TPTP Problem File: SLH0967^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Safe_Range_RC/0021_Relational_Calculus/prob_01165_043863__17115726_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1451 ( 721 unt; 162 typ; 0 def)
% Number of atoms : 3768 (1775 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 11556 ( 427 ~; 30 |; 239 &;9632 @)
% ( 0 <=>;1228 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 921 ( 921 >; 0 *; 0 +; 0 <<)
% Number of symbols : 155 ( 152 usr; 19 con; 0-5 aty)
% Number of variables : 3571 ( 418 ^;3098 !; 55 ?;3571 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:25:53.554
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
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set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (152)
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thf(sy_c_Fun_Omonotone__on_001_Eo_001_Eo,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
top_to225934610606481728_a_b_o: relational_fmla_a_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
top_top_set_o_o: set_o_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
top_to8023889138469157253la_a_b: set_Re381260168593705685la_a_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
top_to5419598668418220987la_a_b: set_se6865892389300016395la_a_b ).
thf(sy_c_Relational__Calculus_Ocov_001tf__a_001tf__b,type,
relational_cov_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocov_H_001tf__a_001tf__b,type,
relational_cov_a_b2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocp_001tf__a_001tf__b,type,
relational_cp_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ocpropagated_001tf__a_001tf__b,type,
relati1591879772219623554ed_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oexists_001tf__a_001tf__b,type,
relati3989891337220013914ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OConj_001tf__a_001tf__b,type,
relational_Conj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OEq_001tf__a_001tf__b,type,
relational_Eq_a_b: nat > relational_term_a > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OExists_001tf__a_001tf__b,type,
relati591517084277583526ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ogen_001tf__a_001tf__b,type,
relational_gen_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogen_H_001tf__a_001tf__b,type,
relational_gen_a_b2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Osubst_001tf__a_001tf__b,type,
relational_subst_a_b: relational_fmla_a_b > nat > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Oterm_OConst_001tf__a,type,
relational_Const_a: a > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_OVar_001tf__a,type,
relational_Var_a: nat > relational_term_a ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
collec3419995626248312948la_a_b: ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
collec2099942116761351594la_a_b: ( set_Re381260168593705685la_a_b > $o ) > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Obind_001_Eo_001_Eo,type,
bind_o_o: set_o > ( $o > set_o ) > set_o ).
thf(sy_c_Set_Obind_001_Eo_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
bind_o5577781867067165604la_a_b: set_o > ( $o > set_Re381260168593705685la_a_b ) > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Obind_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001_Eo,type,
bind_R598982169242168356_a_b_o: set_Re381260168593705685la_a_b > ( relational_fmla_a_b > set_o ) > set_o ).
thf(sy_c_Set_Obind_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
bind_R4911597583902765075la_a_b: set_Re381260168593705685la_a_b > ( relational_fmla_a_b > set_Re381260168593705685la_a_b ) > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001_Eo,type,
image_o_o_o: ( ( $o > $o ) > $o ) > set_o_o > set_o ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_3369214520242522240la_a_b: ( ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b ) > set_Re4210025064442248784_a_b_o > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_6295478110982790130la_a_b: ( $o > relational_fmla_a_b ) > set_o > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_4064131284474593576la_a_b: ( $o > set_Re381260168593705685la_a_b ) > set_o > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001_Eo,type,
image_1316678413157792882_a_b_o: ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b > set_o ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_6790371041703824709la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Set__Oset_I_Eo_J,type,
image_5236934457445505874_set_o: ( relational_fmla_a_b > set_o ) > set_Re381260168593705685la_a_b > set_set_o ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_8209480069293074043la_a_b: ( relational_fmla_a_b > set_Re381260168593705685la_a_b ) > set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_4892062863073139410la_a_b: ( set_o > relational_fmla_a_b ) > set_set_o > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
image_4417937264480358420_a_b_o: ( set_Re381260168593705685la_a_b > relational_fmla_a_b > $o ) > set_se6865892389300016395la_a_b > set_Re4210025064442248784_a_b_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001_Eo,type,
image_7129636462341284284_a_b_o: ( set_Re381260168593705685la_a_b > $o ) > set_se6865892389300016395la_a_b > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_1964146275488721019la_a_b: ( set_Re381260168593705685la_a_b > relational_fmla_a_b ) > set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_I_Eo_J,type,
image_2902226032177518748_set_o: ( set_Re381260168593705685la_a_b > set_o ) > set_se6865892389300016395la_a_b > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_7051608999182166449la_a_b: ( set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ) > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001_062_I_Eo_M_Eo_J,type,
insert_o_o: ( $o > $o ) > set_o_o > set_o_o ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert2023870700798818565la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__empty_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_emp6953259385542938189la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_sin6594375743535830443la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Opairwise_001_Eo,type,
pairwise_o: ( $o > $o > $o ) > set_o > $o ).
thf(sy_c_Set_Opairwise_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
pairwi7405196478771624820la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
remove4261432235257513082la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3481406638322139244la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_v_G,type,
g: set_Re381260168593705685la_a_b ).
thf(sy_v_Gy,type,
gy: set_Re381260168593705685la_a_b ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v_x,type,
x: nat ).
thf(sy_v_y,type,
y: nat ).
% Relevant facts (1278)
thf(fact_0_fmla_Oinject_I3_J,axiom,
! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
( ( ( relational_Eq_a_b @ X31 @ X32 )
= ( relational_Eq_a_b @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fmla.inject(3)
thf(fact_1_fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: relational_fmla_a_b,Y71: nat,Y72: relational_fmla_a_b] :
( ( ( relati591517084277583526ts_a_b @ X71 @ X72 )
= ( relati591517084277583526ts_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fmla.inject(7)
thf(fact_2_term_Oinject_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( relational_Var_a @ X2 )
= ( relational_Var_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% term.inject(2)
thf(fact_3_gen_Ointros_I10_J,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).
% gen.intros(10)
thf(fact_4_fmla_Odistinct_I29_J,axiom,
! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(29)
thf(fact_5_cov_OEqL,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov.EqL
thf(fact_6_cov_OEqR,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov.EqR
thf(fact_7_cov_OExists,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).
% cov.Exists
thf(fact_8_cov_OEq__self,axiom,
! [X: nat] : ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ).
% cov.Eq_self
thf(fact_9_insert__Diff__single,axiom,
! [A: set_o,A2: set_set_o] :
( ( insert_set_o @ A @ ( minus_4899875422681990719_set_o @ A2 @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
= ( insert_set_o @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_10_insert__Diff__single,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ A @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) )
= ( insert2023870700798818565la_a_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_11_insert__Diff__single,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_12_insert__Diff__single,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
= ( insert7010464514620295119la_a_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_13_singleton__conv,axiom,
! [A: set_o] :
( ( collect_set_o
@ ^ [X3: set_o] : ( X3 = A ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singleton_conv
thf(fact_14_singleton__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( collec2099942116761351594la_a_b
@ ^ [X3: set_Re381260168593705685la_a_b] : ( X3 = A ) )
= ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) ).
% singleton_conv
thf(fact_15_singleton__conv,axiom,
! [A: relational_fmla_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( X3 = A ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singleton_conv
thf(fact_16_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X3: $o] : ( X3 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_17_singleton__conv2,axiom,
! [A: set_o] :
( ( collect_set_o
@ ( ^ [Y3: set_o,Z: set_o] : ( Y3 = Z )
@ A ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singleton_conv2
thf(fact_18_singleton__conv2,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( collec2099942116761351594la_a_b
@ ( ^ [Y3: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] : ( Y3 = Z )
@ A ) )
= ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) ).
% singleton_conv2
thf(fact_19_singleton__conv2,axiom,
! [A: relational_fmla_a_b] :
( ( collec3419995626248312948la_a_b
@ ( ^ [Y3: relational_fmla_a_b,Z: relational_fmla_a_b] : ( Y3 = Z )
@ A ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singleton_conv2
thf(fact_20_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y3: $o,Z: $o] : ( Y3 = Z )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_21_Un__Diff__cancel,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( minus_4705846553145473764la_a_b @ B @ A2 ) )
= ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_22_Un__Diff__cancel,axiom,
! [A2: set_set_o,B: set_set_o] :
( ( sup_sup_set_set_o @ A2 @ ( minus_4899875422681990719_set_o @ B @ A2 ) )
= ( sup_sup_set_set_o @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_23_Un__Diff__cancel,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_24_Un__Diff__cancel,axiom,
! [A2: set_o,B: set_o] :
( ( sup_sup_set_o @ A2 @ ( minus_minus_set_o @ B @ A2 ) )
= ( sup_sup_set_o @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_25_Un__Diff__cancel2,axiom,
! [B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( minus_4705846553145473764la_a_b @ B @ A2 ) @ A2 )
= ( sup_su4783144482993978935la_a_b @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_26_Un__Diff__cancel2,axiom,
! [B: set_set_o,A2: set_set_o] :
( ( sup_sup_set_set_o @ ( minus_4899875422681990719_set_o @ B @ A2 ) @ A2 )
= ( sup_sup_set_set_o @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_27_Un__Diff__cancel2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ B @ A2 ) @ A2 )
= ( sup_su5130108678486352897la_a_b @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_28_Un__Diff__cancel2,axiom,
! [B: set_o,A2: set_o] :
( ( sup_sup_set_o @ ( minus_minus_set_o @ B @ A2 ) @ A2 )
= ( sup_sup_set_o @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_29_Diff__insert0,axiom,
! [X: set_o,A2: set_set_o,B: set_set_o] :
( ~ ( member_set_o @ X @ A2 )
=> ( ( minus_4899875422681990719_set_o @ A2 @ ( insert_set_o @ X @ B ) )
= ( minus_4899875422681990719_set_o @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_30_Diff__insert0,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ B ) )
= ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_31_Diff__insert0,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_32_Diff__insert0,axiom,
! [X: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( minus_minus_set_o @ A2 @ ( insert_o @ X @ B ) )
= ( minus_minus_set_o @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_33_insert__Diff1,axiom,
! [X: set_o,B: set_set_o,A2: set_set_o] :
( ( member_set_o @ X @ B )
=> ( ( minus_4899875422681990719_set_o @ ( insert_set_o @ X @ A2 ) @ B )
= ( minus_4899875422681990719_set_o @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_34_insert__Diff1,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) @ B )
= ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_35_insert__Diff1,axiom,
! [X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_36_insert__Diff1,axiom,
! [X: $o,B: set_o,A2: set_o] :
( ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_37_Un__insert__left,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A @ B ) @ C )
= ( insert2023870700798818565la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_38_Un__insert__left,axiom,
! [A: set_o,B: set_set_o,C: set_set_o] :
( ( sup_sup_set_set_o @ ( insert_set_o @ A @ B ) @ C )
= ( insert_set_o @ A @ ( sup_sup_set_set_o @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_39_Un__insert__left,axiom,
! [A: $o,B: set_o,C: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B ) @ C )
= ( insert_o @ A @ ( sup_sup_set_o @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_40_Un__insert__left,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( insert7010464514620295119la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_41_Un__insert__right,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( insert2023870700798818565la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_42_Un__insert__right,axiom,
! [A2: set_set_o,A: set_o,B: set_set_o] :
( ( sup_sup_set_set_o @ A2 @ ( insert_set_o @ A @ B ) )
= ( insert_set_o @ A @ ( sup_sup_set_set_o @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_43_Un__insert__right,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_44_Un__insert__right,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_45_Diff__empty,axiom,
! [A2: set_set_o] :
( ( minus_4899875422681990719_set_o @ A2 @ bot_bot_set_set_o )
= A2 ) ).
% Diff_empty
thf(fact_46_Diff__empty,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ bot_bo2891247006866115487la_a_b )
= A2 ) ).
% Diff_empty
thf(fact_47_Diff__empty,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% Diff_empty
thf(fact_48_Diff__empty,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Diff_empty
thf(fact_49_empty__Diff,axiom,
! [A2: set_set_o] :
( ( minus_4899875422681990719_set_o @ bot_bot_set_set_o @ A2 )
= bot_bot_set_set_o ) ).
% empty_Diff
thf(fact_50_empty__Diff,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ bot_bo2891247006866115487la_a_b @ A2 )
= bot_bo2891247006866115487la_a_b ) ).
% empty_Diff
thf(fact_51_empty__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ bot_bo4495933725496725865la_a_b @ A2 )
= bot_bo4495933725496725865la_a_b ) ).
% empty_Diff
thf(fact_52_empty__Diff,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_53_Diff__cancel,axiom,
! [A2: set_set_o] :
( ( minus_4899875422681990719_set_o @ A2 @ A2 )
= bot_bot_set_set_o ) ).
% Diff_cancel
thf(fact_54_Diff__cancel,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ A2 )
= bot_bo2891247006866115487la_a_b ) ).
% Diff_cancel
thf(fact_55_Diff__cancel,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ A2 )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_cancel
thf(fact_56_Diff__cancel,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ A2 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_57_image__eqI,axiom,
! [B2: $o,F: $o > $o,X: $o,A2: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_58_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: $o > relational_fmla_a_b,X: $o,A2: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6295478110982790130la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_59_image__eqI,axiom,
! [B2: $o,F: relational_fmla_a_b > $o,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member_o @ B2 @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_60_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_61_image__eqI,axiom,
! [B2: set_o,F: $o > set_o,X: $o,A2: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_62_image__eqI,axiom,
! [B2: $o,F: set_o > $o,X: set_o,A2: set_set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_o @ X @ A2 )
=> ( member_o @ B2 @ ( image_set_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_63_image__eqI,axiom,
! [B2: $o,F: ( $o > $o ) > $o,X: $o > $o,A2: set_o_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o_o @ X @ A2 )
=> ( member_o @ B2 @ ( image_o_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_64_image__eqI,axiom,
! [B2: set_o,F: set_o > set_o,X: set_o,A2: set_set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_o @ X @ A2 )
=> ( member_set_o @ B2 @ ( image_set_o_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_65_image__eqI,axiom,
! [B2: set_o,F: ( $o > $o ) > set_o,X: $o > $o,A2: set_o_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o_o @ X @ A2 )
=> ( member_set_o @ B2 @ ( image_o_o_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_66_image__eqI,axiom,
! [B2: set_Re381260168593705685la_a_b,F: $o > set_Re381260168593705685la_a_b,X: $o,A2: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_4064131284474593576la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_67_empty__Collect__eq,axiom,
! [P: set_o > $o] :
( ( bot_bot_set_set_o
= ( collect_set_o @ P ) )
= ( ! [X3: set_o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_68_empty__Collect__eq,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( bot_bo2891247006866115487la_a_b
= ( collec2099942116761351594la_a_b @ P ) )
= ( ! [X3: set_Re381260168593705685la_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_69_empty__Collect__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X3: relational_fmla_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_70_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_71_Collect__empty__eq,axiom,
! [P: set_o > $o] :
( ( ( collect_set_o @ P )
= bot_bot_set_set_o )
= ( ! [X3: set_o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_72_Collect__empty__eq,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( ( collec2099942116761351594la_a_b @ P )
= bot_bo2891247006866115487la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: relational_fmla_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_75_all__not__in__conv,axiom,
! [A2: set_set_o] :
( ( ! [X3: set_o] :
~ ( member_set_o @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_o ) ) ).
% all_not_in_conv
thf(fact_76_all__not__in__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ! [X3: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X3 @ A2 ) )
= ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% all_not_in_conv
thf(fact_77_all__not__in__conv,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_78_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_79_empty__iff,axiom,
! [C2: set_o] :
~ ( member_set_o @ C2 @ bot_bot_set_set_o ) ).
% empty_iff
thf(fact_80_empty__iff,axiom,
! [C2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ C2 @ bot_bo2891247006866115487la_a_b ) ).
% empty_iff
thf(fact_81_empty__iff,axiom,
! [C2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_82_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_83_insert__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= ( insert2023870700798818565la_a_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_84_insert__absorb2,axiom,
! [X: set_o,A2: set_set_o] :
( ( insert_set_o @ X @ ( insert_set_o @ X @ A2 ) )
= ( insert_set_o @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_85_insert__absorb2,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ X @ A2 ) )
= ( insert7010464514620295119la_a_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_86_insert__absorb2,axiom,
! [X: $o,A2: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
= ( insert_o @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_87_insert__iff,axiom,
! [A: set_o,B2: set_o,A2: set_set_o] :
( ( member_set_o @ A @ ( insert_set_o @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_88_insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member3481406638322139244la_a_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_89_insert__iff,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_90_insert__iff,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_91_insertCI,axiom,
! [A: set_o,B: set_set_o,B2: set_o] :
( ( ~ ( member_set_o @ A @ B )
=> ( A = B2 ) )
=> ( member_set_o @ A @ ( insert_set_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_92_insertCI,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ A @ B )
=> ( A = B2 ) )
=> ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_93_insertCI,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A @ B )
=> ( A = B2 ) )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_94_insertCI,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A @ B )
=> ( A = B2 ) )
=> ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_95_Un__iff,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( ( member3481406638322139244la_a_b @ C2 @ A2 )
| ( member3481406638322139244la_a_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_96_Un__iff,axiom,
! [C2: set_o,A2: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ ( sup_sup_set_set_o @ A2 @ B ) )
= ( ( member_set_o @ C2 @ A2 )
| ( member_set_o @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_97_Un__iff,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) )
= ( ( member_o @ C2 @ A2 )
| ( member_o @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_98_Un__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( member4680049679412964150la_a_b @ C2 @ A2 )
| ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_99_UnCI,axiom,
! [C2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ C2 @ B )
=> ( member3481406638322139244la_a_b @ C2 @ A2 ) )
=> ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_100_UnCI,axiom,
! [C2: set_o,B: set_set_o,A2: set_set_o] :
( ( ~ ( member_set_o @ C2 @ B )
=> ( member_set_o @ C2 @ A2 ) )
=> ( member_set_o @ C2 @ ( sup_sup_set_set_o @ A2 @ B ) ) ) ).
% UnCI
thf(fact_101_UnCI,axiom,
! [C2: $o,B: set_o,A2: set_o] :
( ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ A2 ) )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnCI
thf(fact_102_UnCI,axiom,
! [C2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ A2 ) )
=> ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_103_Diff__idemp,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_104_Diff__idemp,axiom,
! [A2: set_o,B: set_o] :
( ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_105_Diff__iff,axiom,
! [C2: set_o,A2: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A2 @ B ) )
= ( ( member_set_o @ C2 @ A2 )
& ~ ( member_set_o @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_106_Diff__iff,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) )
= ( ( member3481406638322139244la_a_b @ C2 @ A2 )
& ~ ( member3481406638322139244la_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_107_Diff__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( ( member4680049679412964150la_a_b @ C2 @ A2 )
& ~ ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_108_Diff__iff,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
= ( ( member_o @ C2 @ A2 )
& ~ ( member_o @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_109_DiffI,axiom,
! [C2: set_o,A2: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ A2 )
=> ( ~ ( member_set_o @ C2 @ B )
=> ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_110_DiffI,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ A2 )
=> ( ~ ( member3481406638322139244la_a_b @ C2 @ B )
=> ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_111_DiffI,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_112_DiffI,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ A2 )
=> ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_113_image__ident,axiom,
! [Y4: set_set_o] :
( ( image_set_o_set_o
@ ^ [X3: set_o] : X3
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_114_image__ident,axiom,
! [Y4: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : X3
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_115_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_116_image__is__empty,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o] :
( ( ( image_6295478110982790130la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_117_image__is__empty,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b] :
( ( ( image_1316678413157792882_a_b_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_118_image__is__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_119_image__is__empty,axiom,
! [F: set_o > $o,A2: set_set_o] :
( ( ( image_set_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_120_image__is__empty,axiom,
! [F: $o > set_o,A2: set_o] :
( ( ( image_o_set_o @ F @ A2 )
= bot_bot_set_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_121_image__is__empty,axiom,
! [F: ( $o > $o ) > $o,A2: set_o_o] :
( ( ( image_o_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o_o ) ) ).
% image_is_empty
thf(fact_122_image__is__empty,axiom,
! [F: set_o > set_o,A2: set_set_o] :
( ( ( image_set_o_set_o @ F @ A2 )
= bot_bot_set_set_o )
= ( A2 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_123_image__is__empty,axiom,
! [F: set_o > $o > $o,A2: set_set_o] :
( ( ( image_set_o_o_o @ F @ A2 )
= bot_bot_set_o_o )
= ( A2 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_124_image__is__empty,axiom,
! [F: set_o > relational_fmla_a_b,A2: set_set_o] :
( ( ( image_4892062863073139410la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_125_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_126_empty__is__image,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o] :
( ( bot_bo4495933725496725865la_a_b
= ( image_6295478110982790130la_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_127_empty__is__image,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b] :
( ( bot_bot_set_o
= ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_128_empty__is__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_129_empty__is__image,axiom,
! [F: set_o > $o,A2: set_set_o] :
( ( bot_bot_set_o
= ( image_set_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_130_empty__is__image,axiom,
! [F: $o > set_o,A2: set_o] :
( ( bot_bot_set_set_o
= ( image_o_set_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_131_empty__is__image,axiom,
! [F: ( $o > $o ) > $o,A2: set_o_o] :
( ( bot_bot_set_o
= ( image_o_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o_o ) ) ).
% empty_is_image
thf(fact_132_empty__is__image,axiom,
! [F: set_o > set_o,A2: set_set_o] :
( ( bot_bot_set_set_o
= ( image_set_o_set_o @ F @ A2 ) )
= ( A2 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_133_empty__is__image,axiom,
! [F: set_o > $o > $o,A2: set_set_o] :
( ( bot_bot_set_o_o
= ( image_set_o_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_134_empty__is__image,axiom,
! [F: set_o > relational_fmla_a_b,A2: set_set_o] :
( ( bot_bo4495933725496725865la_a_b
= ( image_4892062863073139410la_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_135_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_136_image__empty,axiom,
! [F: relational_fmla_a_b > $o] :
( ( image_1316678413157792882_a_b_o @ F @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_o ) ).
