TPTP Problem File: SLH0042^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00046_001736__14584214_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1428 ( 554 unt; 147 typ; 0 def)
% Number of atoms : 3899 (1096 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10952 ( 402 ~; 85 |; 240 &;8537 @)
% ( 0 <=>;1688 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 833 ( 833 >; 0 *; 0 +; 0 <<)
% Number of symbols : 143 ( 140 usr; 15 con; 0-6 aty)
% Number of variables : 3588 ( 270 ^;3242 !; 76 ?;3588 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:42:43.419
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Filter__Ofilter_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_o_a: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
filter_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (140)
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001_062_I_Eo_Mtf__a_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Set__Oset_Itf__a_J,type,
condit6315317455391067509_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001tf__a,type,
condit4103000493307248661_bdd_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001tf__a,type,
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thf(sy_c_Elementary__Topology_Oclosure_001tf__a,type,
elementary_closure_a: set_a > set_a ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001tf__a,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
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_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Filter__Ofilter_Itf__a_J,type,
inf_inf_filter_a: filter_a > filter_a > filter_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Filter__Ofilter_Itf__a_J_J,type,
inf_inf_set_filter_a: set_filter_a > set_filter_a > set_filter_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
inf_in1205276777018777868_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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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_Osup__class_Osup_001t__Filter__Ofilter_Itf__a_J,type,
sup_sup_filter_a: filter_a > filter_a > filter_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001tf__a,type,
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thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001tf__a,type,
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interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mtf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mtf__a_J,type,
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set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_Eo_Mtf__a_J,type,
set_ord_atMost_o_a: ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Filter__Ofilter_Itf__a_J,type,
set_or3848680440086517451lter_a: filter_a > set_filter_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or4016371710855203973_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001_062_I_Eo_Mtf__a_J,type,
set_or1794473594002085499st_o_a: ( $o > a ) > ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or6073455820589966094_set_a: set_set_a > set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2503527069484367278_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
set_or4472690218693186638Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001_062_I_Eo_Mtf__a_J,type,
set_or8643659207013295391an_o_a: ( $o > a ) > ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or5544419673604628330_set_a: set_set_a > set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_or6017932776736107018_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__a,type,
set_or5939364468397584554Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_Itf__a_J,type,
set_or460448635090783044_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__a,type,
set_or8632414552788122084Than_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Filter__Ofilter_Itf__a_J,type,
set_or2284990949078461863lter_a: filter_a > set_filter_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or5421148953861284865_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__a,type,
set_ord_lessThan_a: a > set_a ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo8477419352202985285open_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001tf__a,type,
topolo1902352237885396414thin_a: a > set_a > filter_a ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed_001tf__a,type,
topolo784654279908865136osed_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact_001tf__a,type,
topolo8439159285038550427pact_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001tf__a,type,
topolo2370605967727889109cted_a: set_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001t__Set__Oset_Itf__a_J,type,
chain_subset_set_a: set_set_set_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xa,type,
xa: a ).
thf(sy_v_y____,type,
y: a ).
% Relevant facts (1274)
thf(fact_0_assms,axiom,
up_ray_a @ i ).
% assms
thf(fact_1_c,axiom,
member_a @ x @ i ).
% c
thf(fact_2_True,axiom,
? [Y: a] :
( ( ord_less_a @ Y @ x )
& ( member_a @ Y @ i ) ) ).
% True
thf(fact_3_a,axiom,
( ( ord_less_a @ y @ x )
& ( member_a @ y @ i ) ) ).
% a
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_O_Ay_A_060_Ax_A_092_060and_062_Ay_A_092_060in_062_AI_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y: a] :
~ ( ( ord_less_a @ Y @ x )
& ( member_a @ Y @ i ) ) ).
% \<open>\<And>thesis. (\<And>y. y < x \<and> y \<in> I \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_b,axiom,
~ ( topolo784654279908865136osed_a @ i ) ).
% b
thf(fact_6_calculation_I2_J,axiom,
member_a @ x @ ( set_or8632414552788122084Than_a @ y ) ).
% calculation(2)
thf(fact_7_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_8_order__refl,axiom,
! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% order_refl
thf(fact_9_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_10_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_11_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_12_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_13_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_14_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_15_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X2: a,Y2: a] :
( ( member_a @ X2 @ I )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
=> ( member_a @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_16_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ I )
=> ( ( member_a @ Z @ I )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( member_a @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_17_minf_I8_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ~ ( ord_less_eq_a @ T @ X3 ) ) ).
% minf(8)
thf(fact_18_minf_I6_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ( ord_less_eq_a @ X3 @ T ) ) ).
% minf(6)
thf(fact_19_pinf_I8_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( ord_less_eq_a @ T @ X3 ) ) ).
% pinf(8)
thf(fact_20_pinf_I6_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ T ) ) ).
% pinf(6)
thf(fact_21_verit__comp__simplify1_I3_J,axiom,
! [B: a,A2: a] :
( ( ~ ( ord_less_eq_a @ B @ A2 ) )
= ( ord_less_a @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_22_leD,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ~ ( ord_less_set_a @ X @ Y3 ) ) ).
% leD
thf(fact_23_leD,axiom,
! [Y3: $o > a,X: $o > a] :
( ( ord_less_eq_o_a @ Y3 @ X )
=> ~ ( ord_less_o_a @ X @ Y3 ) ) ).
% leD
thf(fact_24_leD,axiom,
! [Y3: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ~ ( ord_less_set_set_a @ X @ Y3 ) ) ).
% leD
thf(fact_25_leD,axiom,
! [Y3: a,X: a] :
( ( ord_less_eq_a @ Y3 @ X )
=> ~ ( ord_less_a @ X @ Y3 ) ) ).
% leD
thf(fact_26_leI,axiom,
! [X: a,Y3: a] :
( ~ ( ord_less_a @ X @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X ) ) ).
% leI
thf(fact_27_order__antisym__conv,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_28_order__antisym__conv,axiom,
! [Y3: $o > a,X: $o > a] :
( ( ord_less_eq_o_a @ Y3 @ X )
=> ( ( ord_less_eq_o_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_29_order__antisym__conv,axiom,
! [Y3: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_30_order__antisym__conv,axiom,
! [Y3: a,X: a] :
( ( ord_less_eq_a @ Y3 @ X )
=> ( ( ord_less_eq_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_31_linorder__le__cases,axiom,
! [X: a,Y3: a] :
( ~ ( ord_less_eq_a @ X @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_32_ord__le__eq__subst,axiom,
! [A: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_33_ord__le__eq__subst,axiom,
! [A: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_34_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_35_ord__le__eq__subst,axiom,
! [A: a,B2: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_36_ord__le__eq__subst,axiom,
! [A: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_37_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_38_ord__le__eq__subst,axiom,
! [A: $o > a,B2: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_42_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B2: a,C: a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_43_ord__eq__le__subst,axiom,
! [A: set_a,F: a > set_a,B2: a,C: a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_44_ord__eq__le__subst,axiom,
! [A: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_45_ord__eq__le__subst,axiom,
! [A: $o > a,F: a > $o > a,B2: a,C: a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_46_ord__eq__le__subst,axiom,
! [A: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_47_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_48_ord__eq__le__subst,axiom,
! [A: a,F: ( $o > a ) > a,B2: $o > a,C: $o > a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_ord__eq__le__subst,axiom,
! [A: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_ord__eq__le__subst,axiom,
! [A: $o > a,F: set_a > $o > a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_linorder__linear,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_a @ X @ Y3 )
| ( ord_less_eq_a @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_53_verit__la__disequality,axiom,
! [A: a,B2: a] :
( ( A = B2 )
| ~ ( ord_less_eq_a @ A @ B2 )
| ~ ( ord_less_eq_a @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_54_order__eq__refl,axiom,
! [X: set_a,Y3: set_a] :
( ( X = Y3 )
=> ( ord_less_eq_set_a @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_55_order__eq__refl,axiom,
! [X: $o > a,Y3: $o > a] :
( ( X = Y3 )
=> ( ord_less_eq_o_a @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_56_order__eq__refl,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( X = Y3 )
=> ( ord_le3724670747650509150_set_a @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_57_order__eq__refl,axiom,
! [X: a,Y3: a] :
( ( X = Y3 )
=> ( ord_less_eq_a @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_58_order__subst2,axiom,
! [A: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_59_order__subst2,axiom,
! [A: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_60_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_61_order__subst2,axiom,
! [A: a,B2: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_62_order__subst2,axiom,
! [A: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_63_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_64_order__subst2,axiom,
! [A: $o > a,B2: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_65_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_66_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_67_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_68_order__subst1,axiom,
! [A: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_69_order__subst1,axiom,
! [A: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_70_order__subst1,axiom,
! [A: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_71_order__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B2: $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_72_order__subst1,axiom,
! [A: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_73_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_74_order__subst1,axiom,
! [A: $o > a,F: a > $o > a,B2: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_order__subst1,axiom,
! [A: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_76_order__subst1,axiom,
! [A: set_a,F: ( $o > a ) > set_a,B2: $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst1,axiom,
! [A: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_78_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_79_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > a,Z3: $o > a] : ( Y4 = Z3 ) )
= ( ^ [A3: $o > a,B3: $o > a] :
( ( ord_less_eq_o_a @ A3 @ B3 )
& ( ord_less_eq_o_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_80_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_81_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_82_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G: $o > a] :
! [X2: $o] : ( ord_less_eq_a @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_83_le__funI,axiom,
! [F: $o > a,G2: $o > a] :
( ! [X4: $o] : ( ord_less_eq_a @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq_o_a @ F @ G2 ) ) ).
% le_funI
thf(fact_84_le__funE,axiom,
! [F: $o > a,G2: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_85_le__funD,axiom,
! [F: $o > a,G2: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_86_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_87_antisym,axiom,
! [A: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_88_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_89_antisym,axiom,
! [A: a,B2: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_eq_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_90_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_91_dual__order_Otrans,axiom,
! [B2: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A )
=> ( ( ord_less_eq_o_a @ C @ B2 )
=> ( ord_less_eq_o_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_92_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_93_dual__order_Otrans,axiom,
! [B2: a,A: a,C: a] :
( ( ord_less_eq_a @ B2 @ A )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_94_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_95_dual__order_Oantisym,axiom,
! [B2: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A )
=> ( ( ord_less_eq_o_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_96_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_97_dual__order_Oantisym,axiom,
! [B2: a,A: a] :
( ( ord_less_eq_a @ B2 @ A )
=> ( ( ord_less_eq_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_98_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_99_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > a,Z3: $o > a] : ( Y4 = Z3 ) )
= ( ^ [A3: $o > a,B3: $o > a] :
( ( ord_less_eq_o_a @ B3 @ A3 )
& ( ord_less_eq_o_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_100_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_101_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_102_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B2: a] :
( ! [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_103_order__trans,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z4 )
=> ( ord_less_eq_set_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_104_order__trans,axiom,
! [X: $o > a,Y3: $o > a,Z4: $o > a] :
( ( ord_less_eq_o_a @ X @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ Z4 )
=> ( ord_less_eq_o_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_105_order__trans,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z4 )
=> ( ord_le3724670747650509150_set_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_106_order__trans,axiom,
! [X: a,Y3: a,Z4: a] :
( ( ord_less_eq_a @ X @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z4 )
=> ( ord_less_eq_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_107_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_108_order_Otrans,axiom,
! [A: $o > a,B2: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% order.trans
thf(fact_109_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_110_order_Otrans,axiom,
! [A: a,B2: a,C: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_111_order__antisym,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_112_order__antisym,axiom,
! [X: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_113_order__antisym,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_114_order__antisym,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_a @ X @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_115_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_116_ord__le__eq__trans,axiom,
! [A: $o > a,B2: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_117_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_118_ord__le__eq__trans,axiom,
! [A: a,B2: a,C: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_119_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_120_ord__eq__le__trans,axiom,
! [A: $o > a,B2: $o > a,C: $o > a] :
( ( A = B2 )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_121_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_122_ord__eq__le__trans,axiom,
! [A: a,B2: a,C: a] :
( ( A = B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_123_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_124_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > a,Z3: $o > a] : ( Y4 = Z3 ) )
= ( ^ [X2: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y2 )
& ( ord_less_eq_o_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_125_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
& ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_126_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 ) )
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( ord_less_eq_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_127_le__cases3,axiom,
! [X: a,Y3: a,Z4: a] :
( ( ( ord_less_eq_a @ X @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ Z4 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ X )
=> ~ ( ord_less_eq_a @ X @ Z4 ) )
=> ( ( ( ord_less_eq_a @ X @ Z4 )
=> ~ ( ord_less_eq_a @ Z4 @ Y3 ) )
=> ( ( ( ord_less_eq_a @ Z4 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ X ) )
=> ( ( ( ord_less_eq_a @ Y3 @ Z4 )
=> ~ ( ord_less_eq_a @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z4 @ X )
=> ~ ( ord_less_eq_a @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_128_nle__le,axiom,
! [A: a,B2: a] :
( ( ~ ( ord_less_eq_a @ A @ B2 ) )
= ( ( ord_less_eq_a @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_129_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A5: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_133_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_134_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_135_verit__comp__simplify1_I2_J,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_136_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_137_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_138_order__less__imp__not__less,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_139_order__less__imp__not__less,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_140_order__less__imp__not__eq2,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_141_order__less__imp__not__eq2,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_142_order__less__imp__not__eq,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_143_order__less__imp__not__eq,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_144_linorder__less__linear,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_a @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_145_order__less__imp__triv,axiom,
! [X: set_a,Y3: set_a,P: $o] :
( ( ord_less_set_a @ X @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_146_order__less__imp__triv,axiom,
! [X: a,Y3: a,P: $o] :
( ( ord_less_a @ X @ Y3 )
=> ( ( ord_less_a @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_147_order__less__not__sym,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_148_order__less__not__sym,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_149_order__less__subst2,axiom,
! [A: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_150_order__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_151_order__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_152_order__less__subst2,axiom,
! [A: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_153_order__less__subst1,axiom,
! [A: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_154_order__less__subst1,axiom,
! [A: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_155_order__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_156_order__less__subst1,axiom,
! [A: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_157_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_158_order__less__irrefl,axiom,
! [X: a] :
~ ( ord_less_a @ X @ X ) ).
% order_less_irrefl
thf(fact_159_ord__less__eq__subst,axiom,
! [A: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_160_ord__less__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_161_ord__less__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_162_ord__less__eq__subst,axiom,
! [A: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_163_ord__eq__less__subst,axiom,
! [A: set_a,F: a > set_a,B2: a,C: a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_164_ord__eq__less__subst,axiom,
! [A: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_165_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_166_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B2: a,C: a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_167_order__less__trans,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z4 )
=> ( ord_less_set_a @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_168_order__less__trans,axiom,
! [X: a,Y3: a,Z4: a] :
( ( ord_less_a @ X @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z4 )
=> ( ord_less_a @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_169_order__less__asym_H,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_170_order__less__asym_H,axiom,
! [A: a,B2: a] :
( ( ord_less_a @ A @ B2 )
=> ~ ( ord_less_a @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_171_linorder__neq__iff,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
= ( ( ord_less_a @ X @ Y3 )
| ( ord_less_a @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_172_order__less__asym,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_173_order__less__asym,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_174_linorder__neqE,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
=> ( ~ ( ord_less_a @ X @ Y3 )
=> ( ord_less_a @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_175_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_176_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A: a] :
( ( ord_less_a @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_177_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_178_order_Ostrict__implies__not__eq,axiom,
! [A: a,B2: a] :
( ( ord_less_a @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_179_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_180_dual__order_Ostrict__trans,axiom,
! [B2: a,A: a,C: a] :
( ( ord_less_a @ B2 @ A )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_181_not__less__iff__gr__or__eq,axiom,
! [X: a,Y3: a] :
( ( ~ ( ord_less_a @ X @ Y3 ) )
= ( ( ord_less_a @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_182_order_Ostrict__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_183_order_Ostrict__trans,axiom,
! [A: a,B2: a,C: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_184_linorder__less__wlog,axiom,
! [P: a > a > $o,A: a,B2: a] :
( ! [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a] : ( P @ A4 @ A4 )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_185_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_186_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_187_dual__order_Oasym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ~ ( ord_less_set_a @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_188_dual__order_Oasym,axiom,
! [B2: a,A: a] :
( ( ord_less_a @ B2 @ A )
=> ~ ( ord_less_a @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_189_linorder__cases,axiom,
! [X: a,Y3: a] :
( ~ ( ord_less_a @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_a @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_190_antisym__conv3,axiom,
! [Y3: a,X: a] :
( ~ ( ord_less_a @ Y3 @ X )
=> ( ( ~ ( ord_less_a @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_191_ord__less__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_192_ord__less__eq__trans,axiom,
! [A: a,B2: a,C: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_193_ord__eq__less__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_194_ord__eq__less__trans,axiom,
! [A: a,B2: a,C: a] :
( ( A = B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_195_order_Oasym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A ) ) ).
% order.asym
thf(fact_196_order_Oasym,axiom,
! [A: a,B2: a] :
( ( ord_less_a @ A @ B2 )
=> ~ ( ord_less_a @ B2 @ A ) ) ).
% order.asym
thf(fact_197_less__imp__neq,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_198_less__imp__neq,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_199_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_200_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_201_pinf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_202_pinf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_203_pinf_I3_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_204_pinf_I4_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_205_pinf_I5_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ~ ( ord_less_a @ X3 @ T ) ) ).
% pinf(5)
thf(fact_206_pinf_I7_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ Z2 @ X3 )
=> ( ord_less_a @ T @ X3 ) ) ).
% pinf(7)
thf(fact_207_minf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_208_minf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_209_minf_I3_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_210_minf_I4_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_211_minf_I5_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ( ord_less_a @ X3 @ T ) ) ).
% minf(5)
thf(fact_212_minf_I7_J,axiom,
! [T: a] :
? [Z2: a] :
! [X3: a] :
( ( ord_less_a @ X3 @ Z2 )
=> ~ ( ord_less_a @ T @ X3 ) ) ).
