TPTP Problem File: ITP115^1.p
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%------------------------------------------------------------------------------
% File : ITP115^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_96__6247558_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Lower_Semicontinuous/prob_96__6247558_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 393 ( 202 unt; 36 typ; 0 def)
% Number of atoms : 865 ( 374 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2540 ( 120 ~; 5 |; 74 &;2074 @)
% ( 0 <=>; 267 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% Number of symbols : 33 ( 32 usr; 6 con; 0-2 aty)
% Number of variables : 878 ( 62 ^; 796 !; 20 ?; 878 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:40:18.025
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (32)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001tf__b,type,
inf_inf_b: b > b > b ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001tf__b,type,
lower_464587817at_a_b: a > ( a > b ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001tf__b,type,
bot_bot_b: b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__b,type,
is_empty_b: set_b > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo1276428101open_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__b,type,
topolo1276428102open_b: set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_A____,type,
a2: b ).
thf(sy_v_f,type,
f: a > b ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x0,type,
x0: a ).
% Relevant facts (355)
thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062,axiom,
( a2
!= ( f @ x0 ) ) ).
% \<open>A \<noteq> f x0\<close>
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,
( ( ( f @ x0 )
!= a2 )
=> ? [U: set_b,V: set_b] :
( ( topolo1276428102open_b @ U )
& ( topolo1276428102open_b @ V )
& ( member_b @ ( f @ x0 ) @ U )
& ( member_b @ a2 @ V )
& ( ( inf_inf_set_b @ U @ V )
= bot_bot_set_b ) ) ) ).
% \<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>
thf(fact_2_open__Int,axiom,
! [S: set_b,T: set_b] :
( ( topolo1276428102open_b @ S )
=> ( ( topolo1276428102open_b @ T )
=> ( topolo1276428102open_b @ ( inf_inf_set_b @ S @ T ) ) ) ) ).
% open_Int
thf(fact_3_open__Int,axiom,
! [S: set_a,T: set_a] :
( ( topolo1276428101open_a @ S )
=> ( ( topolo1276428101open_a @ T )
=> ( topolo1276428101open_a @ ( inf_inf_set_a @ S @ T ) ) ) ) ).
% open_Int
thf(fact_4_open__empty,axiom,
topolo1276428102open_b @ bot_bot_set_b ).
% open_empty
thf(fact_5_open__empty,axiom,
topolo1276428101open_a @ bot_bot_set_a ).
% open_empty
thf(fact_6_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_7_inf__bot__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_8_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_9_inf__bot__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_10_hausdorff,axiom,
! [X: b,Y: b] :
( ( X != Y )
=> ? [U: set_b,V: set_b] :
( ( topolo1276428102open_b @ U )
& ( topolo1276428102open_b @ V )
& ( member_b @ X @ U )
& ( member_b @ Y @ V )
& ( ( inf_inf_set_b @ U @ V )
= bot_bot_set_b ) ) ) ).
% hausdorff
thf(fact_11_separation__t2,axiom,
! [X: b,Y: b] :
( ( X != Y )
= ( ? [U2: set_b,V2: set_b] :
( ( topolo1276428102open_b @ U2 )
& ( topolo1276428102open_b @ V2 )
& ( member_b @ X @ U2 )
& ( member_b @ Y @ V2 )
& ( ( inf_inf_set_b @ U2 @ V2 )
= bot_bot_set_b ) ) ) ) ).
% separation_t2
thf(fact_12_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_13_IntI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ( member_b @ C @ B )
=> ( member_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_14_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_15_Int__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ( member_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_16_inf_Oidem,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_17_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_18_inf__idem,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ X )
= X ) ).
% inf_idem
thf(fact_19_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_20_inf_Oleft__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_21_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_22_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_23_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_24_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_25_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_26_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_27_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_28_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_29_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_30_inf__right__idem,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y )
= ( inf_inf_set_b @ X @ Y ) ) ).
% inf_right_idem
thf(fact_31_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_32_inf_Oright__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_b @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_33_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_34_inf__left__idem,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
= ( inf_inf_set_b @ X @ Y ) ) ).