% image_empty
thf(fact_137_image__empty,axiom,
! [F: $o > relational_fmla_a_b] :
( ( image_6295478110982790130la_a_b @ F @ bot_bot_set_o )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_138_image__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b] :
( ( image_6790371041703824709la_a_b @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_139_image__empty,axiom,
! [F: $o > set_o] :
( ( image_o_set_o @ F @ bot_bot_set_o )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_140_image__empty,axiom,
! [F: set_o > $o] :
( ( image_set_o_o @ F @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_141_image__empty,axiom,
! [F: ( $o > $o ) > $o] :
( ( image_o_o_o @ F @ bot_bot_set_o_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_142_image__empty,axiom,
! [F: set_o > set_o] :
( ( image_set_o_set_o @ F @ bot_bot_set_set_o )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_143_image__empty,axiom,
! [F: ( $o > $o ) > set_o] :
( ( image_o_o_set_o @ F @ bot_bot_set_o_o )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_144_image__empty,axiom,
! [F: relational_fmla_a_b > set_o] :
( ( image_5236934457445505874_set_o @ F @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_145_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) )
= ( image_o_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_146_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > relational_fmla_a_b] :
( ( member_o @ X @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X ) @ ( image_6295478110982790130la_a_b @ F @ A2 ) )
= ( image_6295478110982790130la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_147_insert__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_148_insert__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_149_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > set_o] :
( ( member_o @ X @ A2 )
=> ( ( insert_set_o @ ( F @ X ) @ ( image_o_set_o @ F @ A2 ) )
= ( image_o_set_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_150_insert__image,axiom,
! [X: set_o,A2: set_set_o,F: set_o > $o] :
( ( member_set_o @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_set_o_o @ F @ A2 ) )
= ( image_set_o_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_151_insert__image,axiom,
! [X: $o > $o,A2: set_o_o,F: ( $o > $o ) > $o] :
( ( member_o_o @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_o_o_o @ F @ A2 ) )
= ( image_o_o_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_152_insert__image,axiom,
! [X: set_o,A2: set_set_o,F: set_o > set_o] :
( ( member_set_o @ X @ A2 )
=> ( ( insert_set_o @ ( F @ X ) @ ( image_set_o_set_o @ F @ A2 ) )
= ( image_set_o_set_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_153_insert__image,axiom,
! [X: $o > $o,A2: set_o_o,F: ( $o > $o ) > set_o] :
( ( member_o_o @ X @ A2 )
=> ( ( insert_set_o @ ( F @ X ) @ ( image_o_o_set_o @ F @ A2 ) )
= ( image_o_o_set_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_154_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > set_Re381260168593705685la_a_b] :
( ( member_o @ X @ A2 )
=> ( ( insert2023870700798818565la_a_b @ ( F @ X ) @ ( image_4064131284474593576la_a_b @ F @ A2 ) )
= ( image_4064131284474593576la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_155_image__insert,axiom,
! [F: $o > $o,A: $o,B: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B ) ) ) ).
% image_insert
thf(fact_156_image__insert,axiom,
! [F: relational_fmla_a_b > $o,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_1316678413157792882_a_b_o @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_1316678413157792882_a_b_o @ F @ B ) ) ) ).
% image_insert
thf(fact_157_image__insert,axiom,
! [F: $o > relational_fmla_a_b,A: $o,B: set_o] :
( ( image_6295478110982790130la_a_b @ F @ ( insert_o @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_6295478110982790130la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_158_image__insert,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_159_image__insert,axiom,
! [F: $o > set_o,A: $o,B: set_o] :
( ( image_o_set_o @ F @ ( insert_o @ A @ B ) )
= ( insert_set_o @ ( F @ A ) @ ( image_o_set_o @ F @ B ) ) ) ).
% image_insert
thf(fact_160_image__insert,axiom,
! [F: set_o > $o,A: set_o,B: set_set_o] :
( ( image_set_o_o @ F @ ( insert_set_o @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_set_o_o @ F @ B ) ) ) ).
% image_insert
thf(fact_161_image__insert,axiom,
! [F: ( $o > $o ) > $o,A: $o > $o,B: set_o_o] :
( ( image_o_o_o @ F @ ( insert_o_o @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o_o @ F @ B ) ) ) ).
% image_insert
thf(fact_162_image__insert,axiom,
! [F: set_o > set_o,A: set_o,B: set_set_o] :
( ( image_set_o_set_o @ F @ ( insert_set_o @ A @ B ) )
= ( insert_set_o @ ( F @ A ) @ ( image_set_o_set_o @ F @ B ) ) ) ).
% image_insert
thf(fact_163_image__insert,axiom,
! [F: ( $o > $o ) > set_o,A: $o > $o,B: set_o_o] :
( ( image_o_o_set_o @ F @ ( insert_o_o @ A @ B ) )
= ( insert_set_o @ ( F @ A ) @ ( image_o_o_set_o @ F @ B ) ) ) ).
% image_insert
thf(fact_164_image__insert,axiom,
! [F: relational_fmla_a_b > set_o,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_5236934457445505874_set_o @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert_set_o @ ( F @ A ) @ ( image_5236934457445505874_set_o @ F @ B ) ) ) ).
% image_insert
thf(fact_165_singletonI,axiom,
! [A: set_o] : ( member_set_o @ A @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singletonI
thf(fact_166_singletonI,axiom,
! [A: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) ).
% singletonI
thf(fact_167_singletonI,axiom,
! [A: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_168_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_169_Un__empty,axiom,
! [A2: set_set_o,B: set_set_o] :
( ( ( sup_sup_set_set_o @ A2 @ B )
= bot_bot_set_set_o )
= ( ( A2 = bot_bot_set_set_o )
& ( B = bot_bot_set_set_o ) ) ) ).
% Un_empty
thf(fact_170_Un__empty,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A2 @ B )
= bot_bo2891247006866115487la_a_b )
= ( ( A2 = bot_bo2891247006866115487la_a_b )
& ( B = bot_bo2891247006866115487la_a_b ) ) ) ).
% Un_empty
thf(fact_171_Un__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B = bot_bo4495933725496725865la_a_b ) ) ) ).
% Un_empty
thf(fact_172_Un__empty,axiom,
! [A2: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_173_rev__image__eqI,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B2: $o,F: set_Re381260168593705685la_a_b > $o] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_7129636462341284284_a_b_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_174_rev__image__eqI,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B2: relational_fmla_a_b,F: set_Re381260168593705685la_a_b > relational_fmla_a_b] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_1964146275488721019la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_175_rev__image__eqI,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B2: set_o,F: set_Re381260168593705685la_a_b > set_o] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_o @ B2 @ ( image_2902226032177518748_set_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_176_rev__image__eqI,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_7051608999182166449la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_177_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_178_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B2: relational_fmla_a_b,F: $o > relational_fmla_a_b] :
( ( member_o @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6295478110982790130la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_179_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: $o,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_180_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_181_ball__imageD,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_182_image__cong,axiom,
! [M: set_Re381260168593705685la_a_b,N: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,G2: relational_fmla_a_b > relational_fmla_a_b] :
( ( M = N )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_6790371041703824709la_a_b @ F @ M )
= ( image_6790371041703824709la_a_b @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_183_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_184_mem__Collect__eq,axiom,
! [A: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_185_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_186_Collect__mem__eq,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_187_bex__imageD,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_188_image__iff,axiom,
! [Z2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Z2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_189_imageI,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_190_imageI,axiom,
! [X: $o,A2: set_o,F: $o > relational_fmla_a_b] :
( ( member_o @ X @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_6295478110982790130la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_191_imageI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_192_imageI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_193_ex__in__conv,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ? [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A2 ) )
= ( A2 != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_194_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_195_equals0I,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ! [Y5: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y5 @ A2 )
=> ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_196_equals0I,axiom,
! [A2: set_o] :
( ! [Y5: $o] :
~ ( member_o @ Y5 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_197_equals0D,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( A2 = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_198_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_199_emptyE,axiom,
! [A: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_200_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_201_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B3: set_o] :
( ( A2
= ( insert_o @ A @ B3 ) )
& ~ ( member_o @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_202_mk__disjoint__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ? [B3: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ A @ B3 ) )
& ~ ( member4680049679412964150la_a_b @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_203_insert__commute,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ Y @ A2 ) )
= ( insert7010464514620295119la_a_b @ Y @ ( insert7010464514620295119la_a_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_204_insert__commute,axiom,
! [X: $o,Y: $o,A2: set_o] :
( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
= ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_205_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A = (~ B2) )
=> ? [C3: set_o] :
( ( A2
= ( insert_o @ B2 @ C3 ) )
& ~ ( member_o @ B2 @ C3 )
& ( B
= ( insert_o @ A @ C3 ) )
& ~ ( member_o @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_206_insert__eq__iff,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ B2 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ A @ A2 )
= ( insert7010464514620295119la_a_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ B2 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ B2 @ C3 )
& ( B
= ( insert7010464514620295119la_a_b @ A @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_207_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_208_insert__absorb,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( insert7010464514620295119la_a_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_209_insert__ident,axiom,
! [X: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ~ ( member_o @ X @ B )
=> ( ( ( insert_o @ X @ A2 )
= ( insert_o @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_210_insert__ident,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ A2 )
= ( insert7010464514620295119la_a_b @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_211_Set_Oset__insert,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ~ ! [B3: set_o] :
( ( A2
= ( insert_o @ X @ B3 ) )
=> ( member_o @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_212_Set_Oset__insert,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ~ ! [B3: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ X @ B3 ) )
=> ( member4680049679412964150la_a_b @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_213_insertI2,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( member_o @ A @ B )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_214_insertI2,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A @ B )
=> ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_215_insertI1,axiom,
! [A: $o,B: set_o] : ( member_o @ A @ ( insert_o @ A @ B ) ) ).
% insertI1
thf(fact_216_insertI1,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ B ) ) ).
% insertI1
thf(fact_217_insertE,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
=> ( ( A = (~ B2) )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_218_insertE,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_219_Un__left__commute,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) )
= ( sup_su5130108678486352897la_a_b @ B @ ( sup_su5130108678486352897la_a_b @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_220_Un__left__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_221_Un__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_222_Un__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_223_Un__assoc,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% Un_assoc
thf(fact_224_ball__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_225_bex__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_226_UnI2,axiom,
! [C2: $o,B: set_o,A2: set_o] :
( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnI2
thf(fact_227_UnI2,axiom,
! [C2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% UnI2
thf(fact_228_UnI1,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnI1
thf(fact_229_UnI1,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% UnI1
thf(fact_230_UnE,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) )
=> ( ~ ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B ) ) ) ).
% UnE
thf(fact_231_UnE,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( ~ ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_232_DiffD2,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
=> ~ ( member4680049679412964150la_a_b @ C2 @ B ) ) ).
% DiffD2
thf(fact_233_DiffD2,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( member_o @ C2 @ B ) ) ).
% DiffD2
thf(fact_234_DiffD1,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
=> ( member4680049679412964150la_a_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_235_DiffD1,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
=> ( member_o @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_236_DiffE,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% DiffE
thf(fact_237_DiffE,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B ) ) ) ).
% DiffE
thf(fact_238_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_o_o @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_239_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_1316678413157792882_a_b_o @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_240_Compr__image__eq,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_6295478110982790130la_a_b @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_6295478110982790130la_a_b @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_241_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_6790371041703824709la_a_b @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_242_image__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,G2: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( image_6790371041703824709la_a_b @ G2 @ A2 ) )
= ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : ( F @ ( G2 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_243_imageE,axiom,
! [B2: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_244_imageE,axiom,
! [B2: $o,F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b] :
( ( member_o @ B2 @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ~ ! [X4: relational_fmla_a_b] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member4680049679412964150la_a_b @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_245_imageE,axiom,
! [B2: relational_fmla_a_b,F: $o > relational_fmla_a_b,A2: set_o] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_6295478110982790130la_a_b @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_246_imageE,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ~ ! [X4: relational_fmla_a_b] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member4680049679412964150la_a_b @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_247_empty__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : $false ) ) ).
% empty_def
thf(fact_248_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X3: $o] : $false ) ) ).
% empty_def
thf(fact_249_insert__Collect,axiom,
! [A: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( insert7010464514620295119la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) )
= ( collec3419995626248312948la_a_b
@ ^ [U: relational_fmla_a_b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_250_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_251_insert__compr,axiom,
( insert_o
= ( ^ [A4: $o,B4: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( X3 = A4 )
| ( member_o @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_252_insert__compr,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A4: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( X3 = A4 )
| ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_253_Collect__disj__eq,axiom,
! [P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( collec3419995626248312948la_a_b @ P ) @ ( collec3419995626248312948la_a_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_254_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A3: set_o,B4: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A3 )
| ( member_o @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_255_Un__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
| ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_256_set__diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
& ~ ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_257_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B4: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A3 )
& ~ ( member_o @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_258_singleton__inject,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_259_singleton__inject,axiom,
! [A: $o,B2: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_260_insert__not__empty,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A @ A2 )
!= bot_bo4495933725496725865la_a_b ) ).
% insert_not_empty
thf(fact_261_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_262_doubleton__eq__iff,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,C2: relational_fmla_a_b,D: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C2 @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_263_doubleton__eq__iff,axiom,
! [A: $o,B2: $o,C2: $o,D: $o] :
( ( ( insert_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) )
= ( insert_o @ C2 @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_264_singleton__iff,axiom,
! [B2: relational_fmla_a_b,A: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_265_singleton__iff,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_266_singletonD,axiom,
! [B2: relational_fmla_a_b,A: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_267_singletonD,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_268_image__Un,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_269_Un__empty__right,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% Un_empty_right
thf(fact_270_Un__empty__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Un_empty_right
thf(fact_271_Un__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_272_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_273_insert__Diff__if,axiom,
! [X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_274_insert__Diff__if,axiom,
! [X: $o,B: set_o,A2: set_o] :
( ( ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) )
& ( ~ ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B )
= ( insert_o @ X @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_275_Un__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ C ) @ ( minus_4077726661957047470la_a_b @ B @ C ) ) ) ).
% Un_Diff
thf(fact_276_Un__Diff,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( minus_minus_set_o @ ( sup_sup_set_o @ A2 @ B ) @ C )
= ( sup_sup_set_o @ ( minus_minus_set_o @ A2 @ C ) @ ( minus_minus_set_o @ B @ C ) ) ) ).
% Un_Diff
thf(fact_277_Collect__conv__if2,axiom,
! [P: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
( ( ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
& ( ~ ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_conv_if2
thf(fact_278_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_279_Collect__conv__if,axiom,
! [P: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
( ( ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
& ( ~ ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_conv_if
thf(fact_280_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_281_insert__def,axiom,
( insert_o
= ( ^ [A4: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X3: $o] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_282_insert__def,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A4: relational_fmla_a_b] :
( sup_su5130108678486352897la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_283_singleton__Un__iff,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_284_singleton__Un__iff,axiom,
! [X: $o,A2: set_o,B: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_285_Un__singleton__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_286_Un__singleton__iff,axiom,
! [A2: set_o,B: set_o,X: $o] :
( ( ( sup_sup_set_o @ A2 @ B )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_287_insert__is__Un,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A4: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_288_insert__is__Un,axiom,
( insert_o
= ( ^ [A4: $o] : ( sup_sup_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_289_Diff__insert__absorb,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_290_Diff__insert__absorb,axiom,
! [X: $o,A2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_291_Diff__insert2,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_292_Diff__insert2,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B ) ) ).
% Diff_insert2
thf(fact_293_insert__Diff,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( insert7010464514620295119la_a_b @ A @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_294_insert__Diff,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_295_Diff__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ).
% Diff_insert
thf(fact_296_Diff__insert,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_297_image__constant__conv,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: relational_fmla_a_b] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_298_image__constant__conv,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: $o] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_299_image__constant__conv,axiom,
! [A2: set_o,C2: relational_fmla_a_b] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_6295478110982790130la_a_b
@ ^ [X3: $o] : C2
@ A2 )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_6295478110982790130la_a_b
@ ^ [X3: $o] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_300_image__constant__conv,axiom,
! [A2: set_o,C2: $o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X3: $o] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X3: $o] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_301_image__constant,axiom,
! [X: $o,A2: set_o,C2: relational_fmla_a_b] :
( ( member_o @ X @ A2 )
=> ( ( image_6295478110982790130la_a_b
@ ^ [X3: $o] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_302_image__constant,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_303_image__constant,axiom,
! [X: $o,A2: set_o,C2: $o] :
( ( member_o @ X @ A2 )
=> ( ( image_o_o
@ ^ [X3: $o] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_304_image__constant,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: $o] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_305_sup__bot_Oright__neutral,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_306_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_307_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_308_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B2 ) )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_309_sup__bot_Oleft__neutral,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_310_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_311_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= bot_bo4495933725496725865la_a_b )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_312_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_313_sup__eq__bot__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_314_sup__eq__bot__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( sup_sup_set_o @ X @ Y )
= bot_bot_set_o )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_315_bot__eq__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y = bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_316_bot__eq__sup__iff,axiom,
! [X: set_o,Y: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X @ Y ) )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_317_sup__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% sup_bot_right
thf(fact_318_sup__bot__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% sup_bot_right
thf(fact_319_sup__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_320_sup__bot__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X )
= X ) ).
% sup_bot_left
thf(fact_321_cov_H_OExists__gen,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X )
@ Gy ) ) ) ) ) ) ).
% cov'.Exists_gen
thf(fact_322_cov_OExists__gen,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov.Exists_gen
thf(fact_323_sup_Oidem,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_324_sup__idem,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_325_sup_Oleft__idem,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_326_sup__left__idem,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_327_sup_Oright__idem,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ B2 )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_328_cp__exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
= ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q ) ) ) ).
% cp_exists
thf(fact_329_bot__set__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ bot_bo8852203127187332700_a_b_o ) ) ).
% bot_set_def
thf(fact_330_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_331_Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_1 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_12 ) ) ).
% Gen_cp
thf(fact_332_gen__Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_12 ) ) ).
% gen_Gen_cp
thf(fact_333_sup__set__def,axiom,
( sup_sup_set_o
= ( ^ [A3: set_o,B4: set_o] :
( collect_o
@ ( sup_sup_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A3 )
@ ^ [X3: $o] : ( member_o @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_334_sup__set__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( sup_su1471977682094119364_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_335_minus__set__def,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( minus_9215201808853403479_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_336_minus__set__def,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B4: set_o] :
( collect_o
@ ( minus_minus_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A3 )
@ ^ [X3: $o] : ( member_o @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_337_Gen__cp__subst,axiom,
! [Z2: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z2 @ Q @ X_1 )
=> ( ( Z2 != X )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z2 @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) @ X_12 ) ) ) ).
% Gen_cp_subst
thf(fact_338_cp_Osimps_I5_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q ) ) ) ).
% cp.simps(5)
thf(fact_339_exists__cp__subst,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b] :
( ( X != Y )
=> ( ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) )
= ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).
% exists_cp_subst
thf(fact_340_in__image__insert__iff,axiom,
! [B: set_se6865892389300016395la_a_b,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ! [C4: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C4 @ B )
=> ~ ( member4680049679412964150la_a_b @ X @ C4 ) )
=> ( ( member3481406638322139244la_a_b @ A2 @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X ) @ B ) )
= ( ( member4680049679412964150la_a_b @ X @ A2 )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_341_in__image__insert__iff,axiom,
! [B: set_set_o,X: $o,A2: set_o] :
( ! [C4: set_o] :
( ( member_set_o @ C4 @ B )
=> ~ ( member_o @ X @ C4 ) )
=> ( ( member_set_o @ A2 @ ( image_set_o_set_o @ ( insert_o @ X ) @ B ) )
= ( ( member_o @ X @ A2 )
& ( member_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_342_inf__sup__aci_I8_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_343_inf__sup__aci_I7_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) )
= ( sup_su5130108678486352897la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_344_inf__sup__aci_I6_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z2 )
= ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_345_inf__sup__aci_I5_J,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ Y6 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_346_sup_Oassoc,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ C2 )
= ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_347_sup__assoc,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z2 )
= ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_348_sup_Ocommute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_349_sup__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ Y6 @ X3 ) ) ) ).
% sup_commute
thf(fact_350_sup_Oleft__commute,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ B2 @ ( sup_su5130108678486352897la_a_b @ A @ C2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_351_sup__left__commute,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) )
= ( sup_su5130108678486352897la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_352_cov_H_OEq__self,axiom,
! [X: nat] : ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ).
% cov'.Eq_self
thf(fact_353_cov_H_OEqL,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov'.EqL
thf(fact_354_cov_H_OEqR,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov'.EqR
thf(fact_355_cov_H_OExists,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).
% cov'.Exists
thf(fact_356_insert__remove__id,axiom,
! [X: relational_fmla_a_b,X6: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ X6 )
=> ( X6
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ X6 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_357_insert__remove__id,axiom,
! [X: $o,X6: set_o] :
( ( member_o @ X @ X6 )
=> ( X6
= ( insert_o @ X @ ( minus_minus_set_o @ X6 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% insert_remove_id
thf(fact_358_the__elem__eq,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_359_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_360_is__singletonI,axiom,
! [X: relational_fmla_a_b] : ( is_sin6594375743535830443la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% is_singletonI
thf(fact_361_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_362_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_363_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_364_gen_Ointros_I9_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X ) )
@ G ) ) ) ).
% gen.intros(9)
thf(fact_365_gen_Ointros_I8_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X ) )
@ G ) ) ) ).
% gen.intros(8)
thf(fact_366_remove__def,axiom,
( remove4261432235257513082la_a_b
= ( ^ [X3: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] : ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% remove_def
thf(fact_367_remove__def,axiom,
( remove_o
= ( ^ [X3: $o,A3: set_o] : ( minus_minus_set_o @ A3 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_368_gen__nocp__intros_I2_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X )
@ G ) ) ) ).
% gen_nocp_intros(2)
thf(fact_369_member__remove,axiom,
! [X: $o,Y: $o,A2: set_o] :
( ( member_o @ X @ ( remove_o @ Y @ A2 ) )
= ( ( member_o @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_370_member__remove,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( remove4261432235257513082la_a_b @ Y @ A2 ) )
= ( ( member4680049679412964150la_a_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_371_fmla_Odistinct_I25_J,axiom,
! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(25)
thf(fact_372_fmla_Odistinct_I39_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(39)
thf(fact_373_subst_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X @ Y )
= ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y ) @ ( relational_subst_a_b @ Q22 @ X @ Y ) ) ) ).
% subst.simps(5)
thf(fact_374_gen_Ointros_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( ( relational_gen_a_b @ X @ Q1 @ G )
| ( relational_gen_a_b @ X @ Q22 @ G ) )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ).
