TPTP Problem File: SLH0481^1.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 : Combinable_Wands/0001_Mask/prob_00220_006359__7129188_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1621 ( 745 unt; 335 typ; 0 def)
% Number of atoms : 3208 (1848 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 9832 ( 424 ~; 29 |; 240 &;8078 @)
% ( 0 <=>;1061 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 1833 (1833 >; 0 *; 0 +; 0 <<)
% Number of symbols : 286 ( 283 usr; 24 con; 0-5 aty)
% Number of variables : 3642 ( 542 ^;2987 !; 113 ?;3642 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:05:21.601
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
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set_set_set_o: $tType ).
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product_prod_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__PosRat__Oprat,type,
prat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (283)
thf(sy_c_BNF__Greatest__Fixpoint_OfromCard_001tf__b_001t__PosRat__Oprat,type,
bNF_Gr6838450293404232211b_prat: set_b > set_Pr3298805672376727137t_prat > prat > b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_062_Itf__b_Mt__PosRat__Oprat_J_001_062_Itf__b_Mt__PosRat__Oprat_J_001tf__b,type,
bNF_Gr1003551443158410149prat_b: set_b_prat > ( ( b > prat ) > b > prat ) > ( ( b > prat ) > b ) > set_Pr8957522257986161222prat_b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_062_Itf__b_Mt__PosRat__Oprat_J_001_Eo_001_Eo,type,
bNF_Gr3951398729878943206at_o_o: set_b_prat > ( ( b > prat ) > $o ) > ( ( b > prat ) > $o ) > set_Product_prod_o_o ).
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bNF_Gr8394351763884875876prat_b: set_o > ( $o > b > prat ) > ( $o > produc7326829233880046182prat_b ) > set_Pr2868754470712218413prat_b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001_062_Itf__b_Mt__PosRat__Oprat_J_001tf__b,type,
bNF_Gr3107160545168091645prat_b: set_o > ( $o > b > prat ) > ( $o > b ) > set_Pr8957522257986161222prat_b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001_Eo_001_Eo,type,
bNF_Gr6698568825248092222_o_o_o: set_o > ( $o > $o ) > ( $o > $o ) > set_Product_prod_o_o ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001_062_Itf__b_Mt__PosRat__Oprat_J_001tf__b,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001_Eo_001_Eo,type,
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bNF_Gr3559723194386784068prat_b: set_set_o > ( set_o > b > prat ) > ( set_o > produc7326829233880046182prat_b ) > set_Pr2868754470712218413prat_b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_I_Eo_J_001_062_Itf__b_Mt__PosRat__Oprat_J_001tf__b,type,
bNF_Gr917350875349088477prat_b: set_set_o > ( set_o > b > prat ) > ( set_o > b ) > set_Pr8957522257986161222prat_b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_I_Eo_J_001_Eo_001_Eo,type,
bNF_Gr8091873193056382750_o_o_o: set_set_o > ( set_o > $o ) > ( set_o > $o ) > set_Product_prod_o_o ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard_001tf__b_001t__PosRat__Oprat,type,
bNF_Gr5391733101386555298b_prat: set_b > set_Pr3298805672376727137t_prat > b > prat ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred_001tf__b_001t__PosRat__Oprat,type,
bNF_Gr8946146825857109601b_prat: set_b > set_Pr3298805672376727137t_prat > ( b > prat ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_I_Eo_J,type,
boolea379910186789422830_set_o: ( set_o > set_o > set_o ) > ( set_o > set_o > set_o ) > ( set_o > set_o ) > set_o > set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_Itf__b_Mt__PosRat__Oprat_J_J,type,
comple8554269323607207949b_prat: set_set_b_prat > set_b_prat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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comple5785745244134051700_set_o: set_se6599315173249254119_set_o > set_Pr4577759397028426247_set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
comple4436988014476444997_set_o: set_set_set_o > set_set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_I_Eo_J_J_J,type,
comple7771960130403061797_set_o: set_set_set_set_o > set_set_set_o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_Eo,type,
condit5488710616941104124bove_o: set_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001_Eo,type,
condit5413489452508810728elow_o: set_o > $o ).
thf(sy_c_Equiv__Relations_Oquotient_001_Eo,type,
equiv_quotient_o: set_o > set_Product_prod_o_o > set_set_o ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001_Eo,type,
fun_upd_o_o: ( $o > $o ) > $o > $o > $o > $o ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Set__Oset_I_Eo_J,type,
fun_upd_o_set_o: ( $o > set_o ) > $o > set_o > $o > set_o ).
thf(sy_c_Fun_Ofun__upd_001tf__b_001t__PosRat__Oprat,type,
fun_upd_b_prat: ( b > prat ) > b > prat > b > prat ).
thf(sy_c_Fun_Ooverride__on_001_Eo_001_Eo,type,
override_on_o_o: ( $o > $o ) > ( $o > $o ) > set_o > $o > $o ).
thf(sy_c_Fun_Ooverride__on_001_Eo_001t__Set__Oset_I_Eo_J,type,
override_on_o_set_o: ( $o > set_o ) > ( $o > set_o ) > set_o > $o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
minus_7052513330764642158od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
uminus_uminus_set_o: set_o > set_o ).
thf(sy_c_HOL_OThe_001_062_Itf__b_Mt__PosRat__Oprat_J,type,
the_b_prat: ( ( b > prat ) > $o ) > b > prat ).
thf(sy_c_HOL_OThe_001_Eo,type,
the_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_HOL_OThe_001t__Set__Oset_I_Eo_J,type,
the_set_o: ( set_o > $o ) > set_o ).
thf(sy_c_If_001t__Set__Oset_I_Eo_J,type,
if_set_o: $o > set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
inf_in4898592226082374645od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
sup_sup_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
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lex_prod_b_prat_b: set_Pr525462856049757959b_prat > set_Product_prod_b_b > set_Pr3106385887818360711prat_b ).
thf(sy_c_Wellfounded_Olex__prod_001_Eo_001_Eo,type,
lex_prod_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr7011910401827753159od_o_o ).
thf(sy_c_Wellfounded_Owf_001_Eo,type,
wf_o: set_Product_prod_o_o > $o ).
thf(sy_c_Wfrec_Osame__fst_001_062_Itf__b_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J,type,
same_f7802962448890730258prat_b: ( ( b > prat ) > $o ) > ( ( b > prat ) > set_Pr3106385887818360711prat_b ) > set_Pr3102753524365197639prat_b ).
thf(sy_c_Wfrec_Osame__fst_001_062_Itf__b_Mt__PosRat__Oprat_J_001tf__b,type,
same_fst_b_prat_b: ( ( b > prat ) > $o ) > ( ( b > prat ) > set_Product_prod_b_b ) > set_Pr3106385887818360711prat_b ).
thf(sy_c_Wfrec_Osame__fst_001_Eo_001_Eo,type,
same_fst_o_o: ( $o > $o ) > ( $o > set_Product_prod_o_o ) > set_Pr7011910401827753159od_o_o ).
thf(sy_c_Zorn_OChains_001_Eo,type,
chains_o: set_Product_prod_o_o > set_set_o ).
thf(sy_c_Zorn_Opred__on_Ochain_001_Eo,type,
pred_chain_o: set_o > ( $o > $o > $o ) > set_o > $o ).
thf(sy_c_fChoice_001_062_Itf__b_Mt__PosRat__Oprat_J,type,
fChoice_b_prat: ( ( b > prat ) > $o ) > b > prat ).
thf(sy_c_fChoice_001_Eo,type,
fChoice_o: ( $o > $o ) > $o ).
thf(sy_c_fChoice_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
fChoic50381644832149627od_o_o: ( product_prod_o_o > $o ) > product_prod_o_o ).
thf(sy_c_fChoice_001t__Set__Oset_I_Eo_J,type,
fChoice_set_o: ( set_o > $o ) > set_o ).
thf(sy_c_fChoice_001tf__b,type,
fChoice_b: ( b > $o ) > 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_062_Itf__b_Mt__PosRat__Oprat_J,type,
member_b_prat: ( b > prat ) > set_b_prat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_M_062_Itf__b_Mt__PosRat__Oprat_J_J,type,
member6376894174024132944b_prat: produc7696030361869858855b_prat > set_Pr525462856049757959b_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_M_Eo_J,type,
member5777084206023828648prat_o: produc5398857101462582399prat_o > set_Pr5078033069183545951prat_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J_J,type,
member412304827439910006prat_b: produc8919727841856482381prat_b > set_Pr2868754470712218413prat_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mt__Set__Oset_I_Eo_J_J,type,
member8295766130977777544_set_o: produc8934563419688180575_set_o > set_Pr5661576500205409599_set_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J,type,
member5302574828040144399prat_b: produc7326829233880046182prat_b > set_Pr8957522257986161222prat_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_062_Itf__b_Mt__PosRat__Oprat_J_J,type,
member7584064603762627832b_prat: produc7205837499201381583b_prat > set_Pr5930873434710784687b_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
member3692453414038245343od_o_o: produc2310465149704154422od_o_o > set_Pr5237325540968292374od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Set__Oset_I_Eo_J_J,type,
member8938118433653312304_set_o: product_prod_o_set_o > set_Pr3508996721749391591_set_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J_J_Mt__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J_J_J,type,
member6556583152774367120prat_b: produc2703347610102075239prat_b > set_Pr3102753524365197639prat_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J_Mt__Product____Type__Oprod_I_062_Itf__b_Mt__PosRat__Oprat_J_Mtf__b_J_J,type,
member3390411761967995344prat_b: produc7488436305439230631prat_b > set_Pr3106385887818360711prat_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_M_Eo_J,type,
member5943846676503041025_o_o_o: produc4561858412168950104_o_o_o > set_Pr2246928807395521336_o_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
member3169104723136097040od_o_o: produc4465838814804890855od_o_o > set_Pr7011910401827753159od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
member264548248930972385_set_o: produc9030379227580347448_set_o > set_Pr1571070227385496088_set_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_M_062_Itf__b_Mt__PosRat__Oprat_J_J,type,
member9157886055108046360b_prat: produc573311306963673583b_prat > set_Pr3021494175023059919b_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_M_Eo_J,type,
member6599968234716717424et_o_o: product_prod_set_o_o > set_Pr5232024646447722279et_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
member9116954335612470352_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_Itf__b_Mt__PosRat__Oprat_J_J,type,
member_set_b_prat: set_b_prat > set_set_b_prat > $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__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
member_set_set_o: set_set_o > set_set_set_o > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_A,type,
a2: a > b > prat ).
thf(sy_v_B,type,
b2: a > b > prat ).
% Relevant facts (1277)
thf(fact_0__092_060open_062big__greater__mask_AA_AB_092_060close_062,axiom,
big_greater_mask_a_b @ a2 @ b2 ).
% \<open>big_greater_mask A B\<close>
thf(fact_1_add__masks__asso,axiom,
! [A: b > prat,B: b > prat,C: b > prat] :
( ( add_masks_b @ ( add_masks_b @ A @ B ) @ C )
= ( add_masks_b @ A @ ( add_masks_b @ B @ C ) ) ) ).
% add_masks_asso
thf(fact_2_add__masks__comm,axiom,
( add_masks_b
= ( ^ [A2: b > prat,B2: b > prat] : ( add_masks_b @ B2 @ A2 ) ) ) ).
% add_masks_comm
thf(fact_3_big__add__masks__def,axiom,
( big_add_masks_a_b
= ( ^ [Pi: a > b > prat,Pi2: a > b > prat,H: a] : ( add_masks_b @ ( Pi @ H ) @ ( Pi2 @ H ) ) ) ) ).
% big_add_masks_def
thf(fact_4_some__equality,axiom,
! [P: $o > $o,A: $o] :
( ( P @ A )
=> ( ! [X: $o] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_o @ P )
= A ) ) ) ).
% some_equality
thf(fact_5_some__equality,axiom,
! [P: ( b > prat ) > $o,A: b > prat] :
( ( P @ A )
=> ( ! [X: b > prat] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_b_prat @ P )
= A ) ) ) ).
% some_equality
thf(fact_6_some__eq__trivial,axiom,
! [X2: $o] :
( ( fChoice_o
@ ^ [Y: $o] : ( Y = X2 ) )
= X2 ) ).
% some_eq_trivial
thf(fact_7_some__eq__trivial,axiom,
! [X2: b > prat] :
( ( fChoice_b_prat
@ ^ [Y: b > prat] : ( Y = X2 ) )
= X2 ) ).
% some_eq_trivial
thf(fact_8_some__sym__eq__trivial,axiom,
! [X2: $o] :
( ( fChoice_o
@ ( ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_9_some__sym__eq__trivial,axiom,
! [X2: b > prat] :
( ( fChoice_b_prat
@ ( ^ [Y2: b > prat,Z: b > prat] : ( Y2 = Z )
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_10_big__add__greater,axiom,
! [A3: a > b > prat,B3: a > b > prat] : ( big_greater_mask_a_b @ ( big_add_masks_a_b @ A3 @ B3 ) @ B3 ) ).
% big_add_greater
thf(fact_11_verit__sko__ex_H,axiom,
! [P: $o > $o,A3: $o] :
( ( ( P @ ( fChoice_o @ P ) )
= A3 )
=> ( ( ? [X3: $o] : ( P @ X3 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_12_verit__sko__ex_H,axiom,
! [P: ( b > prat ) > $o,A3: $o] :
( ( ( P @ ( fChoice_b_prat @ P ) )
= A3 )
=> ( ( ? [X3: b > prat] : ( P @ X3 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_13_verit__sko__forall,axiom,
( ( ^ [P2: $o > $o] :
! [X4: $o] : ( P2 @ X4 ) )
= ( ^ [P3: $o > $o] :
( P3
@ ( fChoice_o
@ ^ [X5: $o] :
~ ( P3 @ X5 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_14_verit__sko__forall,axiom,
( ( ^ [P2: ( b > prat ) > $o] :
! [X4: b > prat] : ( P2 @ X4 ) )
= ( ^ [P3: ( b > prat ) > $o] :
( P3
@ ( fChoice_b_prat
@ ^ [X5: b > prat] :
~ ( P3 @ X5 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_15_someI2,axiom,
! [P: $o > $o,A: $o,Q: $o > $o] :
( ( P @ A )
=> ( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_o @ P ) ) ) ) ).
% someI2
thf(fact_16_someI2,axiom,
! [P: ( b > prat ) > $o,A: b > prat,Q: ( b > prat ) > $o] :
( ( P @ A )
=> ( ! [X: b > prat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_b_prat @ P ) ) ) ) ).
% someI2
thf(fact_17_verit__sko__forall_H,axiom,
! [P: $o > $o,A3: $o] :
( ( ( P
@ ( fChoice_o
@ ^ [X5: $o] :
~ ( P @ X5 ) ) )
= A3 )
=> ( ( ! [X3: $o] : ( P @ X3 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_18_verit__sko__forall_H,axiom,
! [P: ( b > prat ) > $o,A3: $o] :
( ( ( P
@ ( fChoice_b_prat
@ ^ [X5: b > prat] :
~ ( P @ X5 ) ) )
= A3 )
=> ( ( ! [X3: b > prat] : ( P @ X3 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_19_verit__sko__forall_H_H,axiom,
! [B3: $o,A3: $o,P: $o > $o] :
( ( B3 = A3 )
=> ( ( ( fChoice_o @ P )
= A3 )
= ( ( fChoice_o @ P )
= B3 ) ) ) ).
% verit_sko_forall''
thf(fact_20_verit__sko__forall_H_H,axiom,
! [B3: b > prat,A3: b > prat,P: ( b > prat ) > $o] :
( ( B3 = A3 )
=> ( ( ( fChoice_b_prat @ P )
= A3 )
= ( ( fChoice_b_prat @ P )
= B3 ) ) ) ).
% verit_sko_forall''
thf(fact_21_someI__ex,axiom,
! [P: $o > $o] :
( ? [X_1: $o] : ( P @ X_1 )
=> ( P @ ( fChoice_o @ P ) ) ) ).
% someI_ex
thf(fact_22_someI__ex,axiom,
! [P: ( b > prat ) > $o] :
( ? [X_1: b > prat] : ( P @ X_1 )
=> ( P @ ( fChoice_b_prat @ P ) ) ) ).
% someI_ex
thf(fact_23_someI2__ex,axiom,
! [P: $o > $o,Q: $o > $o] :
( ? [X_1: $o] : ( P @ X_1 )
=> ( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_o @ P ) ) ) ) ).
% someI2_ex
thf(fact_24_someI2__ex,axiom,
! [P: ( b > prat ) > $o,Q: ( b > prat ) > $o] :
( ? [X_1: b > prat] : ( P @ X_1 )
=> ( ! [X: b > prat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_b_prat @ P ) ) ) ) ).
% someI2_ex
thf(fact_25_someI2__bex,axiom,
! [A3: set_Product_prod_o_o,P: product_prod_o_o > $o,Q: product_prod_o_o > $o] :
( ? [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ A3 )
& ( P @ X6 ) )
=> ( ! [X: product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoic50381644832149627od_o_o
@ ^ [X5: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_26_someI2__bex,axiom,
! [A3: set_set_o,P: set_o > $o,Q: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ A3 )
& ( P @ X6 ) )
=> ( ! [X: set_o] :
( ( ( member_set_o @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_set_o
@ ^ [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_27_someI2__bex,axiom,
! [A3: set_o,P: $o > $o,Q: $o > $o] :
( ? [X6: $o] :
( ( member_o @ X6 @ A3 )
& ( P @ X6 ) )
=> ( ! [X: $o] :
( ( ( member_o @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_28_someI2__bex,axiom,
! [A3: set_b_prat,P: ( b > prat ) > $o,Q: ( b > prat ) > $o] :
( ? [X6: b > prat] :
( ( member_b_prat @ X6 @ A3 )
& ( P @ X6 ) )
=> ( ! [X: b > prat] :
( ( ( member_b_prat @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_b_prat
@ ^ [X5: b > prat] :
( ( member_b_prat @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_29_some__eq__imp,axiom,
! [P: $o > $o,A: $o,B: $o] :
( ( ( fChoice_o @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_30_some__eq__imp,axiom,
! [P: ( b > prat ) > $o,A: b > prat,B: b > prat] :
( ( ( fChoice_b_prat @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_31_tfl__some,axiom,
! [P4: $o > $o,X6: $o] :
( ( P4 @ X6 )
=> ( P4 @ ( fChoice_o @ P4 ) ) ) ).
% tfl_some
thf(fact_32_tfl__some,axiom,
! [P4: ( b > prat ) > $o,X6: b > prat] :
( ( P4 @ X6 )
=> ( P4 @ ( fChoice_b_prat @ P4 ) ) ) ).
% tfl_some
thf(fact_33_Eps__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X: $o] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( fChoice_o @ P )
= ( fChoice_o @ Q ) ) ) ).
% Eps_cong
thf(fact_34_Eps__cong,axiom,
! [P: ( b > prat ) > $o,Q: ( b > prat ) > $o] :
( ! [X: b > prat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( fChoice_b_prat @ P )
= ( fChoice_b_prat @ Q ) ) ) ).
% Eps_cong
thf(fact_35_someI,axiom,
! [P: $o > $o,X2: $o] :
( ( P @ X2 )
=> ( P @ ( fChoice_o @ P ) ) ) ).
% someI
thf(fact_36_someI,axiom,
! [P: ( b > prat ) > $o,X2: b > prat] :
( ( P @ X2 )
=> ( P @ ( fChoice_b_prat @ P ) ) ) ).
% someI
thf(fact_37_verit__sko__forall__indirect2,axiom,
! [X2: $o,P: $o > $o,P5: $o > $o] :
( ( X2
= ( fChoice_o
@ ^ [X5: $o] :
~ ( P @ X5 ) ) )
=> ( ! [X: $o] :
( ( P @ X )
= ( P5 @ X ) )
=> ( ( ! [X3: $o] : ( P5 @ X3 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_38_verit__sko__forall__indirect2,axiom,
! [X2: b > prat,P: ( b > prat ) > $o,P5: ( b > prat ) > $o] :
( ( X2
= ( fChoice_b_prat
@ ^ [X5: b > prat] :
~ ( P @ X5 ) ) )
=> ( ! [X: b > prat] :
( ( P @ X )
= ( P5 @ X ) )
=> ( ( ! [X3: b > prat] : ( P5 @ X3 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_39_verit__sko__forall__indirect,axiom,
! [X2: $o,P: $o > $o] :
( ( X2
= ( fChoice_o
@ ^ [X5: $o] :
~ ( P @ X5 ) ) )
=> ( ( ! [X3: $o] : ( P @ X3 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_40_verit__sko__forall__indirect,axiom,
! [X2: b > prat,P: ( b > prat ) > $o] :
( ( X2
= ( fChoice_b_prat
@ ^ [X5: b > prat] :
~ ( P @ X5 ) ) )
=> ( ( ! [X3: b > prat] : ( P @ X3 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_41_some1__equality,axiom,
! [P: $o > $o,A: $o] :
( ? [X6: $o] :
( ( P @ X6 )
& ! [Y3: $o] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_o @ P )
= A ) ) ) ).
% some1_equality
thf(fact_42_some1__equality,axiom,
! [P: ( b > prat ) > $o,A: b > prat] :
( ? [X6: b > prat] :
( ( P @ X6 )
& ! [Y3: b > prat] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_b_prat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_43_verit__sko__ex__indirect2,axiom,
! [X2: $o,P: $o > $o,P5: $o > $o] :
( ( X2
= ( fChoice_o @ P ) )
=> ( ! [X: $o] :
( ( P @ X )
= ( P5 @ X ) )
=> ( ( ? [X3: $o] : ( P5 @ X3 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_44_verit__sko__ex__indirect2,axiom,
! [X2: b > prat,P: ( b > prat ) > $o,P5: ( b > prat ) > $o] :
( ( X2
= ( fChoice_b_prat @ P ) )
=> ( ! [X: b > prat] :
( ( P @ X )
= ( P5 @ X ) )
=> ( ( ? [X3: b > prat] : ( P5 @ X3 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_45_verit__sko__ex__indirect,axiom,
! [X2: $o,P: $o > $o] :
( ( X2
= ( fChoice_o @ P ) )
=> ( ( ? [X3: $o] : ( P @ X3 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_46_verit__sko__ex__indirect,axiom,
! [X2: b > prat,P: ( b > prat ) > $o] :
( ( X2
= ( fChoice_b_prat @ P ) )
=> ( ( ? [X3: b > prat] : ( P @ X3 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_47_some__eq__ex,axiom,
! [P: $o > $o] :
( ( P @ ( fChoice_o @ P ) )
= ( ? [X3: $o] : ( P @ X3 ) ) ) ).
% some_eq_ex
thf(fact_48_some__eq__ex,axiom,
! [P: ( b > prat ) > $o] :
( ( P @ ( fChoice_b_prat @ P ) )
= ( ? [X3: b > prat] : ( P @ X3 ) ) ) ).
% some_eq_ex
thf(fact_49_Nitpick_OEps__psimp,axiom,
! [P: $o > $o,X2: $o,Y4: $o] :
( ( P @ X2 )
=> ( ~ ( P @ Y4 )
=> ( ( ( fChoice_o @ P )
= Y4 )
=> ( ( fChoice_o @ P )
= X2 ) ) ) ) ).
% Nitpick.Eps_psimp
thf(fact_50_Nitpick_OEps__psimp,axiom,
! [P: ( b > prat ) > $o,X2: b > prat,Y4: b > prat] :
( ( P @ X2 )
=> ( ~ ( P @ Y4 )
=> ( ( ( fChoice_b_prat @ P )
= Y4 )
=> ( ( fChoice_b_prat @ P )
= X2 ) ) ) ) ).
% Nitpick.Eps_psimp
thf(fact_51_big__greater__mask__def,axiom,
( big_greater_mask_a_b
= ( ^ [Pi: a > b > prat,Pi2: a > b > prat] :
! [H: a] : ( greater_mask_b @ ( Pi @ H ) @ ( Pi2 @ H ) ) ) ) ).
% big_greater_mask_def
thf(fact_52_minus__empty,axiom,
! [Pi3: b > prat] :
( Pi3
= ( add_masks_b @ Pi3 @ empty_mask_b ) ) ).
% minus_empty
thf(fact_53_greater__mask__decomp,axiom,
! [A: b > prat,B: b > prat,C: b > prat] :
( ( greater_mask_b @ A @ ( add_masks_b @ B @ C ) )
=> ? [A1: b > prat,A22: b > prat] :
( ( A
= ( add_masks_b @ A1 @ A22 ) )
& ( greater_mask_b @ A1 @ B )
& ( greater_mask_b @ A22 @ C ) ) ) ).
% greater_mask_decomp
thf(fact_54_greater__mask__def,axiom,
( greater_mask_b
= ( ^ [Pi4: b > prat,Pi5: b > prat] :
? [R: b > prat] :
( Pi4
= ( add_masks_b @ Pi5 @ R ) ) ) ) ).
% greater_mask_def
thf(fact_55_toCard__def,axiom,
( bNF_Gr5391733101386555298b_prat
= ( ^ [A4: set_b,R: set_Pr3298805672376727137t_prat] : ( fChoice_b_prat @ ( bNF_Gr8946146825857109601b_prat @ A4 @ R ) ) ) ) ).
% toCard_def
thf(fact_56_add__acc__uni__mask,axiom,
( add_acc_b
= ( ^ [Pi5: b > prat,Hl: b,P6: prat] : ( add_masks_b @ Pi5 @ ( uni_mask_b @ Hl @ P6 ) ) ) ) ).
% add_acc_uni_mask
thf(fact_57_some__in__eq,axiom,
! [A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o
@ ( fChoic50381644832149627od_o_o
@ ^ [X5: product_prod_o_o] : ( member7466972457876170832od_o_o @ X5 @ A3 ) )
@ A3 )
= ( A3 != bot_bo7073875226086086771od_o_o ) ) ).
% some_in_eq
thf(fact_58_some__in__eq,axiom,
! [A3: set_set_o] :
( ( member_set_o
@ ( fChoice_set_o
@ ^ [X5: set_o] : ( member_set_o @ X5 @ A3 ) )
@ A3 )
= ( A3 != bot_bot_set_set_o ) ) ).
% some_in_eq
thf(fact_59_some__in__eq,axiom,
! [A3: set_b_prat] :
( ( member_b_prat
@ ( fChoice_b_prat
@ ^ [X5: b > prat] : ( member_b_prat @ X5 @ A3 ) )
@ A3 )
= ( A3 != bot_bot_set_b_prat ) ) ).
% some_in_eq
thf(fact_60_some__in__eq,axiom,
! [A3: set_o] :
( ( member_o
@ ( fChoice_o
@ ^ [X5: $o] : ( member_o @ X5 @ A3 ) )
@ A3 )
= ( A3 != bot_bot_set_o ) ) ).
% some_in_eq
thf(fact_61_add__masks_Osimps,axiom,
( add_masks_b
= ( ^ [Pi4: b > prat,Pi5: b > prat,Hl: b] : ( padd @ ( Pi4 @ Hl ) @ ( Pi5 @ Hl ) ) ) ) ).
% add_masks.simps
thf(fact_62_greater__mask__properties_I1_J,axiom,
! [Pi3: b > prat] : ( greater_mask_b @ Pi3 @ Pi3 ) ).
% greater_mask_properties(1)
thf(fact_63_greater__mask__properties_I2_J,axiom,
! [A: b > prat,B: b > prat,C: b > prat] :
( ( ( greater_mask_b @ A @ B )
& ( greater_mask_b @ B @ C ) )
=> ( greater_mask_b @ A @ C ) ) ).
% greater_mask_properties(2)
thf(fact_64_greater__mask__properties_I3_J,axiom,
! [Pi6: b > prat,Pi3: b > prat] :
( ( ( greater_mask_b @ Pi6 @ Pi3 )
& ( greater_mask_b @ Pi3 @ Pi6 ) )
=> ( Pi3 = Pi6 ) ) ).
% greater_mask_properties(3)
thf(fact_65_add__masks_Oelims,axiom,
! [X2: b > prat,Xa: b > prat,Xb: b,Y4: prat] :
( ( ( add_masks_b @ X2 @ Xa @ Xb )
= Y4 )
=> ( Y4
= ( padd @ ( X2 @ Xb ) @ ( Xa @ Xb ) ) ) ) ).
% add_masks.elims
thf(fact_66_fromCard__def,axiom,
( bNF_Gr6838450293404232211b_prat
= ( ^ [A4: set_b,R: set_Pr3298805672376727137t_prat,K: prat] :
( fChoice_b
@ ^ [B2: b] :
( ( member_b @ B2 @ A4 )
& ( ( bNF_Gr5391733101386555298b_prat @ A4 @ R @ B2 )
= K ) ) ) ) ) ).