% inf_left_idem
thf(fact_35_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
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,
! [S2: set_b] :
( ( ( topolo1276428102open_b @ S2 )
& ( member_b @ ( f @ x0 ) @ S2 ) )
=> ? [T2: set_a] :
( ( topolo1276428101open_a @ T2 )
& ( member_a @ x0 @ T2 )
& ! [X3: a] :
( ( member_a @ X3 @ T2 )
=> ( ( ord_less_eq_b @ ( f @ X3 ) @ ( f @ x0 ) )
=> ( member_b @ ( f @ X3 ) @ S2 ) ) ) ) ) ).
% \<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>
thf(fact_37_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_38_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_39_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_40_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_41_equals0I,axiom,
! [A: set_b] :
( ! [Y2: b] :
~ ( member_b @ Y2 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_42_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_43_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_44_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_45_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_46_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_47_inf__left__commute,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
= ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_48_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_49_inf_Oleft__commute,axiom,
! [B2: set_b,A2: set_b,C: set_b] :
( ( inf_inf_set_b @ B2 @ ( inf_inf_set_b @ A2 @ C ) )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_50_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_51_inf__commute,axiom,
( inf_inf_set_b
= ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_52_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_53_inf_Ocommute,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_54_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_55_inf__assoc,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
= ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_56_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_57_inf_Oassoc,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_58_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_59_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_b,K: set_b,B2: set_b,A2: set_b] :
( ( B
= ( inf_inf_set_b @ K @ B2 ) )
=> ( ( inf_inf_set_b @ A2 @ B )
= ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_60_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_61_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_b,K: set_b,A2: set_b,B2: set_b] :
( ( A
= ( inf_inf_set_b @ K @ A2 ) )
=> ( ( inf_inf_set_b @ A @ B2 )
= ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_62_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_63_inf__sup__aci_I1_J,axiom,
( inf_inf_set_b
= ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_64_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_65_inf__sup__aci_I2_J,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
= ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_66_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_67_inf__sup__aci_I3_J,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
= ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_68_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_69_inf__sup__aci_I4_J,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
= ( inf_inf_set_b @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_70_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_71_Int__left__commute,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
= ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_72_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_73_Int__left__absorb,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_74_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_75_Int__commute,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_76_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_77_Int__absorb,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_78_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_79_Int__assoc,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_80_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_81_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_85_IntD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ B ) ) ).
% IntD2
thf(fact_86_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_87_IntD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% IntD1
thf(fact_88_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_89_IntE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ~ ( member_b @ C @ B ) ) ) ).
% IntE
thf(fact_90_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_91_separation__t1,axiom,
! [X: b,Y: b] :
( ( X != Y )
= ( ? [U2: set_b] :
( ( topolo1276428102open_b @ U2 )
& ( member_b @ X @ U2 )
& ~ ( member_b @ Y @ U2 ) ) ) ) ).
% separation_t1
thf(fact_92_separation__t0,axiom,
! [X: b,Y: b] :
( ( X != Y )
= ( ? [U2: set_b] :
( ( topolo1276428102open_b @ U2 )
& ( ( member_b @ X @ U2 )
!= ( member_b @ Y @ U2 ) ) ) ) ) ).
% separation_t0
thf(fact_93_t1__space,axiom,
! [X: b,Y: b] :
( ( X != Y )
=> ? [U: set_b] :
( ( topolo1276428102open_b @ U )
& ( member_b @ X @ U )
& ~ ( member_b @ Y @ U ) ) ) ).
% t1_space
thf(fact_94_t0__space,axiom,
! [X: b,Y: b] :
( ( X != Y )
=> ? [U: set_b] :
( ( topolo1276428102open_b @ U )
& ( ( member_b @ X @ U )
!= ( member_b @ Y @ U ) ) ) ) ).
