [B2: b,C: b,A2: b] :
% 2.48/1.48 ( ( ord_less_eq_b @ B2 @ C )
% 2.48/1.48 => ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ C ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_337_Diff__triv,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( ( inf_inf_set_b @ A @ B )
% 2.48/1.48 = bot_bot_set_b )
% 2.48/1.48 => ( ( minus_minus_set_b @ A @ B )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_338_Diff__triv,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( ( inf_inf_set_a @ A @ B )
% 2.48/1.48 = bot_bot_set_a )
% 2.48/1.48 => ( ( minus_minus_set_a @ A @ B )
% 2.48/1.48 = A ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_339_open__delete,axiom,
% 2.48/1.48 ! [S3: set_b,X: b] :
% 2.48/1.48 ( ( topolo1276428102open_b @ S3 )
% 2.48/1.48 => ( topolo1276428102open_b @ ( minus_minus_set_b @ S3 @ ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_340_Int__Diff__disjoint,axiom,
% 2.48/1.48 ! [A: set_b,B: set_b] :
% 2.48/1.48 ( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ B ) )
% 2.48/1.48 = bot_bot_set_b ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_341_Int__Diff__disjoint,axiom,
% 2.48/1.48 ! [A: set_a,B: set_a] :
% 2.48/1.48 ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
% 2.48/1.48 = bot_bot_set_a ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_342_is__singleton__the__elem,axiom,
% 2.48/1.48 ( is_singleton_b
% 2.48/1.48 = ( ^ [A4: set_b] :
% 2.48/1.48 ( A4
% 2.48/1.48 = ( insert_b @ ( the_elem_b @ A4 ) @ bot_bot_set_b ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_343_is__singleton__the__elem,axiom,
% 2.48/1.48 ( is_singleton_a
% 2.48/1.48 = ( ^ [A4: set_a] :
% 2.48/1.48 ( A4
% 2.48/1.48 = ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_344_the__elem__eq,axiom,
% 2.48/1.48 ! [X: b] :
% 2.48/1.48 ( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
% 2.48/1.48 = X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_345_the__elem__eq,axiom,
% 2.48/1.48 ! [X: a] :
% 2.48/1.48 ( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
% 2.48/1.48 = X ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_346_empty__subsetI,axiom,
% 2.48/1.48 ! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_347_empty__subsetI,axiom,
% 2.48/1.48 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_348_subset__empty,axiom,
% 2.48/1.48 ! [A: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
% 2.48/1.48 = ( A = bot_bot_set_b ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_349_subset__empty,axiom,
% 2.48/1.48 ! [A: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 2.48/1.48 = ( A = bot_bot_set_a ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_350_insert__subset,axiom,
% 2.48/1.48 ! [X: b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B )
% 2.48/1.48 = ( ( member_b @ X @ B )
% 2.48/1.48 & ( ord_less_eq_set_b @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_351_insert__subset,axiom,
% 2.48/1.48 ! [X: a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
% 2.48/1.48 = ( ( member_a @ X @ B )
% 2.48/1.48 & ( ord_less_eq_set_a @ A @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_352_Int__subset__iff,axiom,
% 2.48/1.48 ! [C2: set_b,A: set_b,B: set_b] :
% 2.48/1.48 ( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
% 2.48/1.48 = ( ( ord_less_eq_set_b @ C2 @ A )
% 2.48/1.48 & ( ord_less_eq_set_b @ C2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_353_Int__subset__iff,axiom,
% 2.48/1.48 ! [C2: set_a,A: set_a,B: set_a] :
% 2.48/1.48 ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
% 2.48/1.48 = ( ( ord_less_eq_set_a @ C2 @ A )
% 2.48/1.48 & ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(fact_354_singleton__insert__inj__eq,axiom,
% 2.48/1.48 ! [B2: a,A2: a,A: set_a] :
% 2.48/1.48 ( ( ( insert_a @ B2 @ bot_bot_set_a )
% 2.48/1.48 = ( insert_a @ A2 @ A ) )
% 2.48/1.48 = ( ( A2 = B2 )
% 2.48/1.48 & ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% 2.48/1.48
% 2.48/1.48 thf(conj_0,hypothesis,
% 2.48/1.48 ! [S2: set_b,V3: set_b] :
% 2.48/1.48 ( ( ( topolo1276428102open_b @ S2 )
% 2.48/1.48 & ( topolo1276428102open_b @ V3 )
% 2.48/1.48 & ( member_b @ ( f @ x0 ) @ S2 )
% 2.48/1.49 & ( member_b @ a2 @ V3 )
% 2.48/1.49 & ( ( inf_inf_set_b @ S2 @ V3 )
% 2.48/1.49 = bot_bot_set_b ) )
% 2.48/1.49 => thesis ) ).
% 2.48/1.49
% 2.48/1.49 thf(conj_1,conjecture,
% 2.48/1.49 thesis ).