% fromCard_def
thf(fact_67_empty__Collect__eq,axiom,
! [P: ( b > prat ) > $o] :
( ( bot_bot_set_b_prat
= ( collect_b_prat @ P ) )
= ( ! [X5: b > prat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_68_empty__Collect__eq,axiom,
! [P: product_prod_o_o > $o] :
( ( bot_bo7073875226086086771od_o_o
= ( collec3167064739498627218od_o_o @ P ) )
= ( ! [X5: product_prod_o_o] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_69_empty__Collect__eq,axiom,
! [P: set_o > $o] :
( ( bot_bot_set_set_o
= ( collect_set_o @ P ) )
= ( ! [X5: set_o] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_70_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X5: $o] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_71_Collect__empty__eq,axiom,
! [P: ( b > prat ) > $o] :
( ( ( collect_b_prat @ P )
= bot_bot_set_b_prat )
= ( ! [X5: b > prat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_72_Collect__empty__eq,axiom,
! [P: product_prod_o_o > $o] :
( ( ( collec3167064739498627218od_o_o @ P )
= bot_bo7073875226086086771od_o_o )
= ( ! [X5: product_prod_o_o] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P: set_o > $o] :
( ( ( collect_set_o @ P )
= bot_bot_set_set_o )
= ( ! [X5: set_o] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X5: $o] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: b > prat,P: ( b > prat ) > $o] :
( ( member_b_prat @ A @ ( collect_b_prat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: product_prod_o_o,P: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ A @ ( collec3167064739498627218od_o_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: set_o,P: set_o > $o] :
( ( member_set_o @ A @ ( collect_set_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A3: set_b_prat] :
( ( collect_b_prat
@ ^ [X5: b > prat] : ( member_b_prat @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A3: set_Product_prod_o_o] :
( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] : ( member7466972457876170832od_o_o @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [X5: $o] : ( member_o @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A3: set_set_o] :
( ( collect_set_o
@ ^ [X5: set_o] : ( member_set_o @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_83_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X: $o] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_84_Collect__cong,axiom,
! [P: set_o > $o,Q: set_o > $o] :
( ! [X: set_o] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_o @ P )
= ( collect_set_o @ Q ) ) ) ).
% Collect_cong
thf(fact_85_all__not__in__conv,axiom,
! [A3: set_b_prat] :
( ( ! [X5: b > prat] :
~ ( member_b_prat @ X5 @ A3 ) )
= ( A3 = bot_bot_set_b_prat ) ) ).
% all_not_in_conv
thf(fact_86_all__not__in__conv,axiom,
! [A3: set_Product_prod_o_o] :
( ( ! [X5: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ X5 @ A3 ) )
= ( A3 = bot_bo7073875226086086771od_o_o ) ) ).
% all_not_in_conv
thf(fact_87_all__not__in__conv,axiom,
! [A3: set_set_o] :
( ( ! [X5: set_o] :
~ ( member_set_o @ X5 @ A3 ) )
= ( A3 = bot_bot_set_set_o ) ) ).
% all_not_in_conv
thf(fact_88_all__not__in__conv,axiom,
! [A3: set_o] :
( ( ! [X5: $o] :
~ ( member_o @ X5 @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_89_empty__iff,axiom,
! [C: b > prat] :
~ ( member_b_prat @ C @ bot_bot_set_b_prat ) ).
% empty_iff
thf(fact_90_empty__iff,axiom,
! [C: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ C @ bot_bo7073875226086086771od_o_o ) ).
% empty_iff
thf(fact_91_empty__iff,axiom,
! [C: set_o] :
~ ( member_set_o @ C @ bot_bot_set_set_o ) ).
% empty_iff
thf(fact_92_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_93_bot__apply,axiom,
( bot_bot_o_o
= ( ^ [X5: $o] : bot_bot_o ) ) ).
% bot_apply
thf(fact_94_empty__def,axiom,
( bot_bot_set_b_prat
= ( collect_b_prat
@ ^ [X5: b > prat] : $false ) ) ).
% empty_def
thf(fact_95_empty__def,axiom,
( bot_bo7073875226086086771od_o_o
= ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] : $false ) ) ).
% empty_def
thf(fact_96_empty__def,axiom,
( bot_bot_set_set_o
= ( collect_set_o
@ ^ [X5: set_o] : $false ) ) ).
% empty_def
thf(fact_97_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X5: $o] : $false ) ) ).
% empty_def
thf(fact_98_padd__cancellative,axiom,
! [A: prat,X2: prat,B: prat,Y4: prat] :
( ( A
= ( padd @ X2 @ B ) )
=> ( ( A
= ( padd @ Y4 @ B ) )
=> ( X2 = Y4 ) ) ) ).
% padd_cancellative
thf(fact_99_padd__comm,axiom,
( padd
= ( ^ [A2: prat,B2: prat] : ( padd @ B2 @ A2 ) ) ) ).
% padd_comm
thf(fact_100_padd__asso,axiom,
! [A: prat,B: prat,C: prat] :
( ( padd @ ( padd @ A @ B ) @ C )
= ( padd @ A @ ( padd @ B @ C ) ) ) ).
% padd_asso
thf(fact_101_uni__mask_Osimps,axiom,
( uni_mask_b
= ( fun_upd_b_prat @ empty_mask_b ) ) ).
% uni_mask.simps
thf(fact_102_bot__set__def,axiom,
( bot_bot_set_b_prat
= ( collect_b_prat @ bot_bot_b_prat_o ) ) ).
% bot_set_def
thf(fact_103_bot__set__def,axiom,
( bot_bo7073875226086086771od_o_o
= ( collec3167064739498627218od_o_o @ bot_bo6429049293326997034_o_o_o ) ) ).
% bot_set_def
thf(fact_104_bot__set__def,axiom,
( bot_bot_set_set_o
= ( collect_set_o @ bot_bot_set_o_o ) ) ).
% bot_set_def
thf(fact_105_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_106_bot__fun__def,axiom,
( bot_bot_o_o
= ( ^ [X5: $o] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_107_emptyE,axiom,
! [A: b > prat] :
~ ( member_b_prat @ A @ bot_bot_set_b_prat ) ).
% emptyE
thf(fact_108_emptyE,axiom,
! [A: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ).
% emptyE
thf(fact_109_emptyE,axiom,
! [A: set_o] :
~ ( member_set_o @ A @ bot_bot_set_set_o ) ).
% emptyE
thf(fact_110_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_111_equals0D,axiom,
! [A3: set_b_prat,A: b > prat] :
( ( A3 = bot_bot_set_b_prat )
=> ~ ( member_b_prat @ A @ A3 ) ) ).
% equals0D
thf(fact_112_equals0D,axiom,
! [A3: set_Product_prod_o_o,A: product_prod_o_o] :
( ( A3 = bot_bo7073875226086086771od_o_o )
=> ~ ( member7466972457876170832od_o_o @ A @ A3 ) ) ).
% equals0D
thf(fact_113_equals0D,axiom,
! [A3: set_set_o,A: set_o] :
( ( A3 = bot_bot_set_set_o )
=> ~ ( member_set_o @ A @ A3 ) ) ).
% equals0D
thf(fact_114_equals0D,axiom,
! [A3: set_o,A: $o] :
( ( A3 = bot_bot_set_o )
=> ~ ( member_o @ A @ A3 ) ) ).
% equals0D
thf(fact_115_equals0I,axiom,
! [A3: set_b_prat] :
( ! [Y3: b > prat] :
~ ( member_b_prat @ Y3 @ A3 )
=> ( A3 = bot_bot_set_b_prat ) ) ).
% equals0I
thf(fact_116_equals0I,axiom,
! [A3: set_Product_prod_o_o] :
( ! [Y3: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ Y3 @ A3 )
=> ( A3 = bot_bo7073875226086086771od_o_o ) ) ).
% equals0I
thf(fact_117_equals0I,axiom,
! [A3: set_set_o] :
( ! [Y3: set_o] :
~ ( member_set_o @ Y3 @ A3 )
=> ( A3 = bot_bot_set_set_o ) ) ).
% equals0I
thf(fact_118_equals0I,axiom,
! [A3: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A3 )
=> ( A3 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_119_ex__in__conv,axiom,
! [A3: set_b_prat] :
( ( ? [X5: b > prat] : ( member_b_prat @ X5 @ A3 ) )
= ( A3 != bot_bot_set_b_prat ) ) ).
% ex_in_conv
thf(fact_120_ex__in__conv,axiom,
! [A3: set_Product_prod_o_o] :
( ( ? [X5: product_prod_o_o] : ( member7466972457876170832od_o_o @ X5 @ A3 ) )
= ( A3 != bot_bo7073875226086086771od_o_o ) ) ).
% ex_in_conv
thf(fact_121_ex__in__conv,axiom,
! [A3: set_set_o] :
( ( ? [X5: set_o] : ( member_set_o @ X5 @ A3 ) )
= ( A3 != bot_bot_set_set_o ) ) ).
% ex_in_conv
thf(fact_122_ex__in__conv,axiom,
! [A3: set_o] :
( ( ? [X5: $o] : ( member_o @ X5 @ A3 ) )
= ( A3 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_123_add__acc_Oelims,axiom,
! [X2: b > prat,Xa: b,Xb: prat,Y4: b > prat] :
( ( ( add_acc_b @ X2 @ Xa @ Xb )
= Y4 )
=> ( Y4
= ( fun_upd_b_prat @ X2 @ Xa @ ( padd @ ( X2 @ Xa ) @ Xb ) ) ) ) ).
% add_acc.elims
thf(fact_124_add__acc_Osimps,axiom,
( add_acc_b
= ( ^ [Pi5: b > prat,Hl: b,P6: prat] : ( fun_upd_b_prat @ Pi5 @ Hl @ ( padd @ ( Pi5 @ Hl ) @ P6 ) ) ) ) ).
% add_acc.simps
thf(fact_125_uni__mask_Oelims,axiom,
! [X2: b,Xa: prat,Y4: b > prat] :
( ( ( uni_mask_b @ X2 @ Xa )
= Y4 )
=> ( Y4
= ( fun_upd_b_prat @ empty_mask_b @ X2 @ Xa ) ) ) ).
% uni_mask.elims
thf(fact_126_fun__upd__upd,axiom,
! [F: $o > set_o,X2: $o,Y4: set_o,Z2: set_o] :
( ( fun_upd_o_set_o @ ( fun_upd_o_set_o @ F @ X2 @ Y4 ) @ X2 @ Z2 )
= ( fun_upd_o_set_o @ F @ X2 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_127_fun__upd__upd,axiom,
! [F: $o > $o,X2: $o,Y4: $o,Z2: $o] :
( ( fun_upd_o_o @ ( fun_upd_o_o @ F @ X2 @ Y4 ) @ X2 @ Z2 )
= ( fun_upd_o_o @ F @ X2 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_128_fun__upd__triv,axiom,
! [F: $o > set_o,X2: $o] :
( ( fun_upd_o_set_o @ F @ X2 @ ( F @ X2 ) )
= F ) ).
% fun_upd_triv
thf(fact_129_fun__upd__triv,axiom,
! [F: $o > $o,X2: $o] :
( ( fun_upd_o_o @ F @ X2 @ ( F @ X2 ) )
= F ) ).
% fun_upd_triv
thf(fact_130_fun__upd__apply,axiom,
( fun_upd_o_set_o
= ( ^ [F2: $o > set_o,X5: $o,Y: set_o,Z3: $o] : ( if_set_o @ ( Z3 = X5 ) @ Y @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_131_fun__upd__apply,axiom,
( fun_upd_o_o
= ( ^ [F2: $o > $o,X5: $o,Y: $o,Z3: $o] :
( ( ( Z3 = X5 )
=> Y )
& ( ( Z3 = (~ X5) )
=> ( F2 @ Z3 ) ) ) ) ) ).
% fun_upd_apply
thf(fact_132_Set_Ois__empty__def,axiom,
( is_empty_b_prat
= ( ^ [A4: set_b_prat] : ( A4 = bot_bot_set_b_prat ) ) ) ).
% Set.is_empty_def
thf(fact_133_Set_Ois__empty__def,axiom,
( is_emp697939970857919993od_o_o
= ( ^ [A4: set_Product_prod_o_o] : ( A4 = bot_bo7073875226086086771od_o_o ) ) ) ).
% Set.is_empty_def
thf(fact_134_Set_Ois__empty__def,axiom,
( is_empty_set_o
= ( ^ [A4: set_set_o] : ( A4 = bot_bot_set_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_135_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A4: set_o] : ( A4 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_136_fun__upd__def,axiom,
( fun_upd_o_set_o
= ( ^ [F2: $o > set_o,A2: $o,B2: set_o,X5: $o] : ( if_set_o @ ( X5 = A2 ) @ B2 @ ( F2 @ X5 ) ) ) ) ).
% fun_upd_def
thf(fact_137_fun__upd__def,axiom,
( fun_upd_o_o
= ( ^ [F2: $o > $o,A2: $o,B2: $o,X5: $o] :
( ( ( X5 = A2 )
=> B2 )
& ( ( X5 = (~ A2) )
=> ( F2 @ X5 ) ) ) ) ) ).
% fun_upd_def
thf(fact_138_fun__upd__eqD,axiom,
! [F: $o > set_o,X2: $o,Y4: set_o,G: $o > set_o,Z2: set_o] :
( ( ( fun_upd_o_set_o @ F @ X2 @ Y4 )
= ( fun_upd_o_set_o @ G @ X2 @ Z2 ) )
=> ( Y4 = Z2 ) ) ).
% fun_upd_eqD
thf(fact_139_fun__upd__eqD,axiom,
! [F: $o > $o,X2: $o,Y4: $o,G: $o > $o,Z2: $o] :
( ( ( fun_upd_o_o @ F @ X2 @ Y4 )
= ( fun_upd_o_o @ G @ X2 @ Z2 ) )
=> ( Y4 = Z2 ) ) ).
% fun_upd_eqD
thf(fact_140_fun__upd__idem,axiom,
! [F: $o > set_o,X2: $o,Y4: set_o] :
( ( ( F @ X2 )
= Y4 )
=> ( ( fun_upd_o_set_o @ F @ X2 @ Y4 )
= F ) ) ).
% fun_upd_idem
thf(fact_141_fun__upd__idem,axiom,
! [F: $o > $o,X2: $o,Y4: $o] :
( ( ( F @ X2 )
= Y4 )
=> ( ( fun_upd_o_o @ F @ X2 @ Y4 )
= F ) ) ).
% fun_upd_idem
thf(fact_142_fun__upd__idem__iff,axiom,
! [F: $o > set_o,X2: $o,Y4: set_o] :
( ( ( fun_upd_o_set_o @ F @ X2 @ Y4 )
= F )
= ( ( F @ X2 )
= Y4 ) ) ).
% fun_upd_idem_iff
thf(fact_143_fun__upd__idem__iff,axiom,
! [F: $o > $o,X2: $o,Y4: $o] :
( ( ( fun_upd_o_o @ F @ X2 @ Y4 )
= F )
= ( ( F @ X2 )
= Y4 ) ) ).
% fun_upd_idem_iff
thf(fact_144_fun__upd__twist,axiom,
! [A: $o,C: $o,M: $o > set_o,B: set_o,D: set_o] :
( ( A != C )
=> ( ( fun_upd_o_set_o @ ( fun_upd_o_set_o @ M @ A @ B ) @ C @ D )
= ( fun_upd_o_set_o @ ( fun_upd_o_set_o @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_145_fun__upd__twist,axiom,
! [A: $o,C: $o,M: $o > $o,B: $o,D: $o] :
( ( A != C )
=> ( ( fun_upd_o_o @ ( fun_upd_o_o @ M @ A @ B ) @ C @ D )
= ( fun_upd_o_o @ ( fun_upd_o_o @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_146_fun__upd__other,axiom,
! [Z2: $o,X2: $o,F: $o > set_o,Y4: set_o] :
( ( Z2 != X2 )
=> ( ( fun_upd_o_set_o @ F @ X2 @ Y4 @ Z2 )
= ( F @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_147_fun__upd__other,axiom,
! [Z2: $o,X2: $o,F: $o > $o,Y4: $o] :
( ( Z2 != X2 )
=> ( ( fun_upd_o_o @ F @ X2 @ Y4 @ Z2 )
= ( F @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_148_fun__upd__same,axiom,
! [F: $o > set_o,X2: $o,Y4: set_o] :
( ( fun_upd_o_set_o @ F @ X2 @ Y4 @ X2 )
= Y4 ) ).
% fun_upd_same
thf(fact_149_fun__upd__same,axiom,
! [F: $o > $o,X2: $o,Y4: $o] :
( ( fun_upd_o_o @ F @ X2 @ Y4 @ X2 )
= Y4 ) ).
% fun_upd_same
thf(fact_150_Collect__empty__eq__bot,axiom,
! [P: ( b > prat ) > $o] :
( ( ( collect_b_prat @ P )
= bot_bot_set_b_prat )
= ( P = bot_bot_b_prat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_151_Collect__empty__eq__bot,axiom,
! [P: product_prod_o_o > $o] :
( ( ( collec3167064739498627218od_o_o @ P )
= bot_bo7073875226086086771od_o_o )
= ( P = bot_bo6429049293326997034_o_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_152_Collect__empty__eq__bot,axiom,
! [P: set_o > $o] :
( ( ( collect_set_o @ P )
= bot_bot_set_set_o )
= ( P = bot_bot_set_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_153_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_154_bot__empty__eq,axiom,
( bot_bot_b_prat_o
= ( ^ [X5: b > prat] : ( member_b_prat @ X5 @ bot_bot_set_b_prat ) ) ) ).
% bot_empty_eq
thf(fact_155_bot__empty__eq,axiom,
( bot_bo6429049293326997034_o_o_o
= ( ^ [X5: product_prod_o_o] : ( member7466972457876170832od_o_o @ X5 @ bot_bo7073875226086086771od_o_o ) ) ) ).
% bot_empty_eq
thf(fact_156_bot__empty__eq,axiom,
( bot_bot_set_o_o
= ( ^ [X5: set_o] : ( member_set_o @ X5 @ bot_bot_set_set_o ) ) ) ).
% bot_empty_eq
thf(fact_157_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X5: $o] : ( member_o @ X5 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_158_is__singletonI_H,axiom,
! [A3: set_b_prat] :
( ( A3 != bot_bot_set_b_prat )
=> ( ! [X: b > prat,Y3: b > prat] :
( ( member_b_prat @ X @ A3 )
=> ( ( member_b_prat @ Y3 @ A3 )
=> ( X = Y3 ) ) )
=> ( is_singleton_b_prat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_159_is__singletonI_H,axiom,
! [A3: set_Product_prod_o_o] :
( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ! [X: product_prod_o_o,Y3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A3 )
=> ( ( member7466972457876170832od_o_o @ Y3 @ A3 )
=> ( X = Y3 ) ) )
=> ( is_sin2666361102579702811od_o_o @ A3 ) ) ) ).
% is_singletonI'
thf(fact_160_is__singletonI_H,axiom,
! [A3: set_set_o] :
( ( A3 != bot_bot_set_set_o )
=> ( ! [X: set_o,Y3: set_o] :
( ( member_set_o @ X @ A3 )
=> ( ( member_set_o @ Y3 @ A3 )
=> ( X = Y3 ) ) )
=> ( is_singleton_set_o @ A3 ) ) ) ).
% is_singletonI'
thf(fact_161_is__singletonI_H,axiom,
! [A3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ! [X: $o,Y3: $o] :
( ( member_o @ X @ A3 )
=> ( ( member_o @ Y3 @ A3 )
=> ( X = Y3 ) ) )
=> ( is_singleton_o @ A3 ) ) ) ).
% is_singletonI'
thf(fact_162_empty__mask_Osimps,axiom,
( empty_mask_b
= ( ^ [Hl: b] : pnone ) ) ).
% empty_mask.simps
thf(fact_163_empty__mask_Oelims,axiom,
! [X2: b,Y4: prat] :
( ( ( empty_mask_b @ X2 )
= Y4 )
=> ( Y4 = pnone ) ) ).
% empty_mask.elims
thf(fact_164_empty__bind,axiom,
! [F: $o > set_o] :
( ( bind_o_o @ bot_bot_set_o @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_165_empty__bind,axiom,
! [F: $o > set_set_o] :
( ( bind_o_set_o @ bot_bot_set_o @ F )
= bot_bot_set_set_o ) ).
% empty_bind
thf(fact_166_empty__bind,axiom,
! [F: set_o > set_o] :
( ( bind_set_o_o @ bot_bot_set_set_o @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_167_empty__bind,axiom,
! [F: $o > set_b_prat] :
( ( bind_o_b_prat @ bot_bot_set_o @ F )
= bot_bot_set_b_prat ) ).
% empty_bind
thf(fact_168_empty__bind,axiom,
! [F: $o > set_Product_prod_o_o] :
( ( bind_o3663692390807148898od_o_o @ bot_bot_set_o @ F )
= bot_bo7073875226086086771od_o_o ) ).
% empty_bind
thf(fact_169_empty__bind,axiom,
! [F: ( b > prat ) > set_o] :
( ( bind_b_prat_o @ bot_bot_set_b_prat @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_170_empty__bind,axiom,
! [F: product_prod_o_o > set_o] :
( ( bind_P7502988038590723076_o_o_o @ bot_bo7073875226086086771od_o_o @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_171_empty__bind,axiom,
! [F: set_o > set_set_o] :
( ( bind_set_o_set_o @ bot_bot_set_set_o @ F )
= bot_bot_set_set_o ) ).
% empty_bind
thf(fact_172_empty__bind,axiom,
! [F: ( b > prat ) > set_set_o] :
( ( bind_b_prat_set_o @ bot_bot_set_b_prat @ F )
= bot_bot_set_set_o ) ).
% empty_bind
thf(fact_173_empty__bind,axiom,
! [F: product_prod_o_o > set_set_o] :
( ( bind_P3141067987844729572_set_o @ bot_bo7073875226086086771od_o_o @ F )
= bot_bot_set_set_o ) ).
% empty_bind
thf(fact_174_nonempty__bind__const,axiom,
! [A3: set_o,B3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( bind_o_o @ A3
@ ^ [Uu: $o] : B3 )
= B3 ) ) ).
% nonempty_bind_const
thf(fact_175_bind__const,axiom,
! [A3: set_o,B3: set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( bind_o_o @ A3
@ ^ [Uu: $o] : B3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( bind_o_o @ A3
@ ^ [Uu: $o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_176_bind__const,axiom,
! [A3: set_o,B3: set_set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( bind_o_set_o @ A3
@ ^ [Uu: $o] : B3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( bind_o_set_o @ A3
@ ^ [Uu: $o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_177_bind__const,axiom,
! [A3: set_set_o,B3: set_o] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( bind_set_o_o @ A3
@ ^ [Uu: set_o] : B3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( bind_set_o_o @ A3
@ ^ [Uu: set_o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_178_bind__const,axiom,
! [A3: set_o,B3: set_b_prat] :
( ( ( A3 = bot_bot_set_o )
=> ( ( bind_o_b_prat @ A3
@ ^ [Uu: $o] : B3 )
= bot_bot_set_b_prat ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( bind_o_b_prat @ A3
@ ^ [Uu: $o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_179_bind__const,axiom,
! [A3: set_o,B3: set_Product_prod_o_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( bind_o3663692390807148898od_o_o @ A3
@ ^ [Uu: $o] : B3 )
= bot_bo7073875226086086771od_o_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( bind_o3663692390807148898od_o_o @ A3
@ ^ [Uu: $o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_180_bind__const,axiom,
! [A3: set_b_prat,B3: set_o] :
( ( ( A3 = bot_bot_set_b_prat )
=> ( ( bind_b_prat_o @ A3
@ ^ [Uu: b > prat] : B3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_b_prat )
=> ( ( bind_b_prat_o @ A3
@ ^ [Uu: b > prat] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_181_bind__const,axiom,
! [A3: set_Product_prod_o_o,B3: set_o] :
( ( ( A3 = bot_bo7073875226086086771od_o_o )
=> ( ( bind_P7502988038590723076_o_o_o @ A3
@ ^ [Uu: product_prod_o_o] : B3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ( bind_P7502988038590723076_o_o_o @ A3
@ ^ [Uu: product_prod_o_o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_182_bind__const,axiom,
! [A3: set_set_o,B3: set_set_o] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( bind_set_o_set_o @ A3
@ ^ [Uu: set_o] : B3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( bind_set_o_set_o @ A3
@ ^ [Uu: set_o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_183_bind__const,axiom,
! [A3: set_b_prat,B3: set_set_o] :
( ( ( A3 = bot_bot_set_b_prat )
=> ( ( bind_b_prat_set_o @ A3
@ ^ [Uu: b > prat] : B3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_b_prat )
=> ( ( bind_b_prat_set_o @ A3
@ ^ [Uu: b > prat] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_184_bind__const,axiom,
! [A3: set_Product_prod_o_o,B3: set_set_o] :
( ( ( A3 = bot_bo7073875226086086771od_o_o )
=> ( ( bind_P3141067987844729572_set_o @ A3
@ ^ [Uu: product_prod_o_o] : B3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ( bind_P3141067987844729572_set_o @ A3
@ ^ [Uu: product_prod_o_o] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_185_uni__mask_Opelims,axiom,
! [X2: b,Xa: prat,Y4: b > prat] :
( ( ( uni_mask_b @ X2 @ Xa )
= Y4 )
=> ( ( accp_P6182521114033490869b_prat @ uni_mask_rel_b @ ( product_Pair_b_prat @ X2 @ Xa ) )
=> ~ ( ( Y4
= ( fun_upd_b_prat @ empty_mask_b @ X2 @ Xa ) )
=> ~ ( accp_P6182521114033490869b_prat @ uni_mask_rel_b @ ( product_Pair_b_prat @ X2 @ Xa ) ) ) ) ) ).
% uni_mask.pelims
thf(fact_186_bot__empty__eq2,axiom,
( bot_bo335163039370745524at_b_o
= ( ^ [X5: b > prat,Y: produc7326829233880046182prat_b] : ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ X5 @ Y ) @ bot_bo6161294920636105113prat_b ) ) ) ).
% bot_empty_eq2
thf(fact_187_bot__empty__eq2,axiom,
( bot_bot_b_prat_b_o
= ( ^ [X5: b > prat,Y: b] : ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ X5 @ Y ) @ bot_bo6399414329391923890prat_b ) ) ) ).
% bot_empty_eq2
thf(fact_188_bot__empty__eq2,axiom,
( bot_bot_o_o_o
= ( ^ [X5: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ bot_bo7073875226086086771od_o_o ) ) ) ).
% bot_empty_eq2
thf(fact_189_pred__equals__eq2,axiom,
! [R2: set_Pr2868754470712218413prat_b,S: set_Pr2868754470712218413prat_b] :
( ( ( ^ [X5: b > prat,Y: produc7326829233880046182prat_b] : ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ X5 @ Y ) @ R2 ) )
= ( ^ [X5: b > prat,Y: produc7326829233880046182prat_b] : ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ X5 @ Y ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_190_pred__equals__eq2,axiom,
! [R2: set_Pr8957522257986161222prat_b,S: set_Pr8957522257986161222prat_b] :
( ( ( ^ [X5: b > prat,Y: b] : ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ X5 @ Y ) @ R2 ) )
= ( ^ [X5: b > prat,Y: b] : ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ X5 @ Y ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_191_pred__equals__eq2,axiom,
! [R2: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( ( ^ [X5: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ R2 ) )
= ( ^ [X5: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_192_Set_Obind__bind,axiom,
! [A3: set_o,B3: $o > set_o,C2: $o > set_o] :
( ( bind_o_o @ ( bind_o_o @ A3 @ B3 ) @ C2 )
= ( bind_o_o @ A3
@ ^ [X5: $o] : ( bind_o_o @ ( B3 @ X5 ) @ C2 ) ) ) ).
% Set.bind_bind
thf(fact_193_padd__zero,axiom,
! [A: prat,B: prat] :
( ( pnone
= ( padd @ A @ B ) )
= ( ( A = pnone )
& ( B = pnone ) ) ) ).
% padd_zero
thf(fact_194_empty__mask_Opelims,axiom,
! [X2: b,Y4: prat] :
( ( ( empty_mask_b @ X2 )
= Y4 )
=> ( ( accp_b @ empty_mask_rel_b @ X2 )
=> ~ ( ( Y4 = pnone )
=> ~ ( accp_b @ empty_mask_rel_b @ X2 ) ) ) ) ).
% empty_mask.pelims
thf(fact_195_empty__mask_Opelims,axiom,
! [X2: produc8919727841856482381prat_b,Y4: prat] :
( ( ( empty_7098519468969535051prat_b @ X2 )
= Y4 )
=> ( ( accp_P3482027791507970390prat_b @ empty_2515944483442981012prat_b @ X2 )
=> ~ ( ( Y4 = pnone )
=> ~ ( accp_P3482027791507970390prat_b @ empty_2515944483442981012prat_b @ X2 ) ) ) ) ).