% minf(7)
thf(fact_213_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_set_a @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_214_order__le__imp__less__or__eq,axiom,
! [X: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X @ Y3 )
=> ( ( ord_less_o_a @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_215_order__le__imp__less__or__eq,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_less_set_set_a @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_216_order__le__imp__less__or__eq,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_a @ X @ Y3 )
=> ( ( ord_less_a @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_217_linorder__le__less__linear,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_a @ X @ Y3 )
| ( ord_less_a @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_218_order__less__le__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_219_order__less__le__subst2,axiom,
! [A: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_220_order__less__le__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_221_order__less__le__subst2,axiom,
! [A: a,B2: a,F: a > $o > a,C: $o > a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_222_order__less__le__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_223_order__less__le__subst2,axiom,
! [A: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_224_order__less__le__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_225_order__less__le__subst2,axiom,
! [A: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_226_order__less__le__subst1,axiom,
! [A: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_227_order__less__le__subst1,axiom,
! [A: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_228_order__less__le__subst1,axiom,
! [A: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_229_order__less__le__subst1,axiom,
! [A: $o > a,F: a > $o > a,B2: a,C: a] :
( ( ord_less_o_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_230_order__less__le__subst1,axiom,
! [A: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_less_set_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_231_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_232_order__less__le__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B2: $o > a,C: $o > a] :
( ( ord_less_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_233_order__less__le__subst1,axiom,
! [A: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_a @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_234_order__less__le__subst1,axiom,
! [A: $o > a,F: set_a > $o > a,B2: set_a,C: set_a] :
( ( ord_less_o_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_235_order__less__le__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_236_order__le__less__subst2,axiom,
! [A: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_237_order__le__less__subst2,axiom,
! [A: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_238_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_239_order__le__less__subst2,axiom,
! [A: a,B2: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_o_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_240_order__le__less__subst2,axiom,
! [A: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_eq_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_241_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_242_order__le__less__subst2,axiom,
! [A: $o > a,B2: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X4: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_243_order__le__less__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_244_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_o_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_245_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_246_order__le__less__subst1,axiom,
! [A: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_247_order__le__less__subst1,axiom,
! [A: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_248_order__le__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_249_order__le__less__subst1,axiom,
! [A: $o > a,F: a > $o > a,B2: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_250_order__le__less__subst1,axiom,
! [A: $o > a,F: set_a > $o > a,B2: set_a,C: set_a] :
( ( ord_less_eq_o_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_o_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_251_order__le__less__subst1,axiom,
! [A: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_252_order__le__less__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_253_order__le__less__subst1,axiom,
! [A: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X4: a,Y: a] :
( ( ord_less_a @ X4 @ Y )
=> ( ord_less_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_254_order__less__le__trans,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z4 )
=> ( ord_less_set_a @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_255_order__less__le__trans,axiom,
! [X: $o > a,Y3: $o > a,Z4: $o > a] :
( ( ord_less_o_a @ X @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ Z4 )
=> ( ord_less_o_a @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_256_order__less__le__trans,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z4 )
=> ( ord_less_set_set_a @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_257_order__less__le__trans,axiom,
! [X: a,Y3: a,Z4: a] :
( ( ord_less_a @ X @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z4 )
=> ( ord_less_a @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_258_order__le__less__trans,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z4 )
=> ( ord_less_set_a @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_259_order__le__less__trans,axiom,
! [X: $o > a,Y3: $o > a,Z4: $o > a] :
( ( ord_less_eq_o_a @ X @ Y3 )
=> ( ( ord_less_o_a @ Y3 @ Z4 )
=> ( ord_less_o_a @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_260_order__le__less__trans,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ Z4 )
=> ( ord_less_set_set_a @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_261_order__le__less__trans,axiom,
! [X: a,Y3: a,Z4: a] :
( ( ord_less_eq_a @ X @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z4 )
=> ( ord_less_a @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_262_order__neq__le__trans,axiom,
! [A: set_a,B2: set_a] :
( ( A != B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_263_order__neq__le__trans,axiom,
! [A: $o > a,B2: $o > a] :
( ( A != B2 )
=> ( ( ord_less_eq_o_a @ A @ B2 )
=> ( ord_less_o_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_264_order__neq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A != B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ord_less_set_set_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_265_order__neq__le__trans,axiom,
! [A: a,B2: a] :
( ( A != B2 )
=> ( ( ord_less_eq_a @ A @ B2 )
=> ( ord_less_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_266_order__le__neq__trans,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_267_order__le__neq__trans,axiom,
! [A: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_o_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_268_order__le__neq__trans,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_set_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_269_order__le__neq__trans,axiom,
! [A: a,B2: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_270_order__less__imp__le,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_271_order__less__imp__le,axiom,
! [X: $o > a,Y3: $o > a] :
( ( ord_less_o_a @ X @ Y3 )
=> ( ord_less_eq_o_a @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_272_order__less__imp__le,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X @ Y3 )
=> ( ord_le3724670747650509150_set_a @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_273_order__less__imp__le,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ( ord_less_eq_a @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_274_linorder__not__less,axiom,
! [X: a,Y3: a] :
( ( ~ ( ord_less_a @ X @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_275_linorder__not__le,axiom,
! [X: a,Y3: a] :
( ( ~ ( ord_less_eq_a @ X @ Y3 ) )
= ( ord_less_a @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_276_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_277_order__less__le,axiom,
( ord_less_o_a
= ( ^ [X2: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_278_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_279_order__less__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_280_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_281_order__le__less,axiom,
( ord_less_eq_o_a
= ( ^ [X2: $o > a,Y2: $o > a] :
( ( ord_less_o_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_282_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_283_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_284_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_285_dual__order_Ostrict__implies__order,axiom,
! [B2: $o > a,A: $o > a] :
( ( ord_less_o_a @ B2 @ A )
=> ( ord_less_eq_o_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_286_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A )
=> ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_287_dual__order_Ostrict__implies__order,axiom,
! [B2: a,A: a] :
( ( ord_less_a @ B2 @ A )
=> ( ord_less_eq_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_288_order_Ostrict__implies__order,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_289_order_Ostrict__implies__order,axiom,
! [A: $o > a,B2: $o > a] :
( ( ord_less_o_a @ A @ B2 )
=> ( ord_less_eq_o_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_290_order_Ostrict__implies__order,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A @ B2 )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_291_order_Ostrict__implies__order,axiom,
! [A: a,B2: a] :
( ( ord_less_a @ A @ B2 )
=> ( ord_less_eq_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_292_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_293_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [B3: $o > a,A3: $o > a] :
( ( ord_less_eq_o_a @ B3 @ A3 )
& ~ ( ord_less_eq_o_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_294_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ~ ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_295_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ~ ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_296_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_297_dual__order_Ostrict__trans2,axiom,
! [B2: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_o_a @ B2 @ A )
=> ( ( ord_less_eq_o_a @ C @ B2 )
=> ( ord_less_o_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_298_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_299_dual__order_Ostrict__trans2,axiom,
! [B2: a,A: a,C: a] :
( ( ord_less_a @ B2 @ A )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_300_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_301_dual__order_Ostrict__trans1,axiom,
! [B2: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A )
=> ( ( ord_less_o_a @ C @ B2 )
=> ( ord_less_o_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_302_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_303_dual__order_Ostrict__trans1,axiom,
! [B2: a,A: a,C: a] :
( ( ord_less_eq_a @ B2 @ A )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_304_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_305_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [B3: $o > a,A3: $o > a] :
( ( ord_less_eq_o_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_306_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_307_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_308_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_309_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [B3: $o > a,A3: $o > a] :
( ( ord_less_o_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_310_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_less_set_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_311_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_312_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_313_order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [A3: $o > a,B3: $o > a] :
( ( ord_less_eq_o_a @ A3 @ B3 )
& ~ ( ord_less_eq_o_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_314_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ~ ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_315_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ~ ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_316_order_Ostrict__trans2,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_317_order_Ostrict__trans2,axiom,
! [A: $o > a,B2: $o > a,C: $o > a] :
( ( ord_less_o_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ B2 @ C )
=> ( ord_less_o_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_318_order_Ostrict__trans2,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_319_order_Ostrict__trans2,axiom,
! [A: a,B2: a,C: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_320_order_Ostrict__trans1,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_321_order_Ostrict__trans1,axiom,
! [A: $o > a,B2: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ord_less_o_a @ B2 @ C )
=> ( ord_less_o_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_322_order_Ostrict__trans1,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_323_order_Ostrict__trans1,axiom,
! [A: a,B2: a,C: a] :
( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_324_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_325_order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [A3: $o > a,B3: $o > a] :
( ( ord_less_eq_o_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_326_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_327_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_328_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_329_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [A3: $o > a,B3: $o > a] :
( ( ord_less_o_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_330_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_331_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_332_not__le__imp__less,axiom,
! [Y3: a,X: a] :
( ~ ( ord_less_eq_a @ Y3 @ X )
=> ( ord_less_a @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_333_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ~ ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_334_less__le__not__le,axiom,
( ord_less_o_a
= ( ^ [X2: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y2 )
& ~ ( ord_less_eq_o_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_335_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
& ~ ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_336_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
& ~ ( ord_less_eq_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_337_antisym__conv2,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_338_antisym__conv2,axiom,
! [X: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X @ Y3 )
=> ( ( ~ ( ord_less_o_a @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_339_antisym__conv2,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ~ ( ord_less_set_set_a @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_340_antisym__conv2,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_a @ X @ Y3 )
=> ( ( ~ ( ord_less_a @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_341_antisym__conv1,axiom,
! [X: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_342_antisym__conv1,axiom,
! [X: $o > a,Y3: $o > a] :
( ~ ( ord_less_o_a @ X @ Y3 )
=> ( ( ord_less_eq_o_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_343_antisym__conv1,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ~ ( ord_less_set_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_344_antisym__conv1,axiom,
! [X: a,Y3: a] :
( ~ ( ord_less_a @ X @ Y3 )
=> ( ( ord_less_eq_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_345_nless__le,axiom,
! [A: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_346_nless__le,axiom,
! [A: $o > a,B2: $o > a] :
( ( ~ ( ord_less_o_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_o_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_347_nless__le,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_348_nless__le,axiom,
! [A: a,B2: a] :
( ( ~ ( ord_less_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_349_greaterThan__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or460448635090783044_set_a @ K ) )
= ( ord_less_set_a @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_350_greaterThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_351_greaterThan__subset__iff,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X ) @ ( set_or8632414552788122084Than_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X ) ) ).
% greaterThan_subset_iff
thf(fact_352_calculation_I1_J,axiom,
topolo8477419352202985285open_a @ ( set_or8632414552788122084Than_a @ y ) ).
% calculation(1)
thf(fact_353_greaterThan__eq__iff,axiom,
! [X: a,Y3: a] :
( ( ( set_or8632414552788122084Than_a @ X )
= ( set_or8632414552788122084Than_a @ Y3 ) )
= ( X = Y3 ) ) ).
% greaterThan_eq_iff
thf(fact_354_Greatest__equality,axiom,
! [P: set_a > $o,X: set_a] :
( ( P @ X )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( ( order_Greatest_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_355_Greatest__equality,axiom,
! [P: ( $o > a ) > $o,X: $o > a] :
( ( P @ X )
=> ( ! [Y: $o > a] :
( ( P @ Y )
=> ( ord_less_eq_o_a @ Y @ X ) )
=> ( ( order_Greatest_o_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_356_Greatest__equality,axiom,
! [P: set_set_a > $o,X: set_set_a] :
( ( P @ X )
=> ( ! [Y: set_set_a] :
( ( P @ Y )
=> ( ord_le3724670747650509150_set_a @ Y @ X ) )
=> ( ( order_3565860530148683671_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_357_Greatest__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y: a] :
( ( P @ Y )
=> ( ord_less_eq_a @ Y @ X ) )
=> ( ( order_Greatest_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_358_GreatestI2__order,axiom,
! [P: set_a > $o,X: set_a,Q: set_a > $o] :
( ( P @ X )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( ! [X4: set_a] :
( ( P @ X4 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_359_GreatestI2__order,axiom,
! [P: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X )
=> ( ! [Y: $o > a] :
( ( P @ Y )
=> ( ord_less_eq_o_a @ Y @ X ) )
=> ( ! [X4: $o > a] :
( ( P @ X4 )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_360_GreatestI2__order,axiom,
! [P: set_set_a > $o,X: set_set_a,Q: set_set_a > $o] :
( ( P @ X )
=> ( ! [Y: set_set_a] :
( ( P @ Y )
=> ( ord_le3724670747650509150_set_a @ Y @ X ) )
=> ( ! [X4: set_set_a] :
( ( P @ X4 )
=> ( ! [Y5: set_set_a] :
( ( P @ Y5 )
=> ( ord_le3724670747650509150_set_a @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_361_GreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y: a] :
( ( P @ Y )
=> ( ord_less_eq_a @ Y @ X ) )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_362_less__fun__def,axiom,
( ord_less_o_a
= ( ^ [F2: $o > a,G: $o > a] :
( ( ord_less_eq_o_a @ F2 @ G )
& ~ ( ord_less_eq_o_a @ G @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_363_subsetI,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( member_set_a @ X4 @ B5 ) )
=> ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ).
% subsetI
thf(fact_364_subsetI,axiom,
! [A5: set_a,B5: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ X4 @ B5 ) )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) ).
% subsetI
thf(fact_365_psubsetI,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( A5 != B5 )
=> ( ord_less_set_a @ A5 @ B5 ) ) ) ).
% psubsetI
thf(fact_366_psubsetI,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( A5 != B5 )
=> ( ord_less_set_set_a @ A5 @ B5 ) ) ) ).
% psubsetI
thf(fact_367_subset__antisym,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_368_subset__antisym,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_369_open__greaterThan,axiom,
! [A: a] : ( topolo8477419352202985285open_a @ ( set_or8632414552788122084Than_a @ A ) ) ).
% open_greaterThan
thf(fact_370_topological__space__class_OopenI,axiom,
! [S: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ S )
=> ? [T2: set_a] :
( ( topolo8477419352202985285open_a @ T2 )
& ( member_a @ X4 @ T2 )
& ( ord_less_eq_set_a @ T2 @ S ) ) )
=> ( topolo8477419352202985285open_a @ S ) ) ).
% topological_space_class.openI
thf(fact_371_open__subopen,axiom,
( topolo8477419352202985285open_a
= ( ^ [S2: set_a] :
! [X2: a] :
( ( member_a @ X2 @ S2 )
=> ? [T3: set_a] :
( ( topolo8477419352202985285open_a @ T3 )
& ( member_a @ X2 @ T3 )
& ( ord_less_eq_set_a @ T3 @ S2 ) ) ) ) ) ).
% open_subopen
thf(fact_372_separation__t1,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
= ( ? [U: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( member_a @ X @ U )
& ~ ( member_a @ Y3 @ U ) ) ) ) ).
% separation_t1
thf(fact_373_psubset__trans,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A5 @ C2 ) ) ) ).
% psubset_trans
thf(fact_374_psubsetD,axiom,
! [A5: set_set_a,B5: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_375_psubsetD,axiom,
! [A5: set_a,B5: set_a,C: a] :
( ( ord_less_set_a @ A5 @ B5 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_376_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_377_subset__iff__psubset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( ord_less_set_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_378_subset__psubset__trans,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A5 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_379_subset__psubset__trans,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( ord_less_set_set_a @ B5 @ C2 )
=> ( ord_less_set_set_a @ A5 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_380_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_381_subset__not__subset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
& ~ ( ord_le3724670747650509150_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_382_psubset__subset__trans,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A5 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_383_psubset__subset__trans,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ C2 )
=> ( ord_less_set_set_a @ A5 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_384_psubset__imp__subset,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_385_psubset__imp__subset,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_386_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_387_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_388_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_389_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_390_subset__trans,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_eq_set_a @ A5 @ C2 ) ) ) ).
% subset_trans
thf(fact_391_subset__trans,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A5 @ C2 ) ) ) ).
% subset_trans
thf(fact_392_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_393_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_394_subset__refl,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ A5 ) ).
% subset_refl
thf(fact_395_subset__refl,axiom,
! [A5: set_set_a] : ( ord_le3724670747650509150_set_a @ A5 @ A5 ) ).
% subset_refl
thf(fact_396_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [T4: set_a] :
( ( member_set_a @ T4 @ A6 )
=> ( member_set_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_397_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T4: a] :
( ( member_a @ T4 @ A6 )
=> ( member_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_398_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_399_psubset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_400_equalityD2,axiom,
! [A5: set_a,B5: set_a] :
( ( A5 = B5 )
=> ( ord_less_eq_set_a @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_401_equalityD2,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( A5 = B5 )
=> ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_402_equalityD1,axiom,
! [A5: set_a,B5: set_a] :
( ( A5 = B5 )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_403_equalityD1,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( A5 = B5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_404_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( member_set_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_405_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_406_equalityE,axiom,
! [A5: set_a,B5: set_a] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B5 )
=> ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_407_equalityE,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( A5 = B5 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ~ ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_408_psubsetE,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ).
% psubsetE
thf(fact_409_psubsetE,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ).
% psubsetE
thf(fact_410_subsetD,axiom,
! [A5: set_set_a,B5: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_411_subsetD,axiom,
! [A5: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_412_in__mono,axiom,
! [A5: set_set_a,B5: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_413_in__mono,axiom,
! [A5: set_a,B5: set_a,X: a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( member_a @ X @ A5 )
=> ( member_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_414_t0__space,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
=> ? [U2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( ( member_a @ X @ U2 )
!= ( member_a @ Y3 @ U2 ) ) ) ) ).
% t0_space
thf(fact_415_t1__space,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
=> ? [U2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( member_a @ X @ U2 )
& ~ ( member_a @ Y3 @ U2 ) ) ) ).
% t1_space
thf(fact_416_separation__t0,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
= ( ? [U: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( ( member_a @ X @ U )
!= ( member_a @ Y3 @ U ) ) ) ) ) ).
% separation_t0
thf(fact_417_increasing__def,axiom,
( measur632008712001695539ng_a_a
= ( ^ [M: set_set_a,Mu: set_a > a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ M )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_418_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M: set_set_a,Mu: set_a > set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ M )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_419_increasing__def,axiom,
( measur3026350297224463318_a_o_a
= ( ^ [M: set_set_a,Mu: set_a > $o > a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ M )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_o_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_420_increasing__def,axiom,
( measur8202069185322079731_set_a
= ( ^ [M: set_set_a,Mu: set_a > set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ M )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_421_increasing__def,axiom,
( measur8059081411538669011et_a_a
= ( ^ [M: set_set_set_a,Mu: set_set_a > a] :
! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ M )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_422_increasing__def,axiom,
( measur5181028491126448947_set_a
= ( ^ [M: set_set_set_a,Mu: set_set_a > set_a] :
! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ M )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_423_increasing__def,axiom,
( measur7956495957070622006_a_o_a
= ( ^ [M: set_set_set_a,Mu: set_set_a > $o > a] :
! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ M )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_o_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_424_increasing__def,axiom,
( measur2197171192767378579_set_a
= ( ^ [M: set_set_set_a,Mu: set_set_a > set_set_a] :
! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ M )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X2 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_425_increasingD,axiom,
! [M2: set_set_a,F: set_a > a,X: set_a,Y3: set_a] :
( ( measur632008712001695539ng_a_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y3 @ M2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_426_increasingD,axiom,
! [M2: set_set_a,F: set_a > set_a,X: set_a,Y3: set_a] :
( ( measur7842569353079325843_set_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y3 @ M2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_427_increasingD,axiom,
! [M2: set_set_a,F: set_a > $o > a,X: set_a,Y3: set_a] :
( ( measur3026350297224463318_a_o_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y3 @ M2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_428_increasingD,axiom,
! [M2: set_set_a,F: set_a > set_set_a,X: set_a,Y3: set_a] :
( ( measur8202069185322079731_set_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y3 @ M2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_429_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > a,X: set_set_a,Y3: set_set_a] :
( ( measur8059081411538669011et_a_a @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( member_set_set_a @ X @ M2 )
=> ( ( member_set_set_a @ Y3 @ M2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_430_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > set_a,X: set_set_a,Y3: set_set_a] :
( ( measur5181028491126448947_set_a @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( member_set_set_a @ X @ M2 )
=> ( ( member_set_set_a @ Y3 @ M2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_431_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > $o > a,X: set_set_a,Y3: set_set_a] :
( ( measur7956495957070622006_a_o_a @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( member_set_set_a @ X @ M2 )
=> ( ( member_set_set_a @ Y3 @ M2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_432_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > set_set_a,X: set_set_a,Y3: set_set_a] :
( ( measur2197171192767378579_set_a @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( member_set_set_a @ X @ M2 )
=> ( ( member_set_set_a @ Y3 @ M2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_433_bdd__above_Opreordering__bdd__axioms,axiom,
condit6315317455391067509_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_434_bdd__above_Opreordering__bdd__axioms,axiom,
condit3670190255528792436dd_o_a @ ord_less_eq_o_a @ ord_less_o_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_435_bdd__above_Opreordering__bdd__axioms,axiom,
condit7401735437286921941_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_436_bdd__above_Opreordering__bdd__axioms,axiom,
condit4103000493307248661_bdd_a @ ord_less_eq_a @ ord_less_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_437_le__left__mono,axiom,
! [X: set_a,Y3: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ A )
=> ( ord_less_eq_set_a @ X @ A ) ) ) ).
% le_left_mono
thf(fact_438_le__left__mono,axiom,
! [X: $o > a,Y3: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ X @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ A )
=> ( ord_less_eq_o_a @ X @ A ) ) ) ).
% le_left_mono
thf(fact_439_le__left__mono,axiom,
! [X: set_set_a,Y3: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ A )
=> ( ord_le3724670747650509150_set_a @ X @ A ) ) ) ).
% le_left_mono
thf(fact_440_le__left__mono,axiom,
! [X: a,Y3: a,A: a] :
( ( ord_less_eq_a @ X @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ A )
=> ( ord_less_eq_a @ X @ A ) ) ) ).