% gen.intros(7)
thf(fact_375_cov_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q22 @ G22 )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov.Conj
thf(fact_376_cov_H_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b2 @ X @ Q22 @ G22 )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov'.Conj
thf(fact_377_is__singleton__the__elem,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( A3
= ( insert7010464514620295119la_a_b @ ( the_el6350558617753882986la_a_b @ A3 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_378_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
( A3
= ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_379_is__singletonI_H,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( member4680049679412964150la_a_b @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_sin6594375743535830443la_a_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_380_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o,Y5: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( member_o @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_381_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Re381260168593705685la_a_b,K: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( B
= ( sup_su5130108678486352897la_a_b @ K @ B2 ) )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B )
= ( sup_su5130108678486352897la_a_b @ K @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_382_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Re381260168593705685la_a_b,K: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( A2
= ( sup_su5130108678486352897la_a_b @ K @ A ) )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B2 )
= ( sup_su5130108678486352897la_a_b @ K @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_383_is__singleton__def,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
? [X3: relational_fmla_a_b] :
( A3
= ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_384_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
? [X3: $o] :
( A3
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_385_is__singletonE,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( is_sin6594375743535830443la_a_b @ A2 )
=> ~ ! [X4: relational_fmla_a_b] :
( A2
!= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% is_singletonE
thf(fact_386_is__singletonE,axiom,
! [A2: set_o] :
( ( is_singleton_o @ A2 )
=> ~ ! [X4: $o] :
( A2
!= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_387_the__elem__image__unique,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y5 @ A2 )
=> ( ( F @ Y5 )
= ( F @ X ) ) )
=> ( ( the_el6350558617753882986la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_388_gen__nocp__intros_I1_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X )
@ G ) ) ) ).
% gen_nocp_intros(1)
thf(fact_389_sup__Un__eq,axiom,
! [R: set_o,S: set_o] :
( ( sup_sup_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ R )
@ ^ [X3: $o] : ( member_o @ X3 @ S ) )
= ( ^ [X3: $o] : ( member_o @ X3 @ ( sup_sup_set_o @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_390_sup__Un__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( sup_su1471977682094119364_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ R )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_391_Collect__empty__eq__bot,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( P = bot_bo8852203127187332700_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_392_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_393_bot__empty__eq,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_394_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_395_gen_H__cp__intros_I1_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(1)
thf(fact_396_gen_H__cp__intros_I2_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(2)
thf(fact_397_gen_H_Ointros_I8_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X )
@ G ) ) ) ).
% gen'.intros(8)
thf(fact_398_gen_H_Ointros_I9_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X )
@ G ) ) ) ).
% gen'.intros(9)
thf(fact_399_gen__Eq,axiom,
! [Z2: nat,A: nat,T: relational_term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ ( relational_Eq_a_b @ A @ T ) @ G )
= ( ( Z2 = A )
& ? [C5: a] :
( ( T
= ( relational_Const_a @ C5 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ A @ T ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% gen_Eq
thf(fact_400_term_Odistinct_I1_J,axiom,
! [X1: a,X2: nat] :
( ( relational_Const_a @ X1 )
!= ( relational_Var_a @ X2 ) ) ).
% term.distinct(1)
thf(fact_401_Relational__Calculus_Oterm_Oexhaust,axiom,
! [Y: relational_term_a] :
( ! [X12: a] :
( Y
!= ( relational_Const_a @ X12 ) )
=> ~ ! [X22: nat] :
( Y
!= ( relational_Var_a @ X22 ) ) ) ).
% Relational_Calculus.term.exhaust
thf(fact_402_fv__fo__term__list_Ocases,axiom,
! [X: relational_term_a] :
( ! [N2: nat] :
( X
!= ( relational_Var_a @ N2 ) )
=> ~ ! [V: a] :
( X
!= ( relational_Const_a @ V ) ) ) ).
% fv_fo_term_list.cases
thf(fact_403_gen_H__gen,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ Q @ G ) ) ).
% gen'_gen
thf(fact_404_gen__gen_H,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b2 @ X @ Q @ G ) ) ).
% gen_gen'
thf(fact_405_gen__eq__gen_H,axiom,
relational_gen_a_b = relational_gen_a_b2 ).
% gen_eq_gen'
thf(fact_406_gen_H_Ointros_I10_J,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).
% gen'.intros(10)
thf(fact_407_Set_Ois__empty__def,axiom,
( is_emp6953259385542938189la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] : ( A3 = bot_bo4495933725496725865la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_408_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_409_bind__singleton__conv__image,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( bind_R4911597583902765075la_a_b @ A2
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ ( F @ X3 ) @ bot_bo4495933725496725865la_a_b ) )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_410_Inf_OINF__identity__eq,axiom,
! [Inf: set_Re381260168593705685la_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( Inf
@ ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_411_Sup_OSUP__identity__eq,axiom,
! [Sup: set_Re381260168593705685la_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( Sup
@ ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_412_the__elem__def,axiom,
( the_el6350558617753882986la_a_b
= ( ^ [X7: set_Re381260168593705685la_a_b] :
( the_Re7701088767514941656la_a_b
@ ^ [X3: relational_fmla_a_b] :
( X7
= ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% the_elem_def
thf(fact_413_the__elem__def,axiom,
( the_elem_o
= ( ^ [X7: set_o] :
( the_o
@ ^ [X3: $o] :
( X7
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_414_range__constant,axiom,
! [X: relational_fmla_a_b] :
( ( image_6790371041703824709la_a_b
@ ^ [Uu: relational_fmla_a_b] : X
@ top_to8023889138469157253la_a_b )
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% range_constant
thf(fact_415_range__constant,axiom,
! [X: relational_fmla_a_b] :
( ( image_6295478110982790130la_a_b
@ ^ [Uu: $o] : X
@ top_top_set_o )
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% range_constant
thf(fact_416_range__constant,axiom,
! [X: $o] :
( ( image_o_o
@ ^ [Uu: $o] : X
@ top_top_set_o )
= ( insert_o @ X @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_417_cpropagated__simps_I3_J,axiom,
! [X: nat,T: relational_term_a] :
( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X @ T ) )
= ( T
!= ( relational_Var_a @ X ) ) ) ).
% cpropagated_simps(3)
thf(fact_418_fun__upd__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_6790371041703824709la_a_b @ ( fun_up2215209898182204861la_a_b @ F @ X @ Y ) @ A2 )
= ( insert7010464514620295119la_a_b @ Y @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_6790371041703824709la_a_b @ ( fun_up2215209898182204861la_a_b @ F @ X @ Y ) @ A2 )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_419_fun__upd__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,Y: $o] :
( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_1316678413157792882_a_b_o @ ( fun_up77365845640406522_a_b_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_1316678413157792882_a_b_o @ ( fun_up77365845640406522_a_b_o @ F @ X @ Y ) @ A2 )
= ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_420_fun__upd__image,axiom,
! [X: $o,A2: set_o,F: $o > relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( member_o @ X @ A2 )
=> ( ( image_6295478110982790130la_a_b @ ( fun_up5056165543465403770la_a_b @ F @ X @ Y ) @ A2 )
= ( insert7010464514620295119la_a_b @ Y @ ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( image_6295478110982790130la_a_b @ ( fun_up5056165543465403770la_a_b @ F @ X @ Y ) @ A2 )
= ( image_6295478110982790130la_a_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_421_fun__upd__image,axiom,
! [X: $o,A2: set_o,F: $o > $o,Y: $o] :
( ( ( member_o @ X @ A2 )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X @ Y ) @ A2 )
= ( image_o_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_422_UNIV__I,axiom,
! [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ top_to8023889138469157253la_a_b ) ).
% UNIV_I
thf(fact_423_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_424_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b ) ).
% boolean_algebra.disj_one_right
thf(fact_425_boolean__algebra_Odisj__one__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_426_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ top_to8023889138469157253la_a_b @ X )
= top_to8023889138469157253la_a_b ) ).
% boolean_algebra.disj_one_left
thf(fact_427_boolean__algebra_Odisj__one__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_428_sup__top__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b ) ).
% sup_top_right
thf(fact_429_sup__top__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_430_sup__top__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ top_to8023889138469157253la_a_b @ X )
= top_to8023889138469157253la_a_b ) ).
% sup_top_left
thf(fact_431_sup__top__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% sup_top_left
thf(fact_432_empty__bind,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( bind_R4911597583902765075la_a_b @ bot_bo4495933725496725865la_a_b @ F )
= bot_bo4495933725496725865la_a_b ) ).
% empty_bind
thf(fact_433_empty__bind,axiom,
! [F: relational_fmla_a_b > set_o] :
( ( bind_R598982169242168356_a_b_o @ bot_bo4495933725496725865la_a_b @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_434_empty__bind,axiom,
! [F: $o > set_Re381260168593705685la_a_b] :
( ( bind_o5577781867067165604la_a_b @ bot_bot_set_o @ F )
= bot_bo4495933725496725865la_a_b ) ).
% empty_bind
thf(fact_435_empty__bind,axiom,
! [F: $o > set_o] :
( ( bind_o_o @ bot_bot_set_o @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_436_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec3419995626248312948la_a_b
@ ^ [S2: relational_fmla_a_b] : P )
= top_to8023889138469157253la_a_b ) )
& ( ~ P
=> ( ( collec3419995626248312948la_a_b
@ ^ [S2: relational_fmla_a_b] : P )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_const
thf(fact_437_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_438_Diff__UNIV,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ top_to8023889138469157253la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_UNIV
thf(fact_439_Diff__UNIV,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_440_UNIV__witness,axiom,
? [X4: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X4 @ top_to8023889138469157253la_a_b ) ).
% UNIV_witness
thf(fact_441_UNIV__witness,axiom,
? [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_442_UNIV__eq__I,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ! [X4: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( top_to8023889138469157253la_a_b = A2 ) ) ).
% UNIV_eq_I
thf(fact_443_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X4: $o] : ( member_o @ X4 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_444_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X3: $o] : $true ) ) ).
% UNIV_def
thf(fact_445_range__eqI,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b] :
( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b ) ) ) ).
% range_eqI
thf(fact_446_range__eqI,axiom,
! [B2: $o,F: $o > $o,X: $o] :
( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_447_range__eqI,axiom,
! [B2: relational_fmla_a_b,F: $o > relational_fmla_a_b,X: $o] :
( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6295478110982790130la_a_b @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_448_rangeI,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b ) ) ).
% rangeI
thf(fact_449_rangeI,axiom,
! [F: $o > $o,X: $o] : ( member_o @ ( F @ X ) @ ( image_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_450_rangeI,axiom,
! [F: $o > relational_fmla_a_b,X: $o] : ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_6295478110982790130la_a_b @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_451_surj__def,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b )
= ( ! [Y6: relational_fmla_a_b] :
? [X3: relational_fmla_a_b] :
( Y6
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_452_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y6: $o] :
? [X3: $o] :
( Y6
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_453_surjI,axiom,
! [G2: relational_fmla_a_b > relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ! [X4: relational_fmla_a_b] :
( ( G2 @ ( F @ X4 ) )
= X4 )
=> ( ( image_6790371041703824709la_a_b @ G2 @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b ) ) ).
% surjI
thf(fact_454_surjI,axiom,
! [G2: $o > $o,F: $o > $o] :
( ! [X4: $o] :
( ( G2 @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_o @ G2 @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_455_surjE,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b )
=> ~ ! [X4: relational_fmla_a_b] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_456_surjE,axiom,
! [F: $o > $o,Y: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X4: $o] :
( Y
= ( ~ ( F @ X4 ) ) ) ) ).
% surjE
thf(fact_457_surjD,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b )
=> ? [X4: relational_fmla_a_b] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_458_surjD,axiom,
! [F: $o > $o,Y: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X4: $o] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_459_empty__not__UNIV,axiom,
bot_bo4495933725496725865la_a_b != top_to8023889138469157253la_a_b ).
% empty_not_UNIV
thf(fact_460_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_461_insert__UNIV,axiom,
! [X: relational_fmla_a_b] :
( ( insert7010464514620295119la_a_b @ X @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b ) ).
% insert_UNIV
thf(fact_462_insert__UNIV,axiom,
! [X: $o] :
( ( insert_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_463_Un__UNIV__right,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b ) ).
% Un_UNIV_right
thf(fact_464_Un__UNIV__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_465_Un__UNIV__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ top_to8023889138469157253la_a_b @ B )
= top_to8023889138469157253la_a_b ) ).
% Un_UNIV_left
thf(fact_466_Un__UNIV__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_467_range__composition,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,G2: relational_fmla_a_b > relational_fmla_a_b] :
( ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : ( F @ ( G2 @ X3 ) )
@ top_to8023889138469157253la_a_b )
= ( image_6790371041703824709la_a_b @ F @ ( image_6790371041703824709la_a_b @ G2 @ top_to8023889138469157253la_a_b ) ) ) ).
% range_composition
thf(fact_468_range__composition,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,G2: $o > relational_fmla_a_b] :
( ( image_6295478110982790130la_a_b
@ ^ [X3: $o] : ( F @ ( G2 @ X3 ) )
@ top_top_set_o )
= ( image_6790371041703824709la_a_b @ F @ ( image_6295478110982790130la_a_b @ G2 @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_469_rangeE,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b ) )
=> ~ ! [X4: relational_fmla_a_b] :
( B2
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_470_rangeE,axiom,
! [B2: $o,F: $o > $o] :
( ( member_o @ B2 @ ( image_o_o @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B2
= ( ~ ( F @ X4 ) ) ) ) ).
% rangeE
thf(fact_471_rangeE,axiom,
! [B2: relational_fmla_a_b,F: $o > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_6295478110982790130la_a_b @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B2
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_472_bind__const,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( bind_R4911597583902765075la_a_b @ A2
@ ^ [Uu: relational_fmla_a_b] : B )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( bind_R4911597583902765075la_a_b @ A2
@ ^ [Uu: relational_fmla_a_b] : B )
= B ) ) ) ).
% bind_const
thf(fact_473_bind__const,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_o] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( bind_R598982169242168356_a_b_o @ A2
@ ^ [Uu: relational_fmla_a_b] : B )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( bind_R598982169242168356_a_b_o @ A2
@ ^ [Uu: relational_fmla_a_b] : B )
= B ) ) ) ).
% bind_const
thf(fact_474_bind__const,axiom,
! [A2: set_o,B: set_Re381260168593705685la_a_b] :
( ( ( A2 = bot_bot_set_o )
=> ( ( bind_o5577781867067165604la_a_b @ A2
@ ^ [Uu: $o] : B )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( bind_o5577781867067165604la_a_b @ A2
@ ^ [Uu: $o] : B )
= B ) ) ) ).
% bind_const
thf(fact_475_bind__const,axiom,
! [A2: set_o,B: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( bind_o_o @ A2
@ ^ [Uu: $o] : B )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( bind_o_o @ A2
@ ^ [Uu: $o] : B )
= B ) ) ) ).
% bind_const
thf(fact_476_range__eq__singletonD,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b,X: relational_fmla_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_477_range__eq__singletonD,axiom,
! [F: $o > relational_fmla_a_b,A: relational_fmla_a_b,X: $o] :
( ( ( image_6295478110982790130la_a_b @ F @ top_top_set_o )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_478_range__eq__singletonD,axiom,
! [F: $o > $o,A: $o,X: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_479_Sup_OSUP__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > relational_fmla_a_b,D2: relational_fmla_a_b > relational_fmla_a_b,Sup: set_Re381260168593705685la_a_b > relational_fmla_a_b] :
( ( A2 = B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_6790371041703824709la_a_b @ C @ A2 ) )
= ( Sup @ ( image_6790371041703824709la_a_b @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_480_Inf_OINF__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > relational_fmla_a_b,D2: relational_fmla_a_b > relational_fmla_a_b,Inf: set_Re381260168593705685la_a_b > relational_fmla_a_b] :
( ( A2 = B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_6790371041703824709la_a_b @ C @ A2 ) )
= ( Inf @ ( image_6790371041703824709la_a_b @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_481_iso__tuple__UNIV__I,axiom,
! [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ top_to8023889138469157253la_a_b ) ).
% iso_tuple_UNIV_I
thf(fact_482_iso__tuple__UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_483_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_484_top__empty__eq,axiom,
( top_to225934610606481728_a_b_o
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ top_to8023889138469157253la_a_b ) ) ) ).
% top_empty_eq
thf(fact_485_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_486_INT__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( minus_4077726661957047470la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= top_to8023889138469157253la_a_b ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( minus_4077726661957047470la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= ( minus_4077726661957047470la_a_b @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_487_INT__simps_I3_J,axiom,
! [C: set_o,A2: $o > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( minus_4077726661957047470la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= top_to8023889138469157253la_a_b ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( minus_4077726661957047470la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= ( minus_4077726661957047470la_a_b @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_488_INT__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_o,B: set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( minus_minus_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( minus_minus_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_489_INT__simps_I3_J,axiom,
! [C: set_o,A2: $o > set_o,B: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( minus_minus_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( minus_minus_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_490_UNION__fun__upd,axiom,
! [A2: relational_fmla_a_b > set_Re381260168593705685la_a_b,I: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,J: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ ( fun_up452696389778429683la_a_b @ A2 @ I @ B ) @ J ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ J @ ( insert7010464514620295119la_a_b @ I @ bot_bo4495933725496725865la_a_b ) ) ) ) @ ( if_set2835548578466827919la_a_b @ ( member4680049679412964150la_a_b @ I @ J ) @ B @ bot_bo4495933725496725865la_a_b ) ) ) ).
% UNION_fun_upd
thf(fact_491_UNION__fun__upd,axiom,
! [A2: relational_fmla_a_b > set_o,I: relational_fmla_a_b,B: set_o,J: set_Re381260168593705685la_a_b] :
( ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ ( fun_up4186687504794232026_set_o @ A2 @ I @ B ) @ J ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ A2 @ ( minus_4077726661957047470la_a_b @ J @ ( insert7010464514620295119la_a_b @ I @ bot_bo4495933725496725865la_a_b ) ) ) ) @ ( if_set_o @ ( member4680049679412964150la_a_b @ I @ J ) @ B @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_492_UNION__fun__upd,axiom,
! [A2: $o > set_Re381260168593705685la_a_b,I: $o,B: set_Re381260168593705685la_a_b,J: set_o] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ ( fun_up3013884331823319728la_a_b @ A2 @ I @ B ) @ J ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ ( minus_minus_set_o @ J @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set2835548578466827919la_a_b @ ( member_o @ I @ J ) @ B @ bot_bo4495933725496725865la_a_b ) ) ) ).
% UNION_fun_upd
thf(fact_493_UNION__fun__upd,axiom,
! [A2: $o > set_o,I: $o,B: set_o,J: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A2 @ I @ B ) @ J ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ ( minus_minus_set_o @ J @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I @ J ) @ B @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_494_UNION__singleton__eq__range,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ ( F @ X3 ) @ bot_bo4495933725496725865la_a_b )
@ A2 ) )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_495_inj__on__insert,axiom,
! [F: relational_fmla_a_b > $o,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) )
= ( ( inj_on8311361820794748934_a_b_o @ F @ A2 )
& ~ ( member_o @ ( F @ A ) @ ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_496_inj__on__insert,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) )
= ( ( inj_on4502624401354443441la_a_b @ F @ A2 )
& ~ ( member4680049679412964150la_a_b @ ( F @ A ) @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_497_inj__on__insert,axiom,
! [F: $o > $o,A: $o,A2: set_o] :
( ( inj_on_o_o @ F @ ( insert_o @ A @ A2 ) )
= ( ( inj_on_o_o @ F @ A2 )
& ~ ( member_o @ ( F @ A ) @ ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_498_inj__on__insert,axiom,
! [F: $o > relational_fmla_a_b,A: $o,A2: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ ( insert_o @ A @ A2 ) )
= ( ( inj_on4066789481764970374la_a_b @ F @ A2 )
& ~ ( member4680049679412964150la_a_b @ ( F @ A ) @ ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_499_INT__extend__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( minus_4077726661957047470la_a_b @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ C ) ) @ B )
= ( minus_4077726661957047470la_a_b @ top_to8023889138469157253la_a_b @ B ) ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( minus_4077726661957047470la_a_b @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ C ) ) @ B )
= ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( minus_4077726661957047470la_a_b @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_500_INT__extend__simps_I3_J,axiom,
! [C: set_o,A2: $o > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( minus_4077726661957047470la_a_b @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ C ) ) @ B )
= ( minus_4077726661957047470la_a_b @ top_to8023889138469157253la_a_b @ B ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( minus_4077726661957047470la_a_b @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ C ) ) @ B )
= ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( minus_4077726661957047470la_a_b @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_501_INT__extend__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_o,B: set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ A2 @ C ) ) @ B )
= ( minus_minus_set_o @ top_top_set_o @ B ) ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ A2 @ C ) ) @ B )
= ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( minus_minus_set_o @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_502_INT__extend__simps_I3_J,axiom,
! [C: set_o,A2: $o > set_o,B: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B )
= ( minus_minus_set_o @ top_top_set_o @ B ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( minus_minus_set_o @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_503_UN__insert,axiom,
! [B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( B @ A ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) ) ).
% UN_insert
thf(fact_504_UN__insert,axiom,
! [B: $o > set_Re381260168593705685la_a_b,A: $o,A2: set_o] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ ( insert_o @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( B @ A ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) ) ).
% UN_insert
thf(fact_505_pairwise__alt,axiom,
( pairwi7405196478771624820la_a_b
= ( ^ [R2: relational_fmla_a_b > relational_fmla_a_b > $o,S3: set_Re381260168593705685la_a_b] :
! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ S3 )
=> ! [Y6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y6 @ ( minus_4077726661957047470la_a_b @ S3 @ ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) )
=> ( R2 @ X3 @ Y6 ) ) ) ) ) ).
% pairwise_alt
thf(fact_506_pairwise__alt,axiom,
( pairwise_o
= ( ^ [R2: $o > $o > $o,S3: set_o] :
! [X3: $o] :
( ( member_o @ X3 @ S3 )
=> ! [Y6: $o] :
( ( member_o @ Y6 @ ( minus_minus_set_o @ S3 @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
=> ( R2 @ X3 @ Y6 ) ) ) ) ) ).