% 2.48/1.49 ------------------------
% 2.65/1.50 % (12291)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/183Mi)
% 2.65/1.50 % (12293)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_12126 for (2999ds/27Mi)
% 2.65/1.51 % (12295)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_12126 for (2999ds/2Mi)
% 2.65/1.51 % (12296)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_12126 for (2999ds/275Mi)
% 2.65/1.51 % (12297)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/18Mi)
% 2.65/1.51 % (12295)Instruction limit reached!
% 2.65/1.51 % (12295)------------------------------
% 2.65/1.51 % (12295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.51 % (12295)Termination reason: Unknown
% 2.65/1.51 % (12295)Termination phase: shuffling
% 2.65/1.51
% 2.65/1.51 % (12295)Memory used [KB]: 1407
% 2.65/1.51 % (12295)Time elapsed: 0.003 s
% 2.65/1.51 % (12295)Instructions burned: 3 (million)
% 2.65/1.51 % (12295)------------------------------
% 2.65/1.51 % (12295)------------------------------
% 2.65/1.51 % (12294)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/2Mi)
% 2.65/1.51 % (12294)Instruction limit reached!
% 2.65/1.51 % (12294)------------------------------
% 2.65/1.51 % (12294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.51 % (12294)Termination reason: Unknown
% 2.65/1.51 % (12294)Termination phase: shuffling
% 2.65/1.51
% 2.65/1.51 % (12294)Memory used [KB]: 1279
% 2.65/1.51 % (12294)Time elapsed: 0.003 s
% 2.65/1.51 % (12294)Instructions burned: 2 (million)
% 2.65/1.51 % (12294)------------------------------
% 2.65/1.51 % (12294)------------------------------
% 2.65/1.51 % (12293)Instruction limit reached!
% 2.65/1.51 % (12293)------------------------------
% 2.65/1.51 % (12293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.51 % (12293)Termination reason: Unknown
% 2.65/1.51 % (12293)Termination phase: Property scanning
% 2.65/1.51
% 2.65/1.51 % (12293)Memory used [KB]: 1663
% 2.65/1.51 % (12293)Time elapsed: 0.010 s
% 2.65/1.51 % (12293)Instructions burned: 29 (million)
% 2.65/1.51 % (12293)------------------------------
% 2.65/1.51 % (12293)------------------------------
% 2.65/1.52 % (12292)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_12126 for (2999ds/4Mi)
% 2.65/1.52 % (12292)Instruction limit reached!
% 2.65/1.52 % (12292)------------------------------
% 2.65/1.52 % (12292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.52 % (12292)Termination reason: Unknown
% 2.65/1.52 % (12292)Termination phase: shuffling
% 2.65/1.52
% 2.65/1.52 % (12292)Memory used [KB]: 1407
% 2.65/1.52 % (12292)Time elapsed: 0.003 s
% 2.65/1.52 % (12292)Instructions burned: 6 (million)
% 2.65/1.52 % (12292)------------------------------
% 2.65/1.52 % (12292)------------------------------
% 2.65/1.52 % (12297)Instruction limit reached!
% 2.65/1.52 % (12297)------------------------------
% 2.65/1.52 % (12297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.52 % (12297)Termination reason: Unknown
% 2.65/1.52 % (12297)Termination phase: shuffling
% 2.65/1.52
% 2.65/1.52 % (12297)Memory used [KB]: 1663
% 2.65/1.52 % (12297)Time elapsed: 0.012 s
% 2.65/1.52 % (12297)Instructions burned: 19 (million)
% 2.65/1.52 % (12297)------------------------------
% 2.65/1.52 % (12297)------------------------------
% 2.65/1.53 % (12302)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_12126 for (2999ds/37Mi)
% 2.65/1.53 % (12301)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/3Mi)
% 2.65/1.53 % (12301)Instruction limit reached!
% 2.65/1.53 % (12301)------------------------------
% 2.65/1.53 % (12301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.53 % (12301)Termination reason: Unknown
% 2.65/1.53 % (12301)Termination phase: shuffling
% 2.65/1.53
% 2.65/1.53 % (12301)Memory used [KB]: 1407
% 2.65/1.53 % (12301)Time elapsed: 0.003 s
% 2.65/1.53 % (12301)Instructions burned: 3 (million)
% 2.65/1.53 % (12301)------------------------------
% 2.65/1.53 % (12301)------------------------------
% 2.65/1.53 % (12303)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_12126 for (2999ds/15Mi)
% 2.65/1.53 % (12307)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_12126 for (2999ds/3Mi)
% 2.65/1.54 % (12307)Instruction limit reached!