% le_left_mono
thf(fact_441_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X5: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_442_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_a
= ( ^ [X5: $o > $o > a,Y6: $o > $o > a] :
( ( ord_less_eq_o_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_443_le__rel__bool__arg__iff,axiom,
( ord_le2411852534652195563_set_a
= ( ^ [X5: $o > set_set_a,Y6: $o > set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3724670747650509150_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_444_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X5: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_445_Ici__subset__Ioi__iff,axiom,
! [A: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ A ) @ ( set_or8632414552788122084Than_a @ B2 ) )
= ( ord_less_a @ B2 @ A ) ) ).
% Ici_subset_Ioi_iff
thf(fact_446_atLeast__eq__iff,axiom,
! [X: a,Y3: a] :
( ( ( set_ord_atLeast_a @ X )
= ( set_ord_atLeast_a @ Y3 ) )
= ( X = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_447_atLeast__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or8362275514725411625_set_a @ K ) )
= ( ord_less_eq_set_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_448_atLeast__iff,axiom,
! [I2: $o > a,K: $o > a] :
( ( member_o_a @ I2 @ ( set_ord_atLeast_o_a @ K ) )
= ( ord_less_eq_o_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_449_atLeast__iff,axiom,
! [I2: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or3904034815786525833_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_450_atLeast__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atLeast_a @ K ) )
= ( ord_less_eq_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_451_atLeast__subset__iff,axiom,
! [X: $o > a,Y3: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_ord_atLeast_o_a @ X ) @ ( set_ord_atLeast_o_a @ Y3 ) )
= ( ord_less_eq_o_a @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_452_atLeast__subset__iff,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or3904034815786525833_set_a @ X ) @ ( set_or3904034815786525833_set_a @ Y3 ) )
= ( ord_le3724670747650509150_set_a @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_453_atLeast__subset__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X ) @ ( set_or8362275514725411625_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_454_atLeast__subset__iff,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X ) @ ( set_ord_atLeast_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_455_closed__atLeast,axiom,
! [A: a] : ( topolo784654279908865136osed_a @ ( set_ord_atLeast_a @ A ) ) ).
% closed_atLeast
thf(fact_456_Ioi__le__Ico,axiom,
! [A: set_a] : ( ord_le3724670747650509150_set_a @ ( set_or460448635090783044_set_a @ A ) @ ( set_or8362275514725411625_set_a @ A ) ) ).
% Ioi_le_Ico
thf(fact_457_Ioi__le__Ico,axiom,
! [A: a] : ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ A ) @ ( set_ord_atLeast_a @ A ) ) ).
% Ioi_le_Ico
thf(fact_458_sup__lexord,axiom,
! [K: a > a,A5: a,B5: a,P: a > $o,C: a,S3: a] :
( ( ( ord_less_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( P @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( P @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( P @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( P @ S3 ) ) )
=> ( P @ ( measur6235662355876231836rd_a_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% sup_lexord
thf(fact_459_le__sup__lexord,axiom,
! [K: a > a,A5: a,B5: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6235662355876231836rd_a_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_460_le__sup__lexord,axiom,
! [K: a > set_a,A5: a,B5: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur1314364908362886140_set_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_461_le__sup__lexord,axiom,
! [K: set_a > a,A5: set_a,B5: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_set_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur1530876966822229308et_a_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_462_le__sup__lexord,axiom,
! [K: a > $o > a,A5: a,B5: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_o_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_o_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_o_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_o_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur8531965893055442349_a_o_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_463_le__sup__lexord,axiom,
! [K: a > set_set_a,A5: a,B5: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_set_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ B5 ) )
=> ( ( ( ord_less_set_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_le3724670747650509150_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_le3724670747650509150_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur3779986862995642716_set_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_464_le__sup__lexord,axiom,
! [K: set_a > set_a,A5: set_a,B5: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ B5 ) )
=> ( ( ( ord_less_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_set_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur758946168800011932_set_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_465_le__sup__lexord,axiom,
! [K: ( $o > a ) > a,A5: $o > a,B5: $o > a,Ca: $o > a,C: $o > a,S3: $o > a] :
( ( ( ord_less_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_o_a @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_o_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_o_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_o_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_o_a @ Ca @ ( measur2480691472411955797_o_a_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_466_le__sup__lexord,axiom,
! [K: set_set_a > a,A5: set_set_a,B5: set_set_a,Ca: set_set_a,C: set_set_a,S3: set_set_a] :
( ( ( ord_less_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ B5 ) )
=> ( ( ( ord_less_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ S3 ) ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ ( measur7285218718371296220et_a_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_467_le__sup__lexord,axiom,
! [K: set_a > $o > a,A5: set_a,B5: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_o_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ B5 ) )
=> ( ( ( ord_less_o_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_set_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_o_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_less_eq_o_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur6368699961593756429_a_o_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_468_le__sup__lexord,axiom,
! [K: set_a > set_set_a,A5: set_a,B5: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_set_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ B5 ) )
=> ( ( ( ord_less_set_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ord_less_eq_set_a @ Ca @ A5 ) )
=> ( ( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_le3724670747650509150_set_a @ ( K @ B5 ) @ ( K @ A5 ) )
=> ( ~ ( ord_le3724670747650509150_set_a @ ( K @ A5 ) @ ( K @ B5 ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur4867547208650036220_set_a @ A5 @ B5 @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_469_preordering__bdd_Omono,axiom,
! [Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,B5: set_set_a,A5: set_set_a] :
( ( condit6315317455391067509_set_a @ Less_eq @ Less )
=> ( ( condit4774827555938943059_set_a @ Less_eq @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( condit4774827555938943059_set_a @ Less_eq @ A5 ) ) ) ) ).
% preordering_bdd.mono
thf(fact_470_preordering__bdd_Omono,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,B5: set_a,A5: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ B5 )
=> ( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ A5 ) ) ) ) ).
% preordering_bdd.mono
thf(fact_471_Icc__subset__Ici__iff,axiom,
! [L: $o > a,H: $o > a,L2: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ L @ H ) @ ( set_ord_atLeast_o_a @ L2 ) )
= ( ~ ( ord_less_eq_o_a @ L @ H )
| ( ord_less_eq_o_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_472_Icc__subset__Ici__iff,axiom,
! [L: set_set_a,H: set_set_a,L2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ L @ H ) @ ( set_or3904034815786525833_set_a @ L2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
| ( ord_le3724670747650509150_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_473_Icc__subset__Ici__iff,axiom,
! [L: set_a,H: set_a,L2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_or8362275514725411625_set_a @ L2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_474_Icc__subset__Ici__iff,axiom,
! [L: a,H: a,L2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L @ H ) @ ( set_ord_atLeast_a @ L2 ) )
= ( ~ ( ord_less_eq_a @ L @ H )
| ( ord_less_eq_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_475_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C3: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ C3 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ C3 )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
| ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_476_chain__subset__def,axiom,
( chain_subset_set_a
= ( ^ [C3: set_set_set_a] :
! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ C3 )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
| ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_477_open__left,axiom,
! [S: set_a,X: a,Y3: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X @ S )
=> ( ( ord_less_a @ Y3 @ X )
=> ? [B4: a] :
( ( ord_less_a @ B4 @ X )
& ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ B4 @ X ) @ S ) ) ) ) ) ).
% open_left
thf(fact_478_atLeastatMost__psubset__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A @ B2 )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B2 @ D )
& ( ( ord_less_set_a @ C @ A )
| ( ord_less_set_a @ B2 @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_479_atLeastatMost__psubset__iff,axiom,
! [A: $o > a,B2: $o > a,C: $o > a,D: $o > a] :
( ( ord_less_set_o_a @ ( set_or8441445163928040022st_o_a @ A @ B2 ) @ ( set_or8441445163928040022st_o_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_o_a @ A @ B2 )
| ( ( ord_less_eq_o_a @ C @ A )
& ( ord_less_eq_o_a @ B2 @ D )
& ( ( ord_less_o_a @ C @ A )
| ( ord_less_o_a @ B2 @ D ) ) ) )
& ( ord_less_eq_o_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_480_atLeastatMost__psubset__iff,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le8783759336481291722_set_a @ ( set_or4761464488706262899_set_a @ A @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D ) )
= ( ( ~ ( ord_le3724670747650509150_set_a @ A @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A )
& ( ord_le3724670747650509150_set_a @ B2 @ D )
& ( ( ord_less_set_set_a @ C @ A )
| ( ord_less_set_set_a @ B2 @ D ) ) ) )
& ( ord_le3724670747650509150_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_481_atLeastatMost__psubset__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_a @ A @ B2 )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B2 @ D )
& ( ( ord_less_a @ C @ A )
| ( ord_less_a @ B2 @ D ) ) ) )
& ( ord_less_eq_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_482_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U3 ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_483_atLeastAtMost__iff,axiom,
! [I2: $o > a,L: $o > a,U3: $o > a] :
( ( member_o_a @ I2 @ ( set_or8441445163928040022st_o_a @ L @ U3 ) )
= ( ( ord_less_eq_o_a @ L @ I2 )
& ( ord_less_eq_o_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_484_atLeastAtMost__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or4761464488706262899_set_a @ L @ U3 ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I2 )
& ( ord_le3724670747650509150_set_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_485_atLeastAtMost__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or672772299803893939Most_a @ L @ U3 ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_486_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H )
= ( set_or6288561110385358355_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_487_Icc__eq__Icc,axiom,
! [L: $o > a,H: $o > a,L2: $o > a,H2: $o > a] :
( ( ( set_or8441445163928040022st_o_a @ L @ H )
= ( set_or8441445163928040022st_o_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_o_a @ L @ H )
& ~ ( ord_less_eq_o_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_488_Icc__eq__Icc,axiom,
! [L: set_set_a,H: set_set_a,L2: set_set_a,H2: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ L @ H )
= ( set_or4761464488706262899_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
& ~ ( ord_le3724670747650509150_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_489_Icc__eq__Icc,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_or672772299803893939Most_a @ L @ H )
= ( set_or672772299803893939Most_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_a @ L @ H )
& ~ ( ord_less_eq_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_490_greaterThanLessThan__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or6017932776736107018_set_a @ L @ U3 ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_491_greaterThanLessThan__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_492_atLeastatMost__subset__iff,axiom,
! [A: $o > a,B2: $o > a,C: $o > a,D: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ A @ B2 ) @ ( set_or8441445163928040022st_o_a @ C @ D ) )
= ( ~ ( ord_less_eq_o_a @ A @ B2 )
| ( ( ord_less_eq_o_a @ C @ A )
& ( ord_less_eq_o_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_493_atLeastatMost__subset__iff,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A )
& ( ord_le3724670747650509150_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_494_atLeastatMost__subset__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A @ B2 )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_495_atLeastatMost__subset__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ~ ( ord_less_eq_a @ A @ B2 )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_496_greaterThanAtMost__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or2503527069484367278_set_a @ L @ U3 ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_497_greaterThanAtMost__iff,axiom,
! [I2: $o > a,L: $o > a,U3: $o > a] :
( ( member_o_a @ I2 @ ( set_or1794473594002085499st_o_a @ L @ U3 ) )
= ( ( ord_less_o_a @ L @ I2 )
& ( ord_less_eq_o_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_498_greaterThanAtMost__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or6073455820589966094_set_a @ L @ U3 ) )
= ( ( ord_less_set_set_a @ L @ I2 )
& ( ord_le3724670747650509150_set_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_499_greaterThanAtMost__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_500_sup__lexord__commute,axiom,
( measur6235662355876231836rd_a_a
= ( ^ [A6: a,B6: a] : ( measur6235662355876231836rd_a_a @ B6 @ A6 ) ) ) ).
% sup_lexord_commute
thf(fact_501_sup__lexord1,axiom,
! [K: a > a,A5: a,B5: a,S3: a,C: a] :
( ( ( K @ A5 )
= ( K @ B5 ) )
=> ( ( measur6235662355876231836rd_a_a @ A5 @ B5 @ K @ S3 @ C )
= C ) ) ).
% sup_lexord1
thf(fact_502_preordering__bdd_Obdd_Ocong,axiom,
condit6541519642617408243_bdd_a = condit6541519642617408243_bdd_a ).
% preordering_bdd.bdd.cong
thf(fact_503_closed__atLeastAtMost,axiom,
! [A: a,B2: a] : ( topolo784654279908865136osed_a @ ( set_or672772299803893939Most_a @ A @ B2 ) ) ).
% closed_atLeastAtMost
thf(fact_504_Ioc__inj,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( set_or4472690218693186638Most_a @ A @ B2 )
= ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ( ord_less_eq_a @ B2 @ A )
& ( ord_less_eq_a @ D @ C ) )
| ( ( A = C )
& ( B2 = D ) ) ) ) ).
% Ioc_inj
thf(fact_505_open__greaterThanLessThan,axiom,
! [A: a,B2: a] : ( topolo8477419352202985285open_a @ ( set_or5939364468397584554Than_a @ A @ B2 ) ) ).
% open_greaterThanLessThan
thf(fact_506_Ioc__subset__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ord_less_eq_a @ B2 @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_507_preordering__bdd_Ounfold,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A5: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ A5 )
= ( ? [M: a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( Less_eq @ X2 @ M ) ) ) ) ) ).
% preordering_bdd.unfold
thf(fact_508_preordering__bdd_OI,axiom,
! [Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A5: set_set_a,M2: set_a] :
( ( condit6315317455391067509_set_a @ Less_eq @ Less )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( Less_eq @ X4 @ M2 ) )
=> ( condit4774827555938943059_set_a @ Less_eq @ A5 ) ) ) ).
% preordering_bdd.I
thf(fact_509_preordering__bdd_OI,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A5: set_a,M2: a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( Less_eq @ X4 @ M2 ) )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ A5 ) ) ) ).
% preordering_bdd.I
thf(fact_510_preordering__bdd_OE,axiom,
! [Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A5: set_set_a] :
( ( condit6315317455391067509_set_a @ Less_eq @ Less )
=> ( ( condit4774827555938943059_set_a @ Less_eq @ A5 )
=> ~ ! [M3: set_a] :
~ ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( Less_eq @ X3 @ M3 ) ) ) ) ).
% preordering_bdd.E
thf(fact_511_preordering__bdd_OE,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A5: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ A5 )
=> ~ ! [M3: a] :
~ ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( Less_eq @ X3 @ M3 ) ) ) ) ).
% preordering_bdd.E
thf(fact_512_sup__lexord__def,axiom,
( measur6235662355876231836rd_a_a
= ( ^ [A6: a,B6: a,K2: a > a,S4: a,C4: a] :
( if_a
@ ( ( K2 @ A6 )
= ( K2 @ B6 ) )
@ C4
@ ( if_a
@ ( ~ ( ord_less_eq_a @ ( K2 @ A6 ) @ ( K2 @ B6 ) )
& ~ ( ord_less_eq_a @ ( K2 @ B6 ) @ ( K2 @ A6 ) ) )
@ S4
@ ( if_a @ ( ord_less_eq_a @ ( K2 @ B6 ) @ ( K2 @ A6 ) ) @ A6 @ B6 ) ) ) ) ) ).
% sup_lexord_def
thf(fact_513_ivl__disj__un__two_I5_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U3 ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_514_ivl__disj__un__one_I6_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_515_ivl__disj__un__one_I7_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ U3 ) @ ( set_or8632414552788122084Than_a @ U3 ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_516_ivl__disj__un__two_I2_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M4 ) @ ( set_or5939364468397584554Than_a @ M4 @ U3 ) )
= ( set_or5939364468397584554Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_517_ivl__disj__un__two__touch_I3_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U3 ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_518_UnCI,axiom,
! [C: set_a,B5: set_set_a,A5: set_set_a] :
( ( ~ ( member_set_a @ C @ B5 )
=> ( member_set_a @ C @ A5 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_519_UnCI,axiom,
! [C: a,B5: set_a,A5: set_a] :
( ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ A5 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_520_Un__iff,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B5 ) )
= ( ( member_set_a @ C @ A5 )
| ( member_set_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_521_Un__iff,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) )
= ( ( member_a @ C @ A5 )
| ( member_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_522_Un__subset__iff,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ C2 )
= ( ( ord_less_eq_set_a @ A5 @ C2 )
& ( ord_less_eq_set_a @ B5 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_523_Un__subset__iff,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A5 @ B5 ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A5 @ C2 )
& ( ord_le3724670747650509150_set_a @ B5 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_524_open__Un,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo8477419352202985285open_a @ ( sup_sup_set_a @ S @ T5 ) ) ) ) ).
% open_Un
thf(fact_525_closed__Un,axiom,
! [S: set_a,T5: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo784654279908865136osed_a @ ( sup_sup_set_a @ S @ T5 ) ) ) ) ).
% closed_Un
thf(fact_526_atLeastLessThan__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or2348907005316661231_set_a @ L @ U3 ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_527_atLeastLessThan__iff,axiom,
! [I2: $o > a,L: $o > a,U3: $o > a] :
( ( member_o_a @ I2 @ ( set_or4510008498168808314an_o_a @ L @ U3 ) )
= ( ( ord_less_eq_o_a @ L @ I2 )
& ( ord_less_o_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_528_atLeastLessThan__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or7717641506629902671_set_a @ L @ U3 ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I2 )
& ( ord_less_set_set_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_529_atLeastLessThan__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L @ U3 ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_530_ivl__subset,axiom,
! [I2: a,J: a,M4: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I2 @ J ) @ ( set_or5139330845457685135Than_a @ M4 @ N ) )
= ( ( ord_less_eq_a @ J @ I2 )
| ( ( ord_less_eq_a @ M4 @ I2 )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_531_UnE,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B5 ) )
=> ( ~ ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_532_UnE,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) )
=> ( ~ ( member_a @ C @ A5 )
=> ( member_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_533_UnI1,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_534_UnI1,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ A5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_535_UnI2,axiom,
! [C: set_a,B5: set_set_a,A5: set_set_a] :
( ( member_set_a @ C @ B5 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_536_UnI2,axiom,
! [C: a,B5: set_a,A5: set_a] :
( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_537_bex__Un,axiom,
! [A5: set_a,B5: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A5 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B5 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_538_ball__Un,axiom,
! [A5: set_a,B5: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A5 @ B5 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_539_Un__assoc,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ C2 )
= ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C2 ) ) ) ).
% Un_assoc
thf(fact_540_Un__absorb,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_541_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_542_Un__left__absorb,axiom,
! [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ A5 @ B5 ) )
= ( sup_sup_set_a @ A5 @ B5 ) ) ).
% Un_left_absorb
thf(fact_543_Un__left__commute,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C2 ) )
= ( sup_sup_set_a @ B5 @ ( sup_sup_set_a @ A5 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_544_ivl__disj__un__two_I3_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_545_ivl__disj__un__two_I7_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_546_ivl__disj__un__one_I8_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_547_atLeastLessThan__inj_I2_J,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_548_atLeastLessThan__inj_I1_J,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_549_Ico__eq__Ico,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_or5139330845457685135Than_a @ L @ H )
= ( set_or5139330845457685135Than_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_a @ L @ H )
& ~ ( ord_less_a @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_550_atLeastLessThan__eq__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( ( ( set_or5139330845457685135Than_a @ A @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( A = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_551_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_552_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( sup_sup_set_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_553_subset__UnE,axiom,
! [C2: set_a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A5 @ B5 ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A5 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B5 )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_554_subset__UnE,axiom,
! [C2: set_set_a,A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A5 @ B5 ) )
=> ~ ! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ A5 )
=> ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ B5 )
=> ( C2
!= ( sup_sup_set_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_555_Un__absorb2,axiom,
! [B5: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( ( sup_sup_set_a @ A5 @ B5 )
= A5 ) ) ).
% Un_absorb2
thf(fact_556_Un__absorb2,axiom,
! [B5: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( ( sup_sup_set_set_a @ A5 @ B5 )
= A5 ) ) ).
% Un_absorb2
thf(fact_557_Un__absorb1,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_558_Un__absorb1,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( sup_sup_set_set_a @ A5 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_559_Un__upper2,axiom,
! [B5: set_a,A5: set_a] : ( ord_less_eq_set_a @ B5 @ ( sup_sup_set_a @ A5 @ B5 ) ) ).