% empty_mask.pelims
thf(fact_196_prod_Oinject,axiom,
! [X1: b > prat,X22: produc7326829233880046182prat_b,Y1: b > prat,Y22: produc7326829233880046182prat_b] :
( ( ( produc4863932588836083261prat_b @ X1 @ X22 )
= ( produc4863932588836083261prat_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_197_prod_Oinject,axiom,
! [X1: b > prat,X22: b,Y1: b > prat,Y22: b] :
( ( ( produc7508179496679232342prat_b @ X1 @ X22 )
= ( produc7508179496679232342prat_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_198_prod_Oinject,axiom,
! [X1: $o,X22: $o,Y1: $o,Y22: $o] :
( ( ( product_Pair_o_o @ X1 @ X22 )
= ( product_Pair_o_o @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_199_old_Oprod_Oinject,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,A5: b > prat,B4: produc7326829233880046182prat_b] :
( ( ( produc4863932588836083261prat_b @ A @ B )
= ( produc4863932588836083261prat_b @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_200_old_Oprod_Oinject,axiom,
! [A: b > prat,B: b,A5: b > prat,B4: b] :
( ( ( produc7508179496679232342prat_b @ A @ B )
= ( produc7508179496679232342prat_b @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_201_old_Oprod_Oinject,axiom,
! [A: $o,B: $o,A5: $o,B4: $o] :
( ( ( product_Pair_o_o @ A @ B )
= ( product_Pair_o_o @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_202_accp_Ocases,axiom,
! [R3: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,A: produc8919727841856482381prat_b] :
( ( accp_P3482027791507970390prat_b @ R3 @ A )
=> ! [Y5: produc8919727841856482381prat_b] :
( ( R3 @ Y5 @ A )
=> ( accp_P3482027791507970390prat_b @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_203_accp_Osimps,axiom,
( accp_P3482027791507970390prat_b
= ( ^ [R: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,A2: produc8919727841856482381prat_b] :
? [X5: produc8919727841856482381prat_b] :
( ( A2 = X5 )
& ! [Y: produc8919727841856482381prat_b] :
( ( R @ Y @ X5 )
=> ( accp_P3482027791507970390prat_b @ R @ Y ) ) ) ) ) ).
% accp.simps
thf(fact_204_accpI,axiom,
! [R3: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,X2: produc8919727841856482381prat_b] :
( ! [Y3: produc8919727841856482381prat_b] :
( ( R3 @ Y3 @ X2 )
=> ( accp_P3482027791507970390prat_b @ R3 @ Y3 ) )
=> ( accp_P3482027791507970390prat_b @ R3 @ X2 ) ) ).
% accpI
thf(fact_205_accp__induct,axiom,
! [R3: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,A: produc8919727841856482381prat_b,P: produc8919727841856482381prat_b > $o] :
( ( accp_P3482027791507970390prat_b @ R3 @ A )
=> ( ! [X: produc8919727841856482381prat_b] :
( ( accp_P3482027791507970390prat_b @ R3 @ X )
=> ( ! [Y5: produc8919727841856482381prat_b] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_206_accp__downward,axiom,
! [R3: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,B: produc8919727841856482381prat_b,A: produc8919727841856482381prat_b] :
( ( accp_P3482027791507970390prat_b @ R3 @ B )
=> ( ( R3 @ A @ B )
=> ( accp_P3482027791507970390prat_b @ R3 @ A ) ) ) ).
% accp_downward
thf(fact_207_not__accp__down,axiom,
! [R2: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,X2: produc8919727841856482381prat_b] :
( ~ ( accp_P3482027791507970390prat_b @ R2 @ X2 )
=> ~ ! [Z4: produc8919727841856482381prat_b] :
( ( R2 @ Z4 @ X2 )
=> ( accp_P3482027791507970390prat_b @ R2 @ Z4 ) ) ) ).
% not_accp_down
thf(fact_208_prod__induct3,axiom,
! [P: produc8919727841856482381prat_b > $o,X2: produc8919727841856482381prat_b] :
( ! [A6: b > prat,B5: b > prat,C3: b] : ( P @ ( produc4863932588836083261prat_b @ A6 @ ( produc7508179496679232342prat_b @ B5 @ C3 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_209_prod__cases3,axiom,
! [Y4: produc8919727841856482381prat_b] :
~ ! [A6: b > prat,B5: b > prat,C3: b] :
( Y4
!= ( produc4863932588836083261prat_b @ A6 @ ( produc7508179496679232342prat_b @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_210_Pair__inject,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,A5: b > prat,B4: produc7326829233880046182prat_b] :
( ( ( produc4863932588836083261prat_b @ A @ B )
= ( produc4863932588836083261prat_b @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_211_Pair__inject,axiom,
! [A: b > prat,B: b,A5: b > prat,B4: b] :
( ( ( produc7508179496679232342prat_b @ A @ B )
= ( produc7508179496679232342prat_b @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_212_Pair__inject,axiom,
! [A: $o,B: $o,A5: $o,B4: $o] :
( ( ( product_Pair_o_o @ A @ B )
= ( product_Pair_o_o @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B = (~ B4) ) ) ) ).
% Pair_inject
thf(fact_213_prod__cases,axiom,
! [P: produc8919727841856482381prat_b > $o,P7: produc8919727841856482381prat_b] :
( ! [A6: b > prat,B5: produc7326829233880046182prat_b] : ( P @ ( produc4863932588836083261prat_b @ A6 @ B5 ) )
=> ( P @ P7 ) ) ).
% prod_cases
thf(fact_214_prod__cases,axiom,
! [P: produc7326829233880046182prat_b > $o,P7: produc7326829233880046182prat_b] :
( ! [A6: b > prat,B5: b] : ( P @ ( produc7508179496679232342prat_b @ A6 @ B5 ) )
=> ( P @ P7 ) ) ).
% prod_cases
thf(fact_215_prod__cases,axiom,
! [P: product_prod_o_o > $o,P7: product_prod_o_o] :
( ! [A6: $o,B5: $o] : ( P @ ( product_Pair_o_o @ A6 @ B5 ) )
=> ( P @ P7 ) ) ).
% prod_cases
thf(fact_216_surj__pair,axiom,
! [P7: produc8919727841856482381prat_b] :
? [X: b > prat,Y3: produc7326829233880046182prat_b] :
( P7
= ( produc4863932588836083261prat_b @ X @ Y3 ) ) ).
% surj_pair
thf(fact_217_surj__pair,axiom,
! [P7: produc7326829233880046182prat_b] :
? [X: b > prat,Y3: b] :
( P7
= ( produc7508179496679232342prat_b @ X @ Y3 ) ) ).
% surj_pair
thf(fact_218_surj__pair,axiom,
! [P7: product_prod_o_o] :
? [X: $o,Y3: $o] :
( P7
= ( product_Pair_o_o @ X @ Y3 ) ) ).
% surj_pair
thf(fact_219_old_Oprod_Oexhaust,axiom,
! [Y4: produc8919727841856482381prat_b] :
~ ! [A6: b > prat,B5: produc7326829233880046182prat_b] :
( Y4
!= ( produc4863932588836083261prat_b @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_220_old_Oprod_Oexhaust,axiom,
! [Y4: produc7326829233880046182prat_b] :
~ ! [A6: b > prat,B5: b] :
( Y4
!= ( produc7508179496679232342prat_b @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_221_old_Oprod_Oexhaust,axiom,
! [Y4: product_prod_o_o] :
~ ! [A6: $o,B5: $o] :
( Y4
!= ( product_Pair_o_o @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_222_accp__induct__rule,axiom,
! [R3: produc8919727841856482381prat_b > produc8919727841856482381prat_b > $o,A: produc8919727841856482381prat_b,P: produc8919727841856482381prat_b > $o] :
( ( accp_P3482027791507970390prat_b @ R3 @ A )
=> ( ! [X: produc8919727841856482381prat_b] :
( ( accp_P3482027791507970390prat_b @ R3 @ X )
=> ( ! [Y5: produc8919727841856482381prat_b] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_223_add__acc_Opelims,axiom,
! [X2: b > prat,Xa: b,Xb: prat,Y4: b > prat] :
( ( ( add_acc_b @ X2 @ Xa @ Xb )
= Y4 )
=> ( ( accp_P5301208041230799268b_prat @ add_acc_rel_b @ ( produc7983781276421977063b_prat @ X2 @ ( product_Pair_b_prat @ Xa @ Xb ) ) )
=> ~ ( ( Y4
= ( fun_upd_b_prat @ X2 @ Xa @ ( padd @ ( X2 @ Xa ) @ Xb ) ) )
=> ~ ( accp_P5301208041230799268b_prat @ add_acc_rel_b @ ( produc7983781276421977063b_prat @ X2 @ ( product_Pair_b_prat @ Xa @ Xb ) ) ) ) ) ) ).
% add_acc.pelims
thf(fact_224_add__masks_Opelims,axiom,
! [X2: b > prat,Xa: b > prat,Xb: b,Y4: prat] :
( ( ( add_masks_b @ X2 @ Xa @ Xb )
= Y4 )
=> ( ( accp_P3482027791507970390prat_b @ add_masks_rel_b @ ( produc4863932588836083261prat_b @ X2 @ ( produc7508179496679232342prat_b @ Xa @ Xb ) ) )
=> ~ ( ( Y4
= ( padd @ ( X2 @ Xb ) @ ( Xa @ Xb ) ) )
=> ~ ( accp_P3482027791507970390prat_b @ add_masks_rel_b @ ( produc4863932588836083261prat_b @ X2 @ ( produc7508179496679232342prat_b @ Xa @ Xb ) ) ) ) ) ) ).
% add_masks.pelims
thf(fact_225_Id__on__empty,axiom,
( ( id_on_b_prat @ bot_bot_set_b_prat )
= bot_bo6344516899060369779b_prat ) ).
% Id_on_empty
thf(fact_226_Id__on__empty,axiom,
( ( id_on_8897472890386745284od_o_o @ bot_bo7073875226086086771od_o_o )
= bot_bo2881837355931550515od_o_o ) ).
% Id_on_empty
thf(fact_227_Id__on__empty,axiom,
( ( id_on_set_o @ bot_bot_set_set_o )
= bot_bo8143257439550059635_set_o ) ).
% Id_on_empty
thf(fact_228_Id__on__empty,axiom,
( ( id_on_o @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Id_on_empty
thf(fact_229_image2__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,C: $o,G: $o > $o,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( bNF_Gr6698568825248092222_o_o_o @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_230_image2__eqI,axiom,
! [B: $o,F: set_o > $o,X2: set_o,C: $o,G: set_o > $o,A3: set_set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_set_o @ X2 @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( bNF_Gr8091873193056382750_o_o_o @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_231_image2__eqI,axiom,
! [B: b > prat,F: $o > b > prat,X2: $o,C: b,G: $o > b,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ B @ C ) @ ( bNF_Gr3107160545168091645prat_b @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_232_image2__eqI,axiom,
! [B: $o,F: ( b > prat ) > $o,X2: b > prat,C: $o,G: ( b > prat ) > $o,A3: set_b_prat] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_b_prat @ X2 @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( bNF_Gr3951398729878943206at_o_o @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_233_image2__eqI,axiom,
! [B: $o,F: product_prod_o_o > $o,X2: product_prod_o_o,C: $o,G: product_prod_o_o > $o,A3: set_Product_prod_o_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( bNF_Gr6873264481357710925_o_o_o @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_234_image2__eqI,axiom,
! [B: b > prat,F: set_o > b > prat,X2: set_o,C: b,G: set_o > b,A3: set_set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_set_o @ X2 @ A3 )
=> ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ B @ C ) @ ( bNF_Gr917350875349088477prat_b @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_235_image2__eqI,axiom,
! [B: b > prat,F: ( b > prat ) > b > prat,X2: b > prat,C: b,G: ( b > prat ) > b,A3: set_b_prat] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_b_prat @ X2 @ A3 )
=> ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ B @ C ) @ ( bNF_Gr1003551443158410149prat_b @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_236_image2__eqI,axiom,
! [B: b > prat,F: product_prod_o_o > b > prat,X2: product_prod_o_o,C: b,G: product_prod_o_o > b,A3: set_Product_prod_o_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ B @ C ) @ ( bNF_Gr6645471459577855884prat_b @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_237_image2__eqI,axiom,
! [B: b > prat,F: $o > b > prat,X2: $o,C: produc7326829233880046182prat_b,G: $o > produc7326829233880046182prat_b,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ B @ C ) @ ( bNF_Gr8394351763884875876prat_b @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_238_image2__eqI,axiom,
! [B: b > prat,F: set_o > b > prat,X2: set_o,C: produc7326829233880046182prat_b,G: set_o > produc7326829233880046182prat_b,A3: set_set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_set_o @ X2 @ A3 )
=> ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ B @ C ) @ ( bNF_Gr3559723194386784068prat_b @ A3 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_239_ssubst__Pair__rhs,axiom,
! [R3: b > prat,S2: produc7326829233880046182prat_b,R2: set_Pr2868754470712218413prat_b,S3: produc7326829233880046182prat_b] :
( ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ R3 @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ R3 @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_240_ssubst__Pair__rhs,axiom,
! [R3: b > prat,S2: b,R2: set_Pr8957522257986161222prat_b,S3: b] :
( ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ R3 @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ R3 @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_241_ssubst__Pair__rhs,axiom,
! [R3: $o,S2: $o,R2: set_Product_prod_o_o,S3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ R3 @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ R3 @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_242_is__singletonI,axiom,
! [X2: b > prat] : ( is_singleton_b_prat @ ( insert_b_prat @ X2 @ bot_bot_set_b_prat ) ) ).
% is_singletonI
thf(fact_243_is__singletonI,axiom,
! [X2: product_prod_o_o] : ( is_sin2666361102579702811od_o_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) ) ).
% is_singletonI
thf(fact_244_is__singletonI,axiom,
! [X2: set_o] : ( is_singleton_set_o @ ( insert_set_o @ X2 @ bot_bot_set_set_o ) ) ).
% is_singletonI
thf(fact_245_is__singletonI,axiom,
! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_246_curryI,axiom,
! [F: produc8919727841856482381prat_b > $o,A: b > prat,B: produc7326829233880046182prat_b] :
( ( F @ ( produc4863932588836083261prat_b @ A @ B ) )
=> ( produc2056597282251005671at_b_o @ F @ A @ B ) ) ).
% curryI
thf(fact_247_curryI,axiom,
! [F: produc7326829233880046182prat_b > $o,A: b > prat,B: b] :
( ( F @ ( produc7508179496679232342prat_b @ A @ B ) )
=> ( produc653278932388300558at_b_o @ F @ A @ B ) ) ).
% curryI
thf(fact_248_curryI,axiom,
! [F: product_prod_o_o > $o,A: $o,B: $o] :
( ( F @ ( product_Pair_o_o @ A @ B ) )
=> ( product_curry_o_o_o @ F @ A @ B ) ) ).
% curryI
thf(fact_249_insert__absorb2,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( insert6201435330877294327od_o_o @ X2 @ ( insert6201435330877294327od_o_o @ X2 @ A3 ) )
= ( insert6201435330877294327od_o_o @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_250_insert__absorb2,axiom,
! [X2: set_o,A3: set_set_o] :
( ( insert_set_o @ X2 @ ( insert_set_o @ X2 @ A3 ) )
= ( insert_set_o @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_251_insert__absorb2,axiom,
! [X2: $o,A3: set_o] :
( ( insert_o @ X2 @ ( insert_o @ X2 @ A3 ) )
= ( insert_o @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_252_insert__iff,axiom,
! [A: b > prat,B: b > prat,A3: set_b_prat] :
( ( member_b_prat @ A @ ( insert_b_prat @ B @ A3 ) )
= ( ( A = B )
| ( member_b_prat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_253_insert__iff,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ A3 ) )
= ( ( A = B )
| ( member7466972457876170832od_o_o @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_254_insert__iff,axiom,
! [A: set_o,B: set_o,A3: set_set_o] :
( ( member_set_o @ A @ ( insert_set_o @ B @ A3 ) )
= ( ( A = B )
| ( member_set_o @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_255_insert__iff,axiom,
! [A: $o,B: $o,A3: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A3 ) )
= ( ( A = B )
| ( member_o @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_256_insertCI,axiom,
! [A: b > prat,B3: set_b_prat,B: b > prat] :
( ( ~ ( member_b_prat @ A @ B3 )
=> ( A = B ) )
=> ( member_b_prat @ A @ ( insert_b_prat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_257_insertCI,axiom,
! [A: product_prod_o_o,B3: set_Product_prod_o_o,B: product_prod_o_o] :
( ( ~ ( member7466972457876170832od_o_o @ A @ B3 )
=> ( A = B ) )
=> ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ B3 ) ) ) ).
% insertCI
thf(fact_258_insertCI,axiom,
! [A: set_o,B3: set_set_o,B: set_o] :
( ( ~ ( member_set_o @ A @ B3 )
=> ( A = B ) )
=> ( member_set_o @ A @ ( insert_set_o @ B @ B3 ) ) ) ).
% insertCI
thf(fact_259_insertCI,axiom,
! [A: $o,B3: set_o,B: $o] :
( ( ~ ( member_o @ A @ B3 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B3 ) ) ) ).
% insertCI
thf(fact_260_singletonI,axiom,
! [A: b > prat] : ( member_b_prat @ A @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) ).
% singletonI
thf(fact_261_singletonI,axiom,
! [A: product_prod_o_o] : ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).
% singletonI
thf(fact_262_singletonI,axiom,
! [A: set_o] : ( member_set_o @ A @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singletonI
thf(fact_263_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_264_Id__onI,axiom,
! [A: b > prat,A3: set_b_prat] :
( ( member_b_prat @ A @ A3 )
=> ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ A @ A ) @ ( id_on_b_prat @ A3 ) ) ) ).
% Id_onI
thf(fact_265_Id__onI,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ A ) @ ( id_on_8897472890386745284od_o_o @ A3 ) ) ) ).
% Id_onI
thf(fact_266_Id__onI,axiom,
! [A: set_o,A3: set_set_o] :
( ( member_set_o @ A @ A3 )
=> ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ A ) @ ( id_on_set_o @ A3 ) ) ) ).
% Id_onI
thf(fact_267_Id__onI,axiom,
! [A: $o,A3: set_o] :
( ( member_o @ A @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( id_on_o @ A3 ) ) ) ).
% Id_onI
thf(fact_268_singleton__conv,axiom,
! [A: b > prat] :
( ( collect_b_prat
@ ^ [X5: b > prat] : ( X5 = A ) )
= ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) ).
% singleton_conv
thf(fact_269_singleton__conv,axiom,
! [A: product_prod_o_o] :
( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] : ( X5 = A ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).
% singleton_conv
thf(fact_270_singleton__conv,axiom,
! [A: set_o] :
( ( collect_set_o
@ ^ [X5: set_o] : ( X5 = A ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singleton_conv
thf(fact_271_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X5: $o] : ( X5 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_272_singleton__conv2,axiom,
! [A: b > prat] :
( ( collect_b_prat
@ ( ^ [Y2: b > prat,Z: b > prat] : ( Y2 = Z )
@ A ) )
= ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) ).
% singleton_conv2
thf(fact_273_singleton__conv2,axiom,
! [A: product_prod_o_o] :
( ( collec3167064739498627218od_o_o
@ ( ^ [Y2: product_prod_o_o,Z: product_prod_o_o] : ( Y2 = Z )
@ A ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).
% singleton_conv2
thf(fact_274_singleton__conv2,axiom,
! [A: set_o] :
( ( collect_set_o
@ ( ^ [Y2: set_o,Z: set_o] : ( Y2 = Z )
@ A ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singleton_conv2
thf(fact_275_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_276_add__masks_Ocases,axiom,
! [X2: produc8919727841856482381prat_b] :
~ ! [Pi7: b > prat,Pi8: b > prat,Hl2: b] :
( X2
!= ( produc4863932588836083261prat_b @ Pi7 @ ( produc7508179496679232342prat_b @ Pi8 @ Hl2 ) ) ) ).
% add_masks.cases
thf(fact_277_mk__disjoint__insert,axiom,
! [A: b > prat,A3: set_b_prat] :
( ( member_b_prat @ A @ A3 )
=> ? [B6: set_b_prat] :
( ( A3
= ( insert_b_prat @ A @ B6 ) )
& ~ ( member_b_prat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_278_mk__disjoint__insert,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A3 )
=> ? [B6: set_Product_prod_o_o] :
( ( A3
= ( insert6201435330877294327od_o_o @ A @ B6 ) )
& ~ ( member7466972457876170832od_o_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_279_mk__disjoint__insert,axiom,
! [A: set_o,A3: set_set_o] :
( ( member_set_o @ A @ A3 )
=> ? [B6: set_set_o] :
( ( A3
= ( insert_set_o @ A @ B6 ) )
& ~ ( member_set_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_280_mk__disjoint__insert,axiom,
! [A: $o,A3: set_o] :
( ( member_o @ A @ A3 )
=> ? [B6: set_o] :
( ( A3
= ( insert_o @ A @ B6 ) )
& ~ ( member_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_281_insert__commute,axiom,
! [X2: product_prod_o_o,Y4: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( insert6201435330877294327od_o_o @ X2 @ ( insert6201435330877294327od_o_o @ Y4 @ A3 ) )
= ( insert6201435330877294327od_o_o @ Y4 @ ( insert6201435330877294327od_o_o @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_282_insert__commute,axiom,
! [X2: set_o,Y4: set_o,A3: set_set_o] :
( ( insert_set_o @ X2 @ ( insert_set_o @ Y4 @ A3 ) )
= ( insert_set_o @ Y4 @ ( insert_set_o @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_283_insert__commute,axiom,
! [X2: $o,Y4: $o,A3: set_o] :
( ( insert_o @ X2 @ ( insert_o @ Y4 @ A3 ) )
= ( insert_o @ Y4 @ ( insert_o @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_284_insert__eq__iff,axiom,
! [A: b > prat,A3: set_b_prat,B: b > prat,B3: set_b_prat] :
( ~ ( member_b_prat @ A @ A3 )
=> ( ~ ( member_b_prat @ B @ B3 )
=> ( ( ( insert_b_prat @ A @ A3 )
= ( insert_b_prat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C4: set_b_prat] :
( ( A3
= ( insert_b_prat @ B @ C4 ) )
& ~ ( member_b_prat @ B @ C4 )
& ( B3
= ( insert_b_prat @ A @ C4 ) )
& ~ ( member_b_prat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_285_insert__eq__iff,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o,B: product_prod_o_o,B3: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ A @ A3 )
=> ( ~ ( member7466972457876170832od_o_o @ B @ B3 )
=> ( ( ( insert6201435330877294327od_o_o @ A @ A3 )
= ( insert6201435330877294327od_o_o @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C4: set_Product_prod_o_o] :
( ( A3
= ( insert6201435330877294327od_o_o @ B @ C4 ) )
& ~ ( member7466972457876170832od_o_o @ B @ C4 )
& ( B3
= ( insert6201435330877294327od_o_o @ A @ C4 ) )
& ~ ( member7466972457876170832od_o_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_286_insert__eq__iff,axiom,
! [A: set_o,A3: set_set_o,B: set_o,B3: set_set_o] :
( ~ ( member_set_o @ A @ A3 )
=> ( ~ ( member_set_o @ B @ B3 )
=> ( ( ( insert_set_o @ A @ A3 )
= ( insert_set_o @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_o] :
( ( A3
= ( insert_set_o @ B @ C4 ) )
& ~ ( member_set_o @ B @ C4 )
& ( B3
= ( insert_set_o @ A @ C4 ) )
& ~ ( member_set_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_287_insert__eq__iff,axiom,
! [A: $o,A3: set_o,B: $o,B3: set_o] :
( ~ ( member_o @ A @ A3 )
=> ( ~ ( member_o @ B @ B3 )
=> ( ( ( insert_o @ A @ A3 )
= ( insert_o @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A = (~ B) )
=> ? [C4: set_o] :
( ( A3
= ( insert_o @ B @ C4 ) )
& ~ ( member_o @ B @ C4 )
& ( B3
= ( insert_o @ A @ C4 ) )
& ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_288_insert__absorb,axiom,
! [A: b > prat,A3: set_b_prat] :
( ( member_b_prat @ A @ A3 )
=> ( ( insert_b_prat @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_289_insert__absorb,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( ( insert6201435330877294327od_o_o @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_290_insert__absorb,axiom,
! [A: set_o,A3: set_set_o] :
( ( member_set_o @ A @ A3 )
=> ( ( insert_set_o @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_291_insert__absorb,axiom,
! [A: $o,A3: set_o] :
( ( member_o @ A @ A3 )
=> ( ( insert_o @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_292_insert__ident,axiom,
! [X2: b > prat,A3: set_b_prat,B3: set_b_prat] :
( ~ ( member_b_prat @ X2 @ A3 )
=> ( ~ ( member_b_prat @ X2 @ B3 )
=> ( ( ( insert_b_prat @ X2 @ A3 )
= ( insert_b_prat @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_293_insert__ident,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( ~ ( member7466972457876170832od_o_o @ X2 @ B3 )
=> ( ( ( insert6201435330877294327od_o_o @ X2 @ A3 )
= ( insert6201435330877294327od_o_o @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_294_insert__ident,axiom,
! [X2: set_o,A3: set_set_o,B3: set_set_o] :
( ~ ( member_set_o @ X2 @ A3 )
=> ( ~ ( member_set_o @ X2 @ B3 )
=> ( ( ( insert_set_o @ X2 @ A3 )
= ( insert_set_o @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_295_insert__ident,axiom,
! [X2: $o,A3: set_o,B3: set_o] :
( ~ ( member_o @ X2 @ A3 )
=> ( ~ ( member_o @ X2 @ B3 )
=> ( ( ( insert_o @ X2 @ A3 )
= ( insert_o @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_296_Set_Oset__insert,axiom,
! [X2: b > prat,A3: set_b_prat] :
( ( member_b_prat @ X2 @ A3 )
=> ~ ! [B6: set_b_prat] :
( ( A3
= ( insert_b_prat @ X2 @ B6 ) )
=> ( member_b_prat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_297_Set_Oset__insert,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ~ ! [B6: set_Product_prod_o_o] :
( ( A3
= ( insert6201435330877294327od_o_o @ X2 @ B6 ) )
=> ( member7466972457876170832od_o_o @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_298_Set_Oset__insert,axiom,
! [X2: set_o,A3: set_set_o] :
( ( member_set_o @ X2 @ A3 )
=> ~ ! [B6: set_set_o] :
( ( A3
= ( insert_set_o @ X2 @ B6 ) )
=> ( member_set_o @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_299_Set_Oset__insert,axiom,
! [X2: $o,A3: set_o] :
( ( member_o @ X2 @ A3 )
=> ~ ! [B6: set_o] :
( ( A3
= ( insert_o @ X2 @ B6 ) )
=> ( member_o @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_300_insertI2,axiom,
! [A: b > prat,B3: set_b_prat,B: b > prat] :
( ( member_b_prat @ A @ B3 )
=> ( member_b_prat @ A @ ( insert_b_prat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_301_insertI2,axiom,
! [A: product_prod_o_o,B3: set_Product_prod_o_o,B: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ B3 )
=> ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ B3 ) ) ) ).
% insertI2
thf(fact_302_insertI2,axiom,
! [A: set_o,B3: set_set_o,B: set_o] :
( ( member_set_o @ A @ B3 )
=> ( member_set_o @ A @ ( insert_set_o @ B @ B3 ) ) ) ).
% insertI2
thf(fact_303_insertI2,axiom,
! [A: $o,B3: set_o,B: $o] :
( ( member_o @ A @ B3 )
=> ( member_o @ A @ ( insert_o @ B @ B3 ) ) ) ).
% insertI2
thf(fact_304_insertI1,axiom,
! [A: b > prat,B3: set_b_prat] : ( member_b_prat @ A @ ( insert_b_prat @ A @ B3 ) ) ).
% insertI1
thf(fact_305_insertI1,axiom,
! [A: product_prod_o_o,B3: set_Product_prod_o_o] : ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ A @ B3 ) ) ).
% insertI1
thf(fact_306_insertI1,axiom,
! [A: set_o,B3: set_set_o] : ( member_set_o @ A @ ( insert_set_o @ A @ B3 ) ) ).
% insertI1
thf(fact_307_insertI1,axiom,
! [A: $o,B3: set_o] : ( member_o @ A @ ( insert_o @ A @ B3 ) ) ).
% insertI1
thf(fact_308_insertE,axiom,
! [A: b > prat,B: b > prat,A3: set_b_prat] :
( ( member_b_prat @ A @ ( insert_b_prat @ B @ A3 ) )
=> ( ( A != B )
=> ( member_b_prat @ A @ A3 ) ) ) ).
% insertE
thf(fact_309_insertE,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ A3 ) )
=> ( ( A != B )
=> ( member7466972457876170832od_o_o @ A @ A3 ) ) ) ).
% insertE
thf(fact_310_insertE,axiom,
! [A: set_o,B: set_o,A3: set_set_o] :
( ( member_set_o @ A @ ( insert_set_o @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_o @ A @ A3 ) ) ) ).
% insertE
thf(fact_311_insertE,axiom,
! [A: $o,B: $o,A3: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A3 ) )
=> ( ( A = (~ B) )
=> ( member_o @ A @ A3 ) ) ) ).