% pairwise_alt
thf(fact_507_Union__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X3: set_o] :
( ( member_set_o @ X3 @ C )
& ( member_o @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_508_Union__iff,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ C ) )
= ( ? [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ C )
& ( member4680049679412964150la_a_b @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_509_UnionI,axiom,
! [X6: set_o,C: set_set_o,A2: $o] :
( ( member_set_o @ X6 @ C )
=> ( ( member_o @ A2 @ X6 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_510_UnionI,axiom,
! [X6: set_Re381260168593705685la_a_b,C: set_se6865892389300016395la_a_b,A2: relational_fmla_a_b] :
( ( member3481406638322139244la_a_b @ X6 @ C )
=> ( ( member4680049679412964150la_a_b @ A2 @ X6 )
=> ( member4680049679412964150la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ C ) ) ) ) ).
% UnionI
thf(fact_511_Inter__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ C )
=> ( member_o @ A2 @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_512_Inter__iff,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple2571870594864334990la_a_b @ C ) )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ C )
=> ( member4680049679412964150la_a_b @ A2 @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_513_InterI,axiom,
! [C: set_set_o,A2: $o] :
( ! [X8: set_o] :
( ( member_set_o @ X8 @ C )
=> ( member_o @ A2 @ X8 ) )
=> ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) ) ) ).
% InterI
thf(fact_514_InterI,axiom,
! [C: set_se6865892389300016395la_a_b,A2: relational_fmla_a_b] :
( ! [X8: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X8 @ C )
=> ( member4680049679412964150la_a_b @ A2 @ X8 ) )
=> ( member4680049679412964150la_a_b @ A2 @ ( comple2571870594864334990la_a_b @ C ) ) ) ).
% InterI
thf(fact_515_Sup__bot__conv_I2_J,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( comple8442120529048846632la_a_b @ A2 ) )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( X3 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_516_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_517_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( X3 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_518_Sup__bot__conv_I1_J,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ( comple8442120529048846632la_a_b @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( X3 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_519_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_520_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( X3 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_521_Inf__top__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A2 ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = top_top_set_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_522_Inf__top__conv_I2_J,axiom,
! [A2: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( X3 = top_top_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_523_Inf__top__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A2 )
= top_top_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = top_top_set_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_524_Inf__top__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Inf_Inf_o @ A2 )
= top_top_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( X3 = top_top_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_525_Union__Un__distrib,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ).
% Union_Un_distrib
thf(fact_526_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A2 ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_527_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A2 )
= top_top_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_528_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_529_INF__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( complete_Inf_Inf_o @ A2 ) ) ).
% INF_identity_eq
thf(fact_530_UN__I,axiom,
! [A: $o,A2: set_o,B2: $o,B: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B2 @ ( B @ A ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_531_UN__I,axiom,
! [A: $o,A2: set_o,B2: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( member_o @ A @ A2 )
=> ( ( member4680049679412964150la_a_b @ B2 @ ( B @ A ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_532_UN__I,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: $o,B: relational_fmla_a_b > set_o] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( member_o @ B2 @ ( B @ A ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_533_UN__I,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( member4680049679412964150la_a_b @ B2 @ ( B @ A ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_534_INT__I,axiom,
! [A2: set_o,B2: $o,B: $o > set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ B2 @ ( B @ X4 ) ) )
=> ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) ) ) ).
% INT_I
thf(fact_535_INT__I,axiom,
! [A2: set_o,B2: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( B @ X4 ) ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) ) ).
% INT_I
thf(fact_536_INT__I,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: $o,B: relational_fmla_a_b > set_o] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( member_o @ B2 @ ( B @ X4 ) ) )
=> ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) ) ) ).
% INT_I
thf(fact_537_INT__I,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( B @ X4 ) ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) ) ).
% INT_I
thf(fact_538_Sup__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Sup_UNIV
thf(fact_539_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_540_Sup__empty,axiom,
( ( comple8442120529048846632la_a_b @ bot_bo2891247006866115487la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Sup_empty
thf(fact_541_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_542_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_543_Inf__UNIV,axiom,
( ( comple2571870594864334990la_a_b @ top_to5419598668418220987la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Inf_UNIV
thf(fact_544_Inf__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inf_UNIV
thf(fact_545_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_546_Inf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inf_empty
thf(fact_547_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_548_Sup__insert,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ A2 ) ) ) ).
% Sup_insert
thf(fact_549_Sup__insert,axiom,
! [A: $o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( insert_o @ A @ A2 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_insert
thf(fact_550_SUP__const,axiom,
! [A2: set_Re381260168593705685la_a_b,F: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( complete_Sup_Sup_o
@ ( image_1316678413157792882_a_b_o
@ ^ [I2: relational_fmla_a_b] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_551_SUP__const,axiom,
! [A2: set_o,F: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I2: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_552_INF__const,axiom,
! [A2: set_Re381260168593705685la_a_b,F: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [I2: relational_fmla_a_b] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_553_INF__const,axiom,
! [A2: set_o,F: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I2: $o] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_554_UN__constant,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_555_UN__constant,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: set_o] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_556_UN__constant,axiom,
! [A2: set_o,C2: set_Re381260168593705685la_a_b] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [Y6: $o] : C2
@ A2 ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [Y6: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_557_UN__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_558_UN__Un,axiom,
! [M: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_559_UN__singleton,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_560_UN__singleton,axiom,
! [A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( insert_o @ X3 @ bot_bot_set_o )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_561_UN__simps_I1_J,axiom,
! [C: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ C ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ C ) )
= ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_562_UN__simps_I1_J,axiom,
! [C: set_Re381260168593705685la_a_b,A: $o,B: relational_fmla_a_b > set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( insert_o @ A @ ( B @ X3 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( insert_o @ A @ ( B @ X3 ) )
@ C ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_563_UN__simps_I1_J,axiom,
! [C: set_o,A: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ C ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ C ) )
= ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_564_UN__simps_I1_J,axiom,
! [C: set_o,A: $o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
@ C ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_565_UN__simps_I2_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_566_UN__simps_I2_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_o,B: set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( sup_sup_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( sup_sup_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ A2 @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_567_UN__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( sup_su5130108678486352897la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( sup_su5130108678486352897la_a_b @ ( A2 @ X3 ) @ B )
@ C ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_568_UN__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_o,B: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( sup_sup_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( sup_sup_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_569_UN__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_570_UN__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_o,B: relational_fmla_a_b > set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( sup_sup_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( sup_sup_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_571_UN__simps_I3_J,axiom,
! [C: set_o,A2: set_Re381260168593705685la_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( sup_su5130108678486352897la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( sup_su5130108678486352897la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_572_UN__simps_I3_J,axiom,
! [C: set_o,A2: set_o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( sup_sup_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( sup_sup_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_573_INT__constant,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: set_o] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= top_top_set_o ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_574_INT__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= top_top_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_575_INT__simps_I4_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( minus_4077726661957047470la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= top_to8023889138469157253la_a_b ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( minus_4077726661957047470la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= ( minus_4077726661957047470la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_576_INT__simps_I4_J,axiom,
! [C: set_o,A2: set_Re381260168593705685la_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( minus_4077726661957047470la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= top_to8023889138469157253la_a_b ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( minus_4077726661957047470la_a_b @ A2 @ ( B @ X3 ) )
@ C ) )
= ( minus_4077726661957047470la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_577_INT__simps_I4_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_o,B: relational_fmla_a_b > set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( minus_minus_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( minus_minus_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_578_INT__simps_I4_J,axiom,
! [C: set_o,A2: set_o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( minus_minus_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( minus_minus_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_579_inj__on__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_se6865892389300016395la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ ( comple8442120529048846632la_a_b @ A2 ) )
=> ( inj_on6143097416644937501la_a_b @ ( image_6790371041703824709la_a_b @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_580_pairwiseI,axiom,
! [S: set_o,R: $o > $o > $o] :
( ! [X4: $o,Y5: $o] :
( ( member_o @ X4 @ S )
=> ( ( member_o @ Y5 @ S )
=> ( ( X4 != Y5 )
=> ( R @ X4 @ Y5 ) ) ) )
=> ( pairwise_o @ R @ S ) ) ).
% pairwiseI
thf(fact_581_pairwiseI,axiom,
! [S: set_Re381260168593705685la_a_b,R: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ! [X4: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ S )
=> ( ( member4680049679412964150la_a_b @ Y5 @ S )
=> ( ( X4 != Y5 )
=> ( R @ X4 @ Y5 ) ) ) )
=> ( pairwi7405196478771624820la_a_b @ R @ S ) ) ).
% pairwiseI
thf(fact_582_pairwiseD,axiom,
! [R: $o > $o > $o,S: set_o,X: $o,Y: $o] :
( ( pairwise_o @ R @ S )
=> ( ( member_o @ X @ S )
=> ( ( member_o @ Y @ S )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_583_pairwiseD,axiom,
! [R: relational_fmla_a_b > relational_fmla_a_b > $o,S: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( pairwi7405196478771624820la_a_b @ R @ S )
=> ( ( member4680049679412964150la_a_b @ X @ S )
=> ( ( member4680049679412964150la_a_b @ Y @ S )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_584_UnionE,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X8: set_o] :
( ( member_o @ A2 @ X8 )
=> ~ ( member_set_o @ X8 @ C ) ) ) ).
% UnionE
thf(fact_585_UnionE,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ C ) )
=> ~ ! [X8: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ X8 )
=> ~ ( member3481406638322139244la_a_b @ X8 @ C ) ) ) ).
% UnionE
thf(fact_586_InterE,axiom,
! [A2: $o,C: set_set_o,X6: set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) )
=> ( ( member_set_o @ X6 @ C )
=> ( member_o @ A2 @ X6 ) ) ) ).
% InterE
thf(fact_587_InterE,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b,X6: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple2571870594864334990la_a_b @ C ) )
=> ( ( member3481406638322139244la_a_b @ X6 @ C )
=> ( member4680049679412964150la_a_b @ A2 @ X6 ) ) ) ).
% InterE
thf(fact_588_InterD,axiom,
! [A2: $o,C: set_set_o,X6: set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) )
=> ( ( member_set_o @ X6 @ C )
=> ( member_o @ A2 @ X6 ) ) ) ).
% InterD
thf(fact_589_InterD,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b,X6: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple2571870594864334990la_a_b @ C ) )
=> ( ( member3481406638322139244la_a_b @ X6 @ C )
=> ( member4680049679412964150la_a_b @ A2 @ X6 ) ) ) ).
% InterD
thf(fact_590_INF__cong,axiom,
! [A2: set_o,B: set_o,C: $o > $o,D2: $o > $o] :
( ( A2 = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_591_INF__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > $o,D2: relational_fmla_a_b > $o] :
( ( A2 = B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_592_SUP__cong,axiom,
! [A2: set_o,B: set_o,C: $o > $o,D2: $o > $o] :
( ( A2 = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_593_SUP__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > $o,D2: relational_fmla_a_b > $o] :
( ( A2 = B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_594_Inter__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inter_empty
thf(fact_595_Inter__UNIV,axiom,
( ( comple2571870594864334990la_a_b @ top_to5419598668418220987la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Inter_UNIV
thf(fact_596_Inter__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inter_UNIV
thf(fact_597_Union__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Union_UNIV
thf(fact_598_empty__Union__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( comple8442120529048846632la_a_b @ A2 ) )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( X3 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% empty_Union_conv
thf(fact_599_empty__Union__conv,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_600_Union__empty__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ( comple8442120529048846632la_a_b @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( X3 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Union_empty_conv
thf(fact_601_Union__empty__conv,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_602_Union__empty,axiom,
( ( comple8442120529048846632la_a_b @ bot_bo2891247006866115487la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Union_empty
thf(fact_603_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_604_Union__insert,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ B ) ) ) ).
% Union_insert
thf(fact_605_SUP__UNIV__bool__expand,axiom,
! [A2: $o > set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ top_top_set_o ) )
= ( sup_su5130108678486352897la_a_b @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_606_SUP__UNIV__bool__expand,axiom,
! [A2: $o > $o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ top_top_set_o ) )
= ( sup_sup_o @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_607_INT__D,axiom,
! [B2: $o,B: $o > set_o,A2: set_o,A: $o] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_o @ B2 @ ( B @ A ) ) ) ) ).
% INT_D
thf(fact_608_INT__D,axiom,
! [B2: $o,B: relational_fmla_a_b > set_o,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) )
=> ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( member_o @ B2 @ ( B @ A ) ) ) ) ).
% INT_D
thf(fact_609_INT__D,axiom,
! [B2: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b,A2: set_o,A: $o] :
( ( member4680049679412964150la_a_b @ B2 @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( B @ A ) ) ) ) ).
% INT_D
thf(fact_610_INT__D,axiom,
! [B2: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
=> ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( B @ A ) ) ) ) ).
% INT_D
thf(fact_611_INT__E,axiom,
! [B2: $o,B: $o > set_o,A2: set_o,A: $o] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) )
=> ( ~ ( member_o @ B2 @ ( B @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_612_INT__E,axiom,
! [B2: $o,B: relational_fmla_a_b > set_o,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) )
=> ( ~ ( member_o @ B2 @ ( B @ A ) )
=> ~ ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% INT_E
thf(fact_613_INT__E,axiom,
! [B2: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b,A2: set_o,A: $o] :
( ( member4680049679412964150la_a_b @ B2 @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) )
=> ( ~ ( member4680049679412964150la_a_b @ B2 @ ( B @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_614_INT__E,axiom,
! [B2: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
=> ( ~ ( member4680049679412964150la_a_b @ B2 @ ( B @ A ) )
=> ~ ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% INT_E
thf(fact_615_UN__bool__eq,axiom,
! [A2: $o > set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ top_top_set_o ) )
= ( sup_su5130108678486352897la_a_b @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_616_Un__eq__UN,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [B5: $o] : ( if_set2835548578466827919la_a_b @ B5 @ A3 @ B4 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_617_UN__E,axiom,
! [B2: $o,B: $o > set_o,A2: set_o] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_o @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_618_UN__E,axiom,
! [B2: $o,B: relational_fmla_a_b > set_o,A2: set_Re381260168593705685la_a_b] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) )
=> ~ ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ~ ( member_o @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_619_UN__E,axiom,
! [B2: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b,A2: set_o] :
( ( member4680049679412964150la_a_b @ B2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_620_UN__E,axiom,
! [B2: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
=> ~ ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_621_pairwise__imageI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,P: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ! [X4: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( member4680049679412964150la_a_b @ Y5 @ A2 )
=> ( ( X4 != Y5 )
=> ( ( ( F @ X4 )
!= ( F @ Y5 ) )
=> ( P @ ( F @ X4 ) @ ( F @ Y5 ) ) ) ) ) )
=> ( pairwi7405196478771624820la_a_b @ P @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% pairwise_imageI
thf(fact_622_pairwise__empty,axiom,
! [P: relational_fmla_a_b > relational_fmla_a_b > $o] : ( pairwi7405196478771624820la_a_b @ P @ bot_bo4495933725496725865la_a_b ) ).
% pairwise_empty
thf(fact_623_pairwise__empty,axiom,
! [P: $o > $o > $o] : ( pairwise_o @ P @ bot_bot_set_o ) ).
% pairwise_empty
thf(fact_624_pairwise__insert,axiom,
! [R3: $o > $o > $o,X: $o,S4: set_o] :
( ( pairwise_o @ R3 @ ( insert_o @ X @ S4 ) )
= ( ! [Y6: $o] :
( ( ( member_o @ Y6 @ S4 )
& ( Y6 != X ) )
=> ( ( R3 @ X @ Y6 )
& ( R3 @ Y6 @ X ) ) )
& ( pairwise_o @ R3 @ S4 ) ) ) ).
% pairwise_insert
thf(fact_625_pairwise__insert,axiom,
! [R3: relational_fmla_a_b > relational_fmla_a_b > $o,X: relational_fmla_a_b,S4: set_Re381260168593705685la_a_b] :
( ( pairwi7405196478771624820la_a_b @ R3 @ ( insert7010464514620295119la_a_b @ X @ S4 ) )
= ( ! [Y6: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ Y6 @ S4 )
& ( Y6 != X ) )
=> ( ( R3 @ X @ Y6 )
& ( R3 @ Y6 @ X ) ) )
& ( pairwi7405196478771624820la_a_b @ R3 @ S4 ) ) ) ).
% pairwise_insert
thf(fact_626_INT__extend__simps_I4_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) )
= ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( minus_4077726661957047470la_a_b @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_627_INT__extend__simps_I4_J,axiom,
! [C: set_o,A2: set_Re381260168593705685la_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) )
= ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( minus_4077726661957047470la_a_b @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_628_INT__extend__simps_I4_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_o,B: relational_fmla_a_b > set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) )
= ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( minus_minus_set_o @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_629_INT__extend__simps_I4_J,axiom,
! [C: set_o,A2: set_o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( minus_minus_set_o @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_630_inj__image__mem__iff,axiom,
! [F: relational_fmla_a_b > $o,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ top_to8023889138469157253la_a_b )
=> ( ( member_o @ ( F @ A ) @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_631_inj__image__mem__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ( member4680049679412964150la_a_b @ ( F @ A ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_632_inj__image__mem__iff,axiom,
! [F: $o > $o,A: $o,A2: set_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( member_o @ ( F @ A ) @ ( image_o_o @ F @ A2 ) )
= ( member_o @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_633_inj__image__mem__iff,axiom,
! [F: $o > relational_fmla_a_b,A: $o,A2: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ top_top_set_o )
=> ( ( member4680049679412964150la_a_b @ ( F @ A ) @ ( image_6295478110982790130la_a_b @ F @ A2 ) )
= ( member_o @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_634_inj__image__eq__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ( ( image_6790371041703824709la_a_b @ F @ A2 )
= ( image_6790371041703824709la_a_b @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_image_eq_iff
thf(fact_635_range__ex1__eq,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b ) )
= ( ? [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
& ! [Y6: relational_fmla_a_b] :
( ( B2
= ( F @ Y6 ) )
=> ( Y6 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_636_range__ex1__eq,axiom,
! [F: $o > $o,B2: $o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( member_o @ B2 @ ( image_o_o @ F @ top_top_set_o ) )
= ( ? [X3: $o] :
( ( B2
= ( F @ X3 ) )
& ! [Y6: $o] :
( ( B2
= ( F @ Y6 ) )
=> ( Y6 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_637_range__ex1__eq,axiom,
! [F: $o > relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( inj_on4066789481764970374la_a_b @ F @ top_top_set_o )
=> ( ( member4680049679412964150la_a_b @ B2 @ ( image_6295478110982790130la_a_b @ F @ top_top_set_o ) )
= ( ? [X3: $o] :
( ( B2
= ( F @ X3 ) )
& ! [Y6: $o] :
( ( B2
= ( F @ Y6 ) )
=> ( Y6 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_638_inj__img__insertE,axiom,
! [F: $o > $o,A2: set_o,X: $o,B: set_o] :
( ( inj_on_o_o @ F @ A2 )
=> ( ~ ( member_o @ X @ B )
=> ( ( ( insert_o @ X @ B )
= ( image_o_o @ F @ A2 ) )
=> ~ ! [X9: $o,A5: set_o] :
( ~ ( member_o @ X9 @ A5 )
=> ( ( A2
= ( insert_o @ X9 @ A5 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B
!= ( image_o_o @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_639_inj__img__insertE,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,X: $o,B: set_o] :
( ( inj_on8311361820794748934_a_b_o @ F @ A2 )
=> ( ~ ( member_o @ X @ B )
=> ( ( ( insert_o @ X @ B )
= ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ~ ! [X9: relational_fmla_a_b,A5: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X9 @ A5 )
=> ( ( A2
= ( insert7010464514620295119la_a_b @ X9 @ A5 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B
!= ( image_1316678413157792882_a_b_o @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_640_inj__img__insertE,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4066789481764970374la_a_b @ F @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ B )
= ( image_6295478110982790130la_a_b @ F @ A2 ) )
=> ~ ! [X9: $o,A5: set_o] :
( ~ ( member_o @ X9 @ A5 )
=> ( ( A2
= ( insert_o @ X9 @ A5 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B
!= ( image_6295478110982790130la_a_b @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_641_inj__img__insertE,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ B )
= ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ~ ! [X9: relational_fmla_a_b,A5: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X9 @ A5 )
=> ( ( A2
= ( insert7010464514620295119la_a_b @ X9 @ A5 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B
!= ( image_6790371041703824709la_a_b @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_642_inj__on__Un__image__eq__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( ( ( image_6790371041703824709la_a_b @ F @ A2 )
= ( image_6790371041703824709la_a_b @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_643_INF__eq__const,axiom,
! [I3: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,X: $o] :
( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ I3 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_644_INF__eq__const,axiom,
! [I3: set_o,F: $o > $o,X: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I3 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_645_SUP__eq__const,axiom,
! [I3: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,X: $o] :
( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_646_SUP__eq__const,axiom,
! [I3: set_o,F: $o > $o,X: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_647_inj__on__fun__updI,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,Y: relational_fmla_a_b,X: relational_fmla_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ Y @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ( inj_on4502624401354443441la_a_b @ ( fun_up2215209898182204861la_a_b @ F @ X @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_648_Sup__union__distrib,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ).
% Sup_union_distrib
thf(fact_649_Sup__union__distrib,axiom,
! [A2: set_o,B: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A2 @ B ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_union_distrib
thf(fact_650_pairwise__image,axiom,
! [R3: relational_fmla_a_b > relational_fmla_a_b > $o,F: relational_fmla_a_b > relational_fmla_a_b,S4: set_Re381260168593705685la_a_b] :
( ( pairwi7405196478771624820la_a_b @ R3 @ ( image_6790371041703824709la_a_b @ F @ S4 ) )
= ( pairwi7405196478771624820la_a_b
@ ^ [X3: relational_fmla_a_b,Y6: relational_fmla_a_b] :
( ( ( F @ X3 )
!= ( F @ Y6 ) )
=> ( R3 @ ( F @ X3 ) @ ( F @ Y6 ) ) )
@ S4 ) ) ).
% pairwise_image
thf(fact_651_inj__singleton,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( inj_on8414577688008913895la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b )
@ A2 ) ).
% inj_singleton
thf(fact_652_inj__singleton,axiom,
! [A2: set_o] :
( inj_on_o_set_o
@ ^ [X3: $o] : ( insert_o @ X3 @ bot_bot_set_o )
@ A2 ) ).