% Un_upper2
thf(fact_560_Un__upper2,axiom,
! [B5: set_set_a,A5: set_set_a] : ( ord_le3724670747650509150_set_a @ B5 @ ( sup_sup_set_set_a @ A5 @ B5 ) ) ).
% Un_upper2
thf(fact_561_Un__upper1,axiom,
! [A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ A5 @ B5 ) ) ).
% Un_upper1
thf(fact_562_Un__upper1,axiom,
! [A5: set_set_a,B5: set_set_a] : ( ord_le3724670747650509150_set_a @ A5 @ ( sup_sup_set_set_a @ A5 @ B5 ) ) ).
% Un_upper1
thf(fact_563_Un__least,axiom,
! [A5: set_a,C2: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ C2 ) ) ) ).
% Un_least
thf(fact_564_Un__least,axiom,
! [A5: set_set_a,C2: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A5 @ B5 ) @ C2 ) ) ) ).
% Un_least
thf(fact_565_Un__mono,axiom,
! [A5: set_a,C2: set_a,B5: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B5 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_566_Un__mono,axiom,
! [A5: set_set_a,C2: set_set_a,B5: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A5 @ B5 ) @ ( sup_sup_set_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_567_ivl__disj__un__two__touch_I2_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_568_ivl__disj__un__two_I1_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U3 ) )
= ( set_or5939364468397584554Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_569_ivl__disj__un__two__touch_I1_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_a @ L @ M4 )
=> ( ( ord_less_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U3 ) )
= ( set_or5939364468397584554Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_570_ivl__disj__un__two_I4_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M4 ) @ ( set_or5939364468397584554Than_a @ M4 @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_571_atLeastLessThan__subset__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_eq_a @ B2 @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_572_ivl__disj__un__two__touch_I4_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_573_ivl__disj__un__two_I6_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M4 ) @ ( set_or4472690218693186638Most_a @ M4 @ U3 ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_574_open__right,axiom,
! [S: set_a,X: a,Y3: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X @ S )
=> ( ( ord_less_a @ X @ Y3 )
=> ? [B4: a] :
( ( ord_less_a @ X @ B4 )
& ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ X @ B4 ) @ S ) ) ) ) ) ).
% open_right
thf(fact_575_ivl__disj__un__two_I8_J,axiom,
! [L: a,M4: a,U3: a] :
( ( ord_less_eq_a @ L @ M4 )
=> ( ( ord_less_eq_a @ M4 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M4 ) @ ( set_or4472690218693186638Most_a @ M4 @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_576_ivl__disj__un__one_I5_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ U3 ) @ ( set_or8632414552788122084Than_a @ U3 ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_577_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $o > a,B2: $o > a,C: $o > a,D: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ A @ B2 ) @ ( set_or4510008498168808314an_o_a @ C @ D ) )
= ( ( ord_less_eq_o_a @ A @ B2 )
=> ( ( ord_less_eq_o_a @ C @ A )
& ( ord_less_o_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_578_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A @ B2 ) @ ( set_or7717641506629902671_set_a @ C @ D ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A )
& ( ord_less_set_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_579_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B2 ) @ ( set_or2348907005316661231_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_580_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( ord_less_eq_a @ A @ B2 )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_581_le__sup__iff,axiom,
! [X: filter_a,Y3: filter_a,Z4: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ X @ Y3 ) @ Z4 )
= ( ( ord_less_eq_filter_a @ X @ Z4 )
& ( ord_less_eq_filter_a @ Y3 @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_582_le__sup__iff,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ Z4 )
= ( ( ord_less_eq_set_a @ X @ Z4 )
& ( ord_less_eq_set_a @ Y3 @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_583_le__sup__iff,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y3 ) @ Z4 )
= ( ( ord_le3724670747650509150_set_a @ X @ Z4 )
& ( ord_le3724670747650509150_set_a @ Y3 @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_584_sup_Obounded__iff,axiom,
! [B2: filter_a,C: filter_a,A: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_filter_a @ B2 @ A )
& ( ord_less_eq_filter_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_585_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_586_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A )
& ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_587_less__supI1,axiom,
! [X: filter_a,A: filter_a,B2: filter_a] :
( ( ord_less_filter_a @ X @ A )
=> ( ord_less_filter_a @ X @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_588_less__supI1,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_589_less__supI2,axiom,
! [X: filter_a,B2: filter_a,A: filter_a] :
( ( ord_less_filter_a @ X @ B2 )
=> ( ord_less_filter_a @ X @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_590_less__supI2,axiom,
! [X: set_a,B2: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_591_sup_Oabsorb3,axiom,
! [B2: filter_a,A: filter_a] :
( ( ord_less_filter_a @ B2 @ A )
=> ( ( sup_sup_filter_a @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_592_sup_Oabsorb3,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_593_sup_Oabsorb4,axiom,
! [A: filter_a,B2: filter_a] :
( ( ord_less_filter_a @ A @ B2 )
=> ( ( sup_sup_filter_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_594_sup_Oabsorb4,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_595_sup_Ostrict__boundedE,axiom,
! [B2: filter_a,C: filter_a,A: filter_a] :
( ( ord_less_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A )
=> ~ ( ( ord_less_filter_a @ B2 @ A )
=> ~ ( ord_less_filter_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_596_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B2 @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_597_sup_OcoboundedI2,axiom,
! [C: filter_a,B2: filter_a,A: filter_a] :
( ( ord_less_eq_filter_a @ C @ B2 )
=> ( ord_less_eq_filter_a @ C @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_598_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_599_sup_OcoboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_600_sup_OcoboundedI1,axiom,
! [C: filter_a,A: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ C @ A )
=> ( ord_less_eq_filter_a @ C @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_601_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_602_sup_OcoboundedI1,axiom,
! [C: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_603_sup_Oabsorb__iff2,axiom,
( ord_less_eq_filter_a
= ( ^ [A3: filter_a,B3: filter_a] :
( ( sup_sup_filter_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_604_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_605_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_606_sup_Oabsorb__iff1,axiom,
( ord_less_eq_filter_a
= ( ^ [B3: filter_a,A3: filter_a] :
( ( sup_sup_filter_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_607_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_608_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_609_sup_Ocobounded2,axiom,
! [B2: filter_a,A: filter_a] : ( ord_less_eq_filter_a @ B2 @ ( sup_sup_filter_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_610_sup_Ocobounded2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_611_sup_Ocobounded2,axiom,
! [B2: set_set_a,A: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( sup_sup_set_set_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_612_sup_Ocobounded1,axiom,
! [A: filter_a,B2: filter_a] : ( ord_less_eq_filter_a @ A @ ( sup_sup_filter_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_613_sup_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_614_sup_Ocobounded1,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_615_sup_Oorder__iff,axiom,
( ord_less_eq_filter_a
= ( ^ [B3: filter_a,A3: filter_a] :
( A3
= ( sup_sup_filter_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_616_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_617_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( A3
= ( sup_sup_set_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_618_sup_OboundedI,axiom,
! [B2: filter_a,A: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A )
=> ( ( ord_less_eq_filter_a @ C @ A )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_619_sup_OboundedI,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_620_sup_OboundedI,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_621_sup_OboundedE,axiom,
! [B2: filter_a,C: filter_a,A: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_filter_a @ B2 @ A )
=> ~ ( ord_less_eq_filter_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_622_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_623_sup_OboundedE,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A )
=> ~ ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ~ ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_624_sup__absorb2,axiom,
! [X: filter_a,Y3: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y3 )
=> ( ( sup_sup_filter_a @ X @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_625_sup__absorb2,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( sup_sup_set_a @ X @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_626_sup__absorb2,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( sup_sup_set_set_a @ X @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_627_sup__absorb1,axiom,
! [Y3: filter_a,X: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X )
=> ( ( sup_sup_filter_a @ X @ Y3 )
= X ) ) ).
% sup_absorb1
thf(fact_628_sup__absorb1,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( sup_sup_set_a @ X @ Y3 )
= X ) ) ).
% sup_absorb1
thf(fact_629_sup__absorb1,axiom,
! [Y3: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( ( sup_sup_set_set_a @ X @ Y3 )
= X ) ) ).
% sup_absorb1
thf(fact_630_sup_Oabsorb2,axiom,
! [A: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A @ B2 )
=> ( ( sup_sup_filter_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_631_sup_Oabsorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_632_sup_Oabsorb2,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( sup_sup_set_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_633_sup_Oabsorb1,axiom,
! [B2: filter_a,A: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A )
=> ( ( sup_sup_filter_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_634_sup_Oabsorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_635_sup_Oabsorb1,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( sup_sup_set_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_636_sup__unique,axiom,
! [F: filter_a > filter_a > filter_a,X: filter_a,Y3: filter_a] :
( ! [X4: filter_a,Y: filter_a] : ( ord_less_eq_filter_a @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: filter_a,Y: filter_a] : ( ord_less_eq_filter_a @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: filter_a,Y: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ Y @ X4 )
=> ( ( ord_less_eq_filter_a @ Z2 @ X4 )
=> ( ord_less_eq_filter_a @ ( F @ Y @ Z2 ) @ X4 ) ) )
=> ( ( sup_sup_filter_a @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_637_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y3: set_a] :
( ! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( ord_less_eq_set_a @ Z2 @ X4 )
=> ( ord_less_eq_set_a @ ( F @ Y @ Z2 ) @ X4 ) ) )
=> ( ( sup_sup_set_a @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_638_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y3: set_set_a] :
( ! [X4: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ( ( ord_le3724670747650509150_set_a @ Z2 @ X4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y @ Z2 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_a @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_639_sup_OorderI,axiom,
! [A: filter_a,B2: filter_a] :
( ( A
= ( sup_sup_filter_a @ A @ B2 ) )
=> ( ord_less_eq_filter_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_640_sup_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_641_sup_OorderI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A
= ( sup_sup_set_set_a @ A @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_642_sup_OorderE,axiom,
! [B2: filter_a,A: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A )
=> ( A
= ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_643_sup_OorderE,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_644_sup_OorderE,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_645_le__iff__sup,axiom,
( ord_less_eq_filter_a
= ( ^ [X2: filter_a,Y2: filter_a] :
( ( sup_sup_filter_a @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_646_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_647_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_648_sup__least,axiom,
! [Y3: filter_a,X: filter_a,Z4: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X )
=> ( ( ord_less_eq_filter_a @ Z4 @ X )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ Y3 @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_649_sup__least,axiom,
! [Y3: set_a,X: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( ord_less_eq_set_a @ Z4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y3 @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_650_sup__least,axiom,
! [Y3: set_set_a,X: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( ( ord_le3724670747650509150_set_a @ Z4 @ X )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y3 @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_651_sup__mono,axiom,
! [A: filter_a,C: filter_a,B2: filter_a,D: filter_a] :
( ( ord_less_eq_filter_a @ A @ C )
=> ( ( ord_less_eq_filter_a @ B2 @ D )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ A @ B2 ) @ ( sup_sup_filter_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_652_sup__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_653_sup__mono,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_654_sup_Omono,axiom,
! [C: filter_a,A: filter_a,D: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ C @ A )
=> ( ( ord_less_eq_filter_a @ D @ B2 )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ C @ D ) @ ( sup_sup_filter_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_655_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_656_sup_Omono,axiom,
! [C: set_set_a,A: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D ) @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_657_le__supI2,axiom,
! [X: filter_a,B2: filter_a,A: filter_a] :
( ( ord_less_eq_filter_a @ X @ B2 )
=> ( ord_less_eq_filter_a @ X @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_658_le__supI2,axiom,
! [X: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_659_le__supI2,axiom,
! [X: set_set_a,B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_660_le__supI1,axiom,
! [X: filter_a,A: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ X @ A )
=> ( ord_less_eq_filter_a @ X @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_661_le__supI1,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_662_le__supI1,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_663_sup__ge2,axiom,
! [Y3: filter_a,X: filter_a] : ( ord_less_eq_filter_a @ Y3 @ ( sup_sup_filter_a @ X @ Y3 ) ) ).
% sup_ge2
thf(fact_664_sup__ge2,axiom,
! [Y3: set_a,X: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X @ Y3 ) ) ).
% sup_ge2
thf(fact_665_sup__ge2,axiom,
! [Y3: set_set_a,X: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( sup_sup_set_set_a @ X @ Y3 ) ) ).
% sup_ge2
thf(fact_666_sup__ge1,axiom,
! [X: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ X @ ( sup_sup_filter_a @ X @ Y3 ) ) ).
% sup_ge1
thf(fact_667_sup__ge1,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) ) ).
% sup_ge1
thf(fact_668_sup__ge1,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ X @ Y3 ) ) ).
% sup_ge1
thf(fact_669_le__supI,axiom,
! [A: filter_a,X: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A @ X )
=> ( ( ord_less_eq_filter_a @ B2 @ X )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_670_le__supI,axiom,
! [A: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_671_le__supI,axiom,
! [A: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_672_le__supE,axiom,
! [A: filter_a,B2: filter_a,X: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_filter_a @ A @ X )
=> ~ ( ord_less_eq_filter_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_673_le__supE,axiom,
! [A: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_674_le__supE,axiom,
! [A: set_set_a,B2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ X )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ X )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_675_inf__sup__ord_I3_J,axiom,
! [X: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ X @ ( sup_sup_filter_a @ X @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_676_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_677_inf__sup__ord_I3_J,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ X @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_678_inf__sup__ord_I4_J,axiom,
! [Y3: filter_a,X: filter_a] : ( ord_less_eq_filter_a @ Y3 @ ( sup_sup_filter_a @ X @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_679_inf__sup__ord_I4_J,axiom,
! [Y3: set_a,X: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_680_inf__sup__ord_I4_J,axiom,
! [Y3: set_set_a,X: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( sup_sup_set_set_a @ X @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_681_sup_Ostrict__coboundedI2,axiom,
! [C: filter_a,B2: filter_a,A: filter_a] :
( ( ord_less_filter_a @ C @ B2 )
=> ( ord_less_filter_a @ C @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_682_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B2: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_683_sup_Ostrict__coboundedI1,axiom,
! [C: filter_a,A: filter_a,B2: filter_a] :
( ( ord_less_filter_a @ C @ A )
=> ( ord_less_filter_a @ C @ ( sup_sup_filter_a @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_684_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B2: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_685_sup_Ostrict__order__iff,axiom,
( ord_less_filter_a
= ( ^ [B3: filter_a,A3: filter_a] :
( ( A3
= ( sup_sup_filter_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_686_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_687_Icc__subset__Iic__iff,axiom,
! [L: $o > a,H: $o > a,H2: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ L @ H ) @ ( set_ord_atMost_o_a @ H2 ) )
= ( ~ ( ord_less_eq_o_a @ L @ H )
| ( ord_less_eq_o_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_688_Icc__subset__Iic__iff,axiom,
! [L: set_set_a,H: set_set_a,H2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ L @ H ) @ ( set_or4016371710855203973_set_a @ H2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
| ( ord_le3724670747650509150_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_689_Icc__subset__Iic__iff,axiom,
! [L: set_a,H: set_a,H2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_ord_atMost_set_a @ H2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_690_Icc__subset__Iic__iff,axiom,
! [L: a,H: a,H2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L @ H ) @ ( set_ord_atMost_a @ H2 ) )
= ( ~ ( ord_less_eq_a @ L @ H )
| ( ord_less_eq_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_691_ivl__disj__un__one_I3_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= ( set_ord_atMost_a @ U3 ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_692_at__within__Icc__at__right,axiom,
! [A: a,B2: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( topolo1902352237885396414thin_a @ A @ ( set_or672772299803893939Most_a @ A @ B2 ) )
= ( topolo1902352237885396414thin_a @ A @ ( set_or8632414552788122084Than_a @ A ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_693_preordering__bdd_Oempty,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ bot_bot_set_a ) ) ).
% preordering_bdd.empty
thf(fact_694_greaterThanAtMost__empty__iff2,axiom,
! [K: set_a,L: set_a] :
( ( bot_bot_set_set_a
= ( set_or2503527069484367278_set_a @ K @ L ) )
= ( ~ ( ord_less_set_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_695_greaterThanAtMost__empty__iff2,axiom,
! [K: a,L: a] :
( ( bot_bot_set_a
= ( set_or4472690218693186638Most_a @ K @ L ) )
= ( ~ ( ord_less_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_696_greaterThanAtMost__empty__iff,axiom,
! [K: set_a,L: set_a] :
( ( ( set_or2503527069484367278_set_a @ K @ L )
= bot_bot_set_set_a )
= ( ~ ( ord_less_set_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_697_greaterThanAtMost__empty__iff,axiom,
! [K: a,L: a] :
( ( ( set_or4472690218693186638Most_a @ K @ L )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_698_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_699_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_700_all__not__in__conv,axiom,
! [A5: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_701_all__not__in__conv,axiom,
! [A5: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_702_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_703_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_704_atMost__eq__iff,axiom,
! [X: a,Y3: a] :
( ( ( set_ord_atMost_a @ X )
= ( set_ord_atMost_a @ Y3 ) )
= ( X = Y3 ) ) ).
% atMost_eq_iff
thf(fact_705_subset__empty,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_706_subset__empty,axiom,
! [A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ bot_bot_set_set_a )
= ( A5 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_707_empty__subsetI,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A5 ) ).
% empty_subsetI
thf(fact_708_empty__subsetI,axiom,
! [A5: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A5 ) ).
% empty_subsetI
thf(fact_709_open__empty,axiom,
topolo8477419352202985285open_a @ bot_bot_set_a ).
% open_empty
thf(fact_710_Un__empty,axiom,
! [A5: set_a,B5: set_a] :
( ( ( sup_sup_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ( A5 = bot_bot_set_a )
& ( B5 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_711_atMost__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_712_atMost__iff,axiom,
! [I2: $o > a,K: $o > a] :
( ( member_o_a @ I2 @ ( set_ord_atMost_o_a @ K ) )
= ( ord_less_eq_o_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_713_atMost__iff,axiom,
! [I2: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or4016371710855203973_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_714_atMost__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atMost_a @ K ) )
= ( ord_less_eq_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_715_closed__empty,axiom,
topolo784654279908865136osed_a @ bot_bot_set_a ).
% closed_empty
thf(fact_716_atLeastatMost__empty__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( set_or6288561110385358355_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_717_atLeastatMost__empty__iff,axiom,
! [A: $o > a,B2: $o > a] :
( ( ( set_or8441445163928040022st_o_a @ A @ B2 )
= bot_bot_set_o_a )
= ( ~ ( ord_less_eq_o_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_718_atLeastatMost__empty__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ A @ B2 )
= bot_bo3380559777022489994_set_a )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_719_atLeastatMost__empty__iff,axiom,
! [A: a,B2: a] :
( ( ( set_or672772299803893939Most_a @ A @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_eq_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_720_atLeastatMost__empty__iff2,axiom,
! [A: set_a,B2: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_721_atLeastatMost__empty__iff2,axiom,
! [A: $o > a,B2: $o > a] :
( ( bot_bot_set_o_a
= ( set_or8441445163928040022st_o_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_o_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_722_atLeastatMost__empty__iff2,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( bot_bo3380559777022489994_set_a
= ( set_or4761464488706262899_set_a @ A @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_723_atLeastatMost__empty__iff2,axiom,
! [A: a,B2: a] :
( ( bot_bot_set_a
= ( set_or672772299803893939Most_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_a @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_724_atLeastatMost__empty_H,axiom,
! [A: set_a,B2: set_a] :
( ~ ( ord_less_eq_set_a @ A @ B2 )
=> ( ( set_or6288561110385358355_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_725_atLeastatMost__empty_H,axiom,
! [A: $o > a,B2: $o > a] :
( ~ ( ord_less_eq_o_a @ A @ B2 )
=> ( ( set_or8441445163928040022st_o_a @ A @ B2 )
= bot_bot_set_o_a ) ) ).