% t0_space
thf(fact_95_disjoint__iff__not__equal,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ! [Y3: b] :
( ( member_b @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_96_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_97_Int__empty__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_98_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_99_Int__empty__left,axiom,
! [B: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_100_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_101_disjoint__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ~ ( member_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_102_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_103_Int__emptyI,axiom,
! [A: set_b,B: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A )
=> ~ ( member_b @ X4 @ B ) )
=> ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_104_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_105_calculation,axiom,
( ~ ! [S2: set_b] :
( ( ( topolo1276428102open_b @ S2 )
& ( member_b @ ( f @ x0 ) @ S2 ) )
=> ? [T2: set_a] :
( ( topolo1276428101open_a @ T2 )
& ( member_a @ x0 @ T2 )
& ! [X3: a] :
( ( member_a @ X3 @ T2 )
=> ( ( ord_less_eq_b @ ( f @ X3 ) @ ( f @ x0 ) )
=> ( member_b @ ( f @ X3 ) @ S2 ) ) ) ) )
=> ~ ( lower_464587817at_a_b @ x0 @ f ) ) ).
% calculation
thf(fact_106_Set_Ois__empty__def,axiom,
( is_empty_b
= ( ^ [A4: set_b] : A4 = bot_bot_set_b ) ) ).
% Set.is_empty_def
thf(fact_107_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: set_a] : A4 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_108_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_109_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_110_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_111_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_112_is__singletonI_H,axiom,
! [A: set_b] :
( ( A != bot_bot_set_b )
=> ( ! [X4: b,Y2: b] :
( ( member_b @ X4 @ A )
=> ( ( member_b @ Y2 @ A )
=> ( X4 = Y2 ) ) )
=> ( is_singleton_b @ A ) ) ) ).
% is_singletonI'
thf(fact_113_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X4: a,Y2: a] :
( ( member_a @ X4 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X4 = Y2 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_114_Diff__disjoint,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ A ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_115_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_116_insert__disjoint_I1_J,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B )
& ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_117_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_118_insert__disjoint_I2_J,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B ) )
= ( ~ ( member_b @ A2 @ B )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_119_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_120_order__refl,axiom,
! [X: b] : ( ord_less_eq_b @ X @ X ) ).
% order_refl
thf(fact_121_insertCI,axiom,
! [A2: b,B: set_b,B2: b] :
( ( ~ ( member_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_122_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_123_insert__iff,axiom,
! [A2: b,B2: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_124_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_125_DiffI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ~ ( member_b @ C @ B )
=> ( member_b @ C @ ( minus_minus_set_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_126_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_127_Diff__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ~ ( member_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_128_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_129_le__inf__iff,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
= ( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_130_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_131_le__inf__iff,axiom,
! [X: b,Y: b,Z: b] :
( ( ord_less_eq_b @ X @ ( inf_inf_b @ Y @ Z ) )
= ( ( ord_less_eq_b @ X @ Y )
& ( ord_less_eq_b @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_132_inf_Obounded__iff,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
= ( ( ord_less_eq_set_b @ A2 @ B2 )
& ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_133_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_134_inf_Obounded__iff,axiom,
! [A2: b,B2: b,C: b] :
( ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) )
= ( ( ord_less_eq_b @ A2 @ B2 )
& ( ord_less_eq_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_135_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_136_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_137_Int__insert__left__if0,axiom,
! [A2: b,C2: set_b,B: set_b] :
( ~ ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_138_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_139_Int__insert__left__if1,axiom,
! [A2: b,C2: set_b,B: set_b] :
( ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_140_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_141_insert__inter__insert,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ ( insert_b @ A2 @ B ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_142_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_143_Int__insert__right__if0,axiom,
! [A2: b,A: set_b,B: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_144_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_145_Int__insert__right__if1,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_146_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_147_Diff__empty,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ A @ bot_bot_set_b )
= A ) ).
% Diff_empty
thf(fact_148_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_149_empty__Diff,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_150_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_151_Diff__cancel,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ A @ A )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_152_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_153_Diff__insert0,axiom,
! [X: b,A: set_b,B: set_b] :
( ~ ( member_b @ X @ A )
=> ( ( minus_minus_set_b @ A @ ( insert_b @ X @ B ) )
= ( minus_minus_set_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_154_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_155_insert__Diff1,axiom,
! [X: b,B: set_b,A: set_b] :
( ( member_b @ X @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
= ( minus_minus_set_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_156_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_157_disjoint__insert_I2_J,axiom,
! [A: set_b,B2: b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A @ ( insert_b @ B2 @ B ) ) )
= ( ~ ( member_b @ B2 @ A )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_158_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_159_disjoint__insert_I1_J,axiom,
! [B: set_b,A2: b,A: set_b] :
( ( ( inf_inf_set_b @ B @ ( insert_b @ A2 @ A ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B )
& ( ( inf_inf_set_b @ B @ A )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_160_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_161_insert__Diff__single,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= ( insert_b @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_162_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_163_is__singletonI,axiom,
! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_164_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_165_DiffE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% DiffE
thf(fact_166_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_167_DiffD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% DiffD1
thf(fact_168_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_169_DiffD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ~ ( member_b @ C @ B ) ) ).