% 2.65/1.54 % (12307)------------------------------
% 2.65/1.54 % (12307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.54 % (12307)Termination reason: Unknown
% 2.65/1.54 % (12307)Termination phase: shuffling
% 2.65/1.54
% 2.65/1.54 % (12307)Memory used [KB]: 1279
% 2.65/1.54 % (12307)Time elapsed: 0.003 s
% 2.65/1.54 % (12307)Instructions burned: 3 (million)
% 2.65/1.54 % (12307)------------------------------
% 2.65/1.54 % (12307)------------------------------
% 2.65/1.54 % (12303)Instruction limit reached!
% 2.65/1.54 % (12303)------------------------------
% 2.65/1.54 % (12303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.54 % (12303)Termination reason: Unknown
% 2.65/1.54 % (12303)Termination phase: shuffling
% 2.65/1.54
% 2.65/1.54 % (12303)Memory used [KB]: 1663
% 2.65/1.54 % (12303)Time elapsed: 0.010 s
% 2.65/1.54 % (12303)Instructions burned: 16 (million)
% 2.65/1.54 % (12303)------------------------------
% 2.65/1.54 % (12303)------------------------------
% 2.65/1.54 % (12308)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_12126 for (2999ds/1041Mi)
% 2.65/1.54 % (12310)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/7Mi)
% 2.65/1.54 % (12310)Instruction limit reached!
% 2.65/1.54 % (12310)------------------------------
% 2.65/1.54 % (12310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.54 % (12310)Termination reason: Unknown
% 2.65/1.54 % (12310)Termination phase: shuffling
% 2.65/1.54
% 2.65/1.54 % (12310)Memory used [KB]: 1535
% 2.65/1.54 % (12310)Time elapsed: 0.004 s
% 2.65/1.54 % (12310)Instructions burned: 9 (million)
% 2.65/1.54 % (12310)------------------------------
% 2.65/1.54 % (12310)------------------------------
% 2.65/1.54 % (12302)Instruction limit reached!
% 2.65/1.54 % (12302)------------------------------
% 2.65/1.54 % (12302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.54 % (12302)Termination reason: Unknown
% 2.65/1.54 % (12302)Termination phase: shuffling
% 2.65/1.54
% 2.65/1.54 % (12302)Memory used [KB]: 2046
% 2.65/1.54 % (12302)Time elapsed: 0.019 s
% 2.65/1.54 % (12302)Instructions burned: 38 (million)
% 2.65/1.54 % (12302)------------------------------
% 2.65/1.54 % (12302)------------------------------
% 2.65/1.55 % (12318)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/3Mi)
% 2.65/1.55 % (12313)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/16Mi)
% 2.65/1.56 % (12318)Instruction limit reached!
% 2.65/1.56 % (12318)------------------------------
% 2.65/1.56 % (12318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (12318)Termination reason: Unknown
% 2.65/1.56 % (12318)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (12318)Memory used [KB]: 1407
% 2.65/1.56 % (12318)Time elapsed: 0.003 s
% 2.65/1.56 % (12318)Instructions burned: 3 (million)
% 2.65/1.56 % (12318)------------------------------
% 2.65/1.56 % (12318)------------------------------
% 2.65/1.56 % (12316)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_12126 for (2999ds/3Mi)
% 2.65/1.56 % (12316)Instruction limit reached!
% 2.65/1.56 % (12316)------------------------------
% 2.65/1.56 % (12316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (12316)Termination reason: Unknown
% 2.65/1.56 % (12316)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (12316)Memory used [KB]: 1407
% 2.65/1.56 % (12316)Time elapsed: 0.004 s
% 2.65/1.56 % (12316)Instructions burned: 4 (million)
% 2.65/1.56 % (12316)------------------------------
% 2.65/1.56 % (12316)------------------------------
% 2.65/1.56 % (12319)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12126 for (2999ds/7Mi)
% 2.65/1.56 % (12313)Instruction limit reached!
% 2.65/1.56 % (12313)------------------------------
% 2.65/1.56 % (12313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (12313)Termination reason: Unknown
% 2.65/1.56 % (12313)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (12313)Memory used [KB]: 1535
% 2.65/1.56 % (12313)Time elapsed: 0.010 s
% 2.65/1.56 % (12313)Instructions burned: 16 (million)
% 2.65/1.56 % (12313)------------------------------
% 2.65/1.56 % (12313)------------------------------
% 2.65/1.56 % (12319)Instruction limit reached!
% 2.65/1.56 % (12319)------------------------------
% 2.65/1.56 % (12319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (12319)Termination reason: Unknown
% 2.65/1.57 % (12319)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (12319)Memory used [KB]: 1407
% 2.65/1.57 % (12319)Time elapsed: 0.005 s
% 2.65/1.57 % (12319)Instructions burned: 7 (million)
% 2.65/1.57 % (12319)------------------------------
% 2.65/1.57 % (12319)------------------------------
% 2.65/1.57 % (12296)First to succeed.