% atLeastatMost_empty'
thf(fact_726_atLeastatMost__empty_H,axiom,
! [A: set_set_a,B2: set_set_a] :
( ~ ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( set_or4761464488706262899_set_a @ A @ B2 )
= bot_bo3380559777022489994_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_727_atLeastatMost__empty_H,axiom,
! [A: a,B2: a] :
( ~ ( ord_less_eq_a @ A @ B2 )
=> ( ( set_or672772299803893939Most_a @ A @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_728_atLeastatMost__empty,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( set_or6288561110385358355_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastatMost_empty
thf(fact_729_atLeastatMost__empty,axiom,
! [B2: a,A: a] :
( ( ord_less_a @ B2 @ A )
=> ( ( set_or672772299803893939Most_a @ A @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty
thf(fact_730_atLeastLessThan__empty,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( set_or2348907005316661231_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_731_atLeastLessThan__empty,axiom,
! [B2: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A )
=> ( ( set_or4510008498168808314an_o_a @ A @ B2 )
= bot_bot_set_o_a ) ) ).
% atLeastLessThan_empty
thf(fact_732_atLeastLessThan__empty,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( set_or7717641506629902671_set_a @ A @ B2 )
= bot_bo3380559777022489994_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_733_atLeastLessThan__empty,axiom,
! [B2: a,A: a] :
( ( ord_less_eq_a @ B2 @ A )
=> ( ( set_or5139330845457685135Than_a @ A @ B2 )
= bot_bot_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_734_atLeastLessThan__empty__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( set_or2348907005316661231_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ~ ( ord_less_set_a @ A @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_735_atLeastLessThan__empty__iff,axiom,
! [A: a,B2: a] :
( ( ( set_or5139330845457685135Than_a @ A @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ A @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_736_atLeastLessThan__empty__iff2,axiom,
! [A: set_a,B2: set_a] :
( ( bot_bot_set_set_a
= ( set_or2348907005316661231_set_a @ A @ B2 ) )
= ( ~ ( ord_less_set_a @ A @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_737_atLeastLessThan__empty__iff2,axiom,
! [A: a,B2: a] :
( ( bot_bot_set_a
= ( set_or5139330845457685135Than_a @ A @ B2 ) )
= ( ~ ( ord_less_a @ A @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_738_atMost__subset__iff,axiom,
! [X: $o > a,Y3: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_ord_atMost_o_a @ X ) @ ( set_ord_atMost_o_a @ Y3 ) )
= ( ord_less_eq_o_a @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_739_atMost__subset__iff,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4016371710855203973_set_a @ X ) @ ( set_or4016371710855203973_set_a @ Y3 ) )
= ( ord_le3724670747650509150_set_a @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_740_atMost__subset__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_741_atMost__subset__iff,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X ) @ ( set_ord_atMost_a @ Y3 ) )
= ( ord_less_eq_a @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_742_greaterThanLessThan__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or6017932776736107018_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_743_greaterThanLessThan__empty,axiom,
! [L: $o > a,K: $o > a] :
( ( ord_less_eq_o_a @ L @ K )
=> ( ( set_or8643659207013295391an_o_a @ K @ L )
= bot_bot_set_o_a ) ) ).
% greaterThanLessThan_empty
thf(fact_744_greaterThanLessThan__empty,axiom,
! [L: set_set_a,K: set_set_a] :
( ( ord_le3724670747650509150_set_a @ L @ K )
=> ( ( set_or5544419673604628330_set_a @ K @ L )
= bot_bo3380559777022489994_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_745_greaterThanLessThan__empty,axiom,
! [L: a,K: a] :
( ( ord_less_eq_a @ L @ K )
=> ( ( set_or5939364468397584554Than_a @ K @ L )
= bot_bot_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_746_greaterThanAtMost__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or2503527069484367278_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_747_greaterThanAtMost__empty,axiom,
! [L: $o > a,K: $o > a] :
( ( ord_less_eq_o_a @ L @ K )
=> ( ( set_or1794473594002085499st_o_a @ K @ L )
= bot_bot_set_o_a ) ) ).
% greaterThanAtMost_empty
thf(fact_748_greaterThanAtMost__empty,axiom,
! [L: set_set_a,K: set_set_a] :
( ( ord_le3724670747650509150_set_a @ L @ K )
=> ( ( set_or6073455820589966094_set_a @ K @ L )
= bot_bo3380559777022489994_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_749_greaterThanAtMost__empty,axiom,
! [L: a,K: a] :
( ( ord_less_eq_a @ L @ K )
=> ( ( set_or4472690218693186638Most_a @ K @ L )
= bot_bot_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_750_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_751_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_752_equals0D,axiom,
! [A5: set_set_a,A: set_a] :
( ( A5 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A5 ) ) ).
% equals0D
thf(fact_753_equals0D,axiom,
! [A5: set_a,A: a] :
( ( A5 = bot_bot_set_a )
=> ~ ( member_a @ A @ A5 ) ) ).
% equals0D
thf(fact_754_equals0I,axiom,
! [A5: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A5 )
=> ( A5 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_755_equals0I,axiom,
! [A5: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A5 )
=> ( A5 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_756_ex__in__conv,axiom,
! [A5: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A5 ) )
= ( A5 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_757_ex__in__conv,axiom,
! [A5: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A5 ) )
= ( A5 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_758_not__empty__eq__Iic__eq__empty,axiom,
! [H: a] :
( bot_bot_set_a
!= ( set_ord_atMost_a @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_759_bot_Oextremum,axiom,
! [A: filter_a] : ( ord_less_eq_filter_a @ bot_bot_filter_a @ A ) ).
% bot.extremum
thf(fact_760_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_761_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_762_bot_Oextremum__unique,axiom,
! [A: filter_a] :
( ( ord_less_eq_filter_a @ A @ bot_bot_filter_a )
= ( A = bot_bot_filter_a ) ) ).
% bot.extremum_unique
thf(fact_763_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_764_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_765_bot_Oextremum__uniqueI,axiom,
! [A: filter_a] :
( ( ord_less_eq_filter_a @ A @ bot_bot_filter_a )
=> ( A = bot_bot_filter_a ) ) ).
% bot.extremum_uniqueI
thf(fact_766_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_767_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_768_bot_Oextremum__strict,axiom,
! [A: filter_a] :
~ ( ord_less_filter_a @ A @ bot_bot_filter_a ) ).
% bot.extremum_strict
thf(fact_769_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_770_bot_Onot__eq__extremum,axiom,
! [A: filter_a] :
( ( A != bot_bot_filter_a )
= ( ord_less_filter_a @ bot_bot_filter_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_771_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_772_at__le,axiom,
! [S3: set_a,T: set_a,X: a] :
( ( ord_less_eq_set_a @ S3 @ T )
=> ( ord_less_eq_filter_a @ ( topolo1902352237885396414thin_a @ X @ S3 ) @ ( topolo1902352237885396414thin_a @ X @ T ) ) ) ).
% at_le
thf(fact_773_at__within__union,axiom,
! [X: a,S: set_a,T5: set_a] :
( ( topolo1902352237885396414thin_a @ X @ ( sup_sup_set_a @ S @ T5 ) )
= ( sup_sup_filter_a @ ( topolo1902352237885396414thin_a @ X @ S ) @ ( topolo1902352237885396414thin_a @ X @ T5 ) ) ) ).
% at_within_union
thf(fact_774_closed__atMost,axiom,
! [A: a] : ( topolo784654279908865136osed_a @ ( set_ord_atMost_a @ A ) ) ).
% closed_atMost
thf(fact_775_Un__empty__left,axiom,
! [B5: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_776_Un__empty__right,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Un_empty_right
thf(fact_777_not__psubset__empty,axiom,
! [A5: set_a] :
~ ( ord_less_set_a @ A5 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_778_not__empty__eq__Ici__eq__empty,axiom,
! [L: a] :
( bot_bot_set_a
!= ( set_ord_atLeast_a @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_779_subset__emptyI,axiom,
! [A5: set_set_a] :
( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_780_subset__emptyI,axiom,
! [A5: set_a] :
( ! [X4: a] :
~ ( member_a @ X4 @ A5 )
=> ( ord_less_eq_set_a @ A5 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_781_ivl__disj__un__singleton_I4_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U3 ) @ ( insert_a @ U3 @ bot_bot_set_a ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_782_ivl__disj__un__singleton_I3_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_783_ivl__disj__un__singleton_I5_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_784_ivl__disj__un__singleton_I6_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U3 ) @ ( insert_a @ U3 @ bot_bot_set_a ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_785_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: set_a] : ( A6 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_786_insert__absorb2,axiom,
! [X: a,A5: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A5 ) )
= ( insert_a @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_787_insert__iff,axiom,
! [A: set_a,B2: set_a,A5: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A5 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_788_insert__iff,axiom,
! [A: a,B2: a,A5: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A5 ) )
= ( ( A = B2 )
| ( member_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_789_insertCI,axiom,
! [A: set_a,B5: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B5 )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_790_insertCI,axiom,
! [A: a,B5: set_a,B2: a] :
( ( ~ ( member_a @ A @ B5 )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_791_insert__subset,axiom,
! [X: set_a,A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A5 ) @ B5 )
= ( ( member_set_a @ X @ B5 )
& ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_792_insert__subset,axiom,
! [X: a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A5 ) @ B5 )
= ( ( member_a @ X @ B5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_793_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_794_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_795_at__within__empty,axiom,
! [A: a] :
( ( topolo1902352237885396414thin_a @ A @ bot_bot_set_a )
= bot_bot_filter_a ) ).
% at_within_empty
thf(fact_796_Un__insert__right,axiom,
! [A5: set_a,A: a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_797_Un__insert__left,axiom,
! [A: a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B5 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_798_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A5: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A5 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_799_singleton__insert__inj__eq,axiom,
! [B2: set_a,A: set_a,A5: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A5 ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_800_singleton__insert__inj__eq_H,axiom,
! [A: a,A5: set_a,B2: a] :
( ( ( insert_a @ A @ A5 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_801_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A5: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A @ A5 )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_802_atLeastAtMost__singleton,axiom,
! [A: a] :
( ( set_or672772299803893939Most_a @ A @ A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% atLeastAtMost_singleton
thf(fact_803_atLeastAtMost__singleton__iff,axiom,
! [A: a,B2: a,C: a] :
( ( ( set_or672772299803893939Most_a @ A @ B2 )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( B2 = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_804_closed__singleton,axiom,
! [A: a] : ( topolo784654279908865136osed_a @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% closed_singleton
thf(fact_805_insert__subsetI,axiom,
! [X: set_a,A5: set_set_a,X6: set_set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ X6 @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X6 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_806_insert__subsetI,axiom,
! [X: a,A5: set_a,X6: set_a] :
( ( member_a @ X @ A5 )
=> ( ( ord_less_eq_set_a @ X6 @ A5 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X6 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_807_mk__disjoint__insert,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ? [B8: set_set_a] :
( ( A5
= ( insert_set_a @ A @ B8 ) )
& ~ ( member_set_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_808_mk__disjoint__insert,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ? [B8: set_a] :
( ( A5
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_809_insert__commute,axiom,
! [X: a,Y3: a,A5: set_a] :
( ( insert_a @ X @ ( insert_a @ Y3 @ A5 ) )
= ( insert_a @ Y3 @ ( insert_a @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_810_insert__eq__iff,axiom,
! [A: set_a,A5: set_set_a,B2: set_a,B5: set_set_a] :
( ~ ( member_set_a @ A @ A5 )
=> ( ~ ( member_set_a @ B2 @ B5 )
=> ( ( ( insert_set_a @ A @ A5 )
= ( insert_set_a @ B2 @ B5 ) )
= ( ( ( A = B2 )
=> ( A5 = B5 ) )
& ( ( A != B2 )
=> ? [C3: set_set_a] :
( ( A5
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B5
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_811_insert__eq__iff,axiom,
! [A: a,A5: set_a,B2: a,B5: set_a] :
( ~ ( member_a @ A @ A5 )
=> ( ~ ( member_a @ B2 @ B5 )
=> ( ( ( insert_a @ A @ A5 )
= ( insert_a @ B2 @ B5 ) )
= ( ( ( A = B2 )
=> ( A5 = B5 ) )
& ( ( A != B2 )
=> ? [C3: set_a] :
( ( A5
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B5
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_812_insert__absorb,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( insert_set_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_813_insert__absorb,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( insert_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_814_insert__ident,axiom,
! [X: set_a,A5: set_set_a,B5: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ~ ( member_set_a @ X @ B5 )
=> ( ( ( insert_set_a @ X @ A5 )
= ( insert_set_a @ X @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_815_insert__ident,axiom,
! [X: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ~ ( member_a @ X @ B5 )
=> ( ( ( insert_a @ X @ A5 )
= ( insert_a @ X @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_816_Set_Oset__insert,axiom,
! [X: set_a,A5: set_set_a] :
( ( member_set_a @ X @ A5 )
=> ~ ! [B8: set_set_a] :
( ( A5
= ( insert_set_a @ X @ B8 ) )
=> ( member_set_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_817_Set_Oset__insert,axiom,
! [X: a,A5: set_a] :
( ( member_a @ X @ A5 )
=> ~ ! [B8: set_a] :
( ( A5
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_818_insertI2,axiom,
! [A: set_a,B5: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B5 )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_819_insertI2,axiom,
! [A: a,B5: set_a,B2: a] :
( ( member_a @ A @ B5 )
=> ( member_a @ A @ ( insert_a @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_820_insertI1,axiom,
! [A: set_a,B5: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B5 ) ) ).
% insertI1
thf(fact_821_insertI1,axiom,
! [A: a,B5: set_a] : ( member_a @ A @ ( insert_a @ A @ B5 ) ) ).
% insertI1
thf(fact_822_insertE,axiom,
! [A: set_a,B2: set_a,A5: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A5 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_823_insertE,axiom,
! [A: a,B2: a,A5: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A5 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_824_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_825_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_826_insert__mono,axiom,
! [C2: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_827_subset__insert,axiom,
! [X: set_a,A5: set_set_a,B5: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X @ B5 ) )
= ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_828_subset__insert,axiom,
! [X: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ B5 ) )
= ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_829_subset__insertI,axiom,
! [B5: set_a,A: a] : ( ord_less_eq_set_a @ B5 @ ( insert_a @ A @ B5 ) ) ).
% subset_insertI
thf(fact_830_subset__insertI,axiom,
! [B5: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B5 @ ( insert_set_a @ A @ B5 ) ) ).
% subset_insertI
thf(fact_831_subset__insertI2,axiom,
! [A5: set_a,B5: set_a,B2: a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ord_less_eq_set_a @ A5 @ ( insert_a @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_832_subset__insertI2,axiom,
! [A5: set_set_a,B5: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_833_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_834_insert__not__empty,axiom,
! [A: a,A5: set_a] :
( ( insert_a @ A @ A5 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_835_doubleton__eq__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_836_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_837_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_838_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_839_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_840_closed__insert,axiom,
! [S: set_a,A: a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo784654279908865136osed_a @ ( insert_a @ A @ S ) ) ) ).
% closed_insert
thf(fact_841_subset__singletonD,axiom,
! [A5: set_a,X: a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A5 = bot_bot_set_a )
| ( A5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_842_subset__singletonD,axiom,
! [A5: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A5 = bot_bot_set_set_a )
| ( A5
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_843_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_844_subset__singleton__iff,axiom,
! [X6: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X6 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X6 = bot_bot_set_set_a )
| ( X6
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_845_atLeastAtMost__singleton_H,axiom,
! [A: a,B2: a] :
( ( A = B2 )
=> ( ( set_or672772299803893939Most_a @ A @ B2 )
= ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_846_singleton__Un__iff,axiom,
! [X: a,A5: set_a,B5: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A5 @ B5 ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_847_Un__singleton__iff,axiom,
! [A5: set_a,B5: set_a,X: a] :
( ( ( sup_sup_set_a @ A5 @ B5 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_848_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_849_ivl__disj__un__singleton_I1_J,axiom,
! [L: a] :
( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or8632414552788122084Than_a @ L ) )
= ( set_ord_atLeast_a @ L ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_850_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_851_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_852_is__singletonE,axiom,
! [A5: set_a] :
( ( is_singleton_a @ A5 )
=> ~ ! [X4: a] :
( A5
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_853_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X2: a] :
( A6
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_854_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_855_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_856_atMost__UNIV__triv,axiom,
( ( set_ord_atMost_set_a @ top_top_set_a )
= top_top_set_set_a ) ).
% atMost_UNIV_triv
thf(fact_857_atLeast__empty__triv,axiom,
( ( set_or8362275514725411625_set_a @ bot_bot_set_a )
= top_top_set_set_a ) ).
% atLeast_empty_triv
thf(fact_858_lessThan__eq__iff,axiom,
! [X: a,Y3: a] :
( ( ( set_ord_lessThan_a @ X )
= ( set_ord_lessThan_a @ Y3 ) )
= ( X = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_859_open__UNIV,axiom,
topolo8477419352202985285open_a @ top_top_set_a ).
% open_UNIV
thf(fact_860_lessThan__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or5421148953861284865_set_a @ K ) )
= ( ord_less_set_a @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_861_lessThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_862_closed__UNIV,axiom,
topolo784654279908865136osed_a @ top_top_set_a ).
% closed_UNIV
thf(fact_863_lessThan__subset__iff,axiom,
! [X: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X ) @ ( set_ord_lessThan_a @ Y3 ) )
= ( ord_less_eq_a @ X @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_864_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: filter_a,Y3: filter_a] :
( ( ( set_or9026928291982918841lter_a @ X @ Y3 )
= top_top_set_filter_a )
= ( ( X = bot_bot_filter_a )
& ( Y3 = top_top_filter_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_865_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( ( set_or6288561110385358355_set_a @ X @ Y3 )
= top_top_set_set_a )
= ( ( X = bot_bot_set_a )
& ( Y3 = top_top_set_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_866_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_867_top_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
=> ( A = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_868_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_869_top_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
= ( A = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_870_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_871_top__greatest,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% top_greatest
thf(fact_872_top_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != top_top_set_a )
= ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_873_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_874_open__lessThan,axiom,
! [A: a] : ( topolo8477419352202985285open_a @ ( set_ord_lessThan_a @ A ) ) ).
% open_lessThan
thf(fact_875_UNIV__eq__I,axiom,
! [A5: set_set_a] :
( ! [X4: set_a] : ( member_set_a @ X4 @ A5 )
=> ( top_top_set_set_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_876_UNIV__eq__I,axiom,
! [A5: set_a] :
( ! [X4: a] : ( member_a @ X4 @ A5 )
=> ( top_top_set_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_877_UNIV__witness,axiom,
? [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_878_UNIV__witness,axiom,
? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_879_atMost__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= top_top_set_set_a )
= ( X = top_top_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_880_subset__UNIV,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_881_subset__UNIV,axiom,
! [A5: set_set_a] : ( ord_le3724670747650509150_set_a @ A5 @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_882_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_883_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_884_Un__UNIV__left,axiom,
! [B5: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B5 )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_885_Un__UNIV__right,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_886_at__eq__sup__left__right,axiom,
! [X: a] :
( ( topolo1902352237885396414thin_a @ X @ top_top_set_a )
= ( sup_sup_filter_a @ ( topolo1902352237885396414thin_a @ X @ ( set_ord_lessThan_a @ X ) ) @ ( topolo1902352237885396414thin_a @ X @ ( set_or8632414552788122084Than_a @ X ) ) ) ) ).
% at_eq_sup_left_right
thf(fact_887_lessThan__strict__subset__iff,axiom,
! [M4: a,N: a] :
( ( ord_less_set_a @ ( set_ord_lessThan_a @ M4 ) @ ( set_ord_lessThan_a @ N ) )
= ( ord_less_a @ M4 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_888_at__within__open,axiom,
! [A: a,S: set_a] :
( ( member_a @ A @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo1902352237885396414thin_a @ A @ S )
= ( topolo1902352237885396414thin_a @ A @ top_top_set_a ) ) ) ) ).