% DiffD2
thf(fact_170_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_171_insertE,axiom,
! [A2: b,B2: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_172_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_173_insertI1,axiom,
! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B ) ) ).
% insertI1
thf(fact_174_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_175_insertI2,axiom,
! [A2: b,B: set_b,B2: b] :
( ( member_b @ A2 @ B )
=> ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_176_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_177_Set_Oset__insert,axiom,
! [X: b,A: set_b] :
( ( member_b @ X @ A )
=> ~ ! [B5: set_b] :
( ( A
= ( insert_b @ X @ B5 ) )
=> ( member_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_178_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_179_insert__ident,axiom,
! [X: b,A: set_b,B: set_b] :
( ~ ( member_b @ X @ A )
=> ( ~ ( member_b @ X @ B )
=> ( ( ( insert_b @ X @ A )
= ( insert_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_180_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_181_insert__absorb,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_182_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_183_insert__eq__iff,axiom,
! [A2: b,A: set_b,B2: b,B: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ~ ( member_b @ B2 @ B )
=> ( ( ( insert_b @ A2 @ A )
= ( insert_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_b] :
( ( A
= ( insert_b @ B2 @ C3 ) )
& ~ ( member_b @ B2 @ C3 )
& ( B
= ( insert_b @ A2 @ C3 ) )
& ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_184_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_185_insert__Diff__if,axiom,
! [X: b,B: set_b,A: set_b] :
( ( ( member_b @ X @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
= ( minus_minus_set_b @ A @ B ) ) )
& ( ~ ( member_b @ X @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ B )
= ( insert_b @ X @ ( minus_minus_set_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_186_insert__Diff__if,axiom,
! [X: a,B: set_a,A: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_187_mk__disjoint__insert,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ? [B5: set_b] :
( ( A
= ( insert_b @ A2 @ B5 ) )
& ~ ( member_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_188_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_189_dual__order_Oantisym,axiom,
! [B2: b,A2: b] :
( ( ord_less_eq_b @ B2 @ A2 )
=> ( ( ord_less_eq_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_190_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: b,Z2: b] : Y4 = Z2 )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_191_dual__order_Otrans,axiom,
! [B2: b,A2: b,C: b] :
( ( ord_less_eq_b @ B2 @ A2 )
=> ( ( ord_less_eq_b @ C @ B2 )
=> ( ord_less_eq_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_192_linorder__wlog,axiom,
! [P: b > b > $o,A2: b,B2: b] :
( ! [A5: b,B6: b] :
( ( ord_less_eq_b @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: b,B6: b] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_193_dual__order_Orefl,axiom,
! [A2: b] : ( ord_less_eq_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_194_order__trans,axiom,
! [X: b,Y: b,Z: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ Z )
=> ( ord_less_eq_b @ X @ Z ) ) ) ).