% insertE
thf(fact_312_insert__compr,axiom,
( insert_b_prat
= ( ^ [A2: b > prat,B7: set_b_prat] :
( collect_b_prat
@ ^ [X5: b > prat] :
( ( X5 = A2 )
| ( member_b_prat @ X5 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_313_insert__compr,axiom,
( insert6201435330877294327od_o_o
= ( ^ [A2: product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( X5 = A2 )
| ( member7466972457876170832od_o_o @ X5 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_314_insert__compr,axiom,
( insert_set_o
= ( ^ [A2: set_o,B7: set_set_o] :
( collect_set_o
@ ^ [X5: set_o] :
( ( X5 = A2 )
| ( member_set_o @ X5 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_315_insert__compr,axiom,
( insert_o
= ( ^ [A2: $o,B7: set_o] :
( collect_o
@ ^ [X5: $o] :
( ( X5 = A2 )
| ( member_o @ X5 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_316_insert__Collect,axiom,
! [A: product_prod_o_o,P: product_prod_o_o > $o] :
( ( insert6201435330877294327od_o_o @ A @ ( collec3167064739498627218od_o_o @ P ) )
= ( collec3167064739498627218od_o_o
@ ^ [U: product_prod_o_o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_317_insert__Collect,axiom,
! [A: set_o,P: set_o > $o] :
( ( insert_set_o @ A @ ( collect_set_o @ P ) )
= ( collect_set_o
@ ^ [U: set_o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_318_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_319_singletonD,axiom,
! [B: b > prat,A: b > prat] :
( ( member_b_prat @ B @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_320_singletonD,axiom,
! [B: product_prod_o_o,A: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_321_singletonD,axiom,
! [B: set_o,A: set_o] :
( ( member_set_o @ B @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_322_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_323_singleton__iff,axiom,
! [B: b > prat,A: b > prat] :
( ( member_b_prat @ B @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_324_singleton__iff,axiom,
! [B: product_prod_o_o,A: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_325_singleton__iff,axiom,
! [B: set_o,A: set_o] :
( ( member_set_o @ B @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_326_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_327_doubleton__eq__iff,axiom,
! [A: b > prat,B: b > prat,C: b > prat,D: b > prat] :
( ( ( insert_b_prat @ A @ ( insert_b_prat @ B @ bot_bot_set_b_prat ) )
= ( insert_b_prat @ C @ ( insert_b_prat @ D @ bot_bot_set_b_prat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_328_doubleton__eq__iff,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,C: product_prod_o_o,D: product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
= ( insert6201435330877294327od_o_o @ C @ ( insert6201435330877294327od_o_o @ D @ bot_bo7073875226086086771od_o_o ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_329_doubleton__eq__iff,axiom,
! [A: set_o,B: set_o,C: set_o,D: set_o] :
( ( ( insert_set_o @ A @ ( insert_set_o @ B @ bot_bot_set_set_o ) )
= ( insert_set_o @ C @ ( insert_set_o @ D @ bot_bot_set_set_o ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_330_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_331_insert__not__empty,axiom,
! [A: b > prat,A3: set_b_prat] :
( ( insert_b_prat @ A @ A3 )
!= bot_bot_set_b_prat ) ).
% insert_not_empty
thf(fact_332_insert__not__empty,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( insert6201435330877294327od_o_o @ A @ A3 )
!= bot_bo7073875226086086771od_o_o ) ).
% insert_not_empty
thf(fact_333_insert__not__empty,axiom,
! [A: set_o,A3: set_set_o] :
( ( insert_set_o @ A @ A3 )
!= bot_bot_set_set_o ) ).
% insert_not_empty
thf(fact_334_insert__not__empty,axiom,
! [A: $o,A3: set_o] :
( ( insert_o @ A @ A3 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_335_singleton__inject,axiom,
! [A: b > prat,B: b > prat] :
( ( ( insert_b_prat @ A @ bot_bot_set_b_prat )
= ( insert_b_prat @ B @ bot_bot_set_b_prat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_336_singleton__inject,axiom,
! [A: product_prod_o_o,B: product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o )
= ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_337_singleton__inject,axiom,
! [A: set_o,B: set_o] :
( ( ( insert_set_o @ A @ bot_bot_set_set_o )
= ( insert_set_o @ B @ bot_bot_set_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_338_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_339_Collect__conv__if,axiom,
! [P: ( b > prat ) > $o,A: b > prat] :
( ( ( P @ A )
=> ( ( collect_b_prat
@ ^ [X5: b > prat] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b_prat
@ ^ [X5: b > prat] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_b_prat ) ) ) ).
% Collect_conv_if
thf(fact_340_Collect__conv__if,axiom,
! [P: product_prod_o_o > $o,A: product_prod_o_o] :
( ( ( P @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
& ( ~ ( P @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_conv_if
thf(fact_341_Collect__conv__if,axiom,
! [P: set_o > $o,A: set_o] :
( ( ( P @ A )
=> ( ( collect_set_o
@ ^ [X5: set_o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_o
@ ^ [X5: set_o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_set_o ) ) ) ).
% Collect_conv_if
thf(fact_342_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_343_Collect__conv__if2,axiom,
! [P: ( b > prat ) > $o,A: b > prat] :
( ( ( P @ A )
=> ( ( collect_b_prat
@ ^ [X5: b > prat] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b_prat
@ ^ [X5: b > prat] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_b_prat ) ) ) ).
% Collect_conv_if2
thf(fact_344_Collect__conv__if2,axiom,
! [P: product_prod_o_o > $o,A: product_prod_o_o] :
( ( ( P @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
& ( ~ ( P @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_conv_if2
thf(fact_345_Collect__conv__if2,axiom,
! [P: set_o > $o,A: set_o] :
( ( ( P @ A )
=> ( ( collect_set_o
@ ^ [X5: set_o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_o
@ ^ [X5: set_o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_346_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_347_Id__onE,axiom,
! [C: produc7696030361869858855b_prat,A3: set_b_prat] :
( ( member6376894174024132944b_prat @ C @ ( id_on_b_prat @ A3 ) )
=> ~ ! [X: b > prat] :
( ( member_b_prat @ X @ A3 )
=> ( C
!= ( produc6129551849031254551b_prat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_348_Id__onE,axiom,
! [C: produc4465838814804890855od_o_o,A3: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ C @ ( id_on_8897472890386745284od_o_o @ A3 ) )
=> ~ ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A3 )
=> ( C
!= ( produc3331439552374987607od_o_o @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_349_Id__onE,axiom,
! [C: produc7369051934464679207_set_o,A3: set_set_o] :
( ( member9116954335612470352_set_o @ C @ ( id_on_set_o @ A3 ) )
=> ~ ! [X: set_o] :
( ( member_set_o @ X @ A3 )
=> ( C
!= ( produc5838405689764958487_set_o @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_350_Id__onE,axiom,
! [C: product_prod_o_o,A3: set_o] :
( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A3 ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A3 )
=> ( C
!= ( product_Pair_o_o @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_351_Id__on__eqI,axiom,
! [A: b > prat,B: b > prat,A3: set_b_prat] :
( ( A = B )
=> ( ( member_b_prat @ A @ A3 )
=> ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ A @ B ) @ ( id_on_b_prat @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_352_Id__on__eqI,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( A = B )
=> ( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( id_on_8897472890386745284od_o_o @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_353_Id__on__eqI,axiom,
! [A: set_o,B: set_o,A3: set_set_o] :
( ( A = B )
=> ( ( member_set_o @ A @ A3 )
=> ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ B ) @ ( id_on_set_o @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_354_Id__on__eqI,axiom,
! [A: $o,B: $o,A3: set_o] :
( ( A = B )
=> ( ( member_o @ A @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( id_on_o @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_355_Id__on__iff,axiom,
! [X2: b > prat,Y4: b > prat,A3: set_b_prat] :
( ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ X2 @ Y4 ) @ ( id_on_b_prat @ A3 ) )
= ( ( X2 = Y4 )
& ( member_b_prat @ X2 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_356_Id__on__iff,axiom,
! [X2: product_prod_o_o,Y4: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ Y4 ) @ ( id_on_8897472890386745284od_o_o @ A3 ) )
= ( ( X2 = Y4 )
& ( member7466972457876170832od_o_o @ X2 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_357_Id__on__iff,axiom,
! [X2: set_o,Y4: set_o,A3: set_set_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X2 @ Y4 ) @ ( id_on_set_o @ A3 ) )
= ( ( X2 = Y4 )
& ( member_set_o @ X2 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_358_Id__on__iff,axiom,
! [X2: $o,Y4: $o,A3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ ( id_on_o @ A3 ) )
= ( ( X2 = Y4 )
& ( member_o @ X2 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_359_curryD,axiom,
! [F: produc8919727841856482381prat_b > $o,A: b > prat,B: produc7326829233880046182prat_b] :
( ( produc2056597282251005671at_b_o @ F @ A @ B )
=> ( F @ ( produc4863932588836083261prat_b @ A @ B ) ) ) ).
% curryD
thf(fact_360_curryD,axiom,
! [F: produc7326829233880046182prat_b > $o,A: b > prat,B: b] :
( ( produc653278932388300558at_b_o @ F @ A @ B )
=> ( F @ ( produc7508179496679232342prat_b @ A @ B ) ) ) ).
% curryD
thf(fact_361_curryD,axiom,
! [F: product_prod_o_o > $o,A: $o,B: $o] :
( ( product_curry_o_o_o @ F @ A @ B )
=> ( F @ ( product_Pair_o_o @ A @ B ) ) ) ).
% curryD
thf(fact_362_curryE,axiom,
! [F: produc8919727841856482381prat_b > $o,A: b > prat,B: produc7326829233880046182prat_b] :
( ( produc2056597282251005671at_b_o @ F @ A @ B )
=> ( F @ ( produc4863932588836083261prat_b @ A @ B ) ) ) ).
% curryE
thf(fact_363_curryE,axiom,
! [F: produc7326829233880046182prat_b > $o,A: b > prat,B: b] :
( ( produc653278932388300558at_b_o @ F @ A @ B )
=> ( F @ ( produc7508179496679232342prat_b @ A @ B ) ) ) ).
% curryE
thf(fact_364_curryE,axiom,
! [F: product_prod_o_o > $o,A: $o,B: $o] :
( ( product_curry_o_o_o @ F @ A @ B )
=> ( F @ ( product_Pair_o_o @ A @ B ) ) ) ).
% curryE
thf(fact_365_is__singleton__def,axiom,
( is_singleton_b_prat
= ( ^ [A4: set_b_prat] :
? [X5: b > prat] :
( A4
= ( insert_b_prat @ X5 @ bot_bot_set_b_prat ) ) ) ) ).
% is_singleton_def
thf(fact_366_is__singleton__def,axiom,
( is_sin2666361102579702811od_o_o
= ( ^ [A4: set_Product_prod_o_o] :
? [X5: product_prod_o_o] :
( A4
= ( insert6201435330877294327od_o_o @ X5 @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% is_singleton_def
thf(fact_367_is__singleton__def,axiom,
( is_singleton_set_o
= ( ^ [A4: set_set_o] :
? [X5: set_o] :
( A4
= ( insert_set_o @ X5 @ bot_bot_set_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_368_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
? [X5: $o] :
( A4
= ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_369_is__singletonE,axiom,
! [A3: set_b_prat] :
( ( is_singleton_b_prat @ A3 )
=> ~ ! [X: b > prat] :
( A3
!= ( insert_b_prat @ X @ bot_bot_set_b_prat ) ) ) ).
% is_singletonE
thf(fact_370_is__singletonE,axiom,
! [A3: set_Product_prod_o_o] :
( ( is_sin2666361102579702811od_o_o @ A3 )
=> ~ ! [X: product_prod_o_o] :
( A3
!= ( insert6201435330877294327od_o_o @ X @ bot_bo7073875226086086771od_o_o ) ) ) ).
% is_singletonE
thf(fact_371_is__singletonE,axiom,
! [A3: set_set_o] :
( ( is_singleton_set_o @ A3 )
=> ~ ! [X: set_o] :
( A3
!= ( insert_set_o @ X @ bot_bot_set_set_o ) ) ) ).
% is_singletonE
thf(fact_372_is__singletonE,axiom,
! [A3: set_o] :
( ( is_singleton_o @ A3 )
=> ~ ! [X: $o] :
( A3
!= ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_373_override__on__insert_H,axiom,
! [F: $o > set_o,G: $o > set_o,X2: $o,X7: set_o] :
( ( override_on_o_set_o @ F @ G @ ( insert_o @ X2 @ X7 ) )
= ( override_on_o_set_o @ ( fun_upd_o_set_o @ F @ X2 @ ( G @ X2 ) ) @ G @ X7 ) ) ).
% override_on_insert'
thf(fact_374_override__on__insert_H,axiom,
! [F: $o > $o,G: $o > $o,X2: $o,X7: set_o] :
( ( override_on_o_o @ F @ G @ ( insert_o @ X2 @ X7 ) )
= ( override_on_o_o @ ( fun_upd_o_o @ F @ X2 @ ( G @ X2 ) ) @ G @ X7 ) ) ).
% override_on_insert'
thf(fact_375_override__on__insert,axiom,
! [F: $o > set_o,G: $o > set_o,X2: $o,X7: set_o] :
( ( override_on_o_set_o @ F @ G @ ( insert_o @ X2 @ X7 ) )
= ( fun_upd_o_set_o @ ( override_on_o_set_o @ F @ G @ X7 ) @ X2 @ ( G @ X2 ) ) ) ).
% override_on_insert
thf(fact_376_override__on__insert,axiom,
! [F: $o > $o,G: $o > $o,X2: $o,X7: set_o] :
( ( override_on_o_o @ F @ G @ ( insert_o @ X2 @ X7 ) )
= ( fun_upd_o_o @ ( override_on_o_o @ F @ G @ X7 ) @ X2 @ ( G @ X2 ) ) ) ).
% override_on_insert
thf(fact_377_is__singleton__the__elem,axiom,
( is_singleton_b_prat
= ( ^ [A4: set_b_prat] :
( A4
= ( insert_b_prat @ ( the_elem_b_prat @ A4 ) @ bot_bot_set_b_prat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_378_is__singleton__the__elem,axiom,
( is_sin2666361102579702811od_o_o
= ( ^ [A4: set_Product_prod_o_o] :
( A4
= ( insert6201435330877294327od_o_o @ ( the_el301066130825229084od_o_o @ A4 ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_379_is__singleton__the__elem,axiom,
( is_singleton_set_o
= ( ^ [A4: set_set_o] :
( A4
= ( insert_set_o @ ( the_elem_set_o @ A4 ) @ bot_bot_set_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_380_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
( A4
= ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_381_the__elem__eq,axiom,
! [X2: b > prat] :
( ( the_elem_b_prat @ ( insert_b_prat @ X2 @ bot_bot_set_b_prat ) )
= X2 ) ).
% the_elem_eq
thf(fact_382_the__elem__eq,axiom,
! [X2: product_prod_o_o] :
( ( the_el301066130825229084od_o_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_383_the__elem__eq,axiom,
! [X2: set_o] :
( ( the_elem_set_o @ ( insert_set_o @ X2 @ bot_bot_set_set_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_384_the__elem__eq,axiom,
! [X2: $o] :
( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_385_in__lex__prod,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,A5: b > prat,B4: produc7326829233880046182prat_b,R3: set_Pr525462856049757959b_prat,S2: set_Pr3106385887818360711prat_b] :
( ( member6556583152774367120prat_b @ ( produc642807067915301591prat_b @ ( produc4863932588836083261prat_b @ A @ B ) @ ( produc4863932588836083261prat_b @ A5 @ B4 ) ) @ ( lex_pr273246689001183829prat_b @ R3 @ S2 ) )
= ( ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member3390411761967995344prat_b @ ( produc4865444734361148823prat_b @ B @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_386_in__lex__prod,axiom,
! [A: b > prat,B: b,A5: b > prat,B4: b,R3: set_Pr525462856049757959b_prat,S2: set_Product_prod_b_b] :
( ( member3390411761967995344prat_b @ ( produc4865444734361148823prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ ( produc7508179496679232342prat_b @ A5 @ B4 ) ) @ ( lex_prod_b_prat_b @ R3 @ S2 ) )
= ( ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_387_in__lex__prod,axiom,
! [A: $o,B: $o,A5: $o,B4: $o,R3: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Pair_o_o @ A5 @ B4 ) ) @ ( lex_prod_o_o @ R3 @ S2 ) )
= ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_388_same__fstI,axiom,
! [P: ( b > prat ) > $o,X2: b > prat,Y6: produc7326829233880046182prat_b,Y4: produc7326829233880046182prat_b,R2: ( b > prat ) > set_Pr3106385887818360711prat_b] :
( ( P @ X2 )
=> ( ( member3390411761967995344prat_b @ ( produc4865444734361148823prat_b @ Y6 @ Y4 ) @ ( R2 @ X2 ) )
=> ( member6556583152774367120prat_b @ ( produc642807067915301591prat_b @ ( produc4863932588836083261prat_b @ X2 @ Y6 ) @ ( produc4863932588836083261prat_b @ X2 @ Y4 ) ) @ ( same_f7802962448890730258prat_b @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_389_same__fstI,axiom,
! [P: ( b > prat ) > $o,X2: b > prat,Y6: b,Y4: b,R2: ( b > prat ) > set_Product_prod_b_b] :
( ( P @ X2 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y6 @ Y4 ) @ ( R2 @ X2 ) )
=> ( member3390411761967995344prat_b @ ( produc4865444734361148823prat_b @ ( produc7508179496679232342prat_b @ X2 @ Y6 ) @ ( produc7508179496679232342prat_b @ X2 @ Y4 ) ) @ ( same_fst_b_prat_b @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_390_same__fstI,axiom,
! [P: $o > $o,X2: $o,Y6: $o,Y4: $o,R2: $o > set_Product_prod_o_o] :
( ( P @ X2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y6 @ Y4 ) @ ( R2 @ X2 ) )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ X2 @ Y6 ) @ ( product_Pair_o_o @ X2 @ Y4 ) ) @ ( same_fst_o_o @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_391_refl__on__singleton,axiom,
! [X2: b > prat] : ( refl_on_b_prat @ ( insert_b_prat @ X2 @ bot_bot_set_b_prat ) @ ( insert6057866348176205303b_prat @ ( produc6129551849031254551b_prat @ X2 @ X2 ) @ bot_bo6344516899060369779b_prat ) ) ).
% refl_on_singleton
thf(fact_392_refl__on__singleton,axiom,
! [X2: product_prod_o_o] : ( refl_o8108712344181003046od_o_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ X2 ) @ bot_bo2881837355931550515od_o_o ) ) ).
% refl_on_singleton
thf(fact_393_refl__on__singleton,axiom,
! [X2: set_o] : ( refl_on_set_o @ ( insert_set_o @ X2 @ bot_bot_set_set_o ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ X2 @ X2 ) @ bot_bo8143257439550059635_set_o ) ) ).
% refl_on_singleton
thf(fact_394_refl__on__singleton,axiom,
! [X2: $o] : ( refl_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% refl_on_singleton
thf(fact_395_linear__order__on__singleton,axiom,
! [X2: b > prat] : ( order_2115221884338004981b_prat @ ( insert_b_prat @ X2 @ bot_bot_set_b_prat ) @ ( insert6057866348176205303b_prat @ ( produc6129551849031254551b_prat @ X2 @ X2 ) @ bot_bo6344516899060369779b_prat ) ) ).
% linear_order_on_singleton
thf(fact_396_linear__order__on__singleton,axiom,
! [X2: product_prod_o_o] : ( order_7019990371155863132od_o_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ X2 ) @ bot_bo2881837355931550515od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_397_linear__order__on__singleton,axiom,
! [X2: set_o] : ( order_1696139471214784045_set_o @ ( insert_set_o @ X2 @ bot_bot_set_set_o ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ X2 @ X2 ) @ bot_bo8143257439550059635_set_o ) ) ).
% linear_order_on_singleton
thf(fact_398_linear__order__on__singleton,axiom,
! [X2: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_399_bind__singleton__conv__image,axiom,
! [A3: set_o,F: $o > set_set_o] :
( ( bind_o_set_set_o @ A3
@ ^ [X5: $o] : ( insert_set_set_o @ ( F @ X5 ) @ bot_bo3529510225556782244_set_o ) )
= ( image_o_set_set_o @ F @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_400_bind__singleton__conv__image,axiom,
! [A3: set_o,F: $o > $o] :
( ( bind_o_o @ A3
@ ^ [X5: $o] : ( insert_o @ ( F @ X5 ) @ bot_bot_set_o ) )
= ( image_o_o2 @ F @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_401_bind__singleton__conv__image,axiom,
! [A3: set_o,F: $o > set_o] :
( ( bind_o_set_o @ A3
@ ^ [X5: $o] : ( insert_set_o @ ( F @ X5 ) @ bot_bot_set_set_o ) )
= ( image_o_set_o2 @ F @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_402_bind__singleton__conv__image,axiom,
! [A3: set_set_o,F: set_o > set_o] :
( ( bind_set_o_set_o @ A3
@ ^ [X5: set_o] : ( insert_set_o @ ( F @ X5 ) @ bot_bot_set_set_o ) )
= ( image_set_o_set_o2 @ F @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_403_Image__singleton__iff,axiom,
! [B: $o,R3: set_Product_prod_o_o,A: $o] :
( ( member_o @ B @ ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_404_Image__singleton__iff,axiom,
! [B: set_o,R3: set_Pr3508996721749391591_set_o,A: $o] :
( ( member_set_o @ B @ ( image_o_set_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member8938118433653312304_set_o @ ( product_Pair_o_set_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_405_Image__singleton__iff,axiom,
! [B: $o,R3: set_Pr5232024646447722279et_o_o,A: set_o] :
( ( member_o @ B @ ( image_set_o_o @ R3 @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
= ( member6599968234716717424et_o_o @ ( product_Pair_set_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_406_Image__singleton__iff,axiom,
! [B: b > prat,R3: set_Pr5930873434710784687b_prat,A: $o] :
( ( member_b_prat @ B @ ( image_o_b_prat @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member7584064603762627832b_prat @ ( produc6454426705846175679b_prat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_407_Image__singleton__iff,axiom,
! [B: product_prod_o_o,R3: set_Pr5237325540968292374od_o_o,A: $o] :
( ( member7466972457876170832od_o_o @ B @ ( image_5252864891027546466od_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member3692453414038245343od_o_o @ ( produc2690343539598703398od_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_408_Image__singleton__iff,axiom,
! [B: $o,R3: set_Pr5078033069183545951prat_o,A: b > prat] :
( ( member_o @ B @ ( image_b_prat_o @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
= ( member5777084206023828648prat_o @ ( produc6389352210403673967prat_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_409_Image__singleton__iff,axiom,
! [B: b,R3: set_Pr8957522257986161222prat_b,A: b > prat] :
( ( member_b @ B @ ( image_b_prat_b @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
= ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_410_Image__singleton__iff,axiom,
! [B: $o,R3: set_Pr2246928807395521336_o_o_o,A: product_prod_o_o] :
( ( member_o @ B @ ( image_9092160538811120644_o_o_o @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
= ( member5943846676503041025_o_o_o @ ( produc6529639187382277576_o_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_411_Image__singleton__iff,axiom,
! [B: set_o,R3: set_Pr4577759397028426247_set_o,A: set_o] :
( ( member_set_o @ B @ ( image_set_o_set_o @ R3 @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
= ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_412_Image__singleton__iff,axiom,
! [B: set_o,R3: set_Pr5661576500205409599_set_o,A: b > prat] :
( ( member_set_o @ B @ ( image_b_prat_set_o @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
= ( member8295766130977777544_set_o @ ( produc5638353472928494159_set_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_413_image__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member_o @ B @ ( image_o_o2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_414_image__eqI,axiom,
! [B: set_o,F: $o > set_o,X2: $o,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member_set_o @ B @ ( image_o_set_o2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_415_image__eqI,axiom,
! [B: $o,F: set_o > $o,X2: set_o,A3: set_set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_o @ X2 @ A3 )
=> ( member_o @ B @ ( image_set_o_o2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_416_image__eqI,axiom,
! [B: $o,F: ( b > prat ) > $o,X2: b > prat,A3: set_b_prat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b_prat @ X2 @ A3 )
=> ( member_o @ B @ ( image_b_prat_o2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_417_image__eqI,axiom,
! [B: set_set_o,F: $o > set_set_o,X2: $o,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member_set_set_o @ B @ ( image_o_set_set_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_418_image__eqI,axiom,
! [B: b > prat,F: $o > b > prat,X2: $o,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member_b_prat @ B @ ( image_o_b_prat2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_419_image__eqI,axiom,
! [B: product_prod_o_o,F: $o > product_prod_o_o,X2: $o,A3: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A3 )
=> ( member7466972457876170832od_o_o @ B @ ( image_4057150146340385428od_o_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_420_image__eqI,axiom,
! [B: $o,F: product_prod_o_o > $o,X2: product_prod_o_o,A3: set_Product_prod_o_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( member_o @ B @ ( image_7896445794123959606_o_o_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_421_image__eqI,axiom,
! [B: set_o,F: set_o > set_o,X2: set_o,A3: set_set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_o @ X2 @ A3 )
=> ( member_set_o @ B @ ( image_set_o_set_o2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_422_image__eqI,axiom,
! [B: set_o,F: ( b > prat ) > set_o,X2: b > prat,A3: set_b_prat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b_prat @ X2 @ A3 )
=> ( member_set_o @ B @ ( image_b_prat_set_o2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_423_image__ident,axiom,
! [Y7: set_o] :
( ( image_o_o2
@ ^ [X5: $o] : X5
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_424_image__ident,axiom,
! [Y7: set_set_o] :
( ( image_set_o_set_o2
@ ^ [X5: set_o] : X5
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_425_image__is__empty,axiom,
! [F: $o > $o,A3: set_o] :
( ( ( image_o_o2 @ F @ A3 )
= bot_bot_set_o )
= ( A3 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_426_image__is__empty,axiom,
! [F: set_o > $o,A3: set_set_o] :
( ( ( image_set_o_o2 @ F @ A3 )
= bot_bot_set_o )
= ( A3 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_427_image__is__empty,axiom,
! [F: $o > set_o,A3: set_o] :
( ( ( image_o_set_o2 @ F @ A3 )
= bot_bot_set_set_o )
= ( A3 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_428_image__is__empty,axiom,
! [F: $o > set_set_o,A3: set_o] :
( ( ( image_o_set_set_o @ F @ A3 )
= bot_bo3529510225556782244_set_o )
= ( A3 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_429_image__is__empty,axiom,
! [F: ( b > prat ) > $o,A3: set_b_prat] :
( ( ( image_b_prat_o2 @ F @ A3 )
= bot_bot_set_o )
= ( A3 = bot_bot_set_b_prat ) ) ).
% image_is_empty
thf(fact_430_image__is__empty,axiom,
! [F: product_prod_o_o > $o,A3: set_Product_prod_o_o] :
( ( ( image_7896445794123959606_o_o_o @ F @ A3 )
= bot_bot_set_o )
= ( A3 = bot_bo7073875226086086771od_o_o ) ) ).
% image_is_empty
thf(fact_431_image__is__empty,axiom,
! [F: $o > b > prat,A3: set_o] :
( ( ( image_o_b_prat2 @ F @ A3 )
= bot_bot_set_b_prat )
= ( A3 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_432_image__is__empty,axiom,
! [F: $o > product_prod_o_o,A3: set_o] :
( ( ( image_4057150146340385428od_o_o @ F @ A3 )
= bot_bo7073875226086086771od_o_o )
= ( A3 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_433_image__is__empty,axiom,
! [F: set_o > set_o,A3: set_set_o] :
( ( ( image_set_o_set_o2 @ F @ A3 )
= bot_bot_set_set_o )
= ( A3 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_434_image__is__empty,axiom,
! [F: set_o > b > prat,A3: set_set_o] :
( ( ( image_set_o_b_prat2 @ F @ A3 )
= bot_bot_set_b_prat )
= ( A3 = bot_bot_set_set_o ) ) ).
% image_is_empty
thf(fact_435_empty__is__image,axiom,
! [F: $o > $o,A3: set_o] :
( ( bot_bot_set_o
= ( image_o_o2 @ F @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_436_empty__is__image,axiom,
! [F: set_o > $o,A3: set_set_o] :
( ( bot_bot_set_o
= ( image_set_o_o2 @ F @ A3 ) )
= ( A3 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_437_empty__is__image,axiom,
! [F: $o > set_o,A3: set_o] :
( ( bot_bot_set_set_o
= ( image_o_set_o2 @ F @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_438_empty__is__image,axiom,
! [F: $o > set_set_o,A3: set_o] :
( ( bot_bo3529510225556782244_set_o
= ( image_o_set_set_o @ F @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_439_empty__is__image,axiom,
! [F: ( b > prat ) > $o,A3: set_b_prat] :
( ( bot_bot_set_o
= ( image_b_prat_o2 @ F @ A3 ) )
= ( A3 = bot_bot_set_b_prat ) ) ).
% empty_is_image
thf(fact_440_empty__is__image,axiom,
! [F: product_prod_o_o > $o,A3: set_Product_prod_o_o] :
( ( bot_bot_set_o
= ( image_7896445794123959606_o_o_o @ F @ A3 ) )
= ( A3 = bot_bo7073875226086086771od_o_o ) ) ).
% empty_is_image
thf(fact_441_empty__is__image,axiom,
! [F: $o > b > prat,A3: set_o] :
( ( bot_bot_set_b_prat
= ( image_o_b_prat2 @ F @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_442_empty__is__image,axiom,
! [F: $o > product_prod_o_o,A3: set_o] :
( ( bot_bo7073875226086086771od_o_o
= ( image_4057150146340385428od_o_o @ F @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_443_empty__is__image,axiom,
! [F: set_o > set_o,A3: set_set_o] :
( ( bot_bot_set_set_o
= ( image_set_o_set_o2 @ F @ A3 ) )
= ( A3 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_444_empty__is__image,axiom,
! [F: set_o > b > prat,A3: set_set_o] :
( ( bot_bot_set_b_prat
= ( image_set_o_b_prat2 @ F @ A3 ) )
= ( A3 = bot_bot_set_set_o ) ) ).