% at_within_open
thf(fact_889_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_890_atLeast__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_or8362275514725411625_set_a @ X )
= top_top_set_set_a )
= ( X = bot_bot_set_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_891_atLeast__eq__UNIV__iff,axiom,
! [X: filter_a] :
( ( ( set_or1258606792136358095lter_a @ X )
= top_top_set_filter_a )
= ( X = bot_bot_filter_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_892_is__singletonI_H,axiom,
! [A5: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [X4: set_a,Y: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( ( member_set_a @ Y @ A5 )
=> ( X4 = Y ) ) )
=> ( is_singleton_set_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_893_is__singletonI_H,axiom,
! [A5: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ! [X4: a,Y: a] :
( ( member_a @ X4 @ A5 )
=> ( ( member_a @ Y @ A5 )
=> ( X4 = Y ) ) )
=> ( is_singleton_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_894_at__within__Icc__at__left,axiom,
! [A: a,B2: a] :
( ( ord_less_a @ A @ B2 )
=> ( ( topolo1902352237885396414thin_a @ B2 @ ( set_or672772299803893939Most_a @ A @ B2 ) )
= ( topolo1902352237885396414thin_a @ B2 @ ( set_ord_lessThan_a @ B2 ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_895_Iic__subset__Iio__iff,axiom,
! [A: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ A ) @ ( set_ord_lessThan_a @ B2 ) )
= ( ord_less_a @ A @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_896_ivl__disj__un__one_I2_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or5139330845457685135Than_a @ L @ U3 ) )
= ( set_ord_lessThan_a @ U3 ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_897_at__within__Icc__at,axiom,
! [A: a,X: a,B2: a] :
( ( ord_less_a @ A @ X )
=> ( ( ord_less_a @ X @ B2 )
=> ( ( topolo1902352237885396414thin_a @ X @ ( set_or672772299803893939Most_a @ A @ B2 ) )
= ( topolo1902352237885396414thin_a @ X @ top_top_set_a ) ) ) ) ).
% at_within_Icc_at
thf(fact_898_at__within__open__subset,axiom,
! [A: a,S: set_a,T5: set_a] :
( ( member_a @ A @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( ord_less_eq_set_a @ S @ T5 )
=> ( ( topolo1902352237885396414thin_a @ A @ T5 )
= ( topolo1902352237885396414thin_a @ A @ top_top_set_a ) ) ) ) ) ).
% at_within_open_subset
thf(fact_899_ivl__disj__un__one_I4_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or672772299803893939Most_a @ L @ U3 ) )
= ( set_ord_atMost_a @ U3 ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_900_ivl__disj__un__singleton_I2_J,axiom,
! [U3: a] :
( ( sup_sup_set_a @ ( set_ord_lessThan_a @ U3 ) @ ( insert_a @ U3 @ bot_bot_set_a ) )
= ( set_ord_atMost_a @ U3 ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_901_ivl__disj__un__one_I1_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( set_ord_lessThan_a @ U3 ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_902_at__eq__bot__iff,axiom,
! [A: a] :
( ( ( topolo1902352237885396414thin_a @ A @ top_top_set_a )
= bot_bot_filter_a )
= ( topolo8477419352202985285open_a @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% at_eq_bot_iff
thf(fact_903_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_904_top_Oordering__top__axioms,axiom,
orderi5875812994216768367_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a @ top_top_set_set_a ).
% top.ordering_top_axioms
thf(fact_905_Inf__atMostLessThan,axiom,
! [X: filter_a] :
( ( ord_less_filter_a @ top_top_filter_a @ X )
=> ( ( comple1024174712277884299lter_a @ ( set_or2284990949078461863lter_a @ X ) )
= bot_bot_filter_a ) ) ).
% Inf_atMostLessThan
thf(fact_906_Inf__atMostLessThan,axiom,
! [X: set_a] :
( ( ord_less_set_a @ top_top_set_a @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or5421148953861284865_set_a @ X ) )
= bot_bot_set_a ) ) ).
% Inf_atMostLessThan
thf(fact_907_atMost__Int__atLeast,axiom,
! [N: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ N ) @ ( set_ord_atLeast_a @ N ) )
= ( insert_a @ N @ bot_bot_set_a ) ) ).
% atMost_Int_atLeast
thf(fact_908_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or4472690218693186638Most_a
= ( ^ [A3: a,B3: a] : ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A3 @ B3 ) @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_909_Int__iff,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) )
= ( ( member_set_a @ C @ A5 )
& ( member_set_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_910_Int__iff,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( ( member_a @ C @ A5 )
& ( member_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_911_IntI,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ( member_set_a @ C @ B5 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_912_IntI,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ A5 )
=> ( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_913_Diff__idemp,axiom,
! [A5: set_a,B5: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ B5 )
= ( minus_minus_set_a @ A5 @ B5 ) ) ).
% Diff_idemp
thf(fact_914_Diff__iff,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) )
= ( ( member_set_a @ C @ A5 )
& ~ ( member_set_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_915_Diff__iff,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) )
= ( ( member_a @ C @ A5 )
& ~ ( member_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_916_DiffI,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ~ ( member_set_a @ C @ B5 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_917_DiffI,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ A5 )
=> ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_918_closed__Inter,axiom,
! [K3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ K3 )
=> ( topolo784654279908865136osed_a @ X4 ) )
=> ( topolo784654279908865136osed_a @ ( comple6135023378680113637_set_a @ K3 ) ) ) ).
% closed_Inter
thf(fact_919_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_920_inf_Obounded__iff,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B2 )
& ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_921_le__inf__iff,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z4 ) )
= ( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_922_le__inf__iff,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z4 ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
& ( ord_le3724670747650509150_set_a @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_923_Int__UNIV,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= top_top_set_a )
= ( ( A5 = top_top_set_a )
& ( B5 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_924_Int__subset__iff,axiom,
! [C2: set_a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( ( ord_less_eq_set_a @ C2 @ A5 )
& ( ord_less_eq_set_a @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_925_Int__subset__iff,axiom,
! [C2: set_set_a,A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A5 @ B5 ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
& ( ord_le3724670747650509150_set_a @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_926_Int__insert__right__if1,axiom,
! [A: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B5 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_927_Int__insert__right__if1,axiom,
! [A: a,A5: set_a,B5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_928_Int__insert__right__if0,axiom,
! [A: set_a,A5: set_set_a,B5: set_set_a] :
( ~ ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B5 ) )
= ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_929_Int__insert__right__if0,axiom,
! [A: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_930_insert__inter__insert,axiom,
! [A: a,A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_931_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B5: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B5 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_932_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B5: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_933_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B5: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_set_a @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_934_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B5: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_935_Diff__cancel,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ A5 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_936_empty__Diff,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A5 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_937_Diff__empty,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Diff_empty
thf(fact_938_open__Int,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo8477419352202985285open_a @ ( inf_inf_set_a @ S @ T5 ) ) ) ) ).
% open_Int
thf(fact_939_insert__Diff1,axiom,
! [X: set_a,B5: set_set_a,A5: set_set_a] :
( ( member_set_a @ X @ B5 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ B5 )
= ( minus_5736297505244876581_set_a @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_940_insert__Diff1,axiom,
! [X: a,B5: set_a,A5: set_a] :
( ( member_a @ X @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ B5 )
= ( minus_minus_set_a @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_941_Diff__insert0,axiom,
! [X: set_a,A5: set_set_a,B5: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ B5 ) )
= ( minus_5736297505244876581_set_a @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_942_Diff__insert0,axiom,
! [X: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( minus_minus_set_a @ A5 @ ( insert_a @ X @ B5 ) )
= ( minus_minus_set_a @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_943_Un__Int__eq_I1_J,axiom,
! [S: set_a,T5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T5 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_944_Un__Int__eq_I2_J,axiom,
! [S: set_a,T5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T5 ) @ T5 )
= T5 ) ).
% Un_Int_eq(2)
thf(fact_945_Un__Int__eq_I3_J,axiom,
! [S: set_a,T5: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T5 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_946_Un__Int__eq_I4_J,axiom,
! [T5: set_a,S: set_a] :
( ( inf_inf_set_a @ T5 @ ( sup_sup_set_a @ S @ T5 ) )
= T5 ) ).
% Un_Int_eq(4)
thf(fact_947_Int__Un__eq_I1_J,axiom,
! [S: set_a,T5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T5 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_948_Int__Un__eq_I2_J,axiom,
! [S: set_a,T5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T5 ) @ T5 )
= T5 ) ).
% Int_Un_eq(2)
thf(fact_949_Int__Un__eq_I3_J,axiom,
! [S: set_a,T5: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T5 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_950_Int__Un__eq_I4_J,axiom,
! [T5: set_a,S: set_a] :
( ( sup_sup_set_a @ T5 @ ( inf_inf_set_a @ S @ T5 ) )
= T5 ) ).
% Int_Un_eq(4)
thf(fact_951_Un__Diff__cancel2,axiom,
! [B5: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B5 @ A5 ) @ A5 )
= ( sup_sup_set_a @ B5 @ A5 ) ) ).
% Un_Diff_cancel2
thf(fact_952_Un__Diff__cancel,axiom,
! [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ ( minus_minus_set_a @ B5 @ A5 ) )
= ( sup_sup_set_a @ A5 @ B5 ) ) ).
% Un_Diff_cancel
thf(fact_953_closed__Int,axiom,
! [S: set_a,T5: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo784654279908865136osed_a @ ( inf_inf_set_a @ S @ T5 ) ) ) ) ).
% closed_Int
thf(fact_954_cInf__atLeast,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( set_or8362275514725411625_set_a @ X ) )
= X ) ).
% cInf_atLeast
thf(fact_955_Inf__atLeast,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( set_or8362275514725411625_set_a @ X ) )
= X ) ).
% Inf_atLeast
thf(fact_956_disjoint__insert_I2_J,axiom,
! [A5: set_set_a,B2: set_a,B5: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ B2 @ B5 ) ) )
= ( ~ ( member_set_a @ B2 @ A5 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_957_disjoint__insert_I2_J,axiom,
! [A5: set_a,B2: a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ ( insert_a @ B2 @ B5 ) ) )
= ( ~ ( member_a @ B2 @ A5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_958_disjoint__insert_I1_J,axiom,
! [B5: set_set_a,A: set_a,A5: set_set_a] :
( ( ( inf_inf_set_set_a @ B5 @ ( insert_set_a @ A @ A5 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B5 )
& ( ( inf_inf_set_set_a @ B5 @ A5 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_959_disjoint__insert_I1_J,axiom,
! [B5: set_a,A: a,A5: set_a] :
( ( ( inf_inf_set_a @ B5 @ ( insert_a @ A @ A5 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B5 )
& ( ( inf_inf_set_a @ B5 @ A5 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_960_insert__disjoint_I2_J,axiom,
! [A: set_a,A5: set_set_a,B5: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A5 ) @ B5 ) )
= ( ~ ( member_set_a @ A @ B5 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_961_insert__disjoint_I2_J,axiom,
! [A: a,A5: set_a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ B5 ) )
= ( ~ ( member_a @ A @ B5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_962_insert__disjoint_I1_J,axiom,
! [A: set_a,A5: set_set_a,B5: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A5 ) @ B5 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B5 )
& ( ( inf_inf_set_set_a @ A5 @ B5 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_963_insert__disjoint_I1_J,axiom,
! [A: a,A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ B5 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B5 )
& ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_964_Diff__UNIV,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_965_Diff__eq__empty__iff,axiom,
! [A5: set_a,B5: set_a] :
( ( ( minus_minus_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A5 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_966_Diff__eq__empty__iff,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A5 @ B5 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_967_cInf__atLeastAtMost,axiom,
! [Y3: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( ( comple9105089376463352645_set_a @ ( set_or4761464488706262899_set_a @ Y3 @ X ) )
= Y3 ) ) ).
% cInf_atLeastAtMost
thf(fact_968_cInf__atLeastAtMost,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y3 @ X ) )
= Y3 ) ) ).
% cInf_atLeastAtMost
thf(fact_969_Inf__atLeastAtMost,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( comple9105089376463352645_set_a @ ( set_or4761464488706262899_set_a @ X @ Y3 ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_970_Inf__atLeastAtMost,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X @ Y3 ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_971_cInf__singleton,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% cInf_singleton
thf(fact_972_insert__Diff__single,axiom,
! [A: a,A5: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A5 ) ) ).
% insert_Diff_single
thf(fact_973_Diff__disjoint,axiom,
! [A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ A5 @ ( minus_minus_set_a @ B5 @ A5 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_974_ivl__diff,axiom,
! [I2: a,N: a,M4: a] :
( ( ord_less_eq_a @ I2 @ N )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I2 @ M4 ) @ ( set_or5139330845457685135Than_a @ I2 @ N ) )
= ( set_or5139330845457685135Than_a @ N @ M4 ) ) ) ).
% ivl_diff
thf(fact_975_cInf__atLeastLessThan,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_set_a @ Y3 @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or2348907005316661231_set_a @ Y3 @ X ) )
= Y3 ) ) ).
% cInf_atLeastLessThan
thf(fact_976_Inf__atLeastLessThan,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ( ( comple6135023378680113637_set_a @ ( set_or2348907005316661231_set_a @ X @ Y3 ) )
= X ) ) ).
% Inf_atLeastLessThan
thf(fact_977_Inf__atMost,axiom,
! [X: filter_a] :
( ( comple1024174712277884299lter_a @ ( set_or3848680440086517451lter_a @ X ) )
= bot_bot_filter_a ) ).
% Inf_atMost
thf(fact_978_Inf__atMost,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( set_ord_atMost_set_a @ X ) )
= bot_bot_set_a ) ).
% Inf_atMost
thf(fact_979_closed__Diff,axiom,
! [S: set_a,T5: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo784654279908865136osed_a @ ( minus_minus_set_a @ S @ T5 ) ) ) ) ).
% closed_Diff
thf(fact_980_open__Diff,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo8477419352202985285open_a @ ( minus_minus_set_a @ S @ T5 ) ) ) ) ).
% open_Diff
thf(fact_981_lessThan__minus__lessThan,axiom,
! [N: a,M4: a] :
( ( minus_minus_set_a @ ( set_ord_lessThan_a @ N ) @ ( set_ord_lessThan_a @ M4 ) )
= ( set_or5139330845457685135Than_a @ M4 @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_982_single__Diff__lessThan,axiom,
! [K: a] :
( ( minus_minus_set_a @ ( insert_a @ K @ bot_bot_set_a ) @ ( set_ord_lessThan_a @ K ) )
= ( insert_a @ K @ bot_bot_set_a ) ) ).
% single_Diff_lessThan
thf(fact_983_Diff__triv,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A5 @ B5 )
= A5 ) ) ).
% Diff_triv
thf(fact_984_Int__Diff__disjoint,axiom,
! [A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( minus_minus_set_a @ A5 @ B5 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_985_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_986_Int__UNIV__left,axiom,
! [B5: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_987_Int__UNIV__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ top_top_set_a )
= A5 ) ).
% Int_UNIV_right
thf(fact_988_Int__Collect__mono,axiom,
! [A5: set_set_a,B5: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B5 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_989_Int__Collect__mono,axiom,
! [A5: set_a,B5: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B5 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_990_Int__greatest,axiom,
! [C2: set_a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C2 @ A5 )
=> ( ( ord_less_eq_set_a @ C2 @ B5 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_991_Int__greatest,axiom,
! [C2: set_set_a,A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B5 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_992_Int__absorb2,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= A5 ) ) ).
% Int_absorb2
thf(fact_993_Int__absorb2,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( inf_inf_set_set_a @ A5 @ B5 )
= A5 ) ) ).
% Int_absorb2
thf(fact_994_Int__absorb1,axiom,
! [B5: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_995_Int__absorb1,axiom,
! [B5: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_996_Int__lower2,axiom,
! [A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_997_Int__lower2,axiom,
! [A5: set_set_a,B5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_998_Int__lower1,axiom,
! [A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ A5 ) ).
% Int_lower1
thf(fact_999_Int__lower1,axiom,
! [A5: set_set_a,B5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ B5 ) @ A5 ) ).
% Int_lower1
thf(fact_1000_Int__mono,axiom,
! [A5: set_a,C2: set_a,B5: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B5 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1001_Int__mono,axiom,
! [A5: set_set_a,C2: set_set_a,B5: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ B5 ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1002_Int__emptyI,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ~ ( member_set_a @ X4 @ B5 ) )
=> ( ( inf_inf_set_set_a @ A5 @ B5 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_1003_Int__emptyI,axiom,
! [A5: set_a,B5: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ~ ( member_a @ X4 @ B5 ) )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1004_disjoint__iff,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ( inf_inf_set_set_a @ A5 @ B5 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ~ ( member_set_a @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1005_disjoint__iff,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ~ ( member_a @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1006_Int__empty__left,axiom,
! [B5: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B5 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_1007_Int__empty__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_1008_disjoint__iff__not__equal,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B5 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1009_double__diff,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ( minus_minus_set_a @ B5 @ ( minus_minus_set_a @ C2 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_1010_double__diff,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B5 @ ( minus_5736297505244876581_set_a @ C2 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_1011_Diff__subset,axiom,
! [A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ A5 ) ).
% Diff_subset
thf(fact_1012_Diff__subset,axiom,
! [A5: set_set_a,B5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) @ A5 ) ).
% Diff_subset
thf(fact_1013_Diff__mono,axiom,
! [A5: set_a,C2: set_a,D2: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B5 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1014_Diff__mono,axiom,
! [A5: set_set_a,C2: set_set_a,D2: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B5 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) @ ( minus_5736297505244876581_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1015_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B5: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B5 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B5 @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1016_Int__insert__left,axiom,
! [A: a,C2: set_a,B5: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1017_Int__insert__right,axiom,
! [A: set_a,A5: set_set_a,B5: set_set_a] :
( ( ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B5 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A5 @ B5 ) ) ) )
& ( ~ ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B5 ) )
= ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1018_Int__insert__right,axiom,
! [A: a,A5: set_a,B5: set_a] :
( ( ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B5 ) ) ) )
& ( ~ ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1019_insert__Diff__if,axiom,
! [X: set_a,B5: set_set_a,A5: set_set_a] :
( ( ( member_set_a @ X @ B5 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ B5 )
= ( minus_5736297505244876581_set_a @ A5 @ B5 ) ) )
& ( ~ ( member_set_a @ X @ B5 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ B5 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1020_insert__Diff__if,axiom,
! [X: a,B5: set_a,A5: set_a] :
( ( ( member_a @ X @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ B5 )
= ( minus_minus_set_a @ A5 @ B5 ) ) )
& ( ~ ( member_a @ X @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ B5 )
= ( insert_a @ X @ ( minus_minus_set_a @ A5 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1021_Un__Int__crazy,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ B5 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A5 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ ( sup_sup_set_a @ B5 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_1022_Int__Un__distrib,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ A5 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1023_Un__Int__distrib,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ ( sup_sup_set_a @ A5 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1024_Int__Un__distrib2,axiom,
! [B5: set_a,C2: set_a,A5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B5 @ C2 ) @ A5 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B5 @ A5 ) @ ( inf_inf_set_a @ C2 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_1025_Un__Int__distrib2,axiom,
! [B5: set_a,C2: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B5 @ C2 ) @ A5 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B5 @ A5 ) @ ( sup_sup_set_a @ C2 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_1026_Un__Diff,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ C2 ) @ ( minus_minus_set_a @ B5 @ C2 ) ) ) ).
% Un_Diff
thf(fact_1027_Diff__Int__distrib2,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C2 ) @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1028_Int__left__commute,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( inf_inf_set_a @ B5 @ ( inf_inf_set_a @ A5 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1029_Diff__Int__distrib,axiom,
! [C2: set_a,A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A5 @ B5 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A5 ) @ ( inf_inf_set_a @ C2 @ B5 ) ) ) ).
% Diff_Int_distrib
thf(fact_1030_Int__left__absorb,axiom,
! [A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ).
% Int_left_absorb
thf(fact_1031_Diff__Diff__Int,axiom,
! [A5: set_a,B5: set_a] :
( ( minus_minus_set_a @ A5 @ ( minus_minus_set_a @ A5 @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ).
% Diff_Diff_Int
thf(fact_1032_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_1033_Int__absorb,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_1034_Int__assoc,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1035_Diff__Int2,axiom,
! [A5: set_a,C2: set_a,B5: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C2 ) @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C2 ) @ B5 ) ) ).