% inj_singleton
thf(fact_653_SUP__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_Re381260168593705685la_a_b] :
( ( member_o @ K @ I3 )
=> ( ( sup_su5130108678486352897la_a_b @ ( A2 @ K ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ I3 ) ) )
= ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_654_SUP__absorb,axiom,
! [K: relational_fmla_a_b,I3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ K @ I3 )
=> ( ( sup_su5130108678486352897la_a_b @ ( A2 @ K ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ I3 ) ) )
= ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_655_SUP__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > $o] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ I3 ) ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_656_SUP__absorb,axiom,
! [K: relational_fmla_a_b,I3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ K @ I3 )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ A2 @ I3 ) ) )
= ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_657_INT__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( member_o @ U2 @ A2 )
=> ( ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert7010464514620295119la_a_b @ A @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_658_INT__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: $o,B: $o > set_o] :
( ( member_o @ U2 @ A2 )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_659_INT__insert__distrib,axiom,
! [U2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ U2 @ A2 )
=> ( ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert7010464514620295119la_a_b @ A @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_660_INT__insert__distrib,axiom,
! [U2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,A: $o,B: relational_fmla_a_b > set_o] :
( ( member4680049679412964150la_a_b @ U2 @ A2 )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( insert_o @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_661_UN__empty,axiom,
! [B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
= bot_bo4495933725496725865la_a_b ) ).
% UN_empty
thf(fact_662_UN__empty,axiom,
! [B: relational_fmla_a_b > set_o] :
( ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ bot_bo4495933725496725865la_a_b ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_663_UN__empty,axiom,
! [B: $o > set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ bot_bot_set_o ) )
= bot_bo4495933725496725865la_a_b ) ).
% UN_empty
thf(fact_664_UN__empty,axiom,
! [B: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_665_UN__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( member_o @ U2 @ A2 )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_666_UN__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: $o,B: $o > set_o] :
( ( member_o @ U2 @ A2 )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_667_UN__insert__distrib,axiom,
! [U2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ U2 @ A2 )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_668_UN__insert__distrib,axiom,
! [U2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,A: $o,B: relational_fmla_a_b > set_o] :
( ( member4680049679412964150la_a_b @ U2 @ A2 )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( insert_o @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_669_UN__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_Re381260168593705685la_a_b] :
( ( member_o @ K @ I3 )
=> ( ( sup_su5130108678486352897la_a_b @ ( A2 @ K ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ I3 ) ) )
= ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_670_UN__absorb,axiom,
! [K: relational_fmla_a_b,I3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ K @ I3 )
=> ( ( sup_su5130108678486352897la_a_b @ ( A2 @ K ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ I3 ) ) )
= ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_671_Un__Inter,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple2571870594864334990la_a_b @ B ) )
= ( comple2571870594864334990la_a_b @ ( image_7051608999182166449la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 ) @ B ) ) ) ).
% Un_Inter
thf(fact_672_image__Union,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,S: set_se6865892389300016395la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( comple8442120529048846632la_a_b @ S ) )
= ( comple8442120529048846632la_a_b @ ( image_7051608999182166449la_a_b @ ( image_6790371041703824709la_a_b @ F ) @ S ) ) ) ).
% image_Union
thf(fact_673_image__set__diff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_674_image__set__diff,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ top_to8023889138469157253la_a_b )
=> ( ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_minus_set_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) @ ( image_1316678413157792882_a_b_o @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_675_image__set__diff,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o,B: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ top_top_set_o )
=> ( ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( image_6295478110982790130la_a_b @ F @ A2 ) @ ( image_6295478110982790130la_a_b @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_676_image__set__diff,axiom,
! [F: $o > $o,A2: set_o,B: set_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_677_pairwise__singleton,axiom,
! [P: relational_fmla_a_b > relational_fmla_a_b > $o,A2: relational_fmla_a_b] : ( pairwi7405196478771624820la_a_b @ P @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% pairwise_singleton
thf(fact_678_pairwise__singleton,axiom,
! [P: $o > $o > $o,A2: $o] : ( pairwise_o @ P @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% pairwise_singleton
thf(fact_679_INF__constant,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: set_o] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= top_top_set_o ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_680_INF__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= top_top_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_681_INF__constant,axiom,
! [C2: $o,A2: set_Re381260168593705685la_a_b] :
( ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> top_top_o )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> C2 ) ) ) ).
% INF_constant
thf(fact_682_INF__constant,axiom,
! [C2: $o,A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_o )
=> top_top_o )
& ( ( A2 != bot_bot_set_o )
=> C2 ) ) ) ).
% INF_constant
thf(fact_683_INF__empty,axiom,
! [F: relational_fmla_a_b > set_o] :
( ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ F @ bot_bo4495933725496725865la_a_b ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_684_INF__empty,axiom,
! [F: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_685_INF__empty,axiom,
! [F: relational_fmla_a_b > $o] :
( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ bot_bo4495933725496725865la_a_b ) )
= top_top_o ) ).
% INF_empty
thf(fact_686_INF__empty,axiom,
! [F: $o > $o] :
( ( complete_Inf_Inf_o @ ( image_o_o @ F @ bot_bot_set_o ) )
= top_top_o ) ).
% INF_empty
thf(fact_687_SUP__constant,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_688_SUP__constant,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: set_o] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_689_SUP__constant,axiom,
! [A2: set_o,C2: set_Re381260168593705685la_a_b] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [Y6: $o] : C2
@ A2 ) )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [Y6: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_690_SUP__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_691_SUP__constant,axiom,
! [C2: $o,A2: set_Re381260168593705685la_a_b] :
( ( complete_Sup_Sup_o
@ ( image_1316678413157792882_a_b_o
@ ^ [Y6: relational_fmla_a_b] : C2
@ A2 ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> bot_bot_o )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_692_SUP__constant,axiom,
! [C2: $o,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_o )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_o )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_693_SUP__empty,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ bot_bo4495933725496725865la_a_b ) )
= bot_bo4495933725496725865la_a_b ) ).
% SUP_empty
thf(fact_694_SUP__empty,axiom,
! [F: relational_fmla_a_b > set_o] :
( ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ F @ bot_bo4495933725496725865la_a_b ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_695_SUP__empty,axiom,
! [F: $o > set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ bot_bot_set_o ) )
= bot_bo4495933725496725865la_a_b ) ).
% SUP_empty
thf(fact_696_SUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_697_SUP__empty,axiom,
! [F: relational_fmla_a_b > $o] :
( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ bot_bo4495933725496725865la_a_b ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_698_SUP__empty,axiom,
! [F: $o > $o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ bot_bot_set_o ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_699_SUP__insert,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_700_SUP__insert,axiom,
! [F: $o > set_Re381260168593705685la_a_b,A: $o,A2: set_o] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_701_SUP__insert,axiom,
! [F: relational_fmla_a_b > $o,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_702_SUP__insert,axiom,
! [F: $o > $o,A: $o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_703_SUP__union,axiom,
! [M: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_704_SUP__union,axiom,
! [M: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ M @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ M @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_705_INT__empty,axiom,
! [B: relational_fmla_a_b > set_o] :
( ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ B @ bot_bo4495933725496725865la_a_b ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_706_INT__empty,axiom,
! [B: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_707_UN__extend__simps_I1_J,axiom,
! [C: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) )
= ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_708_UN__extend__simps_I1_J,axiom,
! [C: set_Re381260168593705685la_a_b,A: $o,B: relational_fmla_a_b > set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( insert_o @ A @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_709_UN__extend__simps_I1_J,axiom,
! [C: set_o,A: relational_fmla_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( insert7010464514620295119la_a_b @ A @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) )
= ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( insert7010464514620295119la_a_b @ A @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_710_UN__extend__simps_I1_J,axiom,
! [C: set_o,A: $o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_711_UN__extend__simps_I2_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ C ) ) @ B )
= B ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ C ) ) @ B )
= ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_712_UN__extend__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ C ) ) @ B )
= B ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ A2 @ C ) ) @ B )
= ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( sup_su5130108678486352897la_a_b @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_713_UN__extend__simps_I3_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ C ) ) )
= ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_714_UN__extend__simps_I3_J,axiom,
! [C: set_o,A2: set_Re381260168593705685la_a_b,B: $o > set_Re381260168593705685la_a_b] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ C ) ) )
= ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( sup_su5130108678486352897la_a_b @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_715_cINF__const,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ).
% cINF_const
thf(fact_716_cINF__const,axiom,
! [A2: set_o,C2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : C2
@ A2 ) )
= C2 ) ) ).
% cINF_const
thf(fact_717_cSUP__const,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( complete_Sup_Sup_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_718_cSUP__const,axiom,
! [A2: set_o,C2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_719_cInf__singleton,axiom,
! [X: $o] :
( ( complete_Inf_Inf_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cInf_singleton
thf(fact_720_cSup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cSup_singleton
thf(fact_721_ccpo__Sup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_722_SUP__INF,axiom,
! [P: $o > $o > $o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y6: $o] :
( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : ( P @ X3 @ Y6 )
@ top_top_set_o ) )
@ top_top_set_o ) )
= ( complete_Inf_Inf_o
@ ( image_o_o_o
@ ^ [X3: $o > $o] :
( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y6: $o] : ( P @ ( X3 @ Y6 ) @ Y6 )
@ top_top_set_o ) )
@ top_top_set_o_o ) ) ) ).
% SUP_INF
thf(fact_723_INF__SUP,axiom,
! [P: $o > $o > $o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [Y6: $o] :
( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : ( P @ X3 @ Y6 )
@ top_top_set_o ) )
@ top_top_set_o ) )
= ( complete_Sup_Sup_o
@ ( image_o_o_o
@ ^ [F2: $o > $o] :
( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : ( P @ ( F2 @ X3 ) @ X3 )
@ top_top_set_o ) )
@ top_top_set_o_o ) ) ) ).
% INF_SUP
thf(fact_724_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_725_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S3: set_o_o,X3: $o] : ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_726_Sup__SUP__eq,axiom,
( comple3423763510802972509_a_b_o
= ( ^ [S3: set_Re4210025064442248784_a_b_o,X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( comple8442120529048846632la_a_b @ ( image_3369214520242522240la_a_b @ collec3419995626248312948la_a_b @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_727_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S3: set_o_o,X3: $o] : ( member_o @ X3 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_728_Inf__INT__eq,axiom,
( comple4426252245180427127_a_b_o
= ( ^ [S3: set_Re4210025064442248784_a_b_o,X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( comple2571870594864334990la_a_b @ ( image_3369214520242522240la_a_b @ collec3419995626248312948la_a_b @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_729_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I2: set_o,X3: $o] : ( member_o @ X3 @ I2 )
@ S ) )
= ( ^ [X3: $o] : ( member_o @ X3 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_730_SUP__Sup__eq,axiom,
! [S: set_se6865892389300016395la_a_b] :
( ( comple3423763510802972509_a_b_o
@ ( image_4417937264480358420_a_b_o
@ ^ [I2: set_Re381260168593705685la_a_b,X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ I2 )
@ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( comple8442120529048846632la_a_b @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_731_INF__Int__eq,axiom,
! [S: set_set_o] :
( ( complete_Inf_Inf_o_o
@ ( image_set_o_o_o
@ ^ [I2: set_o,X3: $o] : ( member_o @ X3 @ I2 )
@ S ) )
= ( ^ [X3: $o] : ( member_o @ X3 @ ( comple3063163877087187839_set_o @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_732_INF__Int__eq,axiom,
! [S: set_se6865892389300016395la_a_b] :
( ( comple4426252245180427127_a_b_o
@ ( image_4417937264480358420_a_b_o
@ ^ [I2: set_Re381260168593705685la_a_b,X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ I2 )
@ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( comple2571870594864334990la_a_b @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_733_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_734_Sup__set__def,axiom,
( comple8442120529048846632la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( complete_Sup_Sup_o @ ( image_7129636462341284284_a_b_o @ ( member4680049679412964150la_a_b @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_735_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X3: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_736_Inf__set__def,axiom,
( comple2571870594864334990la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( complete_Inf_Inf_o @ ( image_7129636462341284284_a_b_o @ ( member4680049679412964150la_a_b @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_737_Inf__sup__eq__top__iff,axiom,
! [B: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ ( comple2571870594864334990la_a_b @ B ) @ A )
= top_to8023889138469157253la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ X3 @ A )
= top_to8023889138469157253la_a_b ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_738_Inf__sup__eq__top__iff,axiom,
! [B: set_set_o,A: set_o] :
( ( ( sup_sup_set_o @ ( comple3063163877087187839_set_o @ B ) @ A )
= top_top_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ B )
=> ( ( sup_sup_set_o @ X3 @ A )
= top_top_set_o ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_739_Inf__sup__eq__top__iff,axiom,
! [B: set_o,A: $o] :
( ( ( sup_sup_o @ ( complete_Inf_Inf_o @ B ) @ A )
= top_top_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( sup_sup_o @ X3 @ A )
= top_top_o ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_740_sup__Inf,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( comple2571870594864334990la_a_b @ B ) )
= ( comple2571870594864334990la_a_b @ ( image_7051608999182166449la_a_b @ ( sup_su5130108678486352897la_a_b @ A ) @ B ) ) ) ).
% sup_Inf
thf(fact_741_sup__Inf,axiom,
! [A: $o,B: set_o] :
( ( sup_sup_o @ A @ ( complete_Inf_Inf_o @ B ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ ( sup_sup_o @ A ) @ B ) ) ) ).
% sup_Inf
thf(fact_742_Inf__sup,axiom,
! [B: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( comple2571870594864334990la_a_b @ B ) @ A )
= ( comple2571870594864334990la_a_b
@ ( image_7051608999182166449la_a_b
@ ^ [B5: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B5 @ A )
@ B ) ) ) ).
% Inf_sup
thf(fact_743_Inf__sup,axiom,
! [B: set_o,A: $o] :
( ( sup_sup_o @ ( complete_Inf_Inf_o @ B ) @ A )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [B5: $o] : ( sup_sup_o @ B5 @ A )
@ B ) ) ) ).
% Inf_sup
thf(fact_744_Union__image__insert,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ B ) ) ) ) ).
% Union_image_insert
thf(fact_745_Union__image__insert,axiom,
! [F: $o > set_Re381260168593705685la_a_b,A: $o,B: set_o] :
( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ ( insert_o @ A @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ B ) ) ) ) ).
% Union_image_insert
thf(fact_746_Union__image__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ bot_bo4495933725496725865la_a_b ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_747_Union__image__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,F: $o > set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ bot_bot_set_o ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_748_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_749_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_750_sorted__list__of__set_Oinj__on,axiom,
( inj_on_o_o
@ ^ [X3: $o] : X3
@ top_top_set_o ) ).
% sorted_list_of_set.inj_on
thf(fact_751_INT__simps_I1_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > set_o,B: set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( inf_inf_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( inf_inf_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(1)
thf(fact_752_INT__simps_I1_J,axiom,
! [C: set_o,A2: $o > set_o,B: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( inf_inf_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( inf_inf_set_o @ ( A2 @ X3 ) @ B )
@ C ) )
= ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(1)
thf(fact_753_INT__simps_I2_J,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_o,B: relational_fmla_a_b > set_o] :
( ( ( C = bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( inf_inf_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bo4495933725496725865la_a_b )
=> ( ( comple3063163877087187839_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( inf_inf_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= ( inf_inf_set_o @ A2 @ ( comple3063163877087187839_set_o @ ( image_5236934457445505874_set_o @ B @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_754_INT__simps_I2_J,axiom,
! [C: set_o,A2: set_o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( inf_inf_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= top_top_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( inf_inf_set_o @ A2 @ ( B @ X3 ) )
@ C ) )
= ( inf_inf_set_o @ A2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_755_IntI,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ A2 )
=> ( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_756_IntI,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_757_Int__iff,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
= ( ( member_o @ C2 @ A2 )
& ( member_o @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_758_Int__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= ( ( member4680049679412964150la_a_b @ C2 @ A2 )
& ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_759_inf__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= bot_bo4495933725496725865la_a_b ) ).
% inf_bot_left
thf(fact_760_inf__bot__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_761_inf__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% inf_bot_right
thf(fact_762_inf__bot__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_763_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_764_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_765_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_766_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_767_inf__top__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ X )
= X ) ).
% inf_top_left
thf(fact_768_inf__top__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ top_top_set_o )
= X ) ).
% inf_top_right
thf(fact_769_inf__eq__top__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= top_top_set_o )
= ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% inf_eq_top_iff
thf(fact_770_top__eq__inf__iff,axiom,
! [X: set_o,Y: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ X @ Y ) )
= ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% top_eq_inf_iff
thf(fact_771_inf__top_Oeq__neutr__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B2 = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_772_inf__top_Oleft__neutral,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_773_inf__top_Oneutr__eq__iff,axiom,
! [A: set_o,B2: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ A @ B2 ) )
= ( ( A = top_top_set_o )
& ( B2 = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_774_inf__top_Oright__neutral,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% inf_top.right_neutral
thf(fact_775_Int__UNIV,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= top_top_set_o )
= ( ( A2 = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_776_inf__sup__absorb,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_777_sup__inf__absorb,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_778_Int__insert__left__if0,axiom,
! [A: $o,C: set_o,B: set_o] :
( ~ ( member_o @ A @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B ) @ C )
= ( inf_inf_set_o @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_779_Int__insert__left__if0,axiom,
! [A: relational_fmla_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_780_Int__insert__left__if1,axiom,
! [A: $o,C: set_o,B: set_o] :
( ( member_o @ A @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B ) @ C )
= ( insert_o @ A @ ( inf_inf_set_o @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_781_Int__insert__left__if1,axiom,
! [A: relational_fmla_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_782_insert__inter__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_783_insert__inter__insert,axiom,
! [A: $o,A2: set_o,B: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ ( insert_o @ A @ B ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_784_Int__insert__right__if0,axiom,
! [A: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( inf_inf_set_o @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_785_Int__insert__right__if0,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_786_Int__insert__right__if1,axiom,
! [A: $o,A2: set_o,B: set_o] :
( ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_787_Int__insert__right__if1,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_788_Int__Un__eq_I4_J,axiom,
! [T2: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ T2 @ ( inf_in8483230781156617063la_a_b @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_789_Int__Un__eq_I3_J,axiom,
! [S: set_Re381260168593705685la_a_b,T2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ S @ ( inf_in8483230781156617063la_a_b @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_790_Int__Un__eq_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,T2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_791_Int__Un__eq_I1_J,axiom,
! [S: set_Re381260168593705685la_a_b,T2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_792_Un__Int__eq_I4_J,axiom,
! [T2: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ T2 @ ( sup_su5130108678486352897la_a_b @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_793_Un__Int__eq_I3_J,axiom,
! [S: set_Re381260168593705685la_a_b,T2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ S @ ( sup_su5130108678486352897la_a_b @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_794_Un__Int__eq_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,T2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_795_Un__Int__eq_I1_J,axiom,
! [S: set_Re381260168593705685la_a_b,T2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_796_disjoint__insert_I2_J,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) )
= ( ~ ( member4680049679412964150la_a_b @ B2 @ A2 )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_797_disjoint__insert_I2_J,axiom,
! [A2: set_o,B2: $o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ ( insert_o @ B2 @ B ) ) )
= ( ~ ( member_o @ B2 @ A2 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_798_disjoint__insert_I1_J,axiom,
! [B: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ A2 ) )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A @ B )
& ( ( inf_in8483230781156617063la_a_b @ B @ A2 )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_799_disjoint__insert_I1_J,axiom,
! [B: set_o,A: $o,A2: set_o] :
( ( ( inf_inf_set_o @ B @ ( insert_o @ A @ A2 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B )
& ( ( inf_inf_set_o @ B @ A2 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_800_insert__disjoint_I2_J,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) @ B ) )
= ( ~ ( member4680049679412964150la_a_b @ A @ B )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_801_insert__disjoint_I2_J,axiom,
! [A: $o,A2: set_o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B ) )
= ( ~ ( member_o @ A @ B )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_802_insert__disjoint_I1_J,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) @ B )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A @ B )
& ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_803_insert__disjoint_I1_J,axiom,
! [A: $o,A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B )
& ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_804_Inf__insert,axiom,
! [A: $o,A2: set_o] :
( ( complete_Inf_Inf_o @ ( insert_o @ A @ A2 ) )
= ( inf_inf_o @ A @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_insert
thf(fact_805_Diff__disjoint,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ A2 ) )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_disjoint
thf(fact_806_Diff__disjoint,axiom,
! [A2: set_o,B: set_o] :
( ( inf_inf_set_o @ A2 @ ( minus_minus_set_o @ B @ A2 ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_807_if__image__distrib,axiom,
! [P: relational_fmla_a_b > $o,F: relational_fmla_a_b > relational_fmla_a_b,G2: relational_fmla_a_b > relational_fmla_a_b,S: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b
@ ^ [X3: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( P @ X3 ) @ ( F @ X3 ) @ ( G2 @ X3 ) )
@ S )
= ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ F @ ( inf_in8483230781156617063la_a_b @ S @ ( collec3419995626248312948la_a_b @ P ) ) )
@ ( image_6790371041703824709la_a_b @ G2
@ ( inf_in8483230781156617063la_a_b @ S
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_808_Int__emptyI,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ X4 @ B ) )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b ) ) ).
% Int_emptyI
thf(fact_809_Int__emptyI,axiom,
! [A2: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_o @ X4 @ B ) )
=> ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_810_disjoint__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_811_disjoint__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_812_Int__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= bot_bo4495933725496725865la_a_b ) ).
% Int_empty_left
thf(fact_813_Int__empty__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_814_Int__empty__right,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Int_empty_right
thf(fact_815_Int__empty__right,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_816_disjoint__iff__not__equal,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ! [Y6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y6 @ B )
=> ( X3 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_817_disjoint__iff__not__equal,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ! [Y6: $o] :
( ( member_o @ Y6 @ B )
=> ( X3 = (~ Y6) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_818_Int__insert__left,axiom,
! [A: $o,C: set_o,B: set_o] :
( ( ( member_o @ A @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B ) @ C )
= ( insert_o @ A @ ( inf_inf_set_o @ B @ C ) ) ) )
& ( ~ ( member_o @ A @ C )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B ) @ C )
= ( inf_inf_set_o @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_819_Int__insert__left,axiom,
! [A: relational_fmla_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_820_Int__insert__right,axiom,
! [A: $o,A2: set_o,B: set_o] :
( ( ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_821_Int__insert__right,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_822_IntE,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C2 @ A2 )
=> ~ ( member_o @ C2 @ B ) ) ) ).