% 2.65/1.57 % (12296)Refutation found. Thanks to Tanya!
% 2.65/1.57 % SZS status Theorem for DTF2THF_12126
% 2.65/1.57 % SZS output start Proof for DTF2THF_12126
% 2.65/1.57 thf(type_def_5, type, set_a: $tType).
% 2.65/1.57 thf(type_def_7, type, set_b: $tType).
% 2.65/1.57 thf(type_def_8, type, b: $tType).
% 2.65/1.57 thf(type_def_9, type, a: $tType).
% 2.65/1.57 thf(func_def_0, type, set_b: $tType).
% 2.65/1.57 thf(func_def_1, type, set_a: $tType).
% 2.65/1.57 thf(func_def_2, type, b: $tType).
% 2.65/1.57 thf(func_def_3, type, a: $tType).
% 2.65/1.57 thf(func_def_4, type, minus_minus_set_a: set_a > set_a > set_a).
% 2.65/1.57 thf(func_def_5, type, minus_minus_set_b: set_b > set_b > set_b).
% 2.65/1.57 thf(func_def_6, type, inf_inf_set_a: set_a > set_a > set_a).
% 2.65/1.57 thf(func_def_7, type, inf_inf_set_b: set_b > set_b > set_b).
% 2.65/1.57 thf(func_def_8, type, inf_inf_b: b > b > b).
% 2.65/1.57 thf(func_def_9, type, lower_464587817at_a_b: a > (a > b) > $o).
% 2.65/1.57 thf(func_def_10, type, bot_bot_a_o: a > $o).
% 2.65/1.57 thf(func_def_11, type, bot_bot_b_o: b > $o).
% 2.65/1.57 thf(func_def_12, type, bot_bot_set_a: set_a).
% 2.65/1.57 thf(func_def_13, type, bot_bot_set_b: set_b).
% 2.65/1.57 thf(func_def_14, type, bot_bot_b: b).
% 2.65/1.57 thf(func_def_15, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.65/1.57 thf(func_def_16, type, ord_less_eq_set_b: set_b > set_b > $o).
% 2.65/1.57 thf(func_def_17, type, ord_less_eq_b: b > b > $o).
% 2.65/1.57 thf(func_def_18, type, collect_a: (a > $o) > set_a).
% 2.65/1.57 thf(func_def_19, type, collect_b: (b > $o) > set_b).
% 2.65/1.57 thf(func_def_20, type, insert_a: a > set_a > set_a).
% 2.65/1.57 thf(func_def_21, type, insert_b: b > set_b > set_b).
% 2.65/1.57 thf(func_def_22, type, is_empty_a: set_a > $o).
% 2.65/1.57 thf(func_def_23, type, is_empty_b: set_b > $o).
% 2.65/1.57 thf(func_def_24, type, is_singleton_a: set_a > $o).
% 2.65/1.57 thf(func_def_25, type, is_singleton_b: set_b > $o).
% 2.65/1.57 thf(func_def_26, type, the_elem_a: set_a > a).
% 2.65/1.57 thf(func_def_27, type, the_elem_b: set_b > b).
% 2.65/1.57 thf(func_def_28, type, topolo1276428101open_a: set_a > $o).
% 2.65/1.57 thf(func_def_29, type, topolo1276428102open_b: set_b > $o).
% 2.65/1.57 thf(func_def_30, type, member_a: a > set_a > $o).
% 2.65/1.57 thf(func_def_31, type, member_b: b > set_b > $o).
% 2.65/1.57 thf(func_def_32, type, a2: b).
% 2.65/1.57 thf(func_def_33, type, f: a > b).
% 2.65/1.57 thf(func_def_35, type, x0: a).
% 2.65/1.57 thf(func_def_45, type, sP0: set_b > set_b > b > b > $o).
% 2.65/1.57 thf(func_def_46, type, sP1: b > set_b > b > set_b > $o).
% 2.65/1.57 thf(func_def_47, type, sK2: b > set_b > set_b).
% 2.65/1.57 thf(func_def_48, type, sK3: set_b).
% 2.65/1.57 thf(func_def_49, type, sK4: set_b).
% 2.65/1.57 thf(func_def_50, type, sK5: set_b > set_b > b).
% 2.65/1.57 thf(func_def_51, type, sK6: set_b > set_b > b).
% 2.65/1.57 thf(func_def_52, type, sK7: b > b > set_b).
% 2.65/1.57 thf(func_def_53, type, sK8: b > b > set_b).
% 2.65/1.57 thf(func_def_54, type, sK9: b > b > set_b).
% 2.65/1.57 thf(func_def_55, type, sK10: set_b > set_b > b).
% 2.65/1.57 thf(func_def_56, type, sK11: set_b > set_b > b).
% 2.65/1.57 thf(func_def_57, type, sK12: set_b > set_b > b > b > set_b).