% order_trans
thf(fact_195_order__class_Oorder_Oantisym,axiom,
! [A2: b,B2: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( ord_less_eq_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_196_ord__le__eq__trans,axiom,
! [A2: b,B2: b,C: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_197_ord__eq__le__trans,axiom,
! [A2: b,B2: b,C: b] :
( ( A2 = B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_198_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: b,Z2: b] : Y4 = Z2 )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_199_antisym__conv,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ( ( ord_less_eq_b @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_200_le__cases3,axiom,
! [X: b,Y: b,Z: b] :
( ( ( ord_less_eq_b @ X @ Y )
=> ~ ( ord_less_eq_b @ Y @ Z ) )
=> ( ( ( ord_less_eq_b @ Y @ X )
=> ~ ( ord_less_eq_b @ X @ Z ) )
=> ( ( ( ord_less_eq_b @ X @ Z )
=> ~ ( ord_less_eq_b @ Z @ Y ) )
=> ( ( ( ord_less_eq_b @ Z @ Y )
=> ~ ( ord_less_eq_b @ Y @ X ) )
=> ( ( ( ord_less_eq_b @ Y @ Z )
=> ~ ( ord_less_eq_b @ Z @ X ) )
=> ~ ( ( ord_less_eq_b @ Z @ X )
=> ~ ( ord_less_eq_b @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_201_order_Otrans,axiom,
! [A2: b,B2: b,C: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_202_le__cases,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_eq_b @ X @ Y )
=> ( ord_less_eq_b @ Y @ X ) ) ).
% le_cases
thf(fact_203_eq__refl,axiom,
! [X: b,Y: b] :
( ( X = Y )
=> ( ord_less_eq_b @ X @ Y ) ) ).
% eq_refl
thf(fact_204_linear,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
| ( ord_less_eq_b @ Y @ X ) ) ).
% linear
thf(fact_205_antisym,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_206_eq__iff,axiom,
( ( ^ [Y4: b,Z2: b] : Y4 = Z2 )
= ( ^ [X2: b,Y3: b] :
( ( ord_less_eq_b @ X2 @ Y3 )
& ( ord_less_eq_b @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_207_ord__le__eq__subst,axiom,
! [A2: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_208_ord__eq__le__subst,axiom,
! [A2: b,F: b > b,B2: b,C: b] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_order__subst2,axiom,
! [A2: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_210_order__subst1,axiom,
! [A2: b,F: b > b,B2: b,C: b] :
( ( ord_less_eq_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_211_Diff__insert__absorb,axiom,
! [X: b,A: set_b] :
( ~ ( member_b @ X @ A )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A ) @ ( insert_b @ X @ bot_bot_set_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_212_Diff__insert__absorb,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_213_Diff__insert2,axiom,
! [A: set_b,A2: b,B: set_b] :
( ( minus_minus_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_214_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_215_insert__Diff,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_216_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_217_Diff__insert,axiom,
! [A: set_b,A2: b,B: set_b] :
( ( minus_minus_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A @ B ) @ ( insert_b @ A2 @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_218_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_219_bot_Oextremum,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% bot.extremum
thf(fact_220_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_221_bot_Oextremum,axiom,
! [A2: b] : ( ord_less_eq_b @ bot_bot_b @ A2 ) ).
% bot.extremum
thf(fact_222_bot_Oextremum__unique,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_223_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_224_bot_Oextremum__unique,axiom,
! [A2: b] :
( ( ord_less_eq_b @ A2 @ bot_bot_b )
= ( A2 = bot_bot_b ) ) ).
% bot.extremum_unique
thf(fact_225_bot_Oextremum__uniqueI,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
=> ( A2 = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_226_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_227_bot_Oextremum__uniqueI,axiom,
! [A2: b] :
( ( ord_less_eq_b @ A2 @ bot_bot_b )
=> ( A2 = bot_bot_b ) ) ).