% IntE
thf(fact_823_IntE,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_824_IntD1,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
=> ( member_o @ C2 @ A2 ) ) ).
% IntD1
thf(fact_825_IntD1,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
=> ( member4680049679412964150la_a_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_826_IntD2,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
=> ( member_o @ C2 @ B ) ) ).
% IntD2
thf(fact_827_IntD2,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ).
% IntD2
thf(fact_828_Diff__Int__distrib2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
= ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_829_Diff__Int__distrib2,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( inf_inf_set_o @ ( minus_minus_set_o @ A2 @ B ) @ C )
= ( minus_minus_set_o @ ( inf_inf_set_o @ A2 @ C ) @ ( inf_inf_set_o @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_830_Diff__Int__distrib,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ C @ A2 ) @ ( inf_in8483230781156617063la_a_b @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_831_Diff__Int__distrib,axiom,
! [C: set_o,A2: set_o,B: set_o] :
( ( inf_inf_set_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( inf_inf_set_o @ C @ A2 ) @ ( inf_inf_set_o @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_832_Diff__Diff__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_833_Diff__Diff__Int,axiom,
! [A2: set_o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( minus_minus_set_o @ A2 @ B ) )
= ( inf_inf_set_o @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_834_Diff__Int2,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) @ ( inf_in8483230781156617063la_a_b @ B @ C ) )
= ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_835_Diff__Int2,axiom,
! [A2: set_o,C: set_o,B: set_o] :
( ( minus_minus_set_o @ ( inf_inf_set_o @ A2 @ C ) @ ( inf_inf_set_o @ B @ C ) )
= ( minus_minus_set_o @ ( inf_inf_set_o @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_836_Int__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ C ) ) ) ).
% Int_Diff
thf(fact_837_Int__Diff,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( minus_minus_set_o @ ( inf_inf_set_o @ A2 @ B ) @ C )
= ( inf_inf_set_o @ A2 @ ( minus_minus_set_o @ B @ C ) ) ) ).
% Int_Diff
thf(fact_838_Un__Int__distrib2,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ B @ C ) @ A2 )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ A2 ) @ ( sup_su5130108678486352897la_a_b @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_839_Int__Un__distrib2,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ C ) @ A2 )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ B @ A2 ) @ ( inf_in8483230781156617063la_a_b @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_840_Un__Int__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B @ C ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_841_Int__Un__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_842_Un__Int__crazy,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) @ ( inf_in8483230781156617063la_a_b @ C @ A2 ) )
= ( inf_in8483230781156617063la_a_b @ ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) @ ( sup_su5130108678486352897la_a_b @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_843_Int__UNIV__right,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ top_top_set_o )
= A2 ) ).
% Int_UNIV_right
thf(fact_844_Int__UNIV__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B )
= B ) ).
% Int_UNIV_left
thf(fact_845_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_846_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z2 ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_847_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) @ X )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ X ) @ ( inf_in8483230781156617063la_a_b @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_848_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ Z2 ) @ X )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ X ) @ ( sup_su5130108678486352897la_a_b @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_849_sup__inf__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ Z2 ) @ X )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ X ) @ ( sup_su5130108678486352897la_a_b @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_850_sup__inf__distrib1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z2 ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_851_inf__sup__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) @ X )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ X ) @ ( inf_in8483230781156617063la_a_b @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_852_inf__sup__distrib1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_853_distrib__imp2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ! [X4: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b,Z3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X4 @ ( inf_in8483230781156617063la_a_b @ Y5 @ Z3 ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X4 @ Y5 ) @ ( sup_su5130108678486352897la_a_b @ X4 @ Z3 ) ) )
=> ( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_854_distrib__imp1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ! [X4: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b,Z3: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X4 @ ( sup_su5130108678486352897la_a_b @ Y5 @ Z3 ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X4 @ Y5 ) @ ( inf_in8483230781156617063la_a_b @ X4 @ Z3 ) ) )
=> ( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z2 ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_855_boolean__algebra_Oconj__one__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ top_top_set_o )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_856_Int__Collect,axiom,
! [X: $o,A2: set_o,P: $o > $o] :
( ( member_o @ X @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) )
= ( ( member_o @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_857_Int__Collect,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) ) )
= ( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_858_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A3: set_o,B4: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A3 )
& ( member_o @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_859_Int__def,axiom,
( inf_in8483230781156617063la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
& ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_860_Sup__inf__eq__bot__iff,axiom,
! [B: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( comple8442120529048846632la_a_b @ B ) @ A )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ B )
=> ( ( inf_in8483230781156617063la_a_b @ X3 @ A )
= bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_861_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B ) @ A )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ B )
=> ( ( inf_inf_set_o @ X3 @ A )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_862_Sup__inf__eq__bot__iff,axiom,
! [B: set_o,A: $o] :
( ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B ) @ A )
= bot_bot_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( inf_inf_o @ X3 @ A )
= bot_bot_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_863_inf__Sup,axiom,
! [A: $o,B: set_o] :
( ( inf_inf_o @ A @ ( complete_Sup_Sup_o @ B ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ ( inf_inf_o @ A ) @ B ) ) ) ).
% inf_Sup
thf(fact_864_Inf__union__distrib,axiom,
! [A2: set_o,B: set_o] :
( ( complete_Inf_Inf_o @ ( sup_sup_set_o @ A2 @ B ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_union_distrib
thf(fact_865_Diff__triv,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_866_Diff__triv,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_867_Int__Diff__disjoint,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= bot_bo4495933725496725865la_a_b ) ).
% Int_Diff_disjoint
thf(fact_868_Int__Diff__disjoint,axiom,
! [A2: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B ) @ ( minus_minus_set_o @ A2 @ B ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_869_Union__disjoint,axiom,
! [C: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( comple8442120529048846632la_a_b @ C ) @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ C )
=> ( ( inf_in8483230781156617063la_a_b @ X3 @ A2 )
= bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Union_disjoint
thf(fact_870_Union__disjoint,axiom,
! [C: set_set_o,A2: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C ) @ A2 )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ C )
=> ( ( inf_inf_set_o @ X3 @ A2 )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_871_Diff__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) )
= ( inf_in8483230781156617063la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_872_Diff__Un,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( minus_minus_set_o @ A2 @ ( sup_sup_set_o @ B @ C ) )
= ( inf_inf_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( minus_minus_set_o @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_873_Diff__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B @ C ) )
= ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_874_Diff__Int,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( minus_minus_set_o @ A2 @ ( inf_inf_set_o @ B @ C ) )
= ( sup_sup_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( minus_minus_set_o @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_875_Int__Diff__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_876_Int__Diff__Un,axiom,
! [A2: set_o,B: set_o] :
( ( sup_sup_set_o @ ( inf_inf_set_o @ A2 @ B ) @ ( minus_minus_set_o @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_877_Un__Diff__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_878_Un__Diff__Int,axiom,
! [A2: set_o,B: set_o] :
( ( sup_sup_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( inf_inf_set_o @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_879_Sup__inf,axiom,
! [B: set_o,A: $o] :
( ( inf_inf_o @ ( complete_Sup_Sup_o @ B ) @ A )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [B5: $o] : ( inf_inf_o @ B5 @ A )
@ B ) ) ) ).
% Sup_inf
thf(fact_880_INF__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > $o] :
( ( member_o @ K @ I3 )
=> ( ( inf_inf_o @ ( A2 @ K ) @ ( complete_Inf_Inf_o @ ( image_o_o @ A2 @ I3 ) ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ A2 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_881_INF__absorb,axiom,
! [K: relational_fmla_a_b,I3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ K @ I3 )
=> ( ( inf_inf_o @ ( A2 @ K ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ A2 @ I3 ) ) )
= ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ A2 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_882_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A @ X )
= bot_bo4495933725496725865la_a_b )
=> ( ( ( sup_su5130108678486352897la_a_b @ A @ X )
= top_to8023889138469157253la_a_b )
=> ( ( ( inf_in8483230781156617063la_a_b @ A @ Y )
= bot_bo4495933725496725865la_a_b )
=> ( ( ( sup_su5130108678486352897la_a_b @ A @ Y )
= top_to8023889138469157253la_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_883_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_o,X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ A @ X )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ X )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A @ Y )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ Y )
= top_top_set_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_884_image__Int,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ( image_6790371041703824709la_a_b @ F @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ) ).
% image_Int
thf(fact_885_INF__inf__const1,axiom,
! [I3: set_Re381260168593705685la_a_b,X: $o,F: relational_fmla_a_b > $o] :
( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [I2: relational_fmla_a_b] : ( inf_inf_o @ X @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_o @ X @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_886_INF__inf__const1,axiom,
! [I3: set_o,X: $o,F: $o > $o] :
( ( I3 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I2: $o] : ( inf_inf_o @ X @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_o @ X @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_887_INF__inf__const2,axiom,
! [I3: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,X: $o] :
( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [I2: relational_fmla_a_b] : ( inf_inf_o @ ( F @ I2 ) @ X )
@ I3 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ I3 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_888_INF__inf__const2,axiom,
! [I3: set_o,F: $o > $o,X: $o] :
( ( I3 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I2: $o] : ( inf_inf_o @ ( F @ I2 ) @ X )
@ I3 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ I3 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_889_INF__insert,axiom,
! [F: relational_fmla_a_b > $o,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( inf_inf_o @ ( F @ A ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% INF_insert
thf(fact_890_INF__insert,axiom,
! [F: $o > $o,A: $o,A2: set_o] :
( ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( inf_inf_o @ ( F @ A ) @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INF_insert
thf(fact_891_INF__union,axiom,
! [M: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ M @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ M @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ M @ B ) ) ) ) ).
% INF_union
thf(fact_892_INF__UNIV__bool__expand,axiom,
! [A2: $o > $o] :
( ( complete_Inf_Inf_o @ ( image_o_o @ A2 @ top_top_set_o ) )
= ( inf_inf_o @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_893_inj__on__disjoint__Un,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o,G2: $o > relational_fmla_a_b,B: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ A2 )
=> ( ( inj_on4066789481764970374la_a_b @ G2 @ B )
=> ( ( ( inf_in8483230781156617063la_a_b @ ( image_6295478110982790130la_a_b @ F @ A2 ) @ ( image_6295478110982790130la_a_b @ G2 @ B ) )
= bot_bo4495933725496725865la_a_b )
=> ( inj_on4066789481764970374la_a_b
@ ^ [X3: $o] : ( if_Rel1279876242545935705la_a_b @ ( member_o @ X3 @ A2 ) @ ( F @ X3 ) @ ( G2 @ X3 ) )
@ ( sup_sup_set_o @ A2 @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_894_inj__on__disjoint__Un,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ A2 )
=> ( ( inj_on4502624401354443441la_a_b @ G2 @ B )
=> ( ( ( inf_in8483230781156617063la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ G2 @ B ) )
= bot_bo4495933725496725865la_a_b )
=> ( inj_on4502624401354443441la_a_b
@ ^ [X3: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member4680049679412964150la_a_b @ X3 @ A2 ) @ ( F @ X3 ) @ ( G2 @ X3 ) )
@ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_895_inj__on__disjoint__Un,axiom,
! [F: $o > $o,A2: set_o,G2: $o > $o,B: set_o] :
( ( inj_on_o_o @ F @ A2 )
=> ( ( inj_on_o_o @ G2 @ B )
=> ( ( ( inf_inf_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ G2 @ B ) )
= bot_bot_set_o )
=> ( inj_on_o_o
@ ^ [X3: $o] :
( ( ( member_o @ X3 @ A2 )
=> ( F @ X3 ) )
& ( ~ ( member_o @ X3 @ A2 )
=> ( G2 @ X3 ) ) )
@ ( sup_sup_set_o @ A2 @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_896_inj__on__disjoint__Un,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > $o,B: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ A2 )
=> ( ( inj_on8311361820794748934_a_b_o @ G2 @ B )
=> ( ( ( inf_inf_set_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) @ ( image_1316678413157792882_a_b_o @ G2 @ B ) )
= bot_bot_set_o )
=> ( inj_on8311361820794748934_a_b_o
@ ^ [X3: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( F @ X3 ) )
& ( ~ ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( G2 @ X3 ) ) )
@ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_897_inj__on__Un,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( inj_on4502624401354443441la_a_b @ F @ A2 )
& ( inj_on4502624401354443441la_a_b @ F @ B )
& ( ( inf_in8483230781156617063la_a_b @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ B @ A2 ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% inj_on_Un
thf(fact_898_inj__on__Un,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( inj_on8311361820794748934_a_b_o @ F @ A2 )
& ( inj_on8311361820794748934_a_b_o @ F @ B )
& ( ( inf_inf_set_o @ ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) @ ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ B @ A2 ) ) )
= bot_bot_set_o ) ) ) ).
% inj_on_Un
thf(fact_899_inj__on__Un,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o,B: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ ( sup_sup_set_o @ A2 @ B ) )
= ( ( inj_on4066789481764970374la_a_b @ F @ A2 )
& ( inj_on4066789481764970374la_a_b @ F @ B )
& ( ( inf_in8483230781156617063la_a_b @ ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ A2 @ B ) ) @ ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ B @ A2 ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% inj_on_Un
thf(fact_900_inj__on__Un,axiom,
! [F: $o > $o,A2: set_o,B: set_o] :
( ( inj_on_o_o @ F @ ( sup_sup_set_o @ A2 @ B ) )
= ( ( inj_on_o_o @ F @ A2 )
& ( inj_on_o_o @ F @ B )
& ( ( inf_inf_set_o @ ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ B ) ) @ ( image_o_o @ F @ ( minus_minus_set_o @ B @ A2 ) ) )
= bot_bot_set_o ) ) ) ).
% inj_on_Un
thf(fact_901_insert__partition,axiom,
! [X: set_Re381260168593705685la_a_b,F3: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ F3 )
=> ( ! [X4: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ ( insert2023870700798818565la_a_b @ X @ F3 ) )
=> ! [Xa: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Xa @ ( insert2023870700798818565la_a_b @ X @ F3 ) )
=> ( ( X4 != Xa )
=> ( ( inf_in8483230781156617063la_a_b @ X4 @ Xa )
= bot_bo4495933725496725865la_a_b ) ) ) )
=> ( ( inf_in8483230781156617063la_a_b @ X @ ( comple8442120529048846632la_a_b @ F3 ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% insert_partition
thf(fact_902_insert__partition,axiom,
! [X: set_o,F3: set_set_o] :
( ~ ( member_set_o @ X @ F3 )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ ( insert_set_o @ X @ F3 ) )
=> ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( insert_set_o @ X @ F3 ) )
=> ( ( X4 != Xa )
=> ( ( inf_inf_set_o @ X4 @ Xa )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F3 ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_903_cSUP__UNION,axiom,
! [A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bo4495933725496725865la_a_b ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_904_cSUP__UNION,axiom,
! [A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_o,F: $o > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bot_set_o ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( image_o_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_905_cSUP__UNION,axiom,
! [A2: set_o,B: $o > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bo4495933725496725865la_a_b ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_906_cSUP__UNION,axiom,
! [A2: set_o,B: $o > set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bot_set_o ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( image_o_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_907_cINF__UNION,axiom,
! [A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bo4495933725496725865la_a_b ) )
=> ( ( condit5413489452508810728elow_o
@ ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) ) )
= ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_908_cINF__UNION,axiom,
! [A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_o,F: $o > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bot_set_o ) )
=> ( ( condit5413489452508810728elow_o
@ ( comple90263536869209701_set_o
@ ( image_5236934457445505874_set_o
@ ^ [X3: relational_fmla_a_b] : ( image_o_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_5236934457445505874_set_o @ B @ A2 ) ) ) )
= ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_909_cINF__UNION,axiom,
! [A2: set_o,B: $o > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bo4495933725496725865la_a_b ) )
=> ( ( condit5413489452508810728elow_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) ) )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_910_cINF__UNION,axiom,
! [A2: set_o,B: $o > set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( B @ X4 )
!= bot_bot_set_o ) )
=> ( ( condit5413489452508810728elow_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X3: $o] : ( image_o_o @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) ) )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_911_cSUP__union,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
=> ( ( B != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ B ) )
=> ( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_912_cSUP__union,axiom,
! [A2: set_o,F: $o > set_Re381260168593705685la_a_b,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit454741301069713265la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) )
=> ( ( B != bot_bot_set_o )
=> ( ( condit454741301069713265la_a_b @ ( image_4064131284474593576la_a_b @ F @ B ) )
=> ( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ ( sup_sup_set_o @ A2 @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_913_cSUP__union,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,B: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( B != bot_bo4495933725496725865la_a_b )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ B ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_914_cSUP__union,axiom,
! [A2: set_o,F: $o > $o,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( B != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ B ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( sup_sup_set_o @ A2 @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_915_bdd__below__empty,axiom,
condit5413489452508810728elow_o @ bot_bot_set_o ).
% bdd_below_empty
thf(fact_916_bdd__above__empty,axiom,
condit5488710616941104124bove_o @ bot_bot_set_o ).
% bdd_above_empty
thf(fact_917_bdd__below__insert,axiom,
! [A: $o,A2: set_o] :
( ( condit5413489452508810728elow_o @ ( insert_o @ A @ A2 ) )
= ( condit5413489452508810728elow_o @ A2 ) ) ).
% bdd_below_insert
thf(fact_918_bdd__above__insert,axiom,
! [A: $o,A2: set_o] :
( ( condit5488710616941104124bove_o @ ( insert_o @ A @ A2 ) )
= ( condit5488710616941104124bove_o @ A2 ) ) ).
% bdd_above_insert
thf(fact_919_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A3: set_o,B4: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A3 )
@ ^ [X3: $o] : ( member_o @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_920_inf__set__def,axiom,
( inf_in8483230781156617063la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( inf_in2474466416471573982_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_921_inf__Int__eq,axiom,
! [R: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ R )
@ ^ [X3: $o] : ( member_o @ X3 @ S ) )
= ( ^ [X3: $o] : ( member_o @ X3 @ ( inf_inf_set_o @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_922_inf__Int__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( inf_in2474466416471573982_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ R )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( inf_in8483230781156617063la_a_b @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_923_cSUP__eq__cINF__D,axiom,
! [F: $o > $o,A2: set_o,A: $o] :
( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( member_o @ A @ A2 )
=> ( ( F @ A )
= ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_924_cSUP__eq__cINF__D,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( F @ A )
= ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_925_cInf__insert,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ X6 )
=> ( ( complete_Inf_Inf_o @ ( insert_o @ A @ X6 ) )
= ( inf_inf_o @ A @ ( complete_Inf_Inf_o @ X6 ) ) ) ) ) ).
% cInf_insert
thf(fact_926_cInf__insert__If,axiom,
! [X6: set_o,A: $o] :
( ( condit5413489452508810728elow_o @ X6 )
=> ( ( complete_Inf_Inf_o @ ( insert_o @ A @ X6 ) )
= ( ( ( X6 = bot_bot_set_o )
=> A )
& ( ( X6 != bot_bot_set_o )
=> ( inf_inf_o @ A @ ( complete_Inf_Inf_o @ X6 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_927_cSup__insert__If,axiom,
! [X6: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( condit454741301069713265la_a_b @ X6 )
=> ( ( ( X6 = bot_bo2891247006866115487la_a_b )
=> ( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ X6 ) )
= A ) )
& ( ( X6 != bot_bo2891247006866115487la_a_b )
=> ( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ X6 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ X6 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_928_cSup__insert__If,axiom,
! [X6: set_o,A: $o] :
( ( condit5488710616941104124bove_o @ X6 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X6 ) )
= ( ( ( X6 = bot_bot_set_o )
=> A )
& ( ( X6 != bot_bot_set_o )
=> ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X6 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_929_cSup__insert,axiom,
! [X6: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( X6 != bot_bo2891247006866115487la_a_b )
=> ( ( condit454741301069713265la_a_b @ X6 )
=> ( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ X6 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ X6 ) ) ) ) ) ).
% cSup_insert
thf(fact_930_cSup__insert,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ X6 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X6 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X6 ) ) ) ) ) ).
% cSup_insert
thf(fact_931_cInf__union__distrib,axiom,
! [A2: set_o,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A2 )
=> ( ( B != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B )
=> ( ( complete_Inf_Inf_o @ ( sup_sup_set_o @ A2 @ B ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_932_cSup__union__distrib,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ( condit454741301069713265la_a_b @ A2 )
=> ( ( B != bot_bo2891247006866115487la_a_b )
=> ( ( condit454741301069713265la_a_b @ B )
=> ( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_933_cSup__union__distrib,axiom,
! [A2: set_o,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A2 )
=> ( ( B != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B )
=> ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A2 @ B ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_934_cINF__inf__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ G2 @ A2 ) )
=> ( ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ G2 @ A2 ) ) )
= ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [A4: relational_fmla_a_b] : ( inf_inf_o @ ( F @ A4 ) @ ( G2 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_935_cINF__inf__distrib,axiom,
! [A2: set_o,F: $o > $o,G2: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ G2 @ A2 ) )
=> ( ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ A2 ) ) )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [A4: $o] : ( inf_inf_o @ ( F @ A4 ) @ ( G2 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_936_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ G2 @ A2 ) )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ G2 @ A2 ) ) )
= ( comple8442120529048846632la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [A4: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( F @ A4 ) @ ( G2 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_937_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_o,F: $o > set_Re381260168593705685la_a_b,G2: $o > set_Re381260168593705685la_a_b] :
( ( A2 != bot_bot_set_o )
=> ( ( condit454741301069713265la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) )
=> ( ( condit454741301069713265la_a_b @ ( image_4064131284474593576la_a_b @ G2 @ A2 ) )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ G2 @ A2 ) ) )
= ( comple8442120529048846632la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [A4: $o] : ( sup_su5130108678486352897la_a_b @ ( F @ A4 ) @ ( G2 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_938_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ G2 @ A2 ) )
=> ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ G2 @ A2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_1316678413157792882_a_b_o
@ ^ [A4: relational_fmla_a_b] : ( sup_sup_o @ ( F @ A4 ) @ ( G2 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_939_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_o,F: $o > $o,G2: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ G2 @ A2 ) )
=> ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ A2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [A4: $o] : ( sup_sup_o @ ( F @ A4 ) @ ( G2 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_940_cINF__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( inf_inf_o @ ( F @ A ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_941_cINF__insert,axiom,
! [A2: set_o,F: $o > $o,A: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( inf_inf_o @ ( F @ A ) @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_942_cSUP__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
=> ( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_943_cSUP__insert,axiom,
! [A2: set_o,F: $o > set_Re381260168593705685la_a_b,A: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit454741301069713265la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) )
=> ( ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_4064131284474593576la_a_b @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_944_cSUP__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_945_cSUP__insert,axiom,
! [A2: set_o,F: $o > $o,A: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_946_cINF__union,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,B: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( B != bot_bo4495933725496725865la_a_b )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ B ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ B ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_947_cINF__union,axiom,
! [A2: set_o,F: $o > $o,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( B != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ B ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ ( sup_sup_set_o @ A2 @ B ) ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ B ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_948_inj__on__image__Fpow,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ A2 )
=> ( inj_on6143097416644937501la_a_b @ ( image_6790371041703824709la_a_b @ F ) @ ( finite3079993003003454393la_a_b @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_949_comp__fun__commute__Pow__fold,axiom,
( finite2694875507280083197la_a_b
@ ^ [X3: relational_fmla_a_b,A3: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ A3 @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X3 ) @ A3 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_950_comp__fun__commute__Pow__fold,axiom,
( finite6772287081031898001_set_o
@ ^ [X3: $o,A3: set_set_o] : ( sup_sup_set_set_o @ A3 @ ( image_set_o_set_o @ ( insert_o @ X3 ) @ A3 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_951_Compl__disjoint2,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ A2 ) @ A2 )
= bot_bo4495933725496725865la_a_b ) ).