% 2.65/1.57 thf(func_def_58, type, sK13: b > set_b > set_b).
% 2.65/1.57 thf(func_def_59, type, sK14: set_b > b).
% 2.65/1.57 thf(func_def_60, type, sK15: set_b > b).
% 2.65/1.57 thf(func_def_61, type, sK16: b > b > set_b).
% 2.65/1.57 thf(func_def_62, type, sK17: b > b > set_b).
% 2.65/1.57 thf(func_def_63, type, sK18: b > b > set_b).
% 2.65/1.57 thf(func_def_64, type, sK19: b > b > set_b).
% 2.65/1.57 thf(func_def_65, type, sK20: set_b > b).
% 2.65/1.57 thf(func_def_66, type, sK21: b > b > set_b).
% 2.65/1.57 thf(f1496,plain,(
% 2.65/1.57 $false),
% 2.65/1.57 inference(avatar_sat_refutation,[status(thm)],[f1346,f1355,f1417,f1423,f1495])).
% 2.65/1.57 thf(f1495,plain,(
% 2.65/1.57 spl22_7),
% 2.65/1.57 inference(avatar_contradiction_clause,[status(thm)],[f1494])).
% 2.65/1.57 thf(f1494,plain,(
% 2.65/1.57 $false | spl22_7),
% 2.65/1.57 inference(subsumption_resolution,[status(thm)],[f1493,f1261])).
% 2.65/1.57 thf(f1261,plain,(
% 2.65/1.57 (a2 != (f @ x0))),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1])).
% 2.65/1.57 thf(f1,axiom,(
% 2.65/1.57 (a2 != (f @ x0))),
% 2.65/1.57 file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062)).
% 2.65/1.57 thf(f1493,plain,(
% 2.65/1.57 (a2 = (f @ x0)) | spl22_7),
% 2.65/1.57 inference(trivial_inequality_removal,[status(thm)],[f1492])).
% 2.65/1.57 thf(f1492,plain,(
% 2.65/1.57 (bot_bot_set_b != bot_bot_set_b) | (a2 = (f @ x0)) | spl22_7),
% 2.65/1.57 inference(superposition,[status(thm)],[f1412,f1104])).
% 2.65/1.57 thf(f1104,plain,(
% 2.65/1.57 (bot_bot_set_b = (inf_inf_set_b @ sK3 @ sK4)) | (a2 = (f @ x0))),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1026])).
% 2.65/1.57 thf(f1026,plain,(
% 2.65/1.57 (a2 = (f @ x0)) | (($true = (member_b @ (f @ x0) @ sK3)) & ((member_b @ a2 @ sK4) = $true) & ($true = (topolo1276428102open_b @ sK4)) & ($true = (topolo1276428102open_b @ sK3)) & (bot_bot_set_b = (inf_inf_set_b @ sK3 @ sK4)))),
% 2.65/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4])],[f1009,f1025])).
% 2.65/1.57 thf(f1025,plain,(
% 2.65/1.57 ? [X0 : set_b,X1 : set_b] : (((member_b @ (f @ x0) @ X0) = $true) & ((member_b @ a2 @ X1) = $true) & ((topolo1276428102open_b @ X1) = $true) & ((topolo1276428102open_b @ X0) = $true) & ((inf_inf_set_b @ X0 @ X1) = bot_bot_set_b)) => (($true = (member_b @ (f @ x0) @ sK3)) & ((member_b @ a2 @ sK4) = $true) & ($true = (topolo1276428102open_b @ sK4)) & ($true = (topolo1276428102open_b @ sK3)) & (bot_bot_set_b = (inf_inf_set_b @ sK3 @ sK4)))),
% 2.65/1.57 introduced(definition,[],[choice_axiom])).
% 2.65/1.57 thf(f1009,plain,(
% 2.65/1.57 (a2 = (f @ x0)) | ? [X0 : set_b,X1 : set_b] : (((member_b @ (f @ x0) @ X0) = $true) & ((member_b @ a2 @ X1) = $true) & ((topolo1276428102open_b @ X1) = $true) & ((topolo1276428102open_b @ X0) = $true) & ((inf_inf_set_b @ X0 @ X1) = bot_bot_set_b))),
% 2.65/1.57 inference(ennf_transformation,[status(thm)],[f791])).
% 2.65/1.57 thf(f791,plain,(
% 2.65/1.57 (a2 != (f @ x0)) => ? [X0 : set_b,X1 : set_b] : (((member_b @ (f @ x0) @ X0) = $true) & ((member_b @ a2 @ X1) = $true) & ((topolo1276428102open_b @ X1) = $true) & ((topolo1276428102open_b @ X0) = $true) & ((inf_inf_set_b @ X0 @ X1) = bot_bot_set_b))),
% 2.65/1.57 inference(fool_elimination,[status(thm)],[f790])).