% empty_is_image
thf(fact_445_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o2 @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_446_image__empty,axiom,
! [F: $o > set_o] :
( ( image_o_set_o2 @ F @ bot_bot_set_o )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_447_image__empty,axiom,
! [F: set_o > $o] :
( ( image_set_o_o2 @ F @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_448_image__empty,axiom,
! [F: $o > set_set_o] :
( ( image_o_set_set_o @ F @ bot_bot_set_o )
= bot_bo3529510225556782244_set_o ) ).
% image_empty
thf(fact_449_image__empty,axiom,
! [F: $o > b > prat] :
( ( image_o_b_prat2 @ F @ bot_bot_set_o )
= bot_bot_set_b_prat ) ).
% image_empty
thf(fact_450_image__empty,axiom,
! [F: $o > product_prod_o_o] :
( ( image_4057150146340385428od_o_o @ F @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% image_empty
thf(fact_451_image__empty,axiom,
! [F: ( b > prat ) > $o] :
( ( image_b_prat_o2 @ F @ bot_bot_set_b_prat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_452_image__empty,axiom,
! [F: product_prod_o_o > $o] :
( ( image_7896445794123959606_o_o_o @ F @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_453_image__empty,axiom,
! [F: set_o > set_o] :
( ( image_set_o_set_o2 @ F @ bot_bot_set_set_o )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_454_image__empty,axiom,
! [F: ( b > prat ) > set_o] :
( ( image_b_prat_set_o2 @ F @ bot_bot_set_b_prat )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_455_image__insert,axiom,
! [F: $o > set_set_o,A: $o,B3: set_o] :
( ( image_o_set_set_o @ F @ ( insert_o @ A @ B3 ) )
= ( insert_set_set_o @ ( F @ A ) @ ( image_o_set_set_o @ F @ B3 ) ) ) ).
% image_insert
thf(fact_456_image__insert,axiom,
! [F: $o > product_prod_o_o,A: $o,B3: set_o] :
( ( image_4057150146340385428od_o_o @ F @ ( insert_o @ A @ B3 ) )
= ( insert6201435330877294327od_o_o @ ( F @ A ) @ ( image_4057150146340385428od_o_o @ F @ B3 ) ) ) ).
% image_insert
thf(fact_457_image__insert,axiom,
! [F: $o > set_o,A: $o,B3: set_o] :
( ( image_o_set_o2 @ F @ ( insert_o @ A @ B3 ) )
= ( insert_set_o @ ( F @ A ) @ ( image_o_set_o2 @ F @ B3 ) ) ) ).
% image_insert
thf(fact_458_image__insert,axiom,
! [F: product_prod_o_o > $o,A: product_prod_o_o,B3: set_Product_prod_o_o] :
( ( image_7896445794123959606_o_o_o @ F @ ( insert6201435330877294327od_o_o @ A @ B3 ) )
= ( insert_o @ ( F @ A ) @ ( image_7896445794123959606_o_o_o @ F @ B3 ) ) ) ).
% image_insert
thf(fact_459_image__insert,axiom,
! [F: product_prod_o_o > product_prod_o_o,A: product_prod_o_o,B3: set_Product_prod_o_o] :
( ( image_9131363867636255685od_o_o @ F @ ( insert6201435330877294327od_o_o @ A @ B3 ) )
= ( insert6201435330877294327od_o_o @ ( F @ A ) @ ( image_9131363867636255685od_o_o @ F @ B3 ) ) ) ).
% image_insert
thf(fact_460_image__insert,axiom,
! [F: product_prod_o_o > set_o,A: product_prod_o_o,B3: set_Product_prod_o_o] :
( ( image_7870716977482210326_set_o @ F @ ( insert6201435330877294327od_o_o @ A @ B3 ) )
= ( insert_set_o @ ( F @ A ) @ ( image_7870716977482210326_set_o @ F @ B3 ) ) ) ).
% image_insert
thf(fact_461_image__insert,axiom,
! [F: set_o > $o,A: set_o,B3: set_set_o] :
( ( image_set_o_o2 @ F @ ( insert_set_o @ A @ B3 ) )
= ( insert_o @ ( F @ A ) @ ( image_set_o_o2 @ F @ B3 ) ) ) ).
% image_insert
thf(fact_462_image__insert,axiom,
! [F: set_o > product_prod_o_o,A: set_o,B3: set_set_o] :
( ( image_6524031425630931892od_o_o @ F @ ( insert_set_o @ A @ B3 ) )
= ( insert6201435330877294327od_o_o @ ( F @ A ) @ ( image_6524031425630931892od_o_o @ F @ B3 ) ) ) ).
% image_insert
thf(fact_463_image__insert,axiom,
! [F: set_o > set_o,A: set_o,B3: set_set_o] :
( ( image_set_o_set_o2 @ F @ ( insert_set_o @ A @ B3 ) )
= ( insert_set_o @ ( F @ A ) @ ( image_set_o_set_o2 @ F @ B3 ) ) ) ).
% image_insert
thf(fact_464_image__insert,axiom,
! [F: $o > $o,A: $o,B3: set_o] :
( ( image_o_o2 @ F @ ( insert_o @ A @ B3 ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o2 @ F @ B3 ) ) ) ).
% image_insert
thf(fact_465_insert__image,axiom,
! [X2: $o,A3: set_o,F: $o > $o] :
( ( member_o @ X2 @ A3 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ A3 ) )
= ( image_o_o2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_466_insert__image,axiom,
! [X2: $o,A3: set_o,F: $o > set_o] :
( ( member_o @ X2 @ A3 )
=> ( ( insert_set_o @ ( F @ X2 ) @ ( image_o_set_o2 @ F @ A3 ) )
= ( image_o_set_o2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_467_insert__image,axiom,
! [X2: set_o,A3: set_set_o,F: set_o > $o] :
( ( member_set_o @ X2 @ A3 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_set_o_o2 @ F @ A3 ) )
= ( image_set_o_o2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_468_insert__image,axiom,
! [X2: b > prat,A3: set_b_prat,F: ( b > prat ) > $o] :
( ( member_b_prat @ X2 @ A3 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_b_prat_o2 @ F @ A3 ) )
= ( image_b_prat_o2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_469_insert__image,axiom,
! [X2: $o,A3: set_o,F: $o > set_set_o] :
( ( member_o @ X2 @ A3 )
=> ( ( insert_set_set_o @ ( F @ X2 ) @ ( image_o_set_set_o @ F @ A3 ) )
= ( image_o_set_set_o @ F @ A3 ) ) ) ).
% insert_image
thf(fact_470_insert__image,axiom,
! [X2: $o,A3: set_o,F: $o > product_prod_o_o] :
( ( member_o @ X2 @ A3 )
=> ( ( insert6201435330877294327od_o_o @ ( F @ X2 ) @ ( image_4057150146340385428od_o_o @ F @ A3 ) )
= ( image_4057150146340385428od_o_o @ F @ A3 ) ) ) ).
% insert_image
thf(fact_471_insert__image,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o,F: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_7896445794123959606_o_o_o @ F @ A3 ) )
= ( image_7896445794123959606_o_o_o @ F @ A3 ) ) ) ).
% insert_image
thf(fact_472_insert__image,axiom,
! [X2: set_o,A3: set_set_o,F: set_o > set_o] :
( ( member_set_o @ X2 @ A3 )
=> ( ( insert_set_o @ ( F @ X2 ) @ ( image_set_o_set_o2 @ F @ A3 ) )
= ( image_set_o_set_o2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_473_insert__image,axiom,
! [X2: b > prat,A3: set_b_prat,F: ( b > prat ) > set_o] :
( ( member_b_prat @ X2 @ A3 )
=> ( ( insert_set_o @ ( F @ X2 ) @ ( image_b_prat_set_o2 @ F @ A3 ) )
= ( image_b_prat_set_o2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_474_insert__image,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o,F: product_prod_o_o > set_o] :
( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( ( insert_set_o @ ( F @ X2 ) @ ( image_7870716977482210326_set_o @ F @ A3 ) )
= ( image_7870716977482210326_set_o @ F @ A3 ) ) ) ).
% insert_image
thf(fact_475_ImageI,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,A3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member_o @ A @ A3 )
=> ( member_o @ B @ ( image_o_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_476_ImageI,axiom,
! [A: $o,B: set_o,R3: set_Pr3508996721749391591_set_o,A3: set_o] :
( ( member8938118433653312304_set_o @ ( product_Pair_o_set_o @ A @ B ) @ R3 )
=> ( ( member_o @ A @ A3 )
=> ( member_set_o @ B @ ( image_o_set_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_477_ImageI,axiom,
! [A: set_o,B: $o,R3: set_Pr5232024646447722279et_o_o,A3: set_set_o] :
( ( member6599968234716717424et_o_o @ ( product_Pair_set_o_o @ A @ B ) @ R3 )
=> ( ( member_set_o @ A @ A3 )
=> ( member_o @ B @ ( image_set_o_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_478_ImageI,axiom,
! [A: b > prat,B: $o,R3: set_Pr5078033069183545951prat_o,A3: set_b_prat] :
( ( member5777084206023828648prat_o @ ( produc6389352210403673967prat_o @ A @ B ) @ R3 )
=> ( ( member_b_prat @ A @ A3 )
=> ( member_o @ B @ ( image_b_prat_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_479_ImageI,axiom,
! [A: $o,B: b > prat,R3: set_Pr5930873434710784687b_prat,A3: set_o] :
( ( member7584064603762627832b_prat @ ( produc6454426705846175679b_prat @ A @ B ) @ R3 )
=> ( ( member_o @ A @ A3 )
=> ( member_b_prat @ B @ ( image_o_b_prat @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_480_ImageI,axiom,
! [A: $o,B: product_prod_o_o,R3: set_Pr5237325540968292374od_o_o,A3: set_o] :
( ( member3692453414038245343od_o_o @ ( produc2690343539598703398od_o_o @ A @ B ) @ R3 )
=> ( ( member_o @ A @ A3 )
=> ( member7466972457876170832od_o_o @ B @ ( image_5252864891027546466od_o_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_481_ImageI,axiom,
! [A: product_prod_o_o,B: $o,R3: set_Pr2246928807395521336_o_o_o,A3: set_Product_prod_o_o] :
( ( member5943846676503041025_o_o_o @ ( produc6529639187382277576_o_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( member_o @ B @ ( image_9092160538811120644_o_o_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_482_ImageI,axiom,
! [A: set_o,B: set_o,R3: set_Pr4577759397028426247_set_o,A3: set_set_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ B ) @ R3 )
=> ( ( member_set_o @ A @ A3 )
=> ( member_set_o @ B @ ( image_set_o_set_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_483_ImageI,axiom,
! [A: b > prat,B: b,R3: set_Pr8957522257986161222prat_b,A3: set_b_prat] :
( ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 )
=> ( ( member_b_prat @ A @ A3 )
=> ( member_b @ B @ ( image_b_prat_b @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_484_ImageI,axiom,
! [A: b > prat,B: set_o,R3: set_Pr5661576500205409599_set_o,A3: set_b_prat] :
( ( member8295766130977777544_set_o @ ( produc5638353472928494159_set_o @ A @ B ) @ R3 )
=> ( ( member_b_prat @ A @ A3 )
=> ( member_set_o @ B @ ( image_b_prat_set_o @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_485_Image__empty2,axiom,
! [R2: set_Product_prod_o_o] :
( ( image_o_o @ R2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_486_Image__empty2,axiom,
! [R2: set_Pr3508996721749391591_set_o] :
( ( image_o_set_o @ R2 @ bot_bot_set_o )
= bot_bot_set_set_o ) ).
% Image_empty2
thf(fact_487_Image__empty2,axiom,
! [R2: set_Pr5232024646447722279et_o_o] :
( ( image_set_o_o @ R2 @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_488_Image__empty2,axiom,
! [R2: set_Pr5930873434710784687b_prat] :
( ( image_o_b_prat @ R2 @ bot_bot_set_o )
= bot_bot_set_b_prat ) ).
% Image_empty2
thf(fact_489_Image__empty2,axiom,
! [R2: set_Pr5237325540968292374od_o_o] :
( ( image_5252864891027546466od_o_o @ R2 @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Image_empty2
thf(fact_490_Image__empty2,axiom,
! [R2: set_Pr5078033069183545951prat_o] :
( ( image_b_prat_o @ R2 @ bot_bot_set_b_prat )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_491_Image__empty2,axiom,
! [R2: set_Pr2246928807395521336_o_o_o] :
( ( image_9092160538811120644_o_o_o @ R2 @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_492_Image__empty2,axiom,
! [R2: set_Pr4577759397028426247_set_o] :
( ( image_set_o_set_o @ R2 @ bot_bot_set_set_o )
= bot_bot_set_set_o ) ).
% Image_empty2
thf(fact_493_Image__empty2,axiom,
! [R2: set_Pr5661576500205409599_set_o] :
( ( image_b_prat_set_o @ R2 @ bot_bot_set_b_prat )
= bot_bot_set_set_o ) ).
% Image_empty2
thf(fact_494_Image__empty2,axiom,
! [R2: set_Pr1571070227385496088_set_o] :
( ( image_7547793595507069668_set_o @ R2 @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_set_o ) ).
% Image_empty2
thf(fact_495_Image__empty1,axiom,
! [X7: set_o] :
( ( image_o_o @ bot_bo7073875226086086771od_o_o @ X7 )
= bot_bot_set_o ) ).
% Image_empty1
thf(fact_496_imageI,axiom,
! [X2: $o,A3: set_o,F: $o > $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_497_imageI,axiom,
! [X2: $o,A3: set_o,F: $o > set_o] :
( ( member_o @ X2 @ A3 )
=> ( member_set_o @ ( F @ X2 ) @ ( image_o_set_o2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_498_imageI,axiom,
! [X2: set_o,A3: set_set_o,F: set_o > $o] :
( ( member_set_o @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ ( image_set_o_o2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_499_imageI,axiom,
! [X2: b > prat,A3: set_b_prat,F: ( b > prat ) > $o] :
( ( member_b_prat @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ ( image_b_prat_o2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_500_imageI,axiom,
! [X2: $o,A3: set_o,F: $o > set_set_o] :
( ( member_o @ X2 @ A3 )
=> ( member_set_set_o @ ( F @ X2 ) @ ( image_o_set_set_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_501_imageI,axiom,
! [X2: $o,A3: set_o,F: $o > b > prat] :
( ( member_o @ X2 @ A3 )
=> ( member_b_prat @ ( F @ X2 ) @ ( image_o_b_prat2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_502_imageI,axiom,
! [X2: $o,A3: set_o,F: $o > product_prod_o_o] :
( ( member_o @ X2 @ A3 )
=> ( member7466972457876170832od_o_o @ ( F @ X2 ) @ ( image_4057150146340385428od_o_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_503_imageI,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o,F: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ ( image_7896445794123959606_o_o_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_504_imageI,axiom,
! [X2: set_o,A3: set_set_o,F: set_o > set_o] :
( ( member_set_o @ X2 @ A3 )
=> ( member_set_o @ ( F @ X2 ) @ ( image_set_o_set_o2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_505_imageI,axiom,
! [X2: b > prat,A3: set_b_prat,F: ( b > prat ) > set_o] :
( ( member_b_prat @ X2 @ A3 )
=> ( member_set_o @ ( F @ X2 ) @ ( image_b_prat_set_o2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_506_image__iff,axiom,
! [Z2: set_set_o,F: $o > set_set_o,A3: set_o] :
( ( member_set_set_o @ Z2 @ ( image_o_set_set_o @ F @ A3 ) )
= ( ? [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_507_image__iff,axiom,
! [Z2: $o,F: $o > $o,A3: set_o] :
( ( member_o @ Z2 @ ( image_o_o2 @ F @ A3 ) )
= ( ? [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_508_image__iff,axiom,
! [Z2: set_o,F: $o > set_o,A3: set_o] :
( ( member_set_o @ Z2 @ ( image_o_set_o2 @ F @ A3 ) )
= ( ? [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_509_image__iff,axiom,
! [Z2: set_o,F: set_o > set_o,A3: set_set_o] :
( ( member_set_o @ Z2 @ ( image_set_o_set_o2 @ F @ A3 ) )
= ( ? [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_510_bex__imageD,axiom,
! [F: $o > $o,A3: set_o,P: $o > $o] :
( ? [X6: $o] :
( ( member_o @ X6 @ ( image_o_o2 @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_511_bex__imageD,axiom,
! [F: $o > set_o,A3: set_o,P: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ ( image_o_set_o2 @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_512_bex__imageD,axiom,
! [F: $o > set_set_o,A3: set_o,P: set_set_o > $o] :
( ? [X6: set_set_o] :
( ( member_set_set_o @ X6 @ ( image_o_set_set_o @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_513_bex__imageD,axiom,
! [F: set_o > set_o,A3: set_set_o,P: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ ( image_set_o_set_o2 @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: set_o] :
( ( member_set_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_514_image__cong,axiom,
! [M2: set_o,N: set_o,F: $o > $o,G: $o > $o] :
( ( M2 = N )
=> ( ! [X: $o] :
( ( member_o @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_o_o2 @ F @ M2 )
= ( image_o_o2 @ G @ N ) ) ) ) ).
% image_cong
thf(fact_515_image__cong,axiom,
! [M2: set_o,N: set_o,F: $o > set_o,G: $o > set_o] :
( ( M2 = N )
=> ( ! [X: $o] :
( ( member_o @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_o_set_o2 @ F @ M2 )
= ( image_o_set_o2 @ G @ N ) ) ) ) ).
% image_cong
thf(fact_516_image__cong,axiom,
! [M2: set_o,N: set_o,F: $o > set_set_o,G: $o > set_set_o] :
( ( M2 = N )
=> ( ! [X: $o] :
( ( member_o @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_o_set_set_o @ F @ M2 )
= ( image_o_set_set_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_517_image__cong,axiom,
! [M2: set_set_o,N: set_set_o,F: set_o > set_o,G: set_o > set_o] :
( ( M2 = N )
=> ( ! [X: set_o] :
( ( member_set_o @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_set_o_set_o2 @ F @ M2 )
= ( image_set_o_set_o2 @ G @ N ) ) ) ) ).
% image_cong
thf(fact_518_ball__imageD,axiom,
! [F: $o > $o,A3: set_o,P: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ ( image_o_o2 @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_519_ball__imageD,axiom,
! [F: $o > set_o,A3: set_o,P: set_o > $o] :
( ! [X: set_o] :
( ( member_set_o @ X @ ( image_o_set_o2 @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_520_ball__imageD,axiom,
! [F: $o > set_set_o,A3: set_o,P: set_set_o > $o] :
( ! [X: set_set_o] :
( ( member_set_set_o @ X @ ( image_o_set_set_o @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_521_ball__imageD,axiom,
! [F: set_o > set_o,A3: set_set_o,P: set_o > $o] :
( ! [X: set_o] :
( ( member_set_o @ X @ ( image_set_o_set_o2 @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: set_o] :
( ( member_set_o @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_522_rev__image__eqI,axiom,
! [X2: $o,A3: set_o,B: $o,F: $o > $o] :
( ( member_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_523_rev__image__eqI,axiom,
! [X2: $o,A3: set_o,B: set_o,F: $o > set_o] :
( ( member_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_o_set_o2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_524_rev__image__eqI,axiom,
! [X2: set_o,A3: set_set_o,B: $o,F: set_o > $o] :
( ( member_set_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_set_o_o2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_525_rev__image__eqI,axiom,
! [X2: b > prat,A3: set_b_prat,B: $o,F: ( b > prat ) > $o] :
( ( member_b_prat @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_b_prat_o2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_526_rev__image__eqI,axiom,
! [X2: $o,A3: set_o,B: set_set_o,F: $o > set_set_o] :
( ( member_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_set_o @ B @ ( image_o_set_set_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_527_rev__image__eqI,axiom,
! [X2: $o,A3: set_o,B: b > prat,F: $o > b > prat] :
( ( member_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b_prat @ B @ ( image_o_b_prat2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_528_rev__image__eqI,axiom,
! [X2: $o,A3: set_o,B: product_prod_o_o,F: $o > product_prod_o_o] :
( ( member_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member7466972457876170832od_o_o @ B @ ( image_4057150146340385428od_o_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_529_rev__image__eqI,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o,B: $o,F: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_7896445794123959606_o_o_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_530_rev__image__eqI,axiom,
! [X2: set_o,A3: set_set_o,B: set_o,F: set_o > set_o] :
( ( member_set_o @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_set_o_set_o2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_531_rev__image__eqI,axiom,
! [X2: b > prat,A3: set_b_prat,B: set_o,F: ( b > prat ) > set_o] :
( ( member_b_prat @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_b_prat_set_o2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_532_imageE,axiom,
! [B: $o,F: $o > $o,A3: set_o] :
( ( member_o @ B @ ( image_o_o2 @ F @ A3 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_533_imageE,axiom,
! [B: $o,F: set_o > $o,A3: set_set_o] :
( ( member_o @ B @ ( image_set_o_o2 @ F @ A3 ) )
=> ~ ! [X: set_o] :
( ( B
= ( F @ X ) )
=> ~ ( member_set_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_534_imageE,axiom,
! [B: set_o,F: $o > set_o,A3: set_o] :
( ( member_set_o @ B @ ( image_o_set_o2 @ F @ A3 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_535_imageE,axiom,
! [B: set_set_o,F: $o > set_set_o,A3: set_o] :
( ( member_set_set_o @ B @ ( image_o_set_set_o @ F @ A3 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_536_imageE,axiom,
! [B: b > prat,F: $o > b > prat,A3: set_o] :
( ( member_b_prat @ B @ ( image_o_b_prat2 @ F @ A3 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_537_imageE,axiom,
! [B: $o,F: ( b > prat ) > $o,A3: set_b_prat] :
( ( member_o @ B @ ( image_b_prat_o2 @ F @ A3 ) )
=> ~ ! [X: b > prat] :
( ( B
= ( F @ X ) )
=> ~ ( member_b_prat @ X @ A3 ) ) ) ).
% imageE
thf(fact_538_imageE,axiom,
! [B: $o,F: product_prod_o_o > $o,A3: set_Product_prod_o_o] :
( ( member_o @ B @ ( image_7896445794123959606_o_o_o @ F @ A3 ) )
=> ~ ! [X: product_prod_o_o] :
( ( B
= ( F @ X ) )
=> ~ ( member7466972457876170832od_o_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_539_imageE,axiom,
! [B: product_prod_o_o,F: $o > product_prod_o_o,A3: set_o] :
( ( member7466972457876170832od_o_o @ B @ ( image_4057150146340385428od_o_o @ F @ A3 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_540_imageE,axiom,
! [B: set_o,F: set_o > set_o,A3: set_set_o] :
( ( member_set_o @ B @ ( image_set_o_set_o2 @ F @ A3 ) )
=> ~ ! [X: set_o] :
( ( B
= ( F @ X ) )
=> ~ ( member_set_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_541_imageE,axiom,
! [B: b > prat,F: set_o > b > prat,A3: set_set_o] :
( ( member_b_prat @ B @ ( image_set_o_b_prat2 @ F @ A3 ) )
=> ~ ! [X: set_o] :
( ( B
= ( F @ X ) )
=> ~ ( member_set_o @ X @ A3 ) ) ) ).
% imageE
thf(fact_542_image__image,axiom,
! [F: $o > $o,G: $o > $o,A3: set_o] :
( ( image_o_o2 @ F @ ( image_o_o2 @ G @ A3 ) )
= ( image_o_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_543_image__image,axiom,
! [F: set_o > $o,G: $o > set_o,A3: set_o] :
( ( image_set_o_o2 @ F @ ( image_o_set_o2 @ G @ A3 ) )
= ( image_o_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_544_image__image,axiom,
! [F: $o > set_o,G: $o > $o,A3: set_o] :
( ( image_o_set_o2 @ F @ ( image_o_o2 @ G @ A3 ) )
= ( image_o_set_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_545_image__image,axiom,
! [F: set_set_o > $o,G: $o > set_set_o,A3: set_o] :
( ( image_set_set_o_o @ F @ ( image_o_set_set_o @ G @ A3 ) )
= ( image_o_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_546_image__image,axiom,
! [F: $o > set_o,G: set_o > $o,A3: set_set_o] :
( ( image_o_set_o2 @ F @ ( image_set_o_o2 @ G @ A3 ) )
= ( image_set_o_set_o2
@ ^ [X5: set_o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_547_image__image,axiom,
! [F: $o > set_set_o,G: $o > $o,A3: set_o] :
( ( image_o_set_set_o @ F @ ( image_o_o2 @ G @ A3 ) )
= ( image_o_set_set_o
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_548_image__image,axiom,
! [F: set_o > set_o,G: $o > set_o,A3: set_o] :
( ( image_set_o_set_o2 @ F @ ( image_o_set_o2 @ G @ A3 ) )
= ( image_o_set_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_549_image__image,axiom,
! [F: set_o > set_set_o,G: $o > set_o,A3: set_o] :
( ( image_5023573440332574309_set_o @ F @ ( image_o_set_o2 @ G @ A3 ) )
= ( image_o_set_set_o
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_550_image__image,axiom,
! [F: set_set_o > set_o,G: $o > set_set_o,A3: set_o] :
( ( image_1148209357520620709_set_o @ F @ ( image_o_set_set_o @ G @ A3 ) )
= ( image_o_set_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_551_image__image,axiom,
! [F: set_o > set_o,G: set_o > set_o,A3: set_set_o] :
( ( image_set_o_set_o2 @ F @ ( image_set_o_set_o2 @ G @ A3 ) )
= ( image_set_o_set_o2
@ ^ [X5: set_o] : ( F @ ( G @ X5 ) )
@ A3 ) ) ).