% Compl_disjoint2
thf(fact_952_Compl__disjoint2,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
= bot_bot_set_o ) ).
% Compl_disjoint2
thf(fact_953_Compl__disjoint,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ ( uminus235785408959094814la_a_b @ A2 ) )
= bot_bo4495933725496725865la_a_b ) ).
% Compl_disjoint
thf(fact_954_Compl__disjoint,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= bot_bot_set_o ) ).
% Compl_disjoint
thf(fact_955_Diff__Compl,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( uminus235785408959094814la_a_b @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ).
% Diff_Compl
thf(fact_956_Diff__Compl,axiom,
! [A2: set_o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( uminus_uminus_set_o @ B ) )
= ( inf_inf_set_o @ A2 @ B ) ) ).
% Diff_Compl
thf(fact_957_Compl__Diff__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_958_Compl__Diff__eq,axiom,
! [A2: set_o,B: set_o] :
( ( uminus_uminus_set_o @ ( minus_minus_set_o @ A2 @ B ) )
= ( sup_sup_set_o @ ( uminus_uminus_set_o @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_959_boolean__algebra_Ocompl__one,axiom,
( ( uminus235785408959094814la_a_b @ top_to8023889138469157253la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.compl_one
thf(fact_960_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.compl_one
thf(fact_961_boolean__algebra_Ocompl__zero,axiom,
( ( uminus235785408959094814la_a_b @ bot_bo4495933725496725865la_a_b )
= top_to8023889138469157253la_a_b ) ).
% boolean_algebra.compl_zero
thf(fact_962_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% boolean_algebra.compl_zero
thf(fact_963_sup__compl__top__left1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= top_to8023889138469157253la_a_b ) ).
% sup_compl_top_left1
thf(fact_964_sup__compl__top__left1,axiom,
! [X: set_o,Y: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ ( sup_sup_set_o @ X @ Y ) )
= top_top_set_o ) ).
% sup_compl_top_left1
thf(fact_965_sup__compl__top__left2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ Y ) )
= top_to8023889138469157253la_a_b ) ).
% sup_compl_top_left2
thf(fact_966_sup__compl__top__left2,axiom,
! [X: set_o,Y: set_o] :
( ( sup_sup_set_o @ X @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
= top_top_set_o ) ).
% sup_compl_top_left2
thf(fact_967_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ X )
= top_to8023889138469157253la_a_b ) ).
% boolean_algebra.disj_cancel_left
thf(fact_968_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ X )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_left
thf(fact_969_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( uminus235785408959094814la_a_b @ X ) )
= top_to8023889138469157253la_a_b ) ).
% boolean_algebra.disj_cancel_right
thf(fact_970_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ ( uminus_uminus_set_o @ X ) )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_right
thf(fact_971_inf__compl__bot__left1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ ( inf_in8483230781156617063la_a_b @ X @ Y ) )
= bot_bo4495933725496725865la_a_b ) ).
% inf_compl_bot_left1
thf(fact_972_inf__compl__bot__left1,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left1
thf(fact_973_inf__compl__bot__left2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ Y ) )
= bot_bo4495933725496725865la_a_b ) ).
% inf_compl_bot_left2
thf(fact_974_inf__compl__bot__left2,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left2
thf(fact_975_inf__compl__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ ( uminus235785408959094814la_a_b @ X ) ) )
= bot_bo4495933725496725865la_a_b ) ).
% inf_compl_bot_right
thf(fact_976_inf__compl__bot__right,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
= bot_bot_set_o ) ).
% inf_compl_bot_right
thf(fact_977_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ X )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.conj_cancel_left
thf(fact_978_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_979_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( uminus235785408959094814la_a_b @ X ) )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.conj_cancel_right
thf(fact_980_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_981_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ ( uminus235785408959094814la_a_b @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_982_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ ( uminus235785408959094814la_a_b @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_983_Collect__imp__eq,axiom,
! [P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ ( collec3419995626248312948la_a_b @ P ) ) @ ( collec3419995626248312948la_a_b @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_984_Compl__empty__eq,axiom,
( ( uminus235785408959094814la_a_b @ bot_bo4495933725496725865la_a_b )
= top_to8023889138469157253la_a_b ) ).
% Compl_empty_eq
thf(fact_985_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_986_Compl__UNIV__eq,axiom,
( ( uminus235785408959094814la_a_b @ top_to8023889138469157253la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Compl_UNIV_eq
thf(fact_987_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_988_diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] : ( inf_in8483230781156617063la_a_b @ X3 @ ( uminus235785408959094814la_a_b @ Y6 ) ) ) ) ).
% diff_eq
thf(fact_989_diff__eq,axiom,
( minus_minus_set_o
= ( ^ [X3: set_o,Y6: set_o] : ( inf_inf_set_o @ X3 @ ( uminus_uminus_set_o @ Y6 ) ) ) ) ).
% diff_eq
thf(fact_990_sup__cancel__left1,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ A ) @ ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ B2 ) )
= top_to8023889138469157253la_a_b ) ).
% sup_cancel_left1
thf(fact_991_sup__cancel__left1,axiom,
! [X: set_o,A: set_o,B2: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ X @ A ) @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ B2 ) )
= top_top_set_o ) ).
% sup_cancel_left1
thf(fact_992_sup__cancel__left2,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ A ) @ ( sup_su5130108678486352897la_a_b @ X @ B2 ) )
= top_to8023889138469157253la_a_b ) ).
% sup_cancel_left2
thf(fact_993_sup__cancel__left2,axiom,
! [X: set_o,A: set_o,B2: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ A ) @ ( sup_sup_set_o @ X @ B2 ) )
= top_top_set_o ) ).
% sup_cancel_left2
thf(fact_994_inf__cancel__left1,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ A ) @ ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ B2 ) )
= bot_bo4495933725496725865la_a_b ) ).
% inf_cancel_left1
thf(fact_995_inf__cancel__left1,axiom,
! [X: set_o,A: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B2 ) )
= bot_bot_set_o ) ).
% inf_cancel_left1
thf(fact_996_inf__cancel__left2,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ A ) @ ( inf_in8483230781156617063la_a_b @ X @ B2 ) )
= bot_bo4495933725496725865la_a_b ) ).
% inf_cancel_left2
thf(fact_997_inf__cancel__left2,axiom,
! [X: set_o,A: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A ) @ ( inf_inf_set_o @ X @ B2 ) )
= bot_bot_set_o ) ).
% inf_cancel_left2
thf(fact_998_empty__in__Fpow,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ bot_bo4495933725496725865la_a_b @ ( finite3079993003003454393la_a_b @ A2 ) ) ).
% empty_in_Fpow
thf(fact_999_empty__in__Fpow,axiom,
! [A2: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A2 ) ) ).
% empty_in_Fpow
thf(fact_1000_Compl__partition2,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ A2 ) @ A2 )
= top_to8023889138469157253la_a_b ) ).
% Compl_partition2
thf(fact_1001_Compl__partition2,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
= top_top_set_o ) ).
% Compl_partition2
thf(fact_1002_Compl__partition,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( uminus235785408959094814la_a_b @ A2 ) )
= top_to8023889138469157253la_a_b ) ).
% Compl_partition
thf(fact_1003_Compl__partition,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= top_top_set_o ) ).
% Compl_partition
thf(fact_1004_Compl__eq__Diff__UNIV,axiom,
( uminus235785408959094814la_a_b
= ( minus_4077726661957047470la_a_b @ top_to8023889138469157253la_a_b ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1005_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1006_Compl__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ ( uminus235785408959094814la_a_b @ A2 ) @ ( uminus235785408959094814la_a_b @ B ) ) ) ).
% Compl_Un
thf(fact_1007_Compl__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ A2 ) @ ( uminus235785408959094814la_a_b @ B ) ) ) ).
% Compl_Int
thf(fact_1008_Diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] : ( inf_in8483230781156617063la_a_b @ A3 @ ( uminus235785408959094814la_a_b @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1009_Diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B4: set_o] : ( inf_inf_set_o @ A3 @ ( uminus_uminus_set_o @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1010_comp__fun__commute__filter__fold,axiom,
! [P: relational_fmla_a_b > $o] :
( finite7312080361340319431la_a_b
@ ^ [X3: relational_fmla_a_b,A6: set_Re381260168593705685la_a_b] : ( if_set2835548578466827919la_a_b @ ( P @ X3 ) @ ( insert7010464514620295119la_a_b @ X3 @ A6 ) @ A6 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_1011_comp__fun__commute__filter__fold,axiom,
! [P: $o > $o] :
( finite7927022897302369457_set_o
@ ^ [X3: $o,A6: set_o] : ( if_set_o @ ( P @ X3 ) @ ( insert_o @ X3 @ A6 ) @ A6 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_1012_uminus__Inf,axiom,
! [A2: set_o] :
( ( uminus_uminus_o @ ( complete_Inf_Inf_o @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ uminus_uminus_o @ A2 ) ) ) ).
% uminus_Inf
thf(fact_1013_uminus__Sup,axiom,
! [A2: set_o] :
( ( uminus_uminus_o @ ( complete_Sup_Sup_o @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ uminus_uminus_o @ A2 ) ) ) ).
% uminus_Sup
thf(fact_1014_Compl__insert,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) )
= ( minus_4077726661957047470la_a_b @ ( uminus235785408959094814la_a_b @ A2 ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ).
% Compl_insert
thf(fact_1015_Compl__insert,axiom,
! [X: $o,A2: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X @ A2 ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_1016_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
=> ( ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= top_to8023889138469157253la_a_b )
=> ( ( uminus235785408959094814la_a_b @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1017_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ X @ Y )
= top_top_set_o )
=> ( ( uminus_uminus_set_o @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1018_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea8997957016489498079la_a_b @ inf_in8483230781156617063la_a_b @ sup_su5130108678486352897la_a_b @ uminus235785408959094814la_a_b @ bot_bo4495933725496725865la_a_b @ top_to8023889138469157253la_a_b ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1019_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea379910186789422830_set_o @ inf_inf_set_o @ sup_sup_set_o @ uminus_uminus_set_o @ bot_bot_set_o @ top_top_set_o ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1020_ComplI,axiom,
! [C2: $o,A2: set_o] :
( ~ ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ ( uminus_uminus_set_o @ A2 ) ) ) ).
% ComplI
thf(fact_1021_ComplI,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ ( uminus235785408959094814la_a_b @ A2 ) ) ) ).
% ComplI
thf(fact_1022_Compl__iff,axiom,
! [C2: $o,A2: set_o] :
( ( member_o @ C2 @ ( uminus_uminus_set_o @ A2 ) )
= ( ~ ( member_o @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_1023_Compl__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( uminus235785408959094814la_a_b @ A2 ) )
= ( ~ ( member4680049679412964150la_a_b @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_1024_ComplD,axiom,
! [C2: $o,A2: set_o] :
( ( member_o @ C2 @ ( uminus_uminus_set_o @ A2 ) )
=> ~ ( member_o @ C2 @ A2 ) ) ).
% ComplD
thf(fact_1025_ComplD,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( uminus235785408959094814la_a_b @ A2 ) )
=> ~ ( member4680049679412964150la_a_b @ C2 @ A2 ) ) ).
% ComplD
thf(fact_1026_uminus__set__def,axiom,
( uminus_uminus_set_o
= ( ^ [A3: set_o] :
( collect_o
@ ( uminus_uminus_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1027_uminus__set__def,axiom,
( uminus235785408959094814la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( uminus4297745234618001383_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1028_Compl__eq,axiom,
( uminus_uminus_set_o
= ( ^ [A3: set_o] :
( collect_o
@ ^ [X3: $o] :
~ ( member_o @ X3 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_1029_Compl__eq,axiom,
( uminus235785408959094814la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_1030_cInf__le__cSup,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A2 )
=> ( ( condit5413489452508810728elow_o @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1031_less__eq__cInf__inter,axiom,
! [A2: set_o,B: set_o] :
( ( condit5413489452508810728elow_o @ A2 )
=> ( ( condit5413489452508810728elow_o @ B )
=> ( ( ( inf_inf_set_o @ A2 @ B )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( inf_inf_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A2 @ B ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1032_cSup__inter__less__eq,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( condit454741301069713265la_a_b @ A2 )
=> ( ( condit454741301069713265la_a_b @ B )
=> ( ( ( inf_in1353227996657692061la_a_b @ A2 @ B )
!= bot_bo2891247006866115487la_a_b )
=> ( ord_le4112832032246704949la_a_b @ ( comple8442120529048846632la_a_b @ ( inf_in1353227996657692061la_a_b @ A2 @ B ) ) @ ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1033_cSup__inter__less__eq,axiom,
! [A2: set_o,B: set_o] :
( ( condit5488710616941104124bove_o @ A2 )
=> ( ( condit5488710616941104124bove_o @ B )
=> ( ( ( inf_inf_set_o @ A2 @ B )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A2 @ B ) ) @ ( sup_sup_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1034_empty__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ).
% empty_subsetI
thf(fact_1035_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_1036_subset__empty,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% subset_empty
thf(fact_1037_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_1038_insert__subset,axiom,
! [X: $o,A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B )
= ( ( member_o @ X @ B )
& ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_1039_insert__subset,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( ( member4680049679412964150la_a_b @ X @ B )
& ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_1040_Un__subset__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( ( ord_le4112832032246704949la_a_b @ A2 @ C )
& ( ord_le4112832032246704949la_a_b @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_1041_sup_Obounded__iff,axiom,
! [B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) @ A )
= ( ( ord_le4112832032246704949la_a_b @ B2 @ A )
& ( ord_le4112832032246704949la_a_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1042_le__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z2 )
= ( ( ord_le4112832032246704949la_a_b @ X @ Z2 )
& ( ord_le4112832032246704949la_a_b @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1043_singleton__insert__inj__eq,axiom,
! [B2: relational_fmla_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1044_singleton__insert__inj__eq,axiom,
! [B2: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B2 @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1045_singleton__insert__inj__eq_H,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ A2 )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( ( A = B2 )
& ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1046_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B2: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ bot_bot_set_o ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1047_Diff__eq__empty__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1048_Diff__eq__empty__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( minus_minus_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1049_subset__Compl__singleton,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( uminus235785408959094814la_a_b @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) )
= ( ~ ( member4680049679412964150la_a_b @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1050_subset__Compl__singleton,axiom,
! [A2: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1051_cSUP__subset__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > $o,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1052_cSUP__subset__mono,axiom,
! [A2: set_o,G2: $o > $o,B: set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ G2 @ B ) )
=> ( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1053_cINF__superset__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > $o,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1054_cINF__superset__mono,axiom,
! [A2: set_o,G2: $o > $o,B: set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ G2 @ B ) )
=> ( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1055_cSup__subset__mono,axiom,
! [A2: set_o,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B )
=> ( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1056_cInf__superset__mono,axiom,
! [A2: set_o,B: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B )
=> ( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ B ) @ ( complete_Inf_Inf_o @ A2 ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1057_all__subset__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B4 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B4 @ A2 )
=> ( P @ ( image_6790371041703824709la_a_b @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1058_Inf__superset__mono,axiom,
! [B: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1059_Sup__subset__mono,axiom,
! [A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1060_bot_Oextremum,axiom,
! [A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A ) ).
% bot.extremum
thf(fact_1061_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_1062_bot_Oextremum__unique,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_unique
thf(fact_1063_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_1064_bot_Oextremum__uniqueI,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_1065_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1066_image__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_1067_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1068_image__subsetI,axiom,
! [A2: set_o,F: $o > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X4 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6295478110982790130la_a_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1069_image__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,B: set_o] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1070_image__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X4 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1071_subset__imageE,axiom,
! [B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ~ ! [C4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C4 @ A2 )
=> ( B
!= ( image_6790371041703824709la_a_b @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1072_image__subset__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ B )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1073_subset__image__iff,axiom,
! [B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( ? [AA: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ AA @ A2 )
& ( B
= ( image_6790371041703824709la_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1074_SUP__subset__mono,axiom,
! [A2: set_o,B: set_o,F: $o > $o,G2: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1075_SUP__subset__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1076_insert__mono,axiom,
! [C: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ D2 )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ A @ C ) @ ( insert7010464514620295119la_a_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1077_insert__mono,axiom,
! [C: set_o,D2: set_o,A: $o] :
( ( ord_less_eq_set_o @ C @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C ) @ ( insert_o @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1078_subset__insert,axiom,
! [X: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B ) )
= ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1079_subset__insert,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1080_subset__insertI,axiom,
! [B: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ B ) ) ).
% subset_insertI
thf(fact_1081_subset__insertI,axiom,
! [B: set_o,A: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A @ B ) ) ).
% subset_insertI
thf(fact_1082_subset__insertI2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1083_subset__insertI2,axiom,
! [A2: set_o,B: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1084_Sup__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ X ) )
=> ( ! [Y5: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A2 )
=> ( ord_less_eq_o @ Z4 @ Y5 ) )
=> ( ord_less_eq_o @ X @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1085_Sup__mono,axiom,
! [A2: set_o,B: set_o] :
( ! [A7: $o] :
( ( member_o @ A7 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ A7 @ X5 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_1086_Sup__least,axiom,
! [A2: set_o,Z2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1087_Sup__upper,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1088_Sup__le__iff,axiom,
! [A2: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B2 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1089_Sup__upper2,axiom,
! [U2: $o,A2: set_o,V2: $o] :
( ( member_o @ U2 @ A2 )
=> ( ( ord_less_eq_o @ V2 @ U2 )
=> ( ord_less_eq_o @ V2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1090_INF__superset__mono,axiom,
! [B: set_o,A2: set_o,F: $o > $o,G2: $o > $o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1091_INF__superset__mono,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1092_Inf__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ X @ I4 ) )
=> ( ! [Y5: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ I5 ) )
=> ( ord_less_eq_o @ Y5 @ X ) )
=> ( ( complete_Inf_Inf_o @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1093_Inf__mono,axiom,
! [B: set_o,A2: set_o] :
( ! [B6: $o] :
( ( member_o @ B6 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ X5 @ B6 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_mono
thf(fact_1094_Inf__lower,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1095_Inf__lower2,axiom,
! [U2: $o,A2: set_o,V2: $o] :
( ( member_o @ U2 @ A2 )
=> ( ( ord_less_eq_o @ U2 @ V2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1096_le__Inf__iff,axiom,
! [B2: $o,A2: set_o] :
( ( ord_less_eq_o @ B2 @ ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1097_Inf__greatest,axiom,
! [A2: set_o,Z2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ Z2 @ X4 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1098_double__diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ( minus_4077726661957047470la_a_b @ B @ ( minus_4077726661957047470la_a_b @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1099_double__diff,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ( minus_minus_set_o @ B @ ( minus_minus_set_o @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1100_Diff__subset,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1101_Diff__subset,axiom,
! [A2: set_o,B: set_o] : ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1102_Diff__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ D2 @ B )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1103_Diff__mono,axiom,
! [A2: set_o,C: set_o,D2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ C )
=> ( ( ord_less_eq_set_o @ D2 @ B )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( minus_minus_set_o @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1104_subset__Un__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1105_subset__UnE,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ~ ! [A5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A5 @ A2 )
=> ! [B7: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B7 @ B )
=> ( C
!= ( sup_su5130108678486352897la_a_b @ A5 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1106_Un__absorb2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_1107_Un__absorb1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1108_Un__upper2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_upper2
thf(fact_1109_Un__upper1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_upper1
thf(fact_1110_Un__least,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_1111_Un__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ D2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1112_sup_OcoboundedI2,axiom,
! [C2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ B2 )
=> ( ord_le4112832032246704949la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1113_sup_OcoboundedI1,axiom,
! [C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ A )
=> ( ord_le4112832032246704949la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1114_sup_Oabsorb__iff2,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1115_sup_Oabsorb__iff1,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [B5: set_Re381260168593705685la_a_b,A4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1116_sup_Ocobounded2,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1117_sup_Ocobounded1,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1118_sup_Oorder__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [B5: set_Re381260168593705685la_a_b,A4: set_Re381260168593705685la_a_b] :
( A4
= ( sup_su5130108678486352897la_a_b @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1119_sup_OboundedI,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ C2 @ A )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1120_sup_OboundedE,axiom,
! [B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) @ A )
=> ~ ( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ~ ( ord_le4112832032246704949la_a_b @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_1121_sup__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ Y )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1122_sup__absorb1,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y @ X )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1123_sup_Oabsorb2,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1124_sup_Oabsorb1,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1125_sup__unique,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ! [X4: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X4 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y5 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b,Z3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y5 @ X4 )
=> ( ( ord_le4112832032246704949la_a_b @ Z3 @ X4 )
=> ( ord_le4112832032246704949la_a_b @ ( F @ Y5 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1126_sup_OorderI,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( A
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
=> ( ord_le4112832032246704949la_a_b @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1127_sup_OorderE,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( A
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1128_le__iff__sup,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X3 @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_1129_sup__least,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y @ X )
=> ( ( ord_le4112832032246704949la_a_b @ Z2 @ X )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1130_sup__mono,axiom,
! [A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ C2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ D )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ ( sup_su5130108678486352897la_a_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_1131_sup_Omono,axiom,
! [C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ D @ B2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ C2 @ D ) @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1132_le__supI2,axiom,
! [X: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ B2 )
=> ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1133_le__supI1,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1134_sup__ge2,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_1135_sup__ge1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_1136_le__supI,axiom,
! [A: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ X )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ X )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1137_le__supE,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ X )
=> ~ ( ( ord_le4112832032246704949la_a_b @ A @ X )
=> ~ ( ord_le4112832032246704949la_a_b @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1138_inf__sup__ord_I3_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1139_inf__sup__ord_I4_J,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1140_subset__UNIV,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_1141_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_1142_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_1143_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_1144_image__Fpow__mono,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ B )
=> ( ord_le1577343677690852715la_a_b @ ( image_7051608999182166449la_a_b @ ( image_6790371041703824709la_a_b @ F ) @ ( finite3079993003003454393la_a_b @ A2 ) ) @ ( finite3079993003003454393la_a_b @ B ) ) ) ).