% 2.65/1.57 thf(f790,plain,(
% 2.65/1.57 (a2 != (f @ x0)) => ? [X0 : set_b,X1 : set_b] : ((member_b @ a2 @ X1) & (member_b @ (f @ x0) @ X0) & (topolo1276428102open_b @ X1) & (topolo1276428102open_b @ X0) & ((inf_inf_set_b @ X0 @ X1) = bot_bot_set_b))),
% 2.65/1.57 inference(rectify,[status(thm)],[f2])).
% 2.65/1.57 thf(f2,axiom,(
% 2.65/1.57 (a2 != (f @ x0)) => ? [X0 : set_b,X1 : set_b] : ((member_b @ a2 @ X1) & (member_b @ (f @ x0) @ X0) & (topolo1276428102open_b @ X1) & (topolo1276428102open_b @ X0) & ((inf_inf_set_b @ X0 @ X1) = bot_bot_set_b))),
% 2.65/1.57 file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062)).
% 2.65/1.57 thf(f1412,plain,(
% 2.65/1.57 (bot_bot_set_b != (inf_inf_set_b @ sK3 @ sK4)) | spl22_7),
% 2.65/1.57 inference(avatar_component_clause,[status(thm)],[f1410])).
% 2.65/1.57 thf(f1410,definition,(
% 2.65/1.57 spl22_7 <=> (bot_bot_set_b = (inf_inf_set_b @ sK3 @ sK4))),
% 2.65/1.57 introduced(definition,[new_symbols(naming,[spl22_7])],[avatar_definition])).
% 2.65/1.57 thf(f1423,plain,(
% 2.65/1.57 spl22_8),
% 2.65/1.57 inference(avatar_contradiction_clause,[status(thm)],[f1422])).
% 2.65/1.57 thf(f1422,plain,(
% 2.65/1.57 $false | spl22_8),
% 2.65/1.57 inference(subsumption_resolution,[status(thm)],[f1421,f1261])).
% 2.65/1.57 thf(f1421,plain,(
% 2.65/1.57 (a2 = (f @ x0)) | spl22_8),
% 2.65/1.57 inference(trivial_inequality_removal,[status(thm)],[f1420])).
% 2.65/1.57 thf(f1420,plain,(
% 2.65/1.57 ($true != $true) | (a2 = (f @ x0)) | spl22_8),
% 2.65/1.57 inference(superposition,[status(thm)],[f1416,f1106])).
% 2.65/1.57 thf(f1106,plain,(
% 2.65/1.57 ($true = (topolo1276428102open_b @ sK4)) | (a2 = (f @ x0))),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1026])).
% 2.65/1.57 thf(f1416,plain,(
% 2.65/1.57 ($true != (topolo1276428102open_b @ sK4)) | spl22_8),
% 2.65/1.57 inference(avatar_component_clause,[status(thm)],[f1414])).
% 2.65/1.57 thf(f1414,definition,(
% 2.65/1.57 spl22_8 <=> ($true = (topolo1276428102open_b @ sK4))),
% 2.65/1.57 introduced(definition,[new_symbols(naming,[spl22_8])],[avatar_definition])).
% 2.65/1.57 thf(f1417,plain,(
% 2.65/1.57 ~spl22_7 | ~spl22_8 | ~spl22_4),
% 2.65/1.57 inference(avatar_split_clause,[status(thm)],[f1408,f1344,f1414,f1410])).
% 2.65/1.57 thf(f1344,definition,(
% 2.65/1.57 spl22_4 <=> ! [X0 : set_b] : ((bot_bot_set_b != (inf_inf_set_b @ sK3 @ X0)) | ((member_b @ a2 @ X0) != $true) | ((topolo1276428102open_b @ X0) != $true))),
% 2.65/1.57 introduced(definition,[new_symbols(naming,[spl22_4])],[avatar_definition])).
% 2.65/1.57 thf(f1408,plain,(
% 2.65/1.57 ($true != (topolo1276428102open_b @ sK4)) | (bot_bot_set_b != (inf_inf_set_b @ sK3 @ sK4)) | ~spl22_4),
% 2.65/1.57 inference(subsumption_resolution,[status(thm)],[f1384,f1261])).
% 2.65/1.57 thf(f1384,plain,(
% 2.65/1.57 ($true != (topolo1276428102open_b @ sK4)) | (a2 = (f @ x0)) | (bot_bot_set_b != (inf_inf_set_b @ sK3 @ sK4)) | ~spl22_4),
% 2.65/1.57 inference(trivial_inequality_removal,[status(thm)],[f1375])).
% 2.65/1.57 thf(f1375,plain,(
% 2.65/1.57 (bot_bot_set_b != (inf_inf_set_b @ sK3 @ sK4)) | ($true != $true) | (a2 = (f @ x0)) | ($true != (topolo1276428102open_b @ sK4)) | ~spl22_4),
% 2.65/1.57 inference(superposition,[status(thm)],[f1345,f1107])).