% bot.extremum_uniqueI
thf(fact_228_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A4: set_b] :
? [X2: b] :
( A4
= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_229_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X2: a] :
( A4
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_230_is__singletonE,axiom,
! [A: set_b] :
( ( is_singleton_b @ A )
=> ~ ! [X4: b] :
( A
!= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_231_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X4: a] :
( A
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_232_singletonD,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_233_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_234_singleton__iff,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_235_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_236_doubleton__eq__iff,axiom,
! [A2: b,B2: b,C: b,D: b] :
( ( ( insert_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_237_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_238_insert__not__empty,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ A )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_239_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_240_singleton__inject,axiom,
! [A2: b,B2: b] :
( ( ( insert_b @ A2 @ bot_bot_set_b )
= ( insert_b @ B2 @ bot_bot_set_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_241_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_242_Int__insert__left,axiom,
! [A2: b,C2: set_b,B: set_b] :
( ( ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) )
& ( ~ ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_243_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_244_Int__insert__right,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) )
& ( ~ ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_245_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_246_Int__Diff,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_247_Int__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_248_Diff__Int2,axiom,
! [A: set_b,C2: set_b,B: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_249_Diff__Int2,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_250_Diff__Diff__Int,axiom,
! [A: set_b,B: set_b] :
( ( minus_minus_set_b @ A @ ( minus_minus_set_b @ A @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_251_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_252_Diff__Int__distrib,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( inf_inf_set_b @ C2 @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ C2 @ A ) @ ( inf_inf_set_b @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_253_Diff__Int__distrib,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_254_Diff__Int__distrib2,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( minus_minus_set_b @ A @ B ) @ C2 )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_255_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_256_inf__sup__ord_I2_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_257_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_258_inf__sup__ord_I2_J,axiom,
! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_259_inf__sup__ord_I1_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_260_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_261_inf__sup__ord_I1_J,axiom,
! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_262_inf__le1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_263_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_264_inf__le1,axiom,
! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_265_inf__le2,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_266_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_267_inf__le2,axiom,
! [X: b,Y: b] : ( ord_less_eq_b @ ( inf_inf_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_268_le__infE,axiom,
! [X: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_b @ X @ A2 )
=> ~ ( ord_less_eq_set_b @ X @ B2 ) ) ) ).
% le_infE
thf(fact_269_le__infE,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_270_le__infE,axiom,
! [X: b,A2: b,B2: b] :
( ( ord_less_eq_b @ X @ ( inf_inf_b @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_b @ X @ A2 )
=> ~ ( ord_less_eq_b @ X @ B2 ) ) ) ).
% le_infE
thf(fact_271_le__infI,axiom,
! [X: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ X @ B2 )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_272_le__infI,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_273_le__infI,axiom,
! [X: b,A2: b,B2: b] :
( ( ord_less_eq_b @ X @ A2 )
=> ( ( ord_less_eq_b @ X @ B2 )
=> ( ord_less_eq_b @ X @ ( inf_inf_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_274_inf__mono,axiom,
! [A2: set_b,C: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_275_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_276_inf__mono,axiom,
! [A2: b,C: b,B2: b,D: b] :
( ( ord_less_eq_b @ A2 @ C )
=> ( ( ord_less_eq_b @ B2 @ D )
=> ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ ( inf_inf_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_277_le__infI1,axiom,
! [A2: set_b,X: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_278_le__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_279_le__infI1,axiom,
! [A2: b,X: b,B2: b] :
( ( ord_less_eq_b @ A2 @ X )
=> ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_280_le__infI2,axiom,
! [B2: set_b,X: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_281_le__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_282_le__infI2,axiom,
! [B2: b,X: b,A2: b] :
( ( ord_less_eq_b @ B2 @ X )
=> ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_283_inf_OorderE,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_284_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_285_inf_OorderE,axiom,
! [A2: b,B2: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( A2
= ( inf_inf_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_286_inf_OorderI,axiom,
! [A2: set_b,B2: set_b] :
( ( A2
= ( inf_inf_set_b @ A2 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_287_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_288_inf_OorderI,axiom,
! [A2: b,B2: b] :
( ( A2
= ( inf_inf_b @ A2 @ B2 ) )
=> ( ord_less_eq_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_289_inf__unique,axiom,
! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: set_b,Y2: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y2 )
=> ( ( ord_less_eq_set_b @ X4 @ Z3 )
=> ( ord_less_eq_set_b @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_290_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_291_inf__unique,axiom,
! [F: b > b > b,X: b,Y: b] :
( ! [X4: b,Y2: b] : ( ord_less_eq_b @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: b,Y2: b] : ( ord_less_eq_b @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: b,Y2: b,Z3: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ( ord_less_eq_b @ X4 @ Z3 )