% image_Fpow_mono
thf(fact_1145_Int__Collect__mono,axiom,
! [A2: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1146_Int__Collect__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le4112832032246704949la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) ) @ ( inf_in8483230781156617063la_a_b @ B @ ( collec3419995626248312948la_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1147_diff__shunt__var,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1148_diff__shunt__var,axiom,
! [X: set_o,Y: set_o] :
( ( ( minus_minus_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1149_range__subsetD,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b,I: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b ) @ B )
=> ( member4680049679412964150la_a_b @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1150_range__subsetD,axiom,
! [F: $o > $o,B: set_o,I: $o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ top_top_set_o ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1151_range__subsetD,axiom,
! [F: $o > relational_fmla_a_b,B: set_Re381260168593705685la_a_b,I: $o] :
( ( ord_le4112832032246704949la_a_b @ ( image_6295478110982790130la_a_b @ F @ top_top_set_o ) @ B )
=> ( member4680049679412964150la_a_b @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1152_SUP__eq,axiom,
! [A2: set_o,B: set_o,F: $o > $o,G2: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1153_SUP__eq,axiom,
! [A2: set_o,B: set_Re381260168593705685la_a_b,F: $o > $o,G2: relational_fmla_a_b > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1154_SUP__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_o,F: relational_fmla_a_b > $o,G2: $o > $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A2 )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1155_SUP__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ J2 @ B )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A2 )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1156_cSup__least,axiom,
! [X6: set_o,Z2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_1157_cSup__eq__non__empty,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ X4 @ A ) )
=> ( ! [Y5: $o] :
( ! [X5: $o] :
( ( member_o @ X5 @ X6 )
=> ( ord_less_eq_o @ X5 @ Y5 ) )
=> ( ord_less_eq_o @ A @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1158_less__eq__Sup,axiom,
! [A2: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A2 )
=> ( ord_less_eq_o @ U2 @ V ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1159_subset__singletonD,axiom,
! [A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
=> ( ( A2 = bot_bo4495933725496725865la_a_b )
| ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_1160_subset__singletonD,axiom,
! [A2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1161_subset__singleton__iff,axiom,
! [X6: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ X6 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
= ( ( X6 = bot_bo4495933725496725865la_a_b )
| ( X6
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_1162_subset__singleton__iff,axiom,
! [X6: set_o,A: $o] :
( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X6 = bot_bot_set_o )
| ( X6
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1163_INF__eq,axiom,
! [A2: set_o,B: set_o,G2: $o > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( G2 @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ ( F @ X5 ) @ ( G2 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1164_INF__eq,axiom,
! [A2: set_o,B: set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ B )
& ( ord_less_eq_o @ ( G2 @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ ( F @ X5 ) @ ( G2 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1165_INF__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_o,G2: $o > $o,F: relational_fmla_a_b > $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( G2 @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A2 )
& ( ord_less_eq_o @ ( F @ X5 ) @ ( G2 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1166_INF__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > $o,F: relational_fmla_a_b > $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ B )
& ( ord_less_eq_o @ ( G2 @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ J2 @ B )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A2 )
& ( ord_less_eq_o @ ( F @ X5 ) @ ( G2 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1167_distrib__sup__le,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z2 ) ) @ ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1168_distrib__inf__le,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z2 ) ) @ ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1169_cInf__greatest,axiom,
! [X6: set_o,Z2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ Z2 @ X4 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1170_cInf__eq__non__empty,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ A @ X4 ) )
=> ( ! [Y5: $o] :
( ! [X5: $o] :
( ( member_o @ X5 @ X6 )
=> ( ord_less_eq_o @ Y5 @ X5 ) )
=> ( ord_less_eq_o @ Y5 @ A ) )
=> ( ( complete_Inf_Inf_o @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1171_Inf__less__eq,axiom,
! [A2: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A2 )
=> ( ord_less_eq_o @ V @ U2 ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_1172_image__Int__subset,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) @ ( inf_in8483230781156617063la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1173_image__diff__subset,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1174_image__diff__subset,axiom,
! [F: $o > relational_fmla_a_b,A2: set_o,B: set_o] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ ( image_6295478110982790130la_a_b @ F @ A2 ) @ ( image_6295478110982790130la_a_b @ F @ B ) ) @ ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1175_image__diff__subset,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_less_eq_set_o @ ( minus_minus_set_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) @ ( image_1316678413157792882_a_b_o @ F @ B ) ) @ ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1176_image__diff__subset,axiom,
! [F: $o > $o,A2: set_o,B: set_o] : ( ord_less_eq_set_o @ ( minus_minus_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ F @ B ) ) @ ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1177_subset__image__inj,axiom,
! [S: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,T2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ S @ ( image_6790371041703824709la_a_b @ F @ T2 ) )
= ( ? [U3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ U3 @ T2 )
& ( inj_on4502624401354443441la_a_b @ F @ U3 )
& ( S
= ( image_6790371041703824709la_a_b @ F @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1178_inj__on__image__mem__iff,axiom,
! [F: $o > $o,B: set_o,A: $o,A2: set_o] :
( ( inj_on_o_o @ F @ B )
=> ( ( member_o @ A @ B )
=> ( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ ( F @ A ) @ ( image_o_o @ F @ A2 ) )
= ( member_o @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1179_inj__on__image__mem__iff,axiom,
! [F: $o > relational_fmla_a_b,B: set_o,A: $o,A2: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ B )
=> ( ( member_o @ A @ B )
=> ( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member4680049679412964150la_a_b @ ( F @ A ) @ ( image_6295478110982790130la_a_b @ F @ A2 ) )
= ( member_o @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1180_inj__on__image__mem__iff,axiom,
! [F: relational_fmla_a_b > $o,B: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ B )
=> ( ( member4680049679412964150la_a_b @ A @ B )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( member_o @ ( F @ A ) @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1181_inj__on__image__mem__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ B )
=> ( ( member4680049679412964150la_a_b @ A @ B )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( member4680049679412964150la_a_b @ ( F @ A ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1182_inj__on__image__eq__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ C )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ( ( image_6790371041703824709la_a_b @ F @ A2 )
= ( image_6790371041703824709la_a_b @ F @ B ) )
= ( A2 = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1183_subset__Diff__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ ( insert7010464514620295119la_a_b @ X @ C ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ C ) )
& ~ ( member4680049679412964150la_a_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1184_subset__Diff__insert,axiom,
! [A2: set_o,B: set_o,X: $o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B @ ( insert_o @ X @ C ) ) )
= ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B @ C ) )
& ~ ( member_o @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1185_Un__Int__assoc__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) )
= ( ord_le4112832032246704949la_a_b @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1186_Diff__partition,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1187_Diff__partition,axiom,
! [A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( sup_sup_set_o @ A2 @ ( minus_minus_set_o @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1188_Diff__subset__conv,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
= ( ord_le4112832032246704949la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1189_Diff__subset__conv,axiom,
! [A2: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ C )
= ( ord_less_eq_set_o @ A2 @ ( sup_sup_set_o @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1190_subset__Compl__self__eq,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( uminus235785408959094814la_a_b @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% subset_Compl_self_eq
thf(fact_1191_subset__Compl__self__eq,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_1192_Inter__Un__subset,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] : ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ ( comple2571870594864334990la_a_b @ A2 ) @ ( comple2571870594864334990la_a_b @ B ) ) @ ( comple2571870594864334990la_a_b @ ( inf_in1353227996657692061la_a_b @ A2 @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1193_SUP__upper2,axiom,
! [I: $o,A2: set_o,U2: $o,F: $o > $o] :
( ( member_o @ I @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1194_SUP__upper2,axiom,
! [I: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,U2: $o,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ I @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ I ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1195_SUP__upper,axiom,
! [I: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I @ A2 )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1196_SUP__upper,axiom,
! [I: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ I @ A2 )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1197_SUP__least,axiom,
! [A2: set_o,F: $o > $o,U2: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1198_SUP__least,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,U2: $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1199_SUP__eqI,axiom,
! [A2: set_o,F: $o > $o,X: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X ) )
=> ( ! [Y5: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y5 ) )
=> ( ord_less_eq_o @ X @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1200_SUP__eqI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,X: $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X ) )
=> ( ! [Y5: $o] :
( ! [I5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y5 ) )
=> ( ord_less_eq_o @ X @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_1201_INF__greatest,axiom,
! [A2: set_o,U2: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ U2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1202_INF__greatest,axiom,
! [A2: set_Re381260168593705685la_a_b,U2: $o,F: relational_fmla_a_b > $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ( ord_less_eq_o @ U2 @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1203_INF__lower2,axiom,
! [I: $o,A2: set_o,F: $o > $o,U2: $o] :
( ( member_o @ I @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1204_INF__lower2,axiom,
! [I: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,U2: $o] :
( ( member4680049679412964150la_a_b @ I @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1205_INF__lower,axiom,
! [I: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1206_INF__lower,axiom,
! [I: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ I @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1207_INF__eqI,axiom,
! [A2: set_o,X: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ X @ ( F @ I4 ) ) )
=> ( ! [Y5: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y5 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1208_INF__eqI,axiom,
! [A2: set_Re381260168593705685la_a_b,X: $o,F: relational_fmla_a_b > $o] :
( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ A2 )
=> ( ord_less_eq_o @ X @ ( F @ I4 ) ) )
=> ( ! [Y5: $o] :
( ! [I5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y5 @ X ) )
=> ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1209_sup__shunt,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= top_to8023889138469157253la_a_b )
= ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_1210_sup__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( ( sup_sup_set_o @ X @ Y )
= top_top_set_o )
= ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_1211_inf__shunt,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ X @ ( uminus235785408959094814la_a_b @ Y ) ) ) ).
% inf_shunt
thf(fact_1212_inf__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).
% inf_shunt
thf(fact_1213_cSUP__least,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,M: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1214_cSUP__least,axiom,
! [A2: set_o,F: $o > $o,M: $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1215_SUP__eq__iff,axiom,
! [I3: set_Re381260168593705685la_a_b,C2: $o,F: relational_fmla_a_b > $o] :
( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ I3 )
=> ( ord_less_eq_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ I3 ) )
= C2 )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ I3 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1216_SUP__eq__iff,axiom,
! [I3: set_o,C2: $o,F: $o > $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I3 ) )
= C2 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I3 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1217_sup__neg__inf,axiom,
! [P2: set_Re381260168593705685la_a_b,Q3: set_Re381260168593705685la_a_b,R3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ P2 @ ( sup_su5130108678486352897la_a_b @ Q3 @ R3 ) )
= ( ord_le4112832032246704949la_a_b @ ( inf_in8483230781156617063la_a_b @ P2 @ ( uminus235785408959094814la_a_b @ Q3 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_1218_shunt2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ ( uminus235785408959094814la_a_b @ Y ) ) @ Z2 )
= ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z2 ) ) ) ).
% shunt2
thf(fact_1219_shunt1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ Z2 )
= ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ Y ) @ Z2 ) ) ) ).
% shunt1
thf(fact_1220_cINF__greatest,axiom,
! [A2: set_Re381260168593705685la_a_b,M2: $o,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ord_less_eq_o @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_o @ M2 @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1221_cINF__greatest,axiom,
! [A2: set_o,M2: $o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_o @ M2 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1222_INF__eq__iff,axiom,
! [I3: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,C2: $o] :
( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ! [I4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ I4 @ I3 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ I3 ) )
= C2 )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ I3 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1223_INF__eq__iff,axiom,
! [I3: set_o,F: $o > $o,C2: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I3 ) )
= C2 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I3 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1224_Inf__le__Sup,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1225_inj__image__subset__iff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1226_inj__on__iff__surj,axiom,
! [A2: set_Re381260168593705685la_a_b,A8: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( ? [F2: relational_fmla_a_b > relational_fmla_a_b] :
( ( inj_on4502624401354443441la_a_b @ F2 @ A2 )
& ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F2 @ A2 ) @ A8 ) ) )
= ( ? [G3: relational_fmla_a_b > relational_fmla_a_b] :
( ( image_6790371041703824709la_a_b @ G3 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1227_Sup__inter__less__eq,axiom,
! [A2: set_o,B: set_o] : ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A2 @ B ) ) @ ( inf_inf_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1228_subset__insert__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1229_subset__insert__iff,axiom,
! [A2: set_o,X: $o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B ) )
= ( ( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1230_Diff__single__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1231_Diff__single__insert,axiom,
! [A2: set_o,X: $o,B: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1232_less__eq__Inf__inter,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] : ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ ( comple2571870594864334990la_a_b @ A2 ) @ ( comple2571870594864334990la_a_b @ B ) ) @ ( comple2571870594864334990la_a_b @ ( inf_in1353227996657692061la_a_b @ A2 @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1233_less__eq__Inf__inter,axiom,
! [A2: set_o,B: set_o] : ( ord_less_eq_o @ ( sup_sup_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A2 @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1234_inj__on__image__Int,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ C )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ( image_6790371041703824709la_a_b @ F @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1235_inj__on__image__set__diff,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ C )
=> ( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1236_inj__on__image__set__diff,axiom,
! [F: relational_fmla_a_b > $o,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on8311361820794748934_a_b_o @ F @ C )
=> ( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ( image_1316678413157792882_a_b_o @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_minus_set_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) @ ( image_1316678413157792882_a_b_o @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1237_inj__on__image__set__diff,axiom,
! [F: $o > relational_fmla_a_b,C: set_o,A2: set_o,B: set_o] :
( ( inj_on4066789481764970374la_a_b @ F @ C )
=> ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ C )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ( image_6295478110982790130la_a_b @ F @ ( minus_minus_set_o @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( image_6295478110982790130la_a_b @ F @ A2 ) @ ( image_6295478110982790130la_a_b @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1238_inj__on__image__set__diff,axiom,
! [F: $o > $o,C: set_o,A2: set_o,B: set_o] :
( ( inj_on_o_o @ F @ C )
=> ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ C )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1239_surj__Compl__image__subset,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ top_to8023889138469157253la_a_b )
= top_to8023889138469157253la_a_b )
=> ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) @ ( image_6790371041703824709la_a_b @ F @ ( uminus235785408959094814la_a_b @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1240_surj__Compl__image__subset,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ ( image_o_o @ F @ A2 ) ) @ ( image_o_o @ F @ ( uminus_uminus_set_o @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1241_cSUP__upper,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ord_less_eq_o @ ( F @ X ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1242_cSUP__upper,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ord_less_eq_o @ ( F @ X ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1243_cSUP__upper2,axiom,
! [F: $o > $o,A2: set_o,X: $o,U2: $o] :
( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( member_o @ X @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ X ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1244_cSUP__upper2,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,U2: $o] :
( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( ord_less_eq_o @ U2 @ ( F @ X ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1245_cSup__le__iff,axiom,
! [S: set_o,A: $o] :
( ( S != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ S )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S ) @ A )
= ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ord_less_eq_o @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1246_cSup__mono,axiom,
! [B: set_o,A2: set_o] :
( ( B != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A2 )
=> ( ! [B6: $o] :
( ( member_o @ B6 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ B6 @ X5 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B ) @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_1247_cINF__lower,axiom,
! [F: $o > $o,A2: set_o,X: $o] :
( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( F @ X ) ) ) ) ).
% cINF_lower
thf(fact_1248_cINF__lower,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( F @ X ) ) ) ) ).
% cINF_lower
thf(fact_1249_cINF__lower2,axiom,
! [F: $o > $o,A2: set_o,X: $o,U2: $o] :
( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( member_o @ X @ A2 )
=> ( ( ord_less_eq_o @ ( F @ X ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_1250_cINF__lower2,axiom,
! [F: relational_fmla_a_b > $o,A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,U2: $o] :
( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( ord_less_eq_o @ ( F @ X ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_1251_le__cInf__iff,axiom,
! [S: set_o,A: $o] :
( ( S != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ S )
=> ( ( ord_less_eq_o @ A @ ( complete_Inf_Inf_o @ S ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ord_less_eq_o @ A @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1252_cInf__mono,axiom,
! [B: set_o,A2: set_o] :
( ( B != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A2 )
=> ( ! [B6: $o] :
( ( member_o @ B6 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ X5 @ B6 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ) ) ).
% cInf_mono
thf(fact_1253_disjoint__eq__subset__Compl,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ A2 @ ( uminus235785408959094814la_a_b @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1254_disjoint__eq__subset__Compl,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1255_INF__le__SUP,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1256_INF__le__SUP,axiom,
! [A2: set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1257_cSUP__le__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,U2: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5488710616941104124bove_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) @ U2 )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1258_cSUP__le__iff,axiom,
! [A2: set_o,F: $o > $o,U2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ U2 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1259_le__cINF__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > $o,U2: $o] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) )
=> ( ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ F @ A2 ) ) )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ord_less_eq_o @ U2 @ ( F @ X3 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1260_le__cINF__iff,axiom,
! [A2: set_o,F: $o > $o,U2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F @ A2 ) )
=> ( ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ U2 @ ( F @ X3 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1261_inj__image__Compl__subset,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ top_to8023889138469157253la_a_b )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ ( uminus235785408959094814la_a_b @ A2 ) ) @ ( uminus235785408959094814la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1262_image__INT,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,C: set_Re381260168593705685la_a_b,A2: set_o,B: $o > set_Re381260168593705685la_a_b,J3: $o] :
( ( inj_on4502624401354443441la_a_b @ F @ C )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( B @ X4 ) @ C ) )
=> ( ( member_o @ J3 @ A2 )
=> ( ( image_6790371041703824709la_a_b @ F @ ( comple2571870594864334990la_a_b @ ( image_4064131284474593576la_a_b @ B @ A2 ) ) )
= ( comple2571870594864334990la_a_b
@ ( image_4064131284474593576la_a_b
@ ^ [X3: $o] : ( image_6790371041703824709la_a_b @ F @ ( B @ X3 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1263_image__INT,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b > set_Re381260168593705685la_a_b,J3: relational_fmla_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ C )
=> ( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( B @ X4 ) @ C ) )
=> ( ( member4680049679412964150la_a_b @ J3 @ A2 )
=> ( ( image_6790371041703824709la_a_b @ F @ ( comple2571870594864334990la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
= ( comple2571870594864334990la_a_b
@ ( image_8209480069293074043la_a_b
@ ^ [X3: relational_fmla_a_b] : ( image_6790371041703824709la_a_b @ F @ ( B @ X3 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1264_mono__cInf,axiom,
! [F: $o > $o,A2: set_o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( condit5413489452508810728elow_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( F @ ( complete_Inf_Inf_o @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_1265_mono__cINF,axiom,
! [F: $o > $o,A2: relational_fmla_a_b > $o,I3: set_Re381260168593705685la_a_b] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( condit5413489452508810728elow_o @ ( image_1316678413157792882_a_b_o @ A2 @ I3 ) )
=> ( ( I3 != bot_bo4495933725496725865la_a_b )
=> ( ord_less_eq_o @ ( F @ ( complete_Inf_Inf_o @ ( image_1316678413157792882_a_b_o @ A2 @ I3 ) ) )
@ ( complete_Inf_Inf_o
@ ( image_1316678413157792882_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( F @ ( A2 @ X3 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_1266_mono__cINF,axiom,
! [F: $o > $o,A2: $o > $o,I3: set_o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ A2 @ I3 ) )
=> ( ( I3 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( F @ ( complete_Inf_Inf_o @ ( image_o_o @ A2 @ I3 ) ) )
@ ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : ( F @ ( A2 @ X3 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_1267_mono__cSup,axiom,
! [F: $o > $o,A2: set_o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( condit5488710616941104124bove_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( F @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_1268_subsetI,axiom,
! [A2: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ X4 @ B ) )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ).
% subsetI
thf(fact_1269_subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( member4680049679412964150la_a_b @ X4 @ B ) )
=> ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ).
% subsetI
thf(fact_1270_in__mono,axiom,
! [A2: set_o,B: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ X @ B ) ) ) ).
% in_mono
thf(fact_1271_in__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member4680049679412964150la_a_b @ X @ B ) ) ) ).
% in_mono
thf(fact_1272_subsetD,axiom,
! [A2: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_1273_subsetD,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_1274_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B4: set_o] :
! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1275_subset__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
=> ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1276_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B4: set_o] :
! [T3: $o] :
( ( member_o @ T3 @ A3 )
=> ( member_o @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1277_subset__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
! [T3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ T3 @ A3 )
=> ( member4680049679412964150la_a_b @ T3 @ B4 ) ) ) ) ).
% subset_iff
% Helper facts (7)
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X: set_o,Y: set_o] :
( ( if_set_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X: set_o,Y: set_o] :
( ( if_set_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
x != y ).
thf(conj_1,hypothesis,
relational_cov_a_b @ x @ q @ g ).
thf(conj_2,hypothesis,
relational_gen_a_b @ y @ q @ gy ).
thf(conj_3,conjecture,
( relational_cov_a_b @ x @ ( relati591517084277583526ts_a_b @ y @ q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ y ) @ ( minus_4077726661957047470la_a_b @ g @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ x @ ( relational_Var_a @ y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ y @ x )
@ gy ) ) ) ).
%------------------------------------------------------------------------------