% 2.65/1.57 thf(f1107,plain,(
% 2.65/1.57 ((member_b @ a2 @ sK4) = $true) | (a2 = (f @ x0))),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1026])).
% 2.65/1.57 thf(f1345,plain,(
% 2.65/1.57 ( ! [X0 : set_b] : (((member_b @ a2 @ X0) != $true) | ((topolo1276428102open_b @ X0) != $true) | (bot_bot_set_b != (inf_inf_set_b @ sK3 @ X0))) ) | ~spl22_4),
% 2.65/1.57 inference(avatar_component_clause,[status(thm)],[f1344])).
% 2.65/1.57 thf(f1355,plain,(
% 2.65/1.57 spl22_3),
% 2.65/1.57 inference(avatar_contradiction_clause,[status(thm)],[f1354])).
% 2.65/1.57 thf(f1354,plain,(
% 2.65/1.57 $false | spl22_3),
% 2.65/1.57 inference(subsumption_resolution,[status(thm)],[f1353,f1261])).
% 2.65/1.57 thf(f1353,plain,(
% 2.65/1.57 (a2 = (f @ x0)) | spl22_3),
% 2.65/1.57 inference(trivial_inequality_removal,[status(thm)],[f1352])).
% 2.65/1.57 thf(f1352,plain,(
% 2.65/1.57 (a2 = (f @ x0)) | ($true != $true) | spl22_3),
% 2.65/1.57 inference(superposition,[status(thm)],[f1342,f1105])).
% 2.65/1.57 thf(f1105,plain,(
% 2.65/1.57 ($true = (topolo1276428102open_b @ sK3)) | (a2 = (f @ x0))),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1026])).
% 2.65/1.57 thf(f1342,plain,(
% 2.65/1.57 ($true != (topolo1276428102open_b @ sK3)) | spl22_3),
% 2.65/1.57 inference(avatar_component_clause,[status(thm)],[f1340])).
% 2.65/1.57 thf(f1340,definition,(
% 2.65/1.57 spl22_3 <=> ($true = (topolo1276428102open_b @ sK3))),
% 2.65/1.57 introduced(definition,[new_symbols(naming,[spl22_3])],[avatar_definition])).
% 2.65/1.57 thf(f1346,plain,(
% 2.65/1.57 ~spl22_3 | spl22_4),
% 2.65/1.57 inference(avatar_split_clause,[status(thm)],[f1338,f1344,f1340])).
% 2.65/1.57 thf(f1338,plain,(
% 2.65/1.57 ( ! [X0 : set_b] : (($true != (topolo1276428102open_b @ sK3)) | (bot_bot_set_b != (inf_inf_set_b @ sK3 @ X0)) | ((topolo1276428102open_b @ X0) != $true) | ((member_b @ a2 @ X0) != $true)) )),
% 2.65/1.57 inference(subsumption_resolution,[status(thm)],[f1322,f1261])).
% 2.65/1.57 thf(f1322,plain,(
% 2.65/1.57 ( ! [X0 : set_b] : (((topolo1276428102open_b @ X0) != $true) | ((member_b @ a2 @ X0) != $true) | (bot_bot_set_b != (inf_inf_set_b @ sK3 @ X0)) | ($true != (topolo1276428102open_b @ sK3)) | (a2 = (f @ x0))) )),
% 2.65/1.57 inference(trivial_inequality_removal,[status(thm)],[f1308])).
% 2.65/1.57 thf(f1308,plain,(
% 2.65/1.57 ( ! [X0 : set_b] : (((topolo1276428102open_b @ X0) != $true) | (bot_bot_set_b != (inf_inf_set_b @ sK3 @ X0)) | (a2 = (f @ x0)) | ((member_b @ a2 @ X0) != $true) | ($true != $true) | ($true != (topolo1276428102open_b @ sK3))) )),
% 2.65/1.57 inference(superposition,[status(thm)],[f1289,f1108])).
% 2.65/1.57 thf(f1108,plain,(
% 2.65/1.57 ($true = (member_b @ (f @ x0) @ sK3)) | (a2 = (f @ x0))),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1026])).
% 2.65/1.57 thf(f1289,plain,(
% 2.65/1.57 ( ! [X0 : set_b,X1 : set_b] : (($true != (member_b @ (f @ x0) @ X1)) | ((member_b @ a2 @ X0) != $true) | ((topolo1276428102open_b @ X0) != $true) | ((topolo1276428102open_b @ X1) != $true) | (bot_bot_set_b != (inf_inf_set_b @ X1 @ X0))) )),
% 2.65/1.57 inference(subsumption_resolution,[status(thm)],[f1116,f1121])).
% 2.65/1.57 thf(f1121,plain,(
% 2.65/1.57 (thesis != $true)),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f900])).