% Diff_Int2
thf(fact_1036_Int__Diff,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( minus_minus_set_a @ B5 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1037_DiffD2,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) )
=> ~ ( member_set_a @ C @ B5 ) ) ).
% DiffD2
thf(fact_1038_DiffD2,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) )
=> ~ ( member_a @ C @ B5 ) ) ).
% DiffD2
thf(fact_1039_DiffD1,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_1040_DiffD1,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) )
=> ( member_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_1041_IntD2,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) )
=> ( member_set_a @ C @ B5 ) ) ).
% IntD2
thf(fact_1042_IntD2,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
=> ( member_a @ C @ B5 ) ) ).
% IntD2
thf(fact_1043_IntD1,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% IntD1
thf(fact_1044_IntD1,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
=> ( member_a @ C @ A5 ) ) ).
% IntD1
thf(fact_1045_DiffE,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1046_DiffE,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1047_IntE,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ~ ( member_set_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_1048_IntE,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ~ ( member_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_1049_cInf__eq__minimum,axiom,
! [Z4: set_set_a,X6: set_set_set_a] :
( ( member_set_set_a @ Z4 @ X6 )
=> ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ X6 )
=> ( ord_le3724670747650509150_set_a @ Z4 @ X4 ) )
=> ( ( comple9105089376463352645_set_a @ X6 )
= Z4 ) ) ) ).
% cInf_eq_minimum
thf(fact_1050_cInf__eq__minimum,axiom,
! [Z4: set_a,X6: set_set_a] :
( ( member_set_a @ Z4 @ X6 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ Z4 @ X4 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= Z4 ) ) ) ).
% cInf_eq_minimum
thf(fact_1051_less__infI1,axiom,
! [A: set_a,X: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_1052_less__infI2,axiom,
! [B2: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_1053_inf_Oabsorb3,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb3
thf(fact_1054_inf_Oabsorb4,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_1055_inf_Ostrict__boundedE,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_set_a @ A @ B2 )
=> ~ ( ord_less_set_a @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1056_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1057_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1058_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1059_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1060_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1061_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1062_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1063_inf__le1,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_1064_inf__le1,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_1065_inf__le2,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_1066_inf__le2,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_1067_le__infE,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1068_le__infE,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1069_le__infI,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1070_le__infI,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1071_inf__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1072_inf__mono,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1073_le__infI1,axiom,
! [A: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1074_le__infI1,axiom,
! [A: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1075_le__infI2,axiom,
! [B2: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1076_le__infI2,axiom,
! [B2: set_set_a,X: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1077_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1078_inf_OorderE,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1079_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1080_inf_OorderI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A
= ( inf_inf_set_set_a @ A @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1081_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y3: set_a] :
( ! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ X4 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_1082_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y3: set_set_a] :
( ! [X4: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( F @ Y @ Z2 ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_1083_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1084_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1085_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1086_inf_Oabsorb1,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1087_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1088_inf_Oabsorb2,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1089_inf__absorb1,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( inf_inf_set_a @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_1090_inf__absorb1,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( inf_inf_set_set_a @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_1091_inf__absorb2,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( inf_inf_set_a @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_1092_inf__absorb2,axiom,
! [Y3: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( ( inf_inf_set_set_a @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_1093_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1094_inf_OboundedE,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1095_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1096_inf_OboundedI,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1097_inf__greatest,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_1098_inf__greatest,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z4 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_1099_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_1100_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( A3
= ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1101_inf_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_1102_inf_Ocobounded1,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_1103_inf_Ocobounded2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1104_inf_Ocobounded2,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1105_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_1106_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_1107_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_1108_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1109_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1110_inf_OcoboundedI1,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1111_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1112_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1113_Diff__Un,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( minus_minus_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ ( minus_minus_set_a @ A5 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1114_Diff__Int,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( minus_minus_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ ( minus_minus_set_a @ A5 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1115_Int__Diff__Un,axiom,
! [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( minus_minus_set_a @ A5 @ B5 ) )
= A5 ) ).
% Int_Diff_Un
thf(fact_1116_Un__Diff__Int,axiom,
! [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ A5 @ B5 ) )
= A5 ) ).
% Un_Diff_Int
thf(fact_1117_psubset__imp__ex__mem,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
=> ? [B4: set_a] : ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ B5 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1118_psubset__imp__ex__mem,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B5 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1119_at__within__nhd,axiom,
! [X: a,S: set_a,T5: set_a,U4: set_a] :
( ( member_a @ X @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( ( minus_minus_set_a @ ( inf_inf_set_a @ T5 @ S ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ U4 @ S ) @ ( insert_a @ X @ bot_bot_set_a ) ) )
=> ( ( topolo1902352237885396414thin_a @ X @ T5 )
= ( topolo1902352237885396414thin_a @ X @ U4 ) ) ) ) ) ).
% at_within_nhd
thf(fact_1120_cInf__eq__non__empty,axiom,
! [X6: set_set_set_a,A: set_set_a] :
( ( X6 != bot_bo3380559777022489994_set_a )
=> ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ X6 )
=> ( ord_le3724670747650509150_set_a @ A @ X4 ) )
=> ( ! [Y: set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ X6 )
=> ( ord_le3724670747650509150_set_a @ Y @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ Y @ A ) )
=> ( ( comple9105089376463352645_set_a @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1121_cInf__eq__non__empty,axiom,
! [X6: set_set_a,A: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ A @ X4 ) )
=> ( ! [Y: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ X6 )
=> ( ord_less_eq_set_a @ Y @ X3 ) )
=> ( ord_less_eq_set_a @ Y @ A ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1122_cInf__greatest,axiom,
! [X6: set_set_set_a,Z4: set_set_a] :
( ( X6 != bot_bo3380559777022489994_set_a )
=> ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ X6 )
=> ( ord_le3724670747650509150_set_a @ Z4 @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ Z4 @ ( comple9105089376463352645_set_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1123_cInf__greatest,axiom,
! [X6: set_set_a,Z4: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ Z4 @ X4 ) )
=> ( ord_less_eq_set_a @ Z4 @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1124_distrib__sup__le,axiom,
! [X: filter_a,Y3: filter_a,Z4: filter_a] : ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ X @ ( inf_inf_filter_a @ Y3 @ Z4 ) ) @ ( inf_inf_filter_a @ ( sup_sup_filter_a @ X @ Y3 ) @ ( sup_sup_filter_a @ X @ Z4 ) ) ) ).
% distrib_sup_le
thf(fact_1125_distrib__sup__le,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z4 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% distrib_sup_le
thf(fact_1126_distrib__sup__le,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z4 ) ) @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X @ Y3 ) @ ( sup_sup_set_set_a @ X @ Z4 ) ) ) ).
% distrib_sup_le
thf(fact_1127_distrib__inf__le,axiom,
! [X: filter_a,Y3: filter_a,Z4: filter_a] : ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ ( inf_inf_filter_a @ X @ Y3 ) @ ( inf_inf_filter_a @ X @ Z4 ) ) @ ( inf_inf_filter_a @ X @ ( sup_sup_filter_a @ Y3 @ Z4 ) ) ) ).
% distrib_inf_le
thf(fact_1128_distrib__inf__le,axiom,
! [X: set_a,Y3: set_a,Z4: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ ( inf_inf_set_a @ X @ Z4 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z4 ) ) ) ).
% distrib_inf_le
thf(fact_1129_distrib__inf__le,axiom,
! [X: set_set_a,Y3: set_set_a,Z4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ ( inf_inf_set_set_a @ X @ Z4 ) ) @ ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ Y3 @ Z4 ) ) ) ).
% distrib_inf_le
thf(fact_1130_subset__Diff__insert,axiom,
! [A5: set_set_a,B5: set_set_a,X: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B5 @ ( insert_set_a @ X @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B5 @ C2 ) )
& ~ ( member_set_a @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1131_subset__Diff__insert,axiom,
! [A5: set_a,B5: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( minus_minus_set_a @ B5 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A5 @ ( minus_minus_set_a @ B5 @ C2 ) )
& ~ ( member_a @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1132_Diff__insert,axiom,
! [A5: set_a,A: a,B5: set_a] :
( ( minus_minus_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1133_insert__Diff,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1134_insert__Diff,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1135_Diff__insert2,axiom,
! [A5: set_a,A: a,B5: set_a] :
( ( minus_minus_set_a @ A5 @ ( insert_a @ A @ B5 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_1136_Diff__insert__absorb,axiom,
! [X: set_a,A5: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1137_Diff__insert__absorb,axiom,
! [X: a,A5: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1138_hausdorff,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
=> ? [U2: set_a,V: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( topolo8477419352202985285open_a @ V )
& ( member_a @ X @ U2 )
& ( member_a @ Y3 @ V )
& ( ( inf_inf_set_a @ U2 @ V )
= bot_bot_set_a ) ) ) ).
% hausdorff
thf(fact_1139_separation__t2,axiom,
! [X: a,Y3: a] :
( ( X != Y3 )
= ( ? [U: set_a,V2: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( topolo8477419352202985285open_a @ V2 )
& ( member_a @ X @ U )
& ( member_a @ Y3 @ V2 )
& ( ( inf_inf_set_a @ U @ V2 )
= bot_bot_set_a ) ) ) ) ).
% separation_t2
thf(fact_1140_Un__Int__assoc__eq,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_1141_Un__Int__assoc__eq,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A5 @ B5 ) @ C2 )
= ( inf_inf_set_set_a @ A5 @ ( sup_sup_set_set_a @ B5 @ C2 ) ) )
= ( ord_le3724670747650509150_set_a @ C2 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_1142_Diff__partition,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( sup_sup_set_a @ A5 @ ( minus_minus_set_a @ B5 @ A5 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_1143_Diff__partition,axiom,
! [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
=> ( ( sup_sup_set_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B5 @ A5 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_1144_Diff__subset__conv,axiom,
! [A5: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B5 ) @ C2 )
= ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1145_Diff__subset__conv,axiom,
! [A5: set_set_a,B5: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ B5 ) @ C2 )
= ( ord_le3724670747650509150_set_a @ A5 @ ( sup_sup_set_set_a @ B5 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1146_ivl__disj__int__two_I3_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(3)
thf(fact_1147_ivl__disj__int__two_I6_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ M4 ) @ ( set_or4472690218693186638Most_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(6)
thf(fact_1148_preordering__bdd_OInt2,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,B5: set_a,A5: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ B5 )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% preordering_bdd.Int2
thf(fact_1149_preordering__bdd_OInt1,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A5: set_a,B5: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ A5 )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% preordering_bdd.Int1
thf(fact_1150_Diff__single__insert,axiom,
! [A5: set_a,X: a,B5: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B5 )
=> ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_1151_Diff__single__insert,axiom,
! [A5: set_set_a,X: set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_1152_subset__insert__iff,axiom,
! [A5: set_set_a,X: set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X @ B5 ) )
= ( ( ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B5 ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1153_subset__insert__iff,axiom,
! [A5: set_a,X: a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ B5 ) )
= ( ( ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1154_ivl__disj__int__one_I4_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or672772299803893939Most_a @ L @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(4)
thf(fact_1155_ivl__disj__int__two_I7_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(7)
thf(fact_1156_Ioc__disjoint,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ A @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= bot_bot_set_a )
= ( ( ord_less_eq_a @ B2 @ A )
| ( ord_less_eq_a @ D @ C )
| ( ord_less_eq_a @ B2 @ C )
| ( ord_less_eq_a @ D @ A ) ) ) ).
% Ioc_disjoint
thf(fact_1157_ivl__disj__int__one_I7_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ U3 ) @ ( set_or8632414552788122084Than_a @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(7)
thf(fact_1158_ivl__disj__int__one_I2_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or5139330845457685135Than_a @ L @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(2)
thf(fact_1159_ivl__disj__int__two_I4_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ M4 ) @ ( set_or5939364468397584554Than_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(4)
thf(fact_1160_ivl__disj__int__two_I5_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ M4 ) @ ( set_or672772299803893939Most_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(5)
thf(fact_1161_ivl__disj__int__two_I8_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ M4 ) @ ( set_or4472690218693186638Most_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(8)
thf(fact_1162_ivl__disj__int__two_I1_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ M4 ) @ ( set_or5139330845457685135Than_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(1)
thf(fact_1163_ivl__disj__int__one_I1_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(1)
thf(fact_1164_ivl__disj__int__one_I8_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(8)
thf(fact_1165_ivl__disj__int__one_I3_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(3)
thf(fact_1166_ivl__disj__int__one_I5_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ U3 ) @ ( set_or8632414552788122084Than_a @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(5)
thf(fact_1167_ivl__disj__int__one_I6_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(6)
thf(fact_1168_atLeastAtMost__def,axiom,
( set_or672772299803893939Most_a
= ( ^ [L3: a,U5: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_atMost_a @ U5 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_1169_ivl__disj__int__two_I2_J,axiom,
! [L: a,M4: a,U3: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ M4 ) @ ( set_or5939364468397584554Than_a @ M4 @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(2)
thf(fact_1170_greaterThanLessThan__def,axiom,
( set_or5939364468397584554Than_a
= ( ^ [L3: a,U5: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ L3 ) @ ( set_ord_lessThan_a @ U5 ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_1171_greaterThanLessThan__eq,axiom,
( set_or5939364468397584554Than_a
= ( ^ [A3: a,B3: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ A3 ) @ ( set_ord_lessThan_a @ B3 ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_1172_atLeastLessThan__def,axiom,
( set_or5139330845457685135Than_a
= ( ^ [L3: a,U5: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_lessThan_a @ U5 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_1173_greaterThanAtMost__def,axiom,
( set_or4472690218693186638Most_a
= ( ^ [L3: a,U5: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ L3 ) @ ( set_ord_atMost_a @ U5 ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_1174_Iio__Int__singleton,axiom,
! [X: set_a,K: set_a] :
( ( ( ord_less_set_a @ X @ K )
=> ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ K ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
& ( ~ ( ord_less_set_a @ X @ K )
=> ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ K ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_1175_Iio__Int__singleton,axiom,
! [X: a,K: a] :
( ( ( ord_less_a @ X @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( insert_a @ X @ bot_bot_set_a ) ) )
& ( ~ ( ord_less_a @ X @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X @ bot_bot_set_a ) )
= bot_bot_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_1176_psubset__insert__iff,axiom,
! [A5: set_set_a,X: set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ ( insert_set_a @ X @ B5 ) )
= ( ( ( member_set_a @ X @ B5 )
=> ( ord_less_set_set_a @ A5 @ B5 ) )
& ( ~ ( member_set_a @ X @ B5 )
=> ( ( ( member_set_a @ X @ A5 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B5 ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1177_psubset__insert__iff,axiom,
! [A5: set_a,X: a,B5: set_a] :
( ( ord_less_set_a @ A5 @ ( insert_a @ X @ B5 ) )
= ( ( ( member_a @ X @ B5 )
=> ( ord_less_set_a @ A5 @ B5 ) )
& ( ~ ( member_a @ X @ B5 )
=> ( ( ( member_a @ X @ A5 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1178_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or5139330845457685135Than_a
= ( ^ [A3: a,B3: a] : ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A3 @ B3 ) @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1179_less__separate,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
=> ? [A4: set_a,B4: set_a] :
( ( member_set_a @ X @ ( set_or5421148953861284865_set_a @ A4 ) )
& ( member_set_a @ Y3 @ ( set_or460448635090783044_set_a @ B4 ) )
& ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ A4 ) @ ( set_or460448635090783044_set_a @ B4 ) )
= bot_bot_set_set_a ) ) ) ).
% less_separate
thf(fact_1180_less__separate,axiom,
! [X: a,Y3: a] :
( ( ord_less_a @ X @ Y3 )
=> ? [A4: a,B4: a] :
( ( member_a @ X @ ( set_ord_lessThan_a @ A4 ) )
& ( member_a @ Y3 @ ( set_or8632414552788122084Than_a @ B4 ) )
& ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ A4 ) @ ( set_or8632414552788122084Than_a @ B4 ) )
= bot_bot_set_a ) ) ) ).
% less_separate
thf(fact_1181_atLeastAtMost__diff__ends,axiom,
! [A: a,B2: a] :
( ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A @ B2 ) @ ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( set_or5939364468397584554Than_a @ A @ B2 ) ) ).
% atLeastAtMost_diff_ends
thf(fact_1182_open__delete,axiom,
! [S3: set_a,X: a] :
( ( topolo8477419352202985285open_a @ S3 )
=> ( topolo8477419352202985285open_a @ ( minus_minus_set_a @ S3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% open_delete
thf(fact_1183_less__eq__Inf__inter,axiom,
! [A5: set_filter_a,B5: set_filter_a] : ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ ( comple1024174712277884299lter_a @ A5 ) @ ( comple1024174712277884299lter_a @ B5 ) ) @ ( comple1024174712277884299lter_a @ ( inf_inf_set_filter_a @ A5 @ B5 ) ) ) ).
% less_eq_Inf_inter
thf(fact_1184_less__eq__Inf__inter,axiom,
! [A5: set_set_set_a,B5: set_set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( comple9105089376463352645_set_a @ A5 ) @ ( comple9105089376463352645_set_a @ B5 ) ) @ ( comple9105089376463352645_set_a @ ( inf_in1205276777018777868_set_a @ A5 @ B5 ) ) ) ).
% less_eq_Inf_inter
thf(fact_1185_less__eq__Inf__inter,axiom,
! [A5: set_set_a,B5: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B5 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ).
% less_eq_Inf_inter
thf(fact_1186_Inter__anti__mono,axiom,
! [B5: set_set_set_a,A5: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B5 @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A5 ) @ ( comple9105089376463352645_set_a @ B5 ) ) ) ).
% Inter_anti_mono
thf(fact_1187_Inter__anti__mono,axiom,
! [B5: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B5 ) ) ) ).
% Inter_anti_mono
thf(fact_1188_Inter__greatest,axiom,
! [A5: set_set_set_a,C2: set_set_a] :
( ! [X7: set_set_a] :
( ( member_set_set_a @ X7 @ A5 )
=> ( ord_le3724670747650509150_set_a @ C2 @ X7 ) )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( comple9105089376463352645_set_a @ A5 ) ) ) ).
% Inter_greatest
thf(fact_1189_Inter__greatest,axiom,
! [A5: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A5 )
=> ( ord_less_eq_set_a @ C2 @ X7 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A5 ) ) ) ).
% Inter_greatest
thf(fact_1190_Inter__lower,axiom,
! [B5: set_set_a,A5: set_set_set_a] :
( ( member_set_set_a @ B5 @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A5 ) @ B5 ) ) ).
% Inter_lower
thf(fact_1191_Inter__lower,axiom,
! [B5: set_a,A5: set_set_a] :
( ( member_set_a @ B5 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ B5 ) ) ).
% Inter_lower
thf(fact_1192_Inter__subset,axiom,
! [A5: set_set_set_a,B5: set_set_a] :
( ! [X7: set_set_a] :
( ( member_set_set_a @ X7 @ A5 )
=> ( ord_le3724670747650509150_set_a @ X7 @ B5 ) )
=> ( ( A5 != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A5 ) @ B5 ) ) ) ).
% Inter_subset
thf(fact_1193_Inter__subset,axiom,
! [A5: set_set_a,B5: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A5 )
=> ( ord_less_eq_set_a @ X7 @ B5 ) )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ B5 ) ) ) ).
% Inter_subset
thf(fact_1194_Inf__greatest,axiom,
! [A5: set_set_set_a,Z4: set_set_a] :
( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A5 )
=> ( ord_le3724670747650509150_set_a @ Z4 @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ Z4 @ ( comple9105089376463352645_set_a @ A5 ) ) ) ).
% Inf_greatest
thf(fact_1195_Inf__greatest,axiom,
! [A5: set_set_a,Z4: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( ord_less_eq_set_a @ Z4 @ X4 ) )
=> ( ord_less_eq_set_a @ Z4 @ ( comple6135023378680113637_set_a @ A5 ) ) ) ).