=> ( ord_less_eq_b @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_292_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X2: set_b,Y3: set_b] :
( ( inf_inf_set_b @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_293_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_294_le__iff__inf,axiom,
( ord_less_eq_b
= ( ^ [X2: b,Y3: b] :
( ( inf_inf_b @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_295_inf_Oabsorb1,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_296_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_297_inf_Oabsorb1,axiom,
! [A2: b,B2: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( inf_inf_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_298_inf_Oabsorb2,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_299_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_300_inf_Oabsorb2,axiom,
! [B2: b,A2: b] :
( ( ord_less_eq_b @ B2 @ A2 )
=> ( ( inf_inf_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_301_inf__absorb1,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( inf_inf_set_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_302_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_303_inf__absorb1,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( inf_inf_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_304_inf__absorb2,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( inf_inf_set_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_305_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_306_inf__absorb2,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ( ( inf_inf_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_307_inf_OboundedE,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ~ ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_308_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_309_inf_OboundedE,axiom,
! [A2: b,B2: b,C: b] :
( ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) )
=> ~ ( ( ord_less_eq_b @ A2 @ B2 )
=> ~ ( ord_less_eq_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_310_inf_OboundedI,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_311_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_312_inf_OboundedI,axiom,
! [A2: b,B2: b,C: b] :
( ( ord_less_eq_b @ A2 @ B2 )
=> ( ( ord_less_eq_b @ A2 @ C )
=> ( ord_less_eq_b @ A2 @ ( inf_inf_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_313_inf__greatest,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ X @ Z )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_314_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_315_inf__greatest,axiom,
! [X: b,Y: b,Z: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ X @ Z )
=> ( ord_less_eq_b @ X @ ( inf_inf_b @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_316_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( A3
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_317_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_318_inf_Oorder__iff,axiom,
( ord_less_eq_b
= ( ^ [A3: b,B3: b] :
( A3
= ( inf_inf_b @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_319_inf_Ocobounded1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_320_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_321_inf_Ocobounded1,axiom,
! [A2: b,B2: b] : ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_322_inf_Ocobounded2,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_323_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_324_inf_Ocobounded2,axiom,
! [A2: b,B2: b] : ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_325_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( inf_inf_set_b @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_326_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_327_inf_Oabsorb__iff1,axiom,
( ord_less_eq_b
= ( ^ [A3: b,B3: b] :
( ( inf_inf_b @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_328_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( inf_inf_set_b @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_329_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_330_inf_Oabsorb__iff2,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A3: b] :
( ( inf_inf_b @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_331_inf_OcoboundedI1,axiom,
! [A2: set_b,C: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_332_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_333_inf_OcoboundedI1,axiom,
! [A2: b,C: b,B2: b] :
( ( ord_less_eq_b @ A2 @ C )
=> ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_334_inf_OcoboundedI2,axiom,
! [B2: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_335_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_336_inf_OcoboundedI2,axiom,
! [B2: b,C: b,A2: b] :
( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ ( inf_inf_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_337_Diff__triv,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
=> ( ( minus_minus_set_b @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_338_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_339_open__delete,axiom,
! [S3: set_b,X: b] :
( ( topolo1276428102open_b @ S3 )
=> ( topolo1276428102open_b @ ( minus_minus_set_b @ S3 @ ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% open_delete
thf(fact_340_Int__Diff__disjoint,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ B ) )
= bot_bot_set_b ) ).
% Int_Diff_disjoint
thf(fact_341_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_342_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A4: set_b] :
( A4
= ( insert_b @ ( the_elem_b @ A4 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_343_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_344_the__elem__eq,axiom,
! [X: b] :
( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
= X ) ).
% the_elem_eq
thf(fact_345_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_346_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_347_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_348_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_349_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_350_insert__subset,axiom,
! [X: b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B )
= ( ( member_b @ X @ B )
& ( ord_less_eq_set_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_351_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_352_Int__subset__iff,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
= ( ( ord_less_eq_set_b @ C2 @ A )
& ( ord_less_eq_set_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_353_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_354_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
% Conjectures (2)
thf(conj_0,hypothesis,
! [S2: set_b,V3: set_b] :
( ( ( topolo1276428102open_b @ S2 )
& ( topolo1276428102open_b @ V3 )
& ( member_b @ ( f @ x0 ) @ S2 )
& ( member_b @ a2 @ V3 )
& ( ( inf_inf_set_b @ S2 @ V3 )
= bot_bot_set_b ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------