% 2.65/1.57 thf(f900,plain,(
% 2.65/1.57 (thesis != $true)),
% 2.65/1.57 inference(flattening,[status(thm)],[f835])).
% 2.65/1.57 thf(f835,plain,(
% 2.65/1.57 ~(thesis = $true)),
% 2.65/1.57 inference(fool_elimination,[status(thm)],[f834])).
% 2.65/1.57 thf(f834,plain,(
% 2.65/1.57 ~thesis),
% 2.65/1.57 inference(rectify,[status(thm)],[f358])).
% 2.65/1.57 thf(f358,negated_conjecture,(
% 2.65/1.57 ~thesis),
% 2.65/1.57 inference(negated_conjecture,[status(cth)],[f357])).
% 2.65/1.57 thf(f357,conjecture,(
% 2.65/1.57 thesis),
% 2.65/1.57 file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',conj_1)).
% 2.65/1.57 thf(f1116,plain,(
% 2.65/1.57 ( ! [X0 : set_b,X1 : set_b] : (((topolo1276428102open_b @ X1) != $true) | ($true != (member_b @ (f @ x0) @ X1)) | (thesis = $true) | (bot_bot_set_b != (inf_inf_set_b @ X1 @ X0)) | ((member_b @ a2 @ X0) != $true) | ((topolo1276428102open_b @ X0) != $true)) )),
% 2.65/1.57 inference(cnf_transformation,[status(thm)],[f1015])).
% 2.65/1.57 thf(f1015,plain,(
% 2.65/1.57 ! [X0 : set_b,X1 : set_b] : (((member_b @ a2 @ X0) != $true) | ((topolo1276428102open_b @ X1) != $true) | (bot_bot_set_b != (inf_inf_set_b @ X1 @ X0)) | (thesis = $true) | ($true != (member_b @ (f @ x0) @ X1)) | ((topolo1276428102open_b @ X0) != $true))),
% 2.65/1.57 inference(flattening,[status(thm)],[f1014])).
% 2.65/1.57 thf(f1014,plain,(
% 2.65/1.57 ! [X0 : set_b,X1 : set_b] : ((thesis = $true) | (((member_b @ a2 @ X0) != $true) | ((topolo1276428102open_b @ X1) != $true) | (bot_bot_set_b != (inf_inf_set_b @ X1 @ X0)) | ($true != (member_b @ (f @ x0) @ X1)) | ((topolo1276428102open_b @ X0) != $true)))),
% 2.65/1.57 inference(ennf_transformation,[status(thm)],[f423])).
% 2.65/1.57 thf(f423,plain,(
% 2.65/1.57 ! [X0 : set_b,X1 : set_b] : ((((member_b @ a2 @ X0) = $true) & ((topolo1276428102open_b @ X1) = $true) & (bot_bot_set_b = (inf_inf_set_b @ X1 @ X0)) & ($true = (member_b @ (f @ x0) @ X1)) & ((topolo1276428102open_b @ X0) = $true)) => (thesis = $true))),
% 2.65/1.57 inference(fool_elimination,[status(thm)],[f422])).
% 2.65/1.57 thf(f422,plain,(
% 2.65/1.57 ! [X0 : set_b,X1 : set_b] : (((topolo1276428102open_b @ X1) & (topolo1276428102open_b @ X0) & (member_b @ a2 @ X0) & (bot_bot_set_b = (inf_inf_set_b @ X1 @ X0)) & (member_b @ (f @ x0) @ X1)) => thesis)),
% 2.65/1.57 inference(rectify,[status(thm)],[f356])).
% 2.65/1.57 thf(f356,axiom,(
% 2.65/1.57 ! [X38 : set_b,X15 : set_b] : (((topolo1276428102open_b @ X15) & (topolo1276428102open_b @ X38) & (member_b @ a2 @ X38) & (bot_bot_set_b = (inf_inf_set_b @ X15 @ X38)) & (member_b @ (f @ x0) @ X15)) => thesis)),
% 2.65/1.57 file('/export/starexec/sandbox2/tmp/tmp.JJtMEDvWHr/DTF2THF_12126.p',conj_0)).
% 2.65/1.57 % SZS output end Proof for DTF2THF_12126
% 2.65/1.57 % (12296)------------------------------
% 2.65/1.57 % (12296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (12296)Termination reason: Refutation
% 2.65/1.57
% 2.65/1.57 % (12296)Memory used [KB]: 6780
% 2.65/1.57 % (12296)Time elapsed: 0.063 s
% 2.65/1.57 % (12296)Instructions burned: 125 (million)
% 2.65/1.57 % (12296)------------------------------
% 2.65/1.57 % (12296)------------------------------
% 2.65/1.57 % (12283)Success in time 0.081 s
%------------------------------------------------------------------------------