% image_image
thf(fact_552_Compr__image__eq,axiom,
! [F: $o > $o,A3: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_o_o2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_o_o2 @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_553_Compr__image__eq,axiom,
! [F: set_o > $o,A3: set_set_o,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_set_o_o2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_set_o_o2 @ F
@ ( collect_set_o
@ ^ [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_554_Compr__image__eq,axiom,
! [F: $o > set_o,A3: set_o,P: set_o > $o] :
( ( collect_set_o
@ ^ [X5: set_o] :
( ( member_set_o @ X5 @ ( image_o_set_o2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_o_set_o2 @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_555_Compr__image__eq,axiom,
! [F: $o > set_set_o,A3: set_o,P: set_set_o > $o] :
( ( collect_set_set_o
@ ^ [X5: set_set_o] :
( ( member_set_set_o @ X5 @ ( image_o_set_set_o @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_o_set_set_o @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_556_Compr__image__eq,axiom,
! [F: $o > b > prat,A3: set_o,P: ( b > prat ) > $o] :
( ( collect_b_prat
@ ^ [X5: b > prat] :
( ( member_b_prat @ X5 @ ( image_o_b_prat2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_o_b_prat2 @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_557_Compr__image__eq,axiom,
! [F: $o > product_prod_o_o,A3: set_o,P: product_prod_o_o > $o] :
( ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X5 @ ( image_4057150146340385428od_o_o @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_4057150146340385428od_o_o @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_558_Compr__image__eq,axiom,
! [F: ( b > prat ) > $o,A3: set_b_prat,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_b_prat_o2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_b_prat_o2 @ F
@ ( collect_b_prat
@ ^ [X5: b > prat] :
( ( member_b_prat @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_559_Compr__image__eq,axiom,
! [F: product_prod_o_o > $o,A3: set_Product_prod_o_o,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_7896445794123959606_o_o_o @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_7896445794123959606_o_o_o @ F
@ ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_560_Compr__image__eq,axiom,
! [F: set_o > set_o,A3: set_set_o,P: set_o > $o] :
( ( collect_set_o
@ ^ [X5: set_o] :
( ( member_set_o @ X5 @ ( image_set_o_set_o2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_set_o_set_o2 @ F
@ ( collect_set_o
@ ^ [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_561_Compr__image__eq,axiom,
! [F: set_o > b > prat,A3: set_set_o,P: ( b > prat ) > $o] :
( ( collect_b_prat
@ ^ [X5: b > prat] :
( ( member_b_prat @ X5 @ ( image_set_o_b_prat2 @ F @ A3 ) )
& ( P @ X5 ) ) )
= ( image_set_o_b_prat2 @ F
@ ( collect_set_o
@ ^ [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_562_refl__on__domain,axiom,
! [A3: set_b_prat,R3: set_Pr525462856049757959b_prat,A: b > prat,B: b > prat] :
( ( refl_on_b_prat @ A3 @ R3 )
=> ( ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ A @ B ) @ R3 )
=> ( ( member_b_prat @ A @ A3 )
& ( member_b_prat @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_563_refl__on__domain,axiom,
! [A3: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,A: product_prod_o_o,B: product_prod_o_o] :
( ( refl_o8108712344181003046od_o_o @ A3 @ R3 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ A3 )
& ( member7466972457876170832od_o_o @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_564_refl__on__domain,axiom,
! [A3: set_set_o,R3: set_Pr4577759397028426247_set_o,A: set_o,B: set_o] :
( ( refl_on_set_o @ A3 @ R3 )
=> ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ B ) @ R3 )
=> ( ( member_set_o @ A @ A3 )
& ( member_set_o @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_565_refl__on__domain,axiom,
! [A3: set_o,R3: set_Product_prod_o_o,A: $o,B: $o] :
( ( refl_on_o @ A3 @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member_o @ A @ A3 )
& ( member_o @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_566_the__elem__image__unique,axiom,
! [A3: set_o,F: $o > set_o,X2: $o] :
( ( A3 != bot_bot_set_o )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_o @ ( image_o_set_o2 @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_567_the__elem__image__unique,axiom,
! [A3: set_o,F: $o > set_set_o,X2: $o] :
( ( A3 != bot_bot_set_o )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_set_o @ ( image_o_set_set_o @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_568_the__elem__image__unique,axiom,
! [A3: set_o,F: $o > $o,X2: $o] :
( ( A3 != bot_bot_set_o )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_o_o2 @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_569_the__elem__image__unique,axiom,
! [A3: set_b_prat,F: ( b > prat ) > $o,X2: b > prat] :
( ( A3 != bot_bot_set_b_prat )
=> ( ! [Y3: b > prat] :
( ( member_b_prat @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_b_prat_o2 @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_570_the__elem__image__unique,axiom,
! [A3: set_Product_prod_o_o,F: product_prod_o_o > $o,X2: product_prod_o_o] :
( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ! [Y3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_7896445794123959606_o_o_o @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_571_the__elem__image__unique,axiom,
! [A3: set_set_o,F: set_o > set_o,X2: set_o] :
( ( A3 != bot_bot_set_set_o )
=> ( ! [Y3: set_o] :
( ( member_set_o @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_o @ ( image_set_o_set_o2 @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_572_the__elem__image__unique,axiom,
! [A3: set_set_o,F: set_o > $o,X2: set_o] :
( ( A3 != bot_bot_set_set_o )
=> ( ! [Y3: set_o] :
( ( member_set_o @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_set_o_o2 @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_573_rev__ImageI,axiom,
! [A: $o,A3: set_o,B: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ A3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member_o @ B @ ( image_o_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_574_rev__ImageI,axiom,
! [A: $o,A3: set_o,B: set_o,R3: set_Pr3508996721749391591_set_o] :
( ( member_o @ A @ A3 )
=> ( ( member8938118433653312304_set_o @ ( product_Pair_o_set_o @ A @ B ) @ R3 )
=> ( member_set_o @ B @ ( image_o_set_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_575_rev__ImageI,axiom,
! [A: set_o,A3: set_set_o,B: $o,R3: set_Pr5232024646447722279et_o_o] :
( ( member_set_o @ A @ A3 )
=> ( ( member6599968234716717424et_o_o @ ( product_Pair_set_o_o @ A @ B ) @ R3 )
=> ( member_o @ B @ ( image_set_o_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_576_rev__ImageI,axiom,
! [A: b > prat,A3: set_b_prat,B: $o,R3: set_Pr5078033069183545951prat_o] :
( ( member_b_prat @ A @ A3 )
=> ( ( member5777084206023828648prat_o @ ( produc6389352210403673967prat_o @ A @ B ) @ R3 )
=> ( member_o @ B @ ( image_b_prat_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_577_rev__ImageI,axiom,
! [A: $o,A3: set_o,B: b > prat,R3: set_Pr5930873434710784687b_prat] :
( ( member_o @ A @ A3 )
=> ( ( member7584064603762627832b_prat @ ( produc6454426705846175679b_prat @ A @ B ) @ R3 )
=> ( member_b_prat @ B @ ( image_o_b_prat @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_578_rev__ImageI,axiom,
! [A: $o,A3: set_o,B: product_prod_o_o,R3: set_Pr5237325540968292374od_o_o] :
( ( member_o @ A @ A3 )
=> ( ( member3692453414038245343od_o_o @ ( produc2690343539598703398od_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ B @ ( image_5252864891027546466od_o_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_579_rev__ImageI,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o,B: $o,R3: set_Pr2246928807395521336_o_o_o] :
( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( ( member5943846676503041025_o_o_o @ ( produc6529639187382277576_o_o_o @ A @ B ) @ R3 )
=> ( member_o @ B @ ( image_9092160538811120644_o_o_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_580_rev__ImageI,axiom,
! [A: set_o,A3: set_set_o,B: set_o,R3: set_Pr4577759397028426247_set_o] :
( ( member_set_o @ A @ A3 )
=> ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ B ) @ R3 )
=> ( member_set_o @ B @ ( image_set_o_set_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_581_rev__ImageI,axiom,
! [A: b > prat,A3: set_b_prat,B: b,R3: set_Pr8957522257986161222prat_b] :
( ( member_b_prat @ A @ A3 )
=> ( ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 )
=> ( member_b @ B @ ( image_b_prat_b @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_582_rev__ImageI,axiom,
! [A: b > prat,A3: set_b_prat,B: set_o,R3: set_Pr5661576500205409599_set_o] :
( ( member_b_prat @ A @ A3 )
=> ( ( member8295766130977777544_set_o @ ( produc5638353472928494159_set_o @ A @ B ) @ R3 )
=> ( member_set_o @ B @ ( image_b_prat_set_o @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_583_Image__iff,axiom,
! [B: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b,A3: set_b_prat] :
( ( member5302574828040144399prat_b @ B @ ( image_8007196543710844537prat_b @ R3 @ A3 ) )
= ( ? [X5: b > prat] :
( ( member_b_prat @ X5 @ A3 )
& ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ X5 @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_584_Image__iff,axiom,
! [B: b,R3: set_Pr8957522257986161222prat_b,A3: set_b_prat] :
( ( member_b @ B @ ( image_b_prat_b @ R3 @ A3 ) )
= ( ? [X5: b > prat] :
( ( member_b_prat @ X5 @ A3 )
& ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ X5 @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_585_Image__iff,axiom,
! [B: $o,R3: set_Product_prod_o_o,A3: set_o] :
( ( member_o @ B @ ( image_o_o @ R3 @ A3 ) )
= ( ? [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_586_ImageE,axiom,
! [B: $o,R3: set_Product_prod_o_o,A3: set_o] :
( ( member_o @ B @ ( image_o_o @ R3 @ A3 ) )
=> ~ ! [X: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ B ) @ R3 )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_587_ImageE,axiom,
! [B: $o,R3: set_Pr5232024646447722279et_o_o,A3: set_set_o] :
( ( member_o @ B @ ( image_set_o_o @ R3 @ A3 ) )
=> ~ ! [X: set_o] :
( ( member6599968234716717424et_o_o @ ( product_Pair_set_o_o @ X @ B ) @ R3 )
=> ~ ( member_set_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_588_ImageE,axiom,
! [B: set_o,R3: set_Pr3508996721749391591_set_o,A3: set_o] :
( ( member_set_o @ B @ ( image_o_set_o @ R3 @ A3 ) )
=> ~ ! [X: $o] :
( ( member8938118433653312304_set_o @ ( product_Pair_o_set_o @ X @ B ) @ R3 )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_589_ImageE,axiom,
! [B: b > prat,R3: set_Pr5930873434710784687b_prat,A3: set_o] :
( ( member_b_prat @ B @ ( image_o_b_prat @ R3 @ A3 ) )
=> ~ ! [X: $o] :
( ( member7584064603762627832b_prat @ ( produc6454426705846175679b_prat @ X @ B ) @ R3 )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_590_ImageE,axiom,
! [B: $o,R3: set_Pr5078033069183545951prat_o,A3: set_b_prat] :
( ( member_o @ B @ ( image_b_prat_o @ R3 @ A3 ) )
=> ~ ! [X: b > prat] :
( ( member5777084206023828648prat_o @ ( produc6389352210403673967prat_o @ X @ B ) @ R3 )
=> ~ ( member_b_prat @ X @ A3 ) ) ) ).
% ImageE
thf(fact_591_ImageE,axiom,
! [B: $o,R3: set_Pr2246928807395521336_o_o_o,A3: set_Product_prod_o_o] :
( ( member_o @ B @ ( image_9092160538811120644_o_o_o @ R3 @ A3 ) )
=> ~ ! [X: product_prod_o_o] :
( ( member5943846676503041025_o_o_o @ ( produc6529639187382277576_o_o_o @ X @ B ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_592_ImageE,axiom,
! [B: product_prod_o_o,R3: set_Pr5237325540968292374od_o_o,A3: set_o] :
( ( member7466972457876170832od_o_o @ B @ ( image_5252864891027546466od_o_o @ R3 @ A3 ) )
=> ~ ! [X: $o] :
( ( member3692453414038245343od_o_o @ ( produc2690343539598703398od_o_o @ X @ B ) @ R3 )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_593_ImageE,axiom,
! [B: set_o,R3: set_Pr4577759397028426247_set_o,A3: set_set_o] :
( ( member_set_o @ B @ ( image_set_o_set_o @ R3 @ A3 ) )
=> ~ ! [X: set_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X @ B ) @ R3 )
=> ~ ( member_set_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_594_ImageE,axiom,
! [B: b,R3: set_Pr8957522257986161222prat_b,A3: set_b_prat] :
( ( member_b @ B @ ( image_b_prat_b @ R3 @ A3 ) )
=> ~ ! [X: b > prat] :
( ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ X @ B ) @ R3 )
=> ~ ( member_b_prat @ X @ A3 ) ) ) ).
% ImageE
thf(fact_595_ImageE,axiom,
! [B: b > prat,R3: set_Pr3021494175023059919b_prat,A3: set_set_o] :
( ( member_b_prat @ B @ ( image_set_o_b_prat @ R3 @ A3 ) )
=> ~ ! [X: set_o] :
( ( member9157886055108046360b_prat @ ( produc6176071177017395423b_prat @ X @ B ) @ R3 )
=> ~ ( member_set_o @ X @ A3 ) ) ) ).
% ImageE
thf(fact_596_refl__onD2,axiom,
! [A3: set_b_prat,R3: set_Pr525462856049757959b_prat,X2: b > prat,Y4: b > prat] :
( ( refl_on_b_prat @ A3 @ R3 )
=> ( ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ X2 @ Y4 ) @ R3 )
=> ( member_b_prat @ Y4 @ A3 ) ) ) ).
% refl_onD2
thf(fact_597_refl__onD2,axiom,
! [A3: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,X2: product_prod_o_o,Y4: product_prod_o_o] :
( ( refl_o8108712344181003046od_o_o @ A3 @ R3 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ Y4 ) @ R3 )
=> ( member7466972457876170832od_o_o @ Y4 @ A3 ) ) ) ).
% refl_onD2
thf(fact_598_refl__onD2,axiom,
! [A3: set_set_o,R3: set_Pr4577759397028426247_set_o,X2: set_o,Y4: set_o] :
( ( refl_on_set_o @ A3 @ R3 )
=> ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X2 @ Y4 ) @ R3 )
=> ( member_set_o @ Y4 @ A3 ) ) ) ).
% refl_onD2
thf(fact_599_refl__onD2,axiom,
! [A3: set_o,R3: set_Product_prod_o_o,X2: $o,Y4: $o] :
( ( refl_on_o @ A3 @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ R3 )
=> ( member_o @ Y4 @ A3 ) ) ) ).
% refl_onD2
thf(fact_600_refl__onD1,axiom,
! [A3: set_b_prat,R3: set_Pr525462856049757959b_prat,X2: b > prat,Y4: b > prat] :
( ( refl_on_b_prat @ A3 @ R3 )
=> ( ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ X2 @ Y4 ) @ R3 )
=> ( member_b_prat @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_601_refl__onD1,axiom,
! [A3: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,X2: product_prod_o_o,Y4: product_prod_o_o] :
( ( refl_o8108712344181003046od_o_o @ A3 @ R3 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ Y4 ) @ R3 )
=> ( member7466972457876170832od_o_o @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_602_refl__onD1,axiom,
! [A3: set_set_o,R3: set_Pr4577759397028426247_set_o,X2: set_o,Y4: set_o] :
( ( refl_on_set_o @ A3 @ R3 )
=> ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X2 @ Y4 ) @ R3 )
=> ( member_set_o @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_603_refl__onD1,axiom,
! [A3: set_o,R3: set_Product_prod_o_o,X2: $o,Y4: $o] :
( ( refl_on_o @ A3 @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ R3 )
=> ( member_o @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_604_refl__onD,axiom,
! [A3: set_b_prat,R3: set_Pr525462856049757959b_prat,A: b > prat] :
( ( refl_on_b_prat @ A3 @ R3 )
=> ( ( member_b_prat @ A @ A3 )
=> ( member6376894174024132944b_prat @ ( produc6129551849031254551b_prat @ A @ A ) @ R3 ) ) ) ).
% refl_onD
thf(fact_605_refl__onD,axiom,
! [A3: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,A: product_prod_o_o] :
( ( refl_o8108712344181003046od_o_o @ A3 @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ A ) @ R3 ) ) ) ).
% refl_onD
thf(fact_606_refl__onD,axiom,
! [A3: set_set_o,R3: set_Pr4577759397028426247_set_o,A: set_o] :
( ( refl_on_set_o @ A3 @ R3 )
=> ( ( member_set_o @ A @ A3 )
=> ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ A ) @ R3 ) ) ) ).
% refl_onD
thf(fact_607_refl__onD,axiom,
! [A3: set_o,R3: set_Product_prod_o_o,A: $o] :
( ( refl_on_o @ A3 @ R3 )
=> ( ( member_o @ A @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ R3 ) ) ) ).
% refl_onD
thf(fact_608_refl__on__Id__on,axiom,
! [A3: set_o] : ( refl_on_o @ A3 @ ( id_on_o @ A3 ) ) ).
% refl_on_Id_on
thf(fact_609_lnear__order__on__empty,axiom,
order_2115221884338004981b_prat @ bot_bot_set_b_prat @ bot_bo6344516899060369779b_prat ).
% lnear_order_on_empty
thf(fact_610_lnear__order__on__empty,axiom,
order_7019990371155863132od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo2881837355931550515od_o_o ).
% lnear_order_on_empty
thf(fact_611_lnear__order__on__empty,axiom,
order_1696139471214784045_set_o @ bot_bot_set_set_o @ bot_bo8143257439550059635_set_o ).
% lnear_order_on_empty
thf(fact_612_lnear__order__on__empty,axiom,
order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% lnear_order_on_empty
thf(fact_613_image__constant,axiom,
! [X2: $o,A3: set_o,C: $o] :
( ( member_o @ X2 @ A3 )
=> ( ( image_o_o2
@ ^ [X5: $o] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_614_image__constant,axiom,
! [X2: set_o,A3: set_set_o,C: $o] :
( ( member_set_o @ X2 @ A3 )
=> ( ( image_set_o_o2
@ ^ [X5: set_o] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_615_image__constant,axiom,
! [X2: $o,A3: set_o,C: set_o] :
( ( member_o @ X2 @ A3 )
=> ( ( image_o_set_o2
@ ^ [X5: $o] : C
@ A3 )
= ( insert_set_o @ C @ bot_bot_set_set_o ) ) ) ).
% image_constant
thf(fact_616_image__constant,axiom,
! [X2: $o,A3: set_o,C: set_set_o] :
( ( member_o @ X2 @ A3 )
=> ( ( image_o_set_set_o
@ ^ [X5: $o] : C
@ A3 )
= ( insert_set_set_o @ C @ bot_bo3529510225556782244_set_o ) ) ) ).
% image_constant
thf(fact_617_image__constant,axiom,
! [X2: b > prat,A3: set_b_prat,C: $o] :
( ( member_b_prat @ X2 @ A3 )
=> ( ( image_b_prat_o2
@ ^ [X5: b > prat] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_618_image__constant,axiom,
! [X2: product_prod_o_o,A3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ X2 @ A3 )
=> ( ( image_7896445794123959606_o_o_o
@ ^ [X5: product_prod_o_o] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_619_image__constant,axiom,
! [X2: $o,A3: set_o,C: b > prat] :
( ( member_o @ X2 @ A3 )
=> ( ( image_o_b_prat2
@ ^ [X5: $o] : C
@ A3 )
= ( insert_b_prat @ C @ bot_bot_set_b_prat ) ) ) ).
% image_constant
thf(fact_620_image__constant,axiom,
! [X2: $o,A3: set_o,C: product_prod_o_o] :
( ( member_o @ X2 @ A3 )
=> ( ( image_4057150146340385428od_o_o
@ ^ [X5: $o] : C
@ A3 )
= ( insert6201435330877294327od_o_o @ C @ bot_bo7073875226086086771od_o_o ) ) ) ).
% image_constant
thf(fact_621_image__constant,axiom,
! [X2: set_o,A3: set_set_o,C: set_o] :
( ( member_set_o @ X2 @ A3 )
=> ( ( image_set_o_set_o2
@ ^ [X5: set_o] : C
@ A3 )
= ( insert_set_o @ C @ bot_bot_set_set_o ) ) ) ).
% image_constant
thf(fact_622_image__constant,axiom,
! [X2: set_o,A3: set_set_o,C: b > prat] :
( ( member_set_o @ X2 @ A3 )
=> ( ( image_set_o_b_prat2
@ ^ [X5: set_o] : C
@ A3 )
= ( insert_b_prat @ C @ bot_bot_set_b_prat ) ) ) ).
% image_constant
thf(fact_623_image__constant__conv,axiom,
! [A3: set_o,C: $o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( image_o_o2
@ ^ [X5: $o] : C
@ A3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( image_o_o2
@ ^ [X5: $o] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_624_image__constant__conv,axiom,
! [A3: set_o,C: set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( image_o_set_o2
@ ^ [X5: $o] : C
@ A3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( image_o_set_o2
@ ^ [X5: $o] : C
@ A3 )
= ( insert_set_o @ C @ bot_bot_set_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_625_image__constant__conv,axiom,
! [A3: set_set_o,C: $o] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( image_set_o_o2
@ ^ [X5: set_o] : C
@ A3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( image_set_o_o2
@ ^ [X5: set_o] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_626_image__constant__conv,axiom,
! [A3: set_o,C: set_set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( image_o_set_set_o
@ ^ [X5: $o] : C
@ A3 )
= bot_bo3529510225556782244_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( image_o_set_set_o
@ ^ [X5: $o] : C
@ A3 )
= ( insert_set_set_o @ C @ bot_bo3529510225556782244_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_627_image__constant__conv,axiom,
! [A3: set_o,C: b > prat] :
( ( ( A3 = bot_bot_set_o )
=> ( ( image_o_b_prat2
@ ^ [X5: $o] : C
@ A3 )
= bot_bot_set_b_prat ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( image_o_b_prat2
@ ^ [X5: $o] : C
@ A3 )
= ( insert_b_prat @ C @ bot_bot_set_b_prat ) ) ) ) ).
% image_constant_conv
thf(fact_628_image__constant__conv,axiom,
! [A3: set_o,C: product_prod_o_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( image_4057150146340385428od_o_o
@ ^ [X5: $o] : C
@ A3 )
= bot_bo7073875226086086771od_o_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( image_4057150146340385428od_o_o
@ ^ [X5: $o] : C
@ A3 )
= ( insert6201435330877294327od_o_o @ C @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% image_constant_conv
thf(fact_629_image__constant__conv,axiom,
! [A3: set_b_prat,C: $o] :
( ( ( A3 = bot_bot_set_b_prat )
=> ( ( image_b_prat_o2
@ ^ [X5: b > prat] : C
@ A3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_b_prat )
=> ( ( image_b_prat_o2
@ ^ [X5: b > prat] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_630_image__constant__conv,axiom,
! [A3: set_Product_prod_o_o,C: $o] :
( ( ( A3 = bot_bo7073875226086086771od_o_o )
=> ( ( image_7896445794123959606_o_o_o
@ ^ [X5: product_prod_o_o] : C
@ A3 )
= bot_bot_set_o ) )
& ( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ( image_7896445794123959606_o_o_o
@ ^ [X5: product_prod_o_o] : C
@ A3 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_631_image__constant__conv,axiom,
! [A3: set_set_o,C: set_o] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( image_set_o_set_o2
@ ^ [X5: set_o] : C
@ A3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( image_set_o_set_o2
@ ^ [X5: set_o] : C
@ A3 )
= ( insert_set_o @ C @ bot_bot_set_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_632_image__constant__conv,axiom,
! [A3: set_b_prat,C: set_o] :
( ( ( A3 = bot_bot_set_b_prat )
=> ( ( image_b_prat_set_o2
@ ^ [X5: b > prat] : C
@ A3 )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_b_prat )
=> ( ( image_b_prat_set_o2
@ ^ [X5: b > prat] : C
@ A3 )
= ( insert_set_o @ C @ bot_bot_set_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_633_refl__on__empty,axiom,
refl_on_b_prat @ bot_bot_set_b_prat @ bot_bo6344516899060369779b_prat ).
% refl_on_empty
thf(fact_634_refl__on__empty,axiom,
refl_o8108712344181003046od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo2881837355931550515od_o_o ).
% refl_on_empty
thf(fact_635_refl__on__empty,axiom,
refl_on_set_o @ bot_bot_set_set_o @ bot_bo8143257439550059635_set_o ).
% refl_on_empty
thf(fact_636_refl__on__empty,axiom,
refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% refl_on_empty
thf(fact_637_Image__singleton,axiom,
! [R3: set_Pr3508996721749391591_set_o,A: $o] :
( ( image_o_set_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( collect_set_o
@ ^ [B2: set_o] : ( member8938118433653312304_set_o @ ( product_Pair_o_set_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_638_Image__singleton,axiom,
! [R3: set_Product_prod_o_o,A: $o] :
( ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( collect_o
@ ^ [B2: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_639_Image__singleton,axiom,
! [R3: set_Pr5078033069183545951prat_o,A: b > prat] :
( ( image_b_prat_o @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) )
= ( collect_o
@ ^ [B2: $o] : ( member5777084206023828648prat_o @ ( produc6389352210403673967prat_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_640_Image__singleton,axiom,
! [R3: set_Pr5661576500205409599_set_o,A: b > prat] :
( ( image_b_prat_set_o @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) )
= ( collect_set_o
@ ^ [B2: set_o] : ( member8295766130977777544_set_o @ ( produc5638353472928494159_set_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_641_Image__singleton,axiom,
! [R3: set_Pr2868754470712218413prat_b,A: b > prat] :
( ( image_8007196543710844537prat_b @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) )
= ( collec6603381632470654801prat_b
@ ^ [B2: produc7326829233880046182prat_b] : ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_642_Image__singleton,axiom,
! [R3: set_Pr8957522257986161222prat_b,A: b > prat] :
( ( image_b_prat_b @ R3 @ ( insert_b_prat @ A @ bot_bot_set_b_prat ) )
= ( collect_b
@ ^ [B2: b] : ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_643_Image__singleton,axiom,
! [R3: set_Pr2246928807395521336_o_o_o,A: product_prod_o_o] :
( ( image_9092160538811120644_o_o_o @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( collect_o
@ ^ [B2: $o] : ( member5943846676503041025_o_o_o @ ( produc6529639187382277576_o_o_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_644_Image__singleton,axiom,
! [R3: set_Pr1571070227385496088_set_o,A: product_prod_o_o] :
( ( image_7547793595507069668_set_o @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( collect_set_o
@ ^ [B2: set_o] : ( member264548248930972385_set_o @ ( produc4486244610082643112_set_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_645_Image__singleton,axiom,
! [R3: set_Pr5232024646447722279et_o_o,A: set_o] :
( ( image_set_o_o @ R3 @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
= ( collect_o
@ ^ [B2: $o] : ( member6599968234716717424et_o_o @ ( product_Pair_set_o_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_646_Image__singleton,axiom,
! [R3: set_Pr4577759397028426247_set_o,A: set_o] :
( ( image_set_o_set_o @ R3 @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
= ( collect_set_o
@ ^ [B2: set_o] : ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A @ B2 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_647_Id__on__def,axiom,
( id_on_o
= ( ^ [A4: set_o] :
( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X5 @ X5 ) @ bot_bo7073875226086086771od_o_o )
@ A4 ) ) ) ) ).
% Id_on_def
thf(fact_648_aboveS__def,axiom,
( order_aboveS_set_o
= ( ^ [R: set_Pr4577759397028426247_set_o,A2: set_o] :
( collect_set_o
@ ^ [B2: set_o] :
( ( B2 != A2 )
& ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A2 @ B2 ) @ R ) ) ) ) ) ).
% aboveS_def
thf(fact_649_aboveS__def,axiom,
( order_aboveS_o
= ( ^ [R: set_Product_prod_o_o,A2: $o] :
( collect_o
@ ^ [B2: $o] :
( ( B2 != A2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R ) ) ) ) ) ).
% aboveS_def
thf(fact_650_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A3: set_o] :
( ( Sup
@ ( image_o_o2
@ ^ [X5: $o] : X5
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_651_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_o > set_o,A3: set_set_o] :
( ( Sup
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : X5
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_652_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A3: set_o] :
( ( Inf
@ ( image_o_o2
@ ^ [X5: $o] : X5
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_653_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_o > set_o,A3: set_set_o] :
( ( Inf
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : X5
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_654_the__elem__def,axiom,
( the_elem_b_prat
= ( ^ [X3: set_b_prat] :
( the_b_prat
@ ^ [X5: b > prat] :
( X3
= ( insert_b_prat @ X5 @ bot_bot_set_b_prat ) ) ) ) ) ).
% the_elem_def
thf(fact_655_the__elem__def,axiom,
( the_el301066130825229084od_o_o
= ( ^ [X3: set_Product_prod_o_o] :
( the_Product_prod_o_o
@ ^ [X5: product_prod_o_o] :
( X3
= ( insert6201435330877294327od_o_o @ X5 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% the_elem_def
thf(fact_656_the__elem__def,axiom,
( the_elem_set_o
= ( ^ [X3: set_set_o] :
( the_set_o
@ ^ [X5: set_o] :
( X3
= ( insert_set_o @ X5 @ bot_bot_set_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_657_the__elem__def,axiom,
( the_elem_o
= ( ^ [X3: set_o] :
( the_o
@ ^ [X5: $o] :
( X3
= ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_658_Range__insert,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( range_3221829133846087963prat_b @ ( insert8235054386829347357prat_b @ ( produc4863932588836083261prat_b @ A @ B ) @ R3 ) )
= ( insert1729827522656407606prat_b @ B @ ( range_3221829133846087963prat_b @ R3 ) ) ) ).
% Range_insert
thf(fact_659_Range__insert,axiom,
! [A: b > prat,B: b,R3: set_Pr8957522257986161222prat_b] :
( ( range_b_prat_b @ ( insert1729827522656407606prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 ) )
= ( insert_b @ B @ ( range_b_prat_b @ R3 ) ) ) ).
% Range_insert
thf(fact_660_Range__insert,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( range_o_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) )
= ( insert_o @ B @ ( range_o_o @ R3 ) ) ) ).
% Range_insert
thf(fact_661_range__constant,axiom,
! [X2: $o] :
( ( image_o_o2
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_662_range__constant,axiom,
! [X2: set_o] :
( ( image_o_set_o2
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_set_o @ X2 @ bot_bot_set_set_o ) ) ).
% range_constant
thf(fact_663_range__constant,axiom,
! [X2: $o] :
( ( image_set_o_o2
@ ^ [Uu: set_o] : X2
@ top_top_set_set_o )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_664_range__constant,axiom,
! [X2: set_set_o] :
( ( image_o_set_set_o
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_set_set_o @ X2 @ bot_bo3529510225556782244_set_o ) ) ).
% range_constant
thf(fact_665_range__constant,axiom,
! [X2: b > prat] :
( ( image_o_b_prat2
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_b_prat @ X2 @ bot_bot_set_b_prat ) ) ).
% range_constant
thf(fact_666_range__constant,axiom,
! [X2: product_prod_o_o] :
( ( image_4057150146340385428od_o_o
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) ) ).
% range_constant
thf(fact_667_range__constant,axiom,
! [X2: $o] :
( ( image_7896445794123959606_o_o_o
@ ^ [Uu: product_prod_o_o] : X2
@ top_to7721136755696657239od_o_o )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_668_range__constant,axiom,
! [X2: set_o] :
( ( image_set_o_set_o2
@ ^ [Uu: set_o] : X2
@ top_top_set_set_o )
= ( insert_set_o @ X2 @ bot_bot_set_set_o ) ) ).
% range_constant
thf(fact_669_range__constant,axiom,
! [X2: set_o] :
( ( image_7870716977482210326_set_o
@ ^ [Uu: product_prod_o_o] : X2
@ top_to7721136755696657239od_o_o )
= ( insert_set_o @ X2 @ bot_bot_set_set_o ) ) ).
% range_constant
thf(fact_670_range__constant,axiom,
! [X2: b > prat] :
( ( image_set_o_b_prat2
@ ^ [Uu: set_o] : X2
@ top_top_set_set_o )
= ( insert_b_prat @ X2 @ bot_bot_set_b_prat ) ) ).
% range_constant
thf(fact_671_Domain__insert,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( domain4185636366779699890prat_b @ ( insert8235054386829347357prat_b @ ( produc4863932588836083261prat_b @ A @ B ) @ R3 ) )
= ( insert_b_prat @ A @ ( domain4185636366779699890prat_b @ R3 ) ) ) ).