% Inf_greatest
thf(fact_1196_le__Inf__iff,axiom,
! [B2: set_set_a,A5: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( comple9105089376463352645_set_a @ A5 ) )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A5 )
=> ( ord_le3724670747650509150_set_a @ B2 @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_1197_le__Inf__iff,axiom,
! [B2: set_a,A5: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A5 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_1198_Inf__lower2,axiom,
! [U3: set_set_a,A5: set_set_set_a,V3: set_set_a] :
( ( member_set_set_a @ U3 @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ U3 @ V3 )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A5 ) @ V3 ) ) ) ).
% Inf_lower2
thf(fact_1199_Inf__lower2,axiom,
! [U3: set_a,A5: set_set_a,V3: set_a] :
( ( member_set_a @ U3 @ A5 )
=> ( ( ord_less_eq_set_a @ U3 @ V3 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ V3 ) ) ) ).
% Inf_lower2
thf(fact_1200_member__remove,axiom,
! [X: a,Y3: a,A5: set_a] :
( ( member_a @ X @ ( remove_a @ Y3 @ A5 ) )
= ( ( member_a @ X @ A5 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_1201_mono__greaterThan,axiom,
! [B5: a] : ( extend2808419353335425523_set_a @ ( set_or8632414552788122084Than_a @ B5 ) ) ).
% mono_greaterThan
thf(fact_1202_mono__set,axiom,
( extend2808419353335425523_set_a
= ( ^ [S2: set_a] :
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( member_a @ X2 @ S2 )
=> ( member_a @ Y2 @ S2 ) ) ) ) ) ).
% mono_set
thf(fact_1203_connected__closed,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S4: set_a] :
~ ? [A6: set_a,B6: set_a] :
( ( topolo784654279908865136osed_a @ A6 )
& ( topolo784654279908865136osed_a @ B6 )
& ( ord_less_eq_set_a @ S4 @ ( sup_sup_set_a @ A6 @ B6 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A6 @ B6 ) @ S4 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A6 @ S4 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B6 @ S4 )
!= bot_bot_set_a ) ) ) ) ).
% connected_closed
thf(fact_1204_connected__closedD,axiom,
! [S3: set_a,A5: set_a,B5: set_a] :
( ( topolo2370605967727889109cted_a @ S3 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S3 )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ S3 @ ( sup_sup_set_a @ A5 @ B5 ) )
=> ( ( topolo784654279908865136osed_a @ A5 )
=> ( ( topolo784654279908865136osed_a @ B5 )
=> ( ( ( inf_inf_set_a @ A5 @ S3 )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B5 @ S3 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_1205_connected__def,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S2: set_a] :
~ ? [A6: set_a,B6: set_a] :
( ( topolo8477419352202985285open_a @ A6 )
& ( topolo8477419352202985285open_a @ B6 )
& ( ord_less_eq_set_a @ S2 @ ( sup_sup_set_a @ A6 @ B6 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A6 @ B6 ) @ S2 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A6 @ S2 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B6 @ S2 )
!= bot_bot_set_a ) ) ) ) ).
% connected_def
thf(fact_1206_connected__contains__Ioo,axiom,
! [A5: set_a,A: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A5 )
=> ( ( member_a @ A @ A5 )
=> ( ( member_a @ B2 @ A5 )
=> ( ord_less_eq_set_a @ ( set_or5939364468397584554Than_a @ A @ B2 ) @ A5 ) ) ) ) ).
% connected_contains_Ioo
thf(fact_1207_connectedD__interval,axiom,
! [U4: set_a,X: a,Y3: a,Z4: a] :
( ( topolo2370605967727889109cted_a @ U4 )
=> ( ( member_a @ X @ U4 )
=> ( ( member_a @ Y3 @ U4 )
=> ( ( ord_less_eq_a @ X @ Z4 )
=> ( ( ord_less_eq_a @ Z4 @ Y3 )
=> ( member_a @ Z4 @ U4 ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_1208_connected__contains__Icc,axiom,
! [A5: set_a,A: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A5 )
=> ( ( member_a @ A @ A5 )
=> ( ( member_a @ B2 @ A5 )
=> ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B2 ) @ A5 ) ) ) ) ).
% connected_contains_Icc
thf(fact_1209_connected__diff__open__from__closed,axiom,
! [S3: set_a,T: set_a,U3: set_a] :
( ( ord_less_eq_set_a @ S3 @ T )
=> ( ( ord_less_eq_set_a @ T @ U3 )
=> ( ( topolo8477419352202985285open_a @ S3 )
=> ( ( topolo784654279908865136osed_a @ T )
=> ( ( topolo2370605967727889109cted_a @ U3 )
=> ( ( topolo2370605967727889109cted_a @ ( minus_minus_set_a @ T @ S3 ) )
=> ( topolo2370605967727889109cted_a @ ( minus_minus_set_a @ U3 @ S3 ) ) ) ) ) ) ) ) ).
% connected_diff_open_from_closed
thf(fact_1210_connectedD,axiom,
! [A5: set_a,U4: set_a,V4: set_a] :
( ( topolo2370605967727889109cted_a @ A5 )
=> ( ( topolo8477419352202985285open_a @ U4 )
=> ( ( topolo8477419352202985285open_a @ V4 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ U4 @ V4 ) @ A5 )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ U4 @ V4 ) )
=> ( ( ( inf_inf_set_a @ U4 @ A5 )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ V4 @ A5 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connectedD
thf(fact_1211_connectedI,axiom,
! [U4: set_a] :
( ! [A8: set_a] :
( ( topolo8477419352202985285open_a @ A8 )
=> ! [B8: set_a] :
( ( topolo8477419352202985285open_a @ B8 )
=> ( ( ( inf_inf_set_a @ A8 @ U4 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ B8 @ U4 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A8 @ B8 ) @ U4 )
= bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ U4 @ ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ) )
=> ( topolo2370605967727889109cted_a @ U4 ) ) ).
% connectedI
thf(fact_1212_connected__closed__set,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo2370605967727889109cted_a @ S )
= ( ~ ? [A6: set_a,B6: set_a] :
( ( topolo784654279908865136osed_a @ A6 )
& ( topolo784654279908865136osed_a @ B6 )
& ( A6 != bot_bot_set_a )
& ( B6 != bot_bot_set_a )
& ( ( sup_sup_set_a @ A6 @ B6 )
= S )
& ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a ) ) ) ) ) ).
% connected_closed_set
thf(fact_1213_connected__as__closed__union,axiom,
! [C2: set_a,A5: set_a,B5: set_a] :
( ( topolo2370605967727889109cted_a @ C2 )
=> ( ( C2
= ( sup_sup_set_a @ A5 @ B5 ) )
=> ( ( topolo784654279908865136osed_a @ A5 )
=> ( ( topolo784654279908865136osed_a @ B5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( B5 != bot_bot_set_a )
=> ( ( inf_inf_set_a @ A5 @ B5 )
!= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_as_closed_union
thf(fact_1214_Compl__iff,axiom,
! [C: a,A5: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A5 ) )
= ( ~ ( member_a @ C @ A5 ) ) ) ).
% Compl_iff
thf(fact_1215_ComplI,axiom,
! [C: a,A5: set_a] :
( ~ ( member_a @ C @ A5 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A5 ) ) ) ).
% ComplI
thf(fact_1216_bdd__below_OI,axiom,
! [A5: set_a,M2: a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( ord_less_eq_a @ M2 @ X4 ) )
=> ( condit5901475214736682318elow_a @ A5 ) ) ).
% bdd_below.I
thf(fact_1217_bdd__belowI,axiom,
! [A5: set_a,M4: a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( ord_less_eq_a @ M4 @ X4 ) )
=> ( condit5901475214736682318elow_a @ A5 ) ) ).
% bdd_belowI
thf(fact_1218_bdd__below__Ioi,axiom,
! [A: a] : ( condit5901475214736682318elow_a @ ( set_or8632414552788122084Than_a @ A ) ) ).
% bdd_below_Ioi
thf(fact_1219_closed__Compl,axiom,
! [S: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( topolo784654279908865136osed_a @ ( uminus_uminus_set_a @ S ) ) ) ).
% closed_Compl
thf(fact_1220_open__Compl,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo8477419352202985285open_a @ ( uminus_uminus_set_a @ S ) ) ) ).
% open_Compl
thf(fact_1221_Compl__atMost,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_atMost_a @ K ) )
= ( set_or8632414552788122084Than_a @ K ) ) ).
% Compl_atMost
thf(fact_1222_Compl__greaterThan,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_or8632414552788122084Than_a @ K ) )
= ( set_ord_atMost_a @ K ) ) ).
% Compl_greaterThan
thf(fact_1223_subset__Compl__singleton,axiom,
! [A5: set_a,B2: a] :
( ( ord_less_eq_set_a @ A5 @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_1224_closed__open,axiom,
( topolo784654279908865136osed_a
= ( ^ [S2: set_a] : ( topolo8477419352202985285open_a @ ( uminus_uminus_set_a @ S2 ) ) ) ) ).
% closed_open
thf(fact_1225_open__closed,axiom,
( topolo8477419352202985285open_a
= ( ^ [S2: set_a] : ( topolo784654279908865136osed_a @ ( uminus_uminus_set_a @ S2 ) ) ) ) ).
% open_closed
thf(fact_1226_ComplD,axiom,
! [C: a,A5: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A5 ) )
=> ~ ( member_a @ C @ A5 ) ) ).
% ComplD
thf(fact_1227_bdd__below_Ounfold,axiom,
( condit5901475214736682318elow_a
= ( ^ [A6: set_a] :
? [M: a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_less_eq_a @ M @ X2 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1228_bdd__below_OE,axiom,
! [A5: set_a] :
( ( condit5901475214736682318elow_a @ A5 )
=> ~ ! [M3: a] :
~ ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( ord_less_eq_a @ M3 @ X3 ) ) ) ).
% bdd_below.E
thf(fact_1229_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A5: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member_a @ B2 @ ( image_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_1230_bdd__below_OI2,axiom,
! [A5: set_a,M2: a,F: a > a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( ord_less_eq_a @ M2 @ ( F @ X4 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1231_bdd__belowI2,axiom,
! [A5: set_a,M4: a,F: a > a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( ord_less_eq_a @ M4 @ ( F @ X4 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1232_image__subsetI,axiom,
! [A5: set_a,F: a > a,B5: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1233_imageI,axiom,
! [X: a,A5: set_a,F: a > a] :
( ( member_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1234_rev__image__eqI,axiom,
! [X: a,A5: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A5 )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1235_in__image__insert__iff,axiom,
! [B5: set_set_a,X: a,A5: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B5 )
=> ~ ( member_a @ X @ C5 ) )
=> ( ( member_set_a @ A5 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B5 ) )
= ( ( member_a @ X @ A5 )
& ( member_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1236_compact__Int__closed,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo8439159285038550427pact_a @ ( inf_inf_set_a @ S @ T5 ) ) ) ) ).
% compact_Int_closed
thf(fact_1237_closed__Int__compact,axiom,
! [S3: set_a,T: set_a] :
( ( topolo784654279908865136osed_a @ S3 )
=> ( ( topolo8439159285038550427pact_a @ T )
=> ( topolo8439159285038550427pact_a @ ( inf_inf_set_a @ S3 @ T ) ) ) ) ).
% closed_Int_compact
thf(fact_1238_compact__attains__sup,axiom,
! [S: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ S )
& ! [Xa: a] :
( ( member_a @ Xa @ S )
=> ( ord_less_eq_a @ Xa @ X4 ) ) ) ) ) ).
% compact_attains_sup
thf(fact_1239_compact__attains__inf,axiom,
! [S: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ S )
& ! [Xa: a] :
( ( member_a @ Xa @ S )
=> ( ord_less_eq_a @ X4 @ Xa ) ) ) ) ) ).
% compact_attains_inf
thf(fact_1240_compact__diff,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo8439159285038550427pact_a @ ( minus_minus_set_a @ S @ T5 ) ) ) ) ).
% compact_diff
thf(fact_1241_compact__imp__closed,axiom,
! [S3: set_a] :
( ( topolo8439159285038550427pact_a @ S3 )
=> ( topolo784654279908865136osed_a @ S3 ) ) ).
% compact_imp_closed
thf(fact_1242_compact__fip,axiom,
( topolo8439159285038550427pact_a
= ( ^ [U: set_a] :
! [A6: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( topolo784654279908865136osed_a @ X2 ) )
=> ( ! [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ A6 )
=> ( ( finite_finite_set_a @ B6 )
=> ( ( inf_inf_set_a @ U @ ( comple6135023378680113637_set_a @ B6 ) )
!= bot_bot_set_a ) ) )
=> ( ( inf_inf_set_a @ U @ ( comple6135023378680113637_set_a @ A6 ) )
!= bot_bot_set_a ) ) ) ) ) ).
% compact_fip
thf(fact_1243_compact__imp__fip,axiom,
! [S: set_a,F3: set_set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ! [T6: set_a] :
( ( member_set_a @ T6 @ F3 )
=> ( topolo784654279908865136osed_a @ T6 ) )
=> ( ! [F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ F3 )
=> ( ( inf_inf_set_a @ S @ ( comple6135023378680113637_set_a @ F4 ) )
!= bot_bot_set_a ) ) )
=> ( ( inf_inf_set_a @ S @ ( comple6135023378680113637_set_a @ F3 ) )
!= bot_bot_set_a ) ) ) ) ).
% compact_imp_fip
thf(fact_1244_open__Inter,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ S )
=> ( topolo8477419352202985285open_a @ X4 ) )
=> ( topolo8477419352202985285open_a @ ( comple6135023378680113637_set_a @ S ) ) ) ) ).
% open_Inter
thf(fact_1245_finite__has__minimal,axiom,
! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1246_finite__has__maximal,axiom,
! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1247_finite__has__minimal2,axiom,
! [A5: set_a,A: a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A @ A5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( ord_less_eq_a @ X4 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1248_finite__has__maximal2,axiom,
! [A5: set_a,A: a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A @ A5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( ord_less_eq_a @ A @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1249_finite__indexed__bound,axiom,
! [A5: set_a,P: a > a > $o] :
( ( finite_finite_a @ A5 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ? [X_1: a] : ( P @ X4 @ X_1 ) )
=> ? [M5: a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ? [K4: a] :
( ( ord_less_eq_a @ K4 @ M5 )
& ( P @ X3 @ K4 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1250_finite__imp__closed,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( topolo784654279908865136osed_a @ S ) ) ).
% finite_imp_closed
thf(fact_1251_finite__subset__induct,axiom,
! [F3: set_a,A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A4 @ A5 )
=> ( ~ ( member_a @ A4 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a @ A4 @ F5 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1252_finite__subset__induct_H,axiom,
! [F3: set_a,A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A4 @ A5 )
=> ( ( ord_less_eq_set_a @ F5 @ A5 )
=> ( ~ ( member_a @ A4 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a @ A4 @ F5 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1253_finite__remove__induct,axiom,
! [B5: set_a,P: set_a > $o] :
( ( finite_finite_a @ B5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_1254_remove__induct,axiom,
! [P: set_a > $o,B5: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B5 )
=> ( P @ B5 ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_1255_finite__linorder__min__induct,axiom,
! [A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B4: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( ord_less_a @ B4 @ X3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_a @ B4 @ A8 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1256_infinite__growing,axiom,
! [X6: set_a] :
( ( X6 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ X6 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X6 )
& ( ord_less_a @ X4 @ Xa ) ) )
=> ~ ( finite_finite_a @ X6 ) ) ) ).
% infinite_growing
thf(fact_1257_ex__min__if__finite,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ S )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ S )
& ( ord_less_a @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1258_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > a] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,S5: set_a] :
( ( finite_finite_a @ S5 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S5 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X4 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_a @ X4 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1259_finite__linorder__max__induct,axiom,
! [A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B4: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( ord_less_a @ X3 @ B4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_a @ B4 @ A8 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1260_arg__min__least,axiom,
! [S: set_a,Y3: a,F: a > a] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y3 @ S )
=> ( ord_less_eq_a @ ( F @ ( lattic3288624042836100505on_a_a @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1261_closure__closed,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( elementary_closure_a @ S )
= S ) ) ).
% closure_closed
thf(fact_1262_closed__closure,axiom,
! [S: set_a] : ( topolo784654279908865136osed_a @ ( elementary_closure_a @ S ) ) ).
% closed_closure
thf(fact_1263_closure__eq,axiom,
! [S: set_a] :
( ( ( elementary_closure_a @ S )
= S )
= ( topolo784654279908865136osed_a @ S ) ) ).
% closure_eq
thf(fact_1264_closure__subset__eq,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ S )
= ( topolo784654279908865136osed_a @ S ) ) ).
% closure_subset_eq
thf(fact_1265_closure__minimal,axiom,
! [S: set_a,T5: set_a] :
( ( ord_less_eq_set_a @ S @ T5 )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ T5 ) ) ) ).
% closure_minimal
thf(fact_1266_closure__unique,axiom,
! [S: set_a,T5: set_a] :
( ( ord_less_eq_set_a @ S @ T5 )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( ! [T7: set_a] :
( ( ord_less_eq_set_a @ S @ T7 )
=> ( ( topolo784654279908865136osed_a @ T7 )
=> ( ord_less_eq_set_a @ T5 @ T7 ) ) )
=> ( ( elementary_closure_a @ S )
= T5 ) ) ) ) ).
% closure_unique
thf(fact_1267_open__Int__closure__eq__empty,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( ( inf_inf_set_a @ S @ ( elementary_closure_a @ T5 ) )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ S @ T5 )
= bot_bot_set_a ) ) ) ).
% open_Int_closure_eq_empty
thf(fact_1268_open__Int__closure__subset,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ S @ ( elementary_closure_a @ T5 ) ) @ ( elementary_closure_a @ ( inf_inf_set_a @ S @ T5 ) ) ) ) ).
% open_Int_closure_subset
thf(fact_1269_closure__open__Int__superset,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( elementary_closure_a @ T5 ) )
=> ( ( elementary_closure_a @ ( inf_inf_set_a @ S @ T5 ) )
= ( elementary_closure_a @ S ) ) ) ) ).
% closure_open_Int_superset
thf(fact_1270_closure__iff__nhds__not__empty,axiom,
! [X: a,X6: set_a] :
( ( member_a @ X @ ( elementary_closure_a @ X6 ) )
= ( ! [A6: set_a,S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ A6 )
=> ( ( topolo8477419352202985285open_a @ S2 )
=> ( ( member_a @ X @ S2 )
=> ( ( inf_inf_set_a @ X6 @ A6 )
!= bot_bot_set_a ) ) ) ) ) ) ).
% closure_iff_nhds_not_empty
thf(fact_1271_compact__eq__Heine__Borel,axiom,
( topolo8439159285038550427pact_a
= ( ^ [S2: set_a] :
! [C3: set_set_a] :
( ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C3 )
=> ( topolo8477419352202985285open_a @ X2 ) )
& ( ord_less_eq_set_a @ S2 @ ( comple2307003609928055243_set_a @ C3 ) ) )
=> ? [D3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ D3 @ C3 )
& ( finite_finite_set_a @ D3 )
& ( ord_less_eq_set_a @ S2 @ ( comple2307003609928055243_set_a @ D3 ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_1272_open__Union,axiom,
! [K3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ K3 )
=> ( topolo8477419352202985285open_a @ X4 ) )
=> ( topolo8477419352202985285open_a @ ( comple2307003609928055243_set_a @ K3 ) ) ) ).
% open_Union
thf(fact_1273_closed__Union,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ S )
=> ( topolo784654279908865136osed_a @ X4 ) )
=> ( topolo784654279908865136osed_a @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).
% closed_Union
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y3: a] :
( ( if_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y3: a] :
( ( if_a @ $true @ X @ Y3 )
= X ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
ord_less_a @ y @ xa ).
thf(conj_1,hypothesis,
$true ).
thf(conj_2,hypothesis,
! [X3: a] :
( ( member_a @ X3 @ i )
=> ! [Y5: a] :
( ( ord_less_eq_a @ X3 @ Y5 )
=> ( member_a @ Y5 @ i ) ) ) ).
thf(conj_3,conjecture,
member_a @ xa @ i ).
%------------------------------------------------------------------------------