% Domain_insert
thf(fact_672_Domain__insert,axiom,
! [A: b > prat,B: b,R3: set_Pr8957522257986161222prat_b] :
( ( domain_b_prat_b @ ( insert1729827522656407606prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 ) )
= ( insert_b_prat @ A @ ( domain_b_prat_b @ R3 ) ) ) ).
% Domain_insert
thf(fact_673_Domain__insert,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( domain_o_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) )
= ( insert_o @ A @ ( domain_o_o @ R3 ) ) ) ).
% Domain_insert
thf(fact_674_top__apply,axiom,
( top_top_o_o
= ( ^ [X5: $o] : top_top_o ) ) ).
% top_apply
thf(fact_675_UNIV__I,axiom,
! [X2: b > prat] : ( member_b_prat @ X2 @ top_top_set_b_prat ) ).
% UNIV_I
thf(fact_676_UNIV__I,axiom,
! [X2: product_prod_o_o] : ( member7466972457876170832od_o_o @ X2 @ top_to7721136755696657239od_o_o ) ).
% UNIV_I
thf(fact_677_UNIV__I,axiom,
! [X2: set_o] : ( member_set_o @ X2 @ top_top_set_set_o ) ).
% UNIV_I
thf(fact_678_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_679_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_b_prat] :
( ( ( comple8554269323607207949b_prat @ A3 )
= bot_bot_set_b_prat )
= ( ! [X5: set_b_prat] :
( ( member_set_b_prat @ X5 @ A3 )
=> ( X5 = bot_bot_set_b_prat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_680_Sup__bot__conv_I1_J,axiom,
! [A3: set_o_o] :
( ( ( complete_Sup_Sup_o_o @ A3 )
= bot_bot_o_o )
= ( ! [X5: $o > $o] :
( ( member_o_o @ X5 @ A3 )
=> ( X5 = bot_bot_o_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_681_Sup__bot__conv_I1_J,axiom,
! [A3: set_o] :
( ( ( complete_Sup_Sup_o @ A3 )
= bot_bot_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( X5 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_682_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_set_o] :
( ( ( comple4436988014476444997_set_o @ A3 )
= bot_bot_set_set_o )
= ( ! [X5: set_set_o] :
( ( member_set_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_683_Sup__bot__conv_I1_J,axiom,
! [A3: set_se491199571785101031od_o_o] :
( ( ( comple2567785453016644468od_o_o @ A3 )
= bot_bo7073875226086086771od_o_o )
= ( ! [X5: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X5 @ A3 )
=> ( X5 = bot_bo7073875226086086771od_o_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_684_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_o] :
( ( ( comple90263536869209701_set_o @ A3 )
= bot_bot_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_685_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_b_prat] :
( ( bot_bot_set_b_prat
= ( comple8554269323607207949b_prat @ A3 ) )
= ( ! [X5: set_b_prat] :
( ( member_set_b_prat @ X5 @ A3 )
=> ( X5 = bot_bot_set_b_prat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_686_Sup__bot__conv_I2_J,axiom,
! [A3: set_o_o] :
( ( bot_bot_o_o
= ( complete_Sup_Sup_o_o @ A3 ) )
= ( ! [X5: $o > $o] :
( ( member_o_o @ X5 @ A3 )
=> ( X5 = bot_bot_o_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_687_Sup__bot__conv_I2_J,axiom,
! [A3: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A3 ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( X5 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_688_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_set_o] :
( ( bot_bot_set_set_o
= ( comple4436988014476444997_set_o @ A3 ) )
= ( ! [X5: set_set_o] :
( ( member_set_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_689_Sup__bot__conv_I2_J,axiom,
! [A3: set_se491199571785101031od_o_o] :
( ( bot_bo7073875226086086771od_o_o
= ( comple2567785453016644468od_o_o @ A3 ) )
= ( ! [X5: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X5 @ A3 )
=> ( X5 = bot_bo7073875226086086771od_o_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_690_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A3 ) )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_691_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_b_prat
@ ^ [S4: b > prat] : P )
= top_top_set_b_prat ) )
& ( ~ P
=> ( ( collect_b_prat
@ ^ [S4: b > prat] : P )
= bot_bot_set_b_prat ) ) ) ).
% Collect_const
thf(fact_692_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec3167064739498627218od_o_o
@ ^ [S4: product_prod_o_o] : P )
= top_to7721136755696657239od_o_o ) )
& ( ~ P
=> ( ( collec3167064739498627218od_o_o
@ ^ [S4: product_prod_o_o] : P )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_const
thf(fact_693_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_set_o
@ ^ [S4: set_o] : P )
= top_top_set_set_o ) )
& ( ~ P
=> ( ( collect_set_o
@ ^ [S4: set_o] : P )
= bot_bot_set_set_o ) ) ) ).
% Collect_const
thf(fact_694_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S4: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S4: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_695_SUP__identity__eq,axiom,
! [A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : X5
@ A3 ) )
= ( comple90263536869209701_set_o @ A3 ) ) ).
% SUP_identity_eq
thf(fact_696_SUP__identity__eq,axiom,
! [A3: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [X5: $o] : X5
@ A3 ) )
= ( complete_Sup_Sup_o @ A3 ) ) ).
% SUP_identity_eq
thf(fact_697_SUP__identity__eq,axiom,
! [A3: set_set_set_o] :
( ( comple4436988014476444997_set_o
@ ( image_3056286201437297541_set_o
@ ^ [X5: set_set_o] : X5
@ A3 ) )
= ( comple4436988014476444997_set_o @ A3 ) ) ).
% SUP_identity_eq
thf(fact_698_SUP__identity__eq,axiom,
! [A3: set_se491199571785101031od_o_o] :
( ( comple2567785453016644468od_o_o
@ ( image_2093575045378028485od_o_o
@ ^ [X5: set_Product_prod_o_o] : X5
@ A3 ) )
= ( comple2567785453016644468od_o_o @ A3 ) ) ).
% SUP_identity_eq
thf(fact_699_Domain__Id__on,axiom,
! [A3: set_o] :
( ( domain_o_o @ ( id_on_o @ A3 ) )
= A3 ) ).
% Domain_Id_on
thf(fact_700_Range__Id__on,axiom,
! [A3: set_o] :
( ( range_o_o @ ( id_on_o @ A3 ) )
= A3 ) ).
% Range_Id_on
thf(fact_701_UN__iff,axiom,
! [B: $o,B3: $o > set_o,A3: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( ? [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( member_o @ B @ ( B3 @ X5 ) ) ) ) ) ).
% UN_iff
thf(fact_702_UN__iff,axiom,
! [B: $o,B3: set_o > set_o,A3: set_set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( ? [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
& ( member_o @ B @ ( B3 @ X5 ) ) ) ) ) ).
% UN_iff
thf(fact_703_UN__iff,axiom,
! [B: set_o,B3: $o > set_set_o,A3: set_o] :
( ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( ? [X5: $o] :
( ( member_o @ X5 @ A3 )
& ( member_set_o @ B @ ( B3 @ X5 ) ) ) ) ) ).
% UN_iff
thf(fact_704_UN__I,axiom,
! [A: $o,A3: set_o,B: $o,B3: $o > set_o] :
( ( member_o @ A @ A3 )
=> ( ( member_o @ B @ ( B3 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_705_UN__I,axiom,
! [A: set_o,A3: set_set_o,B: $o,B3: set_o > set_o] :
( ( member_set_o @ A @ A3 )
=> ( ( member_o @ B @ ( B3 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_706_UN__I,axiom,
! [A: $o,A3: set_o,B: set_o,B3: $o > set_set_o] :
( ( member_o @ A @ A3 )
=> ( ( member_set_o @ B @ ( B3 @ A ) )
=> ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_707_UN__I,axiom,
! [A: $o,A3: set_o,B: b > prat,B3: $o > set_b_prat] :
( ( member_o @ A @ A3 )
=> ( ( member_b_prat @ B @ ( B3 @ A ) )
=> ( member_b_prat @ B @ ( comple8554269323607207949b_prat @ ( image_o_set_b_prat @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_708_UN__I,axiom,
! [A: b > prat,A3: set_b_prat,B: $o,B3: ( b > prat ) > set_o] :
( ( member_b_prat @ A @ A3 )
=> ( ( member_o @ B @ ( B3 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_709_UN__I,axiom,
! [A: product_prod_o_o,A3: set_Product_prod_o_o,B: $o,B3: product_prod_o_o > set_o] :
( ( member7466972457876170832od_o_o @ A @ A3 )
=> ( ( member_o @ B @ ( B3 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_710_UN__I,axiom,
! [A: set_o,A3: set_set_o,B: set_o,B3: set_o > set_set_o] :
( ( member_set_o @ A @ A3 )
=> ( ( member_set_o @ B @ ( B3 @ A ) )
=> ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_711_UN__I,axiom,
! [A: $o,A3: set_o,B: product_prod_o_o,B3: $o > set_Product_prod_o_o] :
( ( member_o @ A @ A3 )
=> ( ( member7466972457876170832od_o_o @ B @ ( B3 @ A ) )
=> ( member7466972457876170832od_o_o @ B @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_712_UN__I,axiom,
! [A: set_o,A3: set_set_o,B: b > prat,B3: set_o > set_b_prat] :
( ( member_set_o @ A @ A3 )
=> ( ( member_b_prat @ B @ ( B3 @ A ) )
=> ( member_b_prat @ B @ ( comple8554269323607207949b_prat @ ( image_2300625924940167469b_prat @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_713_UN__I,axiom,
! [A: b > prat,A3: set_b_prat,B: set_o,B3: ( b > prat ) > set_set_o] :
( ( member_b_prat @ A @ A3 )
=> ( ( member_set_o @ B @ ( B3 @ A ) )
=> ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_714_Sup__empty,axiom,
( ( comple8554269323607207949b_prat @ bot_bo5183958253039183724b_prat )
= bot_bot_set_b_prat ) ).
% Sup_empty
thf(fact_715_Sup__empty,axiom,
( ( complete_Sup_Sup_o_o @ bot_bot_set_o_o2 )
= bot_bot_o_o ) ).
% Sup_empty
thf(fact_716_Sup__empty,axiom,
( ( comple4436988014476444997_set_o @ bot_bo3529510225556782244_set_o )
= bot_bot_set_set_o ) ).
% Sup_empty
thf(fact_717_Sup__empty,axiom,
( ( comple2567785453016644468od_o_o @ bot_bo8006819982598807891od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Sup_empty
thf(fact_718_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_719_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_720_Domain__empty,axiom,
( ( domain_o_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Domain_empty
thf(fact_721_Range__empty,axiom,
( ( range_o_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Range_empty
thf(fact_722_member__bind,axiom,
! [X2: $o,A3: set_o,F: $o > set_o] :
( ( member_o @ X2 @ ( bind_o_o @ A3 @ F ) )
= ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ A3 ) ) ) ) ).
% member_bind
thf(fact_723_member__bind,axiom,
! [X2: $o,A3: set_set_o,F: set_o > set_o] :
( ( member_o @ X2 @ ( bind_set_o_o @ A3 @ F ) )
= ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F @ A3 ) ) ) ) ).
% member_bind
thf(fact_724_member__bind,axiom,
! [X2: set_o,A3: set_o,F: $o > set_set_o] :
( ( member_set_o @ X2 @ ( bind_o_set_o @ A3 @ F ) )
= ( member_set_o @ X2 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ F @ A3 ) ) ) ) ).
% member_bind
thf(fact_725_SUP__bot__conv_I2_J,axiom,
! [B3: $o > set_o,A3: set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_726_SUP__bot__conv_I2_J,axiom,
! [B3: set_o > set_o,A3: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_727_SUP__bot__conv_I2_J,axiom,
! [B3: $o > $o,A3: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ ( image_o_o2 @ B3 @ A3 ) ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_728_SUP__bot__conv_I2_J,axiom,
! [B3: $o > set_set_o,A3: set_o] :
( ( bot_bot_set_set_o
= ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_set_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_729_SUP__bot__conv_I1_J,axiom,
! [B3: $o > set_o,A3: set_o] :
( ( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) )
= bot_bot_set_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_730_SUP__bot__conv_I1_J,axiom,
! [B3: set_o > set_o,A3: set_set_o] :
( ( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) )
= bot_bot_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_731_SUP__bot__conv_I1_J,axiom,
! [B3: $o > $o,A3: set_o] :
( ( ( complete_Sup_Sup_o @ ( image_o_o2 @ B3 @ A3 ) )
= bot_bot_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_732_SUP__bot__conv_I1_J,axiom,
! [B3: $o > set_set_o,A3: set_o] :
( ( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) )
= bot_bot_set_set_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_set_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_733_SUP__bot,axiom,
! [A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : bot_bot_set_o
@ A3 ) )
= bot_bot_set_o ) ).
% SUP_bot
thf(fact_734_SUP__bot,axiom,
! [A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : bot_bot_set_o
@ A3 ) )
= bot_bot_set_o ) ).
% SUP_bot
thf(fact_735_SUP__bot,axiom,
! [A3: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [X5: $o] : bot_bot_o
@ A3 ) )
= bot_bot_o ) ).
% SUP_bot
thf(fact_736_SUP__bot,axiom,
! [A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : bot_bot_set_set_o
@ A3 ) )
= bot_bot_set_set_o ) ).
% SUP_bot
thf(fact_737_SUP__const,axiom,
! [A3: set_o,F: $o] :
( ( A3 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [I: $o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_738_SUP__const,axiom,
! [A3: set_o,F: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [I: $o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_739_SUP__const,axiom,
! [A3: set_set_o,F: $o] :
( ( A3 != bot_bot_set_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_set_o_o2
@ ^ [I: set_o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_740_SUP__const,axiom,
! [A3: set_set_o,F: set_o] :
( ( A3 != bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [I: set_o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_741_SUP__const,axiom,
! [A3: set_b_prat,F: $o] :
( ( A3 != bot_bot_set_b_prat )
=> ( ( complete_Sup_Sup_o
@ ( image_b_prat_o2
@ ^ [I: b > prat] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_742_SUP__const,axiom,
! [A3: set_Product_prod_o_o,F: $o] :
( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ( complete_Sup_Sup_o
@ ( image_7896445794123959606_o_o_o
@ ^ [I: product_prod_o_o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_743_SUP__const,axiom,
! [A3: set_o,F: set_set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [I: $o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_744_SUP__const,axiom,
! [A3: set_b_prat,F: set_o] :
( ( A3 != bot_bot_set_b_prat )
=> ( ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [I: b > prat] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_745_SUP__const,axiom,
! [A3: set_Product_prod_o_o,F: set_o] :
( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [I: product_prod_o_o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_746_SUP__const,axiom,
! [A3: set_set_o,F: set_set_o] :
( ( A3 != bot_bot_set_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [I: set_o] : F
@ A3 ) )
= F ) ) ).
% SUP_const
thf(fact_747_UN__constant,axiom,
! [A3: set_o,C: set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_748_UN__constant,axiom,
! [A3: set_set_o,C: set_o] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : C
@ A3 ) )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_749_UN__constant,axiom,
! [A3: set_o,C: set_set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [Y: $o] : C
@ A3 ) )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_750_UN__constant,axiom,
! [A3: set_o,C: set_b_prat] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_o_set_b_prat
@ ^ [Y: $o] : C
@ A3 ) )
= bot_bot_set_b_prat ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_o_set_b_prat
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_751_UN__constant,axiom,
! [A3: set_b_prat,C: set_o] :
( ( ( A3 = bot_bot_set_b_prat )
=> ( ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [Y: b > prat] : C
@ A3 ) )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_b_prat )
=> ( ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [Y: b > prat] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_752_UN__constant,axiom,
! [A3: set_Product_prod_o_o,C: set_o] :
( ( ( A3 = bot_bo7073875226086086771od_o_o )
=> ( ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [Y: product_prod_o_o] : C
@ A3 ) )
= bot_bot_set_o ) )
& ( ( A3 != bot_bo7073875226086086771od_o_o )
=> ( ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [Y: product_prod_o_o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_753_UN__constant,axiom,
! [A3: set_set_o,C: set_set_o] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [Y: set_o] : C
@ A3 ) )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [Y: set_o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_754_UN__constant,axiom,
! [A3: set_o,C: set_Product_prod_o_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [Y: $o] : C
@ A3 ) )
= bot_bo7073875226086086771od_o_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_755_UN__constant,axiom,
! [A3: set_set_o,C: set_b_prat] :
( ( ( A3 = bot_bot_set_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_2300625924940167469b_prat
@ ^ [Y: set_o] : C
@ A3 ) )
= bot_bot_set_b_prat ) )
& ( ( A3 != bot_bot_set_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_2300625924940167469b_prat
@ ^ [Y: set_o] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_756_UN__constant,axiom,
! [A3: set_b_prat,C: set_set_o] :
( ( ( A3 = bot_bot_set_b_prat )
=> ( ( comple4436988014476444997_set_o
@ ( image_4142778379334362525_set_o
@ ^ [Y: b > prat] : C
@ A3 ) )
= bot_bot_set_set_o ) )
& ( ( A3 != bot_bot_set_b_prat )
=> ( ( comple4436988014476444997_set_o
@ ( image_4142778379334362525_set_o
@ ^ [Y: b > prat] : C
@ A3 ) )
= C ) ) ) ).
% UN_constant
thf(fact_757_UN__simps_I1_J,axiom,
! [C2: set_o,A: $o,B3: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_758_UN__simps_I1_J,axiom,
! [C2: set_set_o,A: $o,B3: set_o > set_o] :
( ( ( C2 = bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_759_UN__simps_I1_J,axiom,
! [C2: set_o,A: set_o,B3: $o > set_set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_760_UN__simps_I1_J,axiom,
! [C2: set_o,A: b > prat,B3: $o > set_b_prat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_o_set_b_prat
@ ^ [X5: $o] : ( insert_b_prat @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_b_prat ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_o_set_b_prat
@ ^ [X5: $o] : ( insert_b_prat @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_b_prat @ A @ ( comple8554269323607207949b_prat @ ( image_o_set_b_prat @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_761_UN__simps_I1_J,axiom,
! [C2: set_b_prat,A: $o,B3: ( b > prat ) > set_o] :
( ( ( C2 = bot_bot_set_b_prat )
=> ( ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [X5: b > prat] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_b_prat )
=> ( ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [X5: b > prat] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_762_UN__simps_I1_J,axiom,
! [C2: set_Product_prod_o_o,A: $o,B3: product_prod_o_o > set_o] :
( ( ( C2 = bot_bo7073875226086086771od_o_o )
=> ( ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [X5: product_prod_o_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bo7073875226086086771od_o_o )
=> ( ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [X5: product_prod_o_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_763_UN__simps_I1_J,axiom,
! [C2: set_set_o,A: set_o,B3: set_o > set_set_o] :
( ( ( C2 = bot_bot_set_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_set_o ) )
& ( ( C2 != bot_bot_set_set_o )
=> ( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_764_UN__simps_I1_J,axiom,
! [C2: set_o,A: product_prod_o_o,B3: $o > set_Product_prod_o_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( insert6201435330877294327od_o_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bo7073875226086086771od_o_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( insert6201435330877294327od_o_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert6201435330877294327od_o_o @ A @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_765_UN__simps_I1_J,axiom,
! [C2: set_set_o,A: b > prat,B3: set_o > set_b_prat] :
( ( ( C2 = bot_bot_set_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_2300625924940167469b_prat
@ ^ [X5: set_o] : ( insert_b_prat @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_b_prat ) )
& ( ( C2 != bot_bot_set_set_o )
=> ( ( comple8554269323607207949b_prat
@ ( image_2300625924940167469b_prat
@ ^ [X5: set_o] : ( insert_b_prat @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_b_prat @ A @ ( comple8554269323607207949b_prat @ ( image_2300625924940167469b_prat @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_766_UN__simps_I1_J,axiom,
! [C2: set_b_prat,A: set_o,B3: ( b > prat ) > set_set_o] :
( ( ( C2 = bot_bot_set_b_prat )
=> ( ( comple4436988014476444997_set_o
@ ( image_4142778379334362525_set_o
@ ^ [X5: b > prat] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_set_o ) )
& ( ( C2 != bot_bot_set_b_prat )
=> ( ( comple4436988014476444997_set_o
@ ( image_4142778379334362525_set_o
@ ^ [X5: b > prat] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_767_UN__singleton,axiom,
! [A3: set_b_prat] :
( ( comple8554269323607207949b_prat
@ ( image_3518374120538949733b_prat
@ ^ [X5: b > prat] : ( insert_b_prat @ X5 @ bot_bot_set_b_prat )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_768_UN__singleton,axiom,
! [A3: set_set_o] :
( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( insert_set_o @ X5 @ bot_bot_set_set_o )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_769_UN__singleton,axiom,
! [A3: set_Product_prod_o_o] :
( ( comple2567785453016644468od_o_o
@ ( image_5464257317878419621od_o_o
@ ^ [X5: product_prod_o_o] : ( insert6201435330877294327od_o_o @ X5 @ bot_bo7073875226086086771od_o_o )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_770_UN__singleton,axiom,
! [A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( insert_o @ X5 @ bot_bot_set_o )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_771_Range__Union,axiom,
! [S: set_se491199571785101031od_o_o] :
( ( range_o_o @ ( comple2567785453016644468od_o_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_1778816826048745270_set_o @ range_o_o @ S ) ) ) ).
% Range_Union
thf(fact_772_Domain__Union,axiom,
! [S: set_se491199571785101031od_o_o] :
( ( domain_o_o @ ( comple2567785453016644468od_o_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_1778816826048745270_set_o @ domain_o_o @ S ) ) ) ).
% Domain_Union
thf(fact_773_refl__on__UNION,axiom,
! [S: set_o,A3: $o > set_o,R3: $o > set_Product_prod_o_o] :
( ! [X: $o] :
( ( member_o @ X @ S )
=> ( refl_on_o @ ( A3 @ X ) @ ( R3 @ X ) ) )
=> ( refl_on_o @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ A3 @ S ) ) @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ R3 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_774_refl__on__UNION,axiom,
! [S: set_set_o,A3: set_o > set_o,R3: set_o > set_Product_prod_o_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ S )
=> ( refl_on_o @ ( A3 @ X ) @ ( R3 @ X ) ) )
=> ( refl_on_o @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ A3 @ S ) ) @ ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ R3 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_775_refl__on__UNION,axiom,
! [S: set_o,A3: $o > set_set_o,R3: $o > set_Pr4577759397028426247_set_o] :
( ! [X: $o] :
( ( member_o @ X @ S )
=> ( refl_on_set_o @ ( A3 @ X ) @ ( R3 @ X ) ) )
=> ( refl_on_set_o @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ A3 @ S ) ) @ ( comple5785745244134051700_set_o @ ( image_1696406616792387444_set_o @ R3 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_776_UN__UN__flatten,axiom,
! [C2: $o > set_o,B3: $o > set_o,A3: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_777_UN__UN__flatten,axiom,
! [C2: $o > set_o,B3: set_o > set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_778_UN__UN__flatten,axiom,
! [C2: set_o > set_o,B3: $o > set_set_o,A3: set_o] :
( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_779_UN__UN__flatten,axiom,
! [C2: $o > set_set_o,B3: $o > set_o,A3: set_o] :
( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [Y: $o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_780_UN__UN__flatten,axiom,
! [C2: set_o > set_o,B3: set_o > set_set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_781_UN__UN__flatten,axiom,
! [C2: product_prod_o_o > set_o,B3: $o > set_Product_prod_o_o,A3: set_o] :
( ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_782_UN__UN__flatten,axiom,
! [C2: $o > set_set_o,B3: set_o > set_o,A3: set_set_o] :
( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) ) )
= ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [Y: set_o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_783_UN__UN__flatten,axiom,
! [C2: set_o > set_set_o,B3: $o > set_set_o,A3: set_o] :
( ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ C2 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [Y: $o] : ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_784_UN__UN__flatten,axiom,
! [C2: $o > set_Product_prod_o_o,B3: $o > set_o,A3: set_o] :
( ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) )
= ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [Y: $o] : ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_785_UN__UN__flatten,axiom,
! [C2: product_prod_o_o > set_o,B3: set_o > set_Product_prod_o_o,A3: set_set_o] :
( ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ B3 @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( B3 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_786_SUP__UNION,axiom,
! [F: $o > $o,G: $o > set_o,A3: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ G @ A3 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_787_SUP__UNION,axiom,
! [F: $o > set_o,G: $o > set_o,A3: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ G @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_788_SUP__UNION,axiom,
! [F: $o > $o,G: set_o > set_o,A3: set_set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ G @ A3 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_set_o_o2
@ ^ [Y: set_o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_789_SUP__UNION,axiom,
! [F: set_o > $o,G: $o > set_set_o,A3: set_o] :
( ( complete_Sup_Sup_o @ ( image_set_o_o2 @ F @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ G @ A3 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_790_SUP__UNION,axiom,
! [F: $o > set_o,G: set_o > set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ G @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_791_SUP__UNION,axiom,
! [F: set_o > set_o,G: $o > set_set_o,A3: set_o] :
( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ G @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_792_SUP__UNION,axiom,
! [F: product_prod_o_o > $o,G: $o > set_Product_prod_o_o,A3: set_o] :
( ( complete_Sup_Sup_o @ ( image_7896445794123959606_o_o_o @ F @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ G @ A3 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_7896445794123959606_o_o_o @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_793_SUP__UNION,axiom,
! [F: $o > set_set_o,G: $o > set_o,A3: set_o] :
( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ G @ A3 ) ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [Y: $o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_794_SUP__UNION,axiom,
! [F: set_o > set_o,G: set_o > set_set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ G @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_795_SUP__UNION,axiom,
! [F: product_prod_o_o > set_o,G: $o > set_Product_prod_o_o,A3: set_o] :
( ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ F @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ G @ A3 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ F @ ( G @ Y ) ) )
@ A3 ) ) ) ).
% SUP_UNION
thf(fact_796_UN__E,axiom,
! [B: $o,B3: $o > set_o,A3: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A3 )
=> ~ ( member_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_797_UN__E,axiom,
! [B: $o,B3: set_o > set_o,A3: set_set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
=> ~ ! [X: set_o] :
( ( member_set_o @ X @ A3 )
=> ~ ( member_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_798_UN__E,axiom,
! [B: set_o,B3: $o > set_set_o,A3: set_o] :
( ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A3 )
=> ~ ( member_set_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_799_UN__E,axiom,
! [B: b > prat,B3: $o > set_b_prat,A3: set_o] :
( ( member_b_prat @ B @ ( comple8554269323607207949b_prat @ ( image_o_set_b_prat @ B3 @ A3 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A3 )
=> ~ ( member_b_prat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_800_UN__E,axiom,
! [B: $o,B3: ( b > prat ) > set_o,A3: set_b_prat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ A3 ) ) )
=> ~ ! [X: b > prat] :
( ( member_b_prat @ X @ A3 )
=> ~ ( member_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_801_UN__E,axiom,
! [B: $o,B3: product_prod_o_o > set_o,A3: set_Product_prod_o_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ A3 ) ) )
=> ~ ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A3 )
=> ~ ( member_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_802_UN__E,axiom,
! [B: set_o,B3: set_o > set_set_o,A3: set_set_o] :
( ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) )
=> ~ ! [X: set_o] :
( ( member_set_o @ X @ A3 )
=> ~ ( member_set_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_803_UN__E,axiom,
! [B: product_prod_o_o,B3: $o > set_Product_prod_o_o,A3: set_o] :
( ( member7466972457876170832od_o_o @ B @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A3 )
=> ~ ( member7466972457876170832od_o_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_804_UN__E,axiom,
! [B: b > prat,B3: set_o > set_b_prat,A3: set_set_o] :
( ( member_b_prat @ B @ ( comple8554269323607207949b_prat @ ( image_2300625924940167469b_prat @ B3 @ A3 ) ) )
=> ~ ! [X: set_o] :
( ( member_set_o @ X @ A3 )
=> ~ ( member_b_prat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_805_UN__E,axiom,
! [B: set_o,B3: ( b > prat ) > set_set_o,A3: set_b_prat] :
( ( member_set_o @ B @ ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ A3 ) ) )
=> ~ ! [X: b > prat] :
( ( member_b_prat @ X @ A3 )
=> ~ ( member_set_o @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_806_UN__extend__simps_I8_J,axiom,
! [B3: $o > set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ Y ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ ( comple90263536869209701_set_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_807_UN__extend__simps_I8_J,axiom,
! [B3: set_o > set_o,A3: set_set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_1148209357520620709_set_o
@ ^ [Y: set_set_o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ Y ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ ( comple4436988014476444997_set_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_808_UN__extend__simps_I8_J,axiom,
! [B3: product_prod_o_o > set_o,A3: set_se491199571785101031od_o_o] :
( ( comple90263536869209701_set_o
@ ( image_1778816826048745270_set_o
@ ^ [Y: set_Product_prod_o_o] : ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ Y ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ ( comple2567785453016644468od_o_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_809_UN__extend__simps_I8_J,axiom,
! [B3: $o > set_set_o,A3: set_set_o] :
( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [Y: set_o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ Y ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ ( comple90263536869209701_set_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_810_UN__extend__simps_I8_J,axiom,
! [B3: set_o > set_set_o,A3: set_set_set_o] :
( ( comple4436988014476444997_set_o
@ ( image_3056286201437297541_set_o
@ ^ [Y: set_set_o] : ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ Y ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ ( comple4436988014476444997_set_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_811_UN__extend__simps_I8_J,axiom,
! [B3: product_prod_o_o > set_set_o,A3: set_se491199571785101031od_o_o] :
( ( comple4436988014476444997_set_o
@ ( image_1898638321153874454_set_o
@ ^ [Y: set_Product_prod_o_o] : ( comple4436988014476444997_set_o @ ( image_5008979335655565046_set_o @ B3 @ Y ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_5008979335655565046_set_o @ B3 @ ( comple2567785453016644468od_o_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_812_UN__extend__simps_I8_J,axiom,
! [B3: $o > set_Product_prod_o_o,A3: set_set_o] :
( ( comple2567785453016644468od_o_o
@ ( image_4627112110306186388od_o_o
@ ^ [Y: set_o] : ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ Y ) )
@ A3 ) )
= ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ ( comple90263536869209701_set_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_813_UN__extend__simps_I8_J,axiom,
! [B3: set_o > set_Product_prod_o_o,A3: set_set_set_o] :
( ( comple2567785453016644468od_o_o
@ ( image_1123327267708678580od_o_o
@ ^ [Y: set_set_o] : ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ B3 @ Y ) )
@ A3 ) )
= ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ B3 @ ( comple4436988014476444997_set_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_814_UN__extend__simps_I8_J,axiom,
! [B3: product_prod_o_o > set_Product_prod_o_o,A3: set_se491199571785101031od_o_o] :
( ( comple2567785453016644468od_o_o
@ ( image_2093575045378028485od_o_o
@ ^ [Y: set_Product_prod_o_o] : ( comple2567785453016644468od_o_o @ ( image_5464257317878419621od_o_o @ B3 @ Y ) )
@ A3 ) )
= ( comple2567785453016644468od_o_o @ ( image_5464257317878419621od_o_o @ B3 @ ( comple2567785453016644468od_o_o @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_815_UN__extend__simps_I9_J,axiom,
! [C2: $o > set_o,B3: $o > set_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_816_UN__extend__simps_I9_J,axiom,
! [C2: $o > set_o,B3: set_o > set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ C2 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_817_UN__extend__simps_I9_J,axiom,
! [C2: set_o > set_o,B3: $o > set_set_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_818_UN__extend__simps_I9_J,axiom,
! [C2: $o > set_set_o,B3: $o > set_o,A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_819_UN__extend__simps_I9_J,axiom,
! [C2: set_o > set_o,B3: set_o > set_set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ C2 @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_820_UN__extend__simps_I9_J,axiom,
! [C2: product_prod_o_o > set_o,B3: $o > set_Product_prod_o_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_821_UN__extend__simps_I9_J,axiom,
! [C2: $o > set_set_o,B3: set_o > set_o,A3: set_set_o] :
( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_822_UN__extend__simps_I9_J,axiom,
! [C2: set_o > set_set_o,B3: $o > set_set_o,A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ C2 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_823_UN__extend__simps_I9_J,axiom,
! [C2: $o > set_Product_prod_o_o,B3: $o > set_o,A3: set_o] :
( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_824_UN__extend__simps_I9_J,axiom,
! [C2: product_prod_o_o > set_o,B3: set_o > set_Product_prod_o_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( B3 @ X5 ) ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ C2 @ ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_825_UNIV__def,axiom,
( top_to7721136755696657239od_o_o
= ( collec3167064739498627218od_o_o
@ ^ [X5: product_prod_o_o] : $true ) ) ).
% UNIV_def
thf(fact_826_UNIV__def,axiom,
( top_top_set_set_o
= ( collect_set_o
@ ^ [X5: set_o] : $true ) ) ).
% UNIV_def
thf(fact_827_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X5: $o] : $true ) ) ).
% UNIV_def
thf(fact_828_Nitpick_OThe__psimp,axiom,
! [P: $o > $o,X2: $o] :
( ( P
= ( ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ X2 ) )
=> ( ( the_o @ P )
= X2 ) ) ).
% Nitpick.The_psimp
thf(fact_829_UNIV__witness,axiom,
? [X: b > prat] : ( member_b_prat @ X @ top_top_set_b_prat ) ).
% UNIV_witness
thf(fact_830_UNIV__witness,axiom,
? [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ top_to7721136755696657239od_o_o ) ).
% UNIV_witness
thf(fact_831_UNIV__witness,axiom,
? [X: set_o] : ( member_set_o @ X @ top_top_set_set_o ) ).
% UNIV_witness
thf(fact_832_UNIV__witness,axiom,
? [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_witness
thf(fact_833_UNIV__eq__I,axiom,
! [A3: set_b_prat] :
( ! [X: b > prat] : ( member_b_prat @ X @ A3 )
=> ( top_top_set_b_prat = A3 ) ) ).
% UNIV_eq_I
thf(fact_834_UNIV__eq__I,axiom,
! [A3: set_Product_prod_o_o] :
( ! [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A3 )
=> ( top_to7721136755696657239od_o_o = A3 ) ) ).
% UNIV_eq_I
thf(fact_835_UNIV__eq__I,axiom,
! [A3: set_set_o] :
( ! [X: set_o] : ( member_set_o @ X @ A3 )
=> ( top_top_set_set_o = A3 ) ) ).
% UNIV_eq_I
thf(fact_836_UNIV__eq__I,axiom,
! [A3: set_o] :
( ! [X: $o] : ( member_o @ X @ A3 )
=> ( top_top_set_o = A3 ) ) ).
% UNIV_eq_I
thf(fact_837_empty__Union__conv,axiom,
! [A3: set_set_b_prat] :
( ( bot_bot_set_b_prat
= ( comple8554269323607207949b_prat @ A3 ) )
= ( ! [X5: set_b_prat] :
( ( member_set_b_prat @ X5 @ A3 )
=> ( X5 = bot_bot_set_b_prat ) ) ) ) ).
% empty_Union_conv
thf(fact_838_empty__Union__conv,axiom,
! [A3: set_set_set_o] :
( ( bot_bot_set_set_o
= ( comple4436988014476444997_set_o @ A3 ) )
= ( ! [X5: set_set_o] :
( ( member_set_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_839_empty__Union__conv,axiom,
! [A3: set_se491199571785101031od_o_o] :
( ( bot_bo7073875226086086771od_o_o
= ( comple2567785453016644468od_o_o @ A3 ) )
= ( ! [X5: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X5 @ A3 )
=> ( X5 = bot_bo7073875226086086771od_o_o ) ) ) ) ).
% empty_Union_conv
thf(fact_840_empty__Union__conv,axiom,
! [A3: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A3 ) )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_841_Union__empty__conv,axiom,
! [A3: set_set_b_prat] :
( ( ( comple8554269323607207949b_prat @ A3 )
= bot_bot_set_b_prat )
= ( ! [X5: set_b_prat] :
( ( member_set_b_prat @ X5 @ A3 )
=> ( X5 = bot_bot_set_b_prat ) ) ) ) ).
% Union_empty_conv
thf(fact_842_Union__empty__conv,axiom,
! [A3: set_set_set_o] :
( ( ( comple4436988014476444997_set_o @ A3 )
= bot_bot_set_set_o )
= ( ! [X5: set_set_o] :
( ( member_set_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_843_Union__empty__conv,axiom,
! [A3: set_se491199571785101031od_o_o] :
( ( ( comple2567785453016644468od_o_o @ A3 )
= bot_bo7073875226086086771od_o_o )
= ( ! [X5: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X5 @ A3 )
=> ( X5 = bot_bo7073875226086086771od_o_o ) ) ) ) ).
% Union_empty_conv
thf(fact_844_Union__empty__conv,axiom,
! [A3: set_set_o] :
( ( ( comple90263536869209701_set_o @ A3 )
= bot_bot_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( X5 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_845_Union__empty,axiom,
( ( comple8554269323607207949b_prat @ bot_bo5183958253039183724b_prat )
= bot_bot_set_b_prat ) ).
% Union_empty
thf(fact_846_Union__empty,axiom,
( ( comple4436988014476444997_set_o @ bot_bo3529510225556782244_set_o )
= bot_bot_set_set_o ) ).
% Union_empty
thf(fact_847_Union__empty,axiom,
( ( comple2567785453016644468od_o_o @ bot_bo8006819982598807891od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Union_empty
thf(fact_848_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_849_UN__Image,axiom,
! [X7: $o > set_Product_prod_o_o,I2: set_o,S: set_o] :
( ( image_o_o @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ X7 @ I2 ) ) @ S )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [I: $o] : ( image_o_o @ ( X7 @ I ) @ S )
@ I2 ) ) ) ).
% UN_Image
thf(fact_850_UN__Image,axiom,
! [X7: set_o > set_Product_prod_o_o,I2: set_set_o,S: set_o] :
( ( image_o_o @ ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ X7 @ I2 ) ) @ S )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [I: set_o] : ( image_o_o @ ( X7 @ I ) @ S )
@ I2 ) ) ) ).
% UN_Image
thf(fact_851_bind__UNION,axiom,
( bind_o_o
= ( ^ [A4: set_o,F2: $o > set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_852_bind__UNION,axiom,
( bind_set_o_o
= ( ^ [A4: set_set_o,F2: set_o > set_o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_853_bind__UNION,axiom,
( bind_o_set_o
= ( ^ [A4: set_o,F2: $o > set_set_o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_854_SUP__commute,axiom,
! [F: $o > $o > set_o,B3: set_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [I: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ ( F @ I ) @ B3 ) )
@ A3 ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [J: $o] :
( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [I: $o] : ( F @ I @ J )
@ A3 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_855_SUP__commute,axiom,
! [F: $o > set_o > set_o,B3: set_set_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [I: $o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ ( F @ I ) @ B3 ) )
@ A3 ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [J: set_o] :
( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [I: $o] : ( F @ I @ J )
@ A3 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_856_SUP__commute,axiom,
! [F: set_o > $o > set_o,B3: set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [I: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o2 @ ( F @ I ) @ B3 ) )
@ A3 ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [J: $o] :
( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [I: set_o] : ( F @ I @ J )
@ A3 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_857_SUP__commute,axiom,
! [F: set_o > set_o > set_o,B3: set_set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [I: set_o] : ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ ( F @ I ) @ B3 ) )
@ A3 ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [J: set_o] :
( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [I: set_o] : ( F @ I @ J )
@ A3 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_858_SUP__commute,axiom,
! [F: $o > $o > $o,B3: set_o,A3: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [I: $o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ ( F @ I ) @ B3 ) )
@ A3 ) )
= ( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [J: $o] :
( complete_Sup_Sup_o
@ ( image_o_o2
@ ^ [I: $o] : ( F @ I @ J )
@ A3 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_859_SUP__commute,axiom,
! [F: $o > $o > set_set_o,B3: set_o,A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [I: $o] : ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ ( F @ I ) @ B3 ) )
@ A3 ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [J: $o] :
( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [I: $o] : ( F @ I @ J )
@ A3 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_860_UN__extend__simps_I10_J,axiom,
! [B3: $o > set_o,F: $o > $o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ ( image_o_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_861_UN__extend__simps_I10_J,axiom,
! [B3: set_o > set_o,F: $o > set_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ ( image_o_set_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_862_UN__extend__simps_I10_J,axiom,
! [B3: $o > set_o,F: set_o > $o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [A2: set_o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ ( image_set_o_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_863_UN__extend__simps_I10_J,axiom,
! [B3: $o > set_set_o,F: $o > $o,A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ ( image_o_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_864_UN__extend__simps_I10_J,axiom,
! [B3: set_set_o > set_o,F: $o > set_set_o,A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_1148209357520620709_set_o @ B3 @ ( image_o_set_set_o @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_865_UN__extend__simps_I10_J,axiom,
! [B3: set_o > set_o,F: set_o > set_o,A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [A2: set_o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ ( image_set_o_set_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_866_UN__extend__simps_I10_J,axiom,
! [B3: set_o > set_set_o,F: $o > set_o,A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ ( image_o_set_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_867_UN__extend__simps_I10_J,axiom,
! [B3: $o > set_Product_prod_o_o,F: $o > $o,A3: set_o] :
( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ ( image_o_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_868_UN__extend__simps_I10_J,axiom,
! [B3: set_o > set_set_o,F: set_o > set_o,A3: set_set_o] :
( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [A2: set_o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ ( image_set_o_set_o2 @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_869_UN__extend__simps_I10_J,axiom,
! [B3: set_set_o > set_set_o,F: $o > set_set_o,A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [A2: $o] : ( B3 @ ( F @ A2 ) )
@ A3 ) )
= ( comple4436988014476444997_set_o @ ( image_3056286201437297541_set_o @ B3 @ ( image_o_set_set_o @ F @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_870_image__UN,axiom,
! [F: $o > $o,B3: $o > set_o,A3: set_o] :
( ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( image_o_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_871_image__UN,axiom,
! [F: $o > $o,B3: set_o > set_o,A3: set_set_o] :
( ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( image_o_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_872_image__UN,axiom,
! [F: $o > set_o,B3: $o > set_o,A3: set_o] :
( ( image_o_set_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( image_o_set_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_873_image__UN,axiom,
! [F: set_o > $o,B3: $o > set_set_o,A3: set_o] :
( ( image_set_o_o2 @ F @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( image_set_o_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_874_image__UN,axiom,
! [F: $o > set_set_o,B3: $o > set_o,A3: set_o] :
( ( image_o_set_set_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple7771960130403061797_set_o
@ ( image_2867644922512576549_set_o
@ ^ [X5: $o] : ( image_o_set_set_o @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_875_image__UN,axiom,
! [F: $o > set_o,B3: set_o > set_o,A3: set_set_o] :
( ( image_o_set_o2 @ F @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( image_o_set_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_876_image__UN,axiom,
! [F: $o > product_prod_o_o,B3: $o > set_o,A3: set_o] :
( ( image_4057150146340385428od_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( image_4057150146340385428od_o_o @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_877_image__UN,axiom,
! [F: set_o > $o,B3: set_o > set_set_o,A3: set_set_o] :
( ( image_set_o_o2 @ F @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( image_set_o_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_878_image__UN,axiom,
! [F: set_o > set_o,B3: $o > set_set_o,A3: set_o] :
( ( image_set_o_set_o2 @ F @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( image_set_o_set_o2 @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_879_image__UN,axiom,
! [F: product_prod_o_o > $o,B3: $o > set_Product_prod_o_o,A3: set_o] :
( ( image_7896445794123959606_o_o_o @ F @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( image_7896445794123959606_o_o_o @ F @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_880_image__Union,axiom,
! [F: $o > set_set_o,S: set_set_o] :
( ( image_o_set_set_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple7771960130403061797_set_o @ ( image_6214471063729551685_set_o @ ( image_o_set_set_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_881_image__Union,axiom,
! [F: $o > $o,S: set_set_o] :
( ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ ( image_o_o2 @ F ) @ S ) ) ) ).
% image_Union
thf(fact_882_image__Union,axiom,
! [F: $o > set_o,S: set_set_o] :
( ( image_o_set_o2 @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ ( image_o_set_o2 @ F ) @ S ) ) ) ).
% image_Union
thf(fact_883_image__Union,axiom,
! [F: $o > product_prod_o_o,S: set_set_o] :
( ( image_4057150146340385428od_o_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ ( image_4057150146340385428od_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_884_image__Union,axiom,
! [F: set_o > $o,S: set_set_set_o] :
( ( image_set_o_o2 @ F @ ( comple4436988014476444997_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_1148209357520620709_set_o @ ( image_set_o_o2 @ F ) @ S ) ) ) ).
% image_Union
thf(fact_885_image__Union,axiom,
! [F: set_o > set_o,S: set_set_set_o] :
( ( image_set_o_set_o2 @ F @ ( comple4436988014476444997_set_o @ S ) )
= ( comple4436988014476444997_set_o @ ( image_3056286201437297541_set_o @ ( image_set_o_set_o2 @ F ) @ S ) ) ) ).
% image_Union
thf(fact_886_image__Union,axiom,
! [F: set_o > product_prod_o_o,S: set_set_set_o] :
( ( image_6524031425630931892od_o_o @ F @ ( comple4436988014476444997_set_o @ S ) )
= ( comple2567785453016644468od_o_o @ ( image_1123327267708678580od_o_o @ ( image_6524031425630931892od_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_887_image__Union,axiom,
! [F: product_prod_o_o > $o,S: set_se491199571785101031od_o_o] :
( ( image_7896445794123959606_o_o_o @ F @ ( comple2567785453016644468od_o_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_1778816826048745270_set_o @ ( image_7896445794123959606_o_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_888_image__Union,axiom,
! [F: product_prod_o_o > set_o,S: set_se491199571785101031od_o_o] :
( ( image_7870716977482210326_set_o @ F @ ( comple2567785453016644468od_o_o @ S ) )
= ( comple4436988014476444997_set_o @ ( image_1898638321153874454_set_o @ ( image_7870716977482210326_set_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_889_image__Union,axiom,
! [F: product_prod_o_o > product_prod_o_o,S: set_se491199571785101031od_o_o] :
( ( image_9131363867636255685od_o_o @ F @ ( comple2567785453016644468od_o_o @ S ) )
= ( comple2567785453016644468od_o_o @ ( image_2093575045378028485od_o_o @ ( image_9131363867636255685od_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_890_UNION__empty__conv_I2_J,axiom,
! [B3: $o > set_o,A3: set_o] :
( ( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) )
= bot_bot_set_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_891_UNION__empty__conv_I2_J,axiom,
! [B3: set_o > set_o,A3: set_set_o] :
( ( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) )
= bot_bot_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_892_UNION__empty__conv_I2_J,axiom,
! [B3: $o > set_set_o,A3: set_o] :
( ( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) )
= bot_bot_set_set_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_set_o ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_893_UNION__empty__conv_I1_J,axiom,
! [B3: $o > set_o,A3: set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_894_UNION__empty__conv_I1_J,axiom,
! [B3: set_o > set_o,A3: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_895_UNION__empty__conv_I1_J,axiom,
! [B3: $o > set_set_o,A3: set_o] :
( ( bot_bot_set_set_o
= ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( B3 @ X5 )
= bot_bot_set_set_o ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_896_UN__empty,axiom,
! [B3: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_897_UN__empty,axiom,
! [B3: set_o > set_o] :
( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ bot_bot_set_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_898_UN__empty,axiom,
! [B3: $o > set_set_o] :
( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ bot_bot_set_o ) )
= bot_bot_set_set_o ) ).
% UN_empty
thf(fact_899_UN__empty,axiom,
! [B3: $o > set_b_prat] :
( ( comple8554269323607207949b_prat @ ( image_o_set_b_prat @ B3 @ bot_bot_set_o ) )
= bot_bot_set_b_prat ) ).
% UN_empty
thf(fact_900_UN__empty,axiom,
! [B3: ( b > prat ) > set_o] :
( ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ bot_bot_set_b_prat ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_901_UN__empty,axiom,
! [B3: product_prod_o_o > set_o] :
( ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ bot_bo7073875226086086771od_o_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_902_UN__empty,axiom,
! [B3: set_o > set_set_o] :
( ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ bot_bot_set_set_o ) )
= bot_bot_set_set_o ) ).
% UN_empty
thf(fact_903_UN__empty,axiom,
! [B3: $o > set_Product_prod_o_o] :
( ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ bot_bot_set_o ) )
= bot_bo7073875226086086771od_o_o ) ).
% UN_empty
thf(fact_904_UN__empty,axiom,
! [B3: set_o > set_b_prat] :
( ( comple8554269323607207949b_prat @ ( image_2300625924940167469b_prat @ B3 @ bot_bot_set_set_o ) )
= bot_bot_set_b_prat ) ).
% UN_empty
thf(fact_905_UN__empty,axiom,
! [B3: ( b > prat ) > set_set_o] :
( ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ bot_bot_set_b_prat ) )
= bot_bot_set_set_o ) ).
% UN_empty
thf(fact_906_UN__empty2,axiom,
! [A3: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : bot_bot_set_o
@ A3 ) )
= bot_bot_set_o ) ).
% UN_empty2
thf(fact_907_UN__empty2,axiom,
! [A3: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : bot_bot_set_o
@ A3 ) )
= bot_bot_set_o ) ).
% UN_empty2
thf(fact_908_UN__empty2,axiom,
! [A3: set_o] :
( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : bot_bot_set_set_o
@ A3 ) )
= bot_bot_set_set_o ) ).
% UN_empty2
thf(fact_909_UN__insert__distrib,axiom,
! [U2: $o,A3: set_o,A: $o,B3: $o > set_o] :
( ( member_o @ U2 @ A3 )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_910_UN__insert__distrib,axiom,
! [U2: set_o,A3: set_set_o,A: $o,B3: set_o > set_o] :
( ( member_set_o @ U2 @ A3 )
=> ( ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_911_UN__insert__distrib,axiom,
! [U2: $o,A3: set_o,A: set_o,B3: $o > set_set_o] :
( ( member_o @ U2 @ A3 )
=> ( ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_912_UN__insert__distrib,axiom,
! [U2: b > prat,A3: set_b_prat,A: $o,B3: ( b > prat ) > set_o] :
( ( member_b_prat @ U2 @ A3 )
=> ( ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [X5: b > prat] : ( insert_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_913_UN__insert__distrib,axiom,
! [U2: product_prod_o_o,A3: set_Product_prod_o_o,A: $o,B3: product_prod_o_o > set_o] :
( ( member7466972457876170832od_o_o @ U2 @ A3 )
=> ( ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [X5: product_prod_o_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_914_UN__insert__distrib,axiom,
! [U2: set_o,A3: set_set_o,A: set_o,B3: set_o > set_set_o] :
( ( member_set_o @ U2 @ A3 )
=> ( ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_915_UN__insert__distrib,axiom,
! [U2: $o,A3: set_o,A: product_prod_o_o,B3: $o > set_Product_prod_o_o] :
( ( member_o @ U2 @ A3 )
=> ( ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( insert6201435330877294327od_o_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert6201435330877294327od_o_o @ A @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_916_UN__insert__distrib,axiom,
! [U2: b > prat,A3: set_b_prat,A: set_o,B3: ( b > prat ) > set_set_o] :
( ( member_b_prat @ U2 @ A3 )
=> ( ( comple4436988014476444997_set_o
@ ( image_4142778379334362525_set_o
@ ^ [X5: b > prat] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_917_UN__insert__distrib,axiom,
! [U2: product_prod_o_o,A3: set_Product_prod_o_o,A: set_o,B3: product_prod_o_o > set_set_o] :
( ( member7466972457876170832od_o_o @ U2 @ A3 )
=> ( ( comple4436988014476444997_set_o
@ ( image_5008979335655565046_set_o
@ ^ [X5: product_prod_o_o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_5008979335655565046_set_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_918_UN__insert__distrib,axiom,
! [U2: set_o,A3: set_set_o,A: product_prod_o_o,B3: set_o > set_Product_prod_o_o] :
( ( member_set_o @ U2 @ A3 )
=> ( ( comple2567785453016644468od_o_o
@ ( image_4627112110306186388od_o_o
@ ^ [X5: set_o] : ( insert6201435330877294327od_o_o @ A @ ( B3 @ X5 ) )
@ A3 ) )
= ( insert6201435330877294327od_o_o @ A @ ( comple2567785453016644468od_o_o @ ( image_4627112110306186388od_o_o @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_919_surj__def,axiom,
! [F: $o > set_set_o] :
( ( ( image_o_set_set_o @ F @ top_top_set_o )
= top_to7699215511130388872_set_o )
= ( ! [Y: set_set_o] :
? [X5: $o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_920_surj__def,axiom,
! [F: $o > product_prod_o_o] :
( ( ( image_4057150146340385428od_o_o @ F @ top_top_set_o )
= top_to7721136755696657239od_o_o )
= ( ! [Y: product_prod_o_o] :
? [X5: $o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_921_surj__def,axiom,
! [F: $o > set_o] :
( ( ( image_o_set_o2 @ F @ top_top_set_o )
= top_top_set_set_o )
= ( ! [Y: set_o] :
? [X5: $o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_922_surj__def,axiom,
! [F: product_prod_o_o > $o] :
( ( ( image_7896445794123959606_o_o_o @ F @ top_to7721136755696657239od_o_o )
= top_top_set_o )
= ( ! [Y: $o] :
? [X5: product_prod_o_o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_923_surj__def,axiom,
! [F: product_prod_o_o > product_prod_o_o] :
( ( ( image_9131363867636255685od_o_o @ F @ top_to7721136755696657239od_o_o )
= top_to7721136755696657239od_o_o )
= ( ! [Y: product_prod_o_o] :
? [X5: product_prod_o_o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_924_surj__def,axiom,
! [F: product_prod_o_o > set_o] :
( ( ( image_7870716977482210326_set_o @ F @ top_to7721136755696657239od_o_o )
= top_top_set_set_o )
= ( ! [Y: set_o] :
? [X5: product_prod_o_o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_925_surj__def,axiom,
! [F: set_o > $o] :
( ( ( image_set_o_o2 @ F @ top_top_set_set_o )
= top_top_set_o )
= ( ! [Y: $o] :
? [X5: set_o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_926_surj__def,axiom,
! [F: set_o > product_prod_o_o] :
( ( ( image_6524031425630931892od_o_o @ F @ top_top_set_set_o )
= top_to7721136755696657239od_o_o )
= ( ! [Y: product_prod_o_o] :
? [X5: set_o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_927_surj__def,axiom,
! [F: set_o > set_o] :
( ( ( image_set_o_set_o2 @ F @ top_top_set_set_o )
= top_top_set_set_o )
= ( ! [Y: set_o] :
? [X5: set_o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_928_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y: $o] :
? [X5: $o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_929_surjI,axiom,
! [G: $o > set_set_o,F: set_set_o > $o] :
( ! [X: set_set_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_o_set_set_o @ G @ top_top_set_o )
= top_to7699215511130388872_set_o ) ) ).
% surjI
thf(fact_930_surjI,axiom,
! [G: $o > product_prod_o_o,F: product_prod_o_o > $o] :
( ! [X: product_prod_o_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_4057150146340385428od_o_o @ G @ top_top_set_o )
= top_to7721136755696657239od_o_o ) ) ).
% surjI
thf(fact_931_surjI,axiom,
! [G: $o > set_o,F: set_o > $o] :
( ! [X: set_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_o_set_o2 @ G @ top_top_set_o )
= top_top_set_set_o ) ) ).
% surjI
thf(fact_932_surjI,axiom,
! [G: product_prod_o_o > $o,F: $o > product_prod_o_o] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_7896445794123959606_o_o_o @ G @ top_to7721136755696657239od_o_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_933_surjI,axiom,
! [G: product_prod_o_o > product_prod_o_o,F: product_prod_o_o > product_prod_o_o] :
( ! [X: product_prod_o_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_9131363867636255685od_o_o @ G @ top_to7721136755696657239od_o_o )
= top_to7721136755696657239od_o_o ) ) ).
% surjI
thf(fact_934_surjI,axiom,
! [G: product_prod_o_o > set_o,F: set_o > product_prod_o_o] :
( ! [X: set_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_7870716977482210326_set_o @ G @ top_to7721136755696657239od_o_o )
= top_top_set_set_o ) ) ).
% surjI
thf(fact_935_surjI,axiom,
! [G: set_o > $o,F: $o > set_o] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_set_o_o2 @ G @ top_top_set_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_936_surjI,axiom,
! [G: set_o > product_prod_o_o,F: product_prod_o_o > set_o] :
( ! [X: product_prod_o_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_6524031425630931892od_o_o @ G @ top_top_set_set_o )
= top_to7721136755696657239od_o_o ) ) ).
% surjI
thf(fact_937_surjI,axiom,
! [G: set_o > set_o,F: set_o > set_o] :
( ! [X: set_o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_set_o_set_o2 @ G @ top_top_set_set_o )
= top_top_set_set_o ) ) ).
% surjI
thf(fact_938_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_939_surjE,axiom,
! [F: $o > set_set_o,Y4: set_set_o] :
( ( ( image_o_set_set_o @ F @ top_top_set_o )
= top_to7699215511130388872_set_o )
=> ~ ! [X: $o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_940_surjE,axiom,
! [F: $o > product_prod_o_o,Y4: product_prod_o_o] :
( ( ( image_4057150146340385428od_o_o @ F @ top_top_set_o )
= top_to7721136755696657239od_o_o )
=> ~ ! [X: $o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_941_surjE,axiom,
! [F: $o > set_o,Y4: set_o] :
( ( ( image_o_set_o2 @ F @ top_top_set_o )
= top_top_set_set_o )
=> ~ ! [X: $o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_942_surjE,axiom,
! [F: product_prod_o_o > $o,Y4: $o] :
( ( ( image_7896445794123959606_o_o_o @ F @ top_to7721136755696657239od_o_o )
= top_top_set_o )
=> ~ ! [X: product_prod_o_o] :
( Y4
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_943_surjE,axiom,
! [F: product_prod_o_o > product_prod_o_o,Y4: product_prod_o_o] :
( ( ( image_9131363867636255685od_o_o @ F @ top_to7721136755696657239od_o_o )
= top_to7721136755696657239od_o_o )
=> ~ ! [X: product_prod_o_o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_944_surjE,axiom,
! [F: product_prod_o_o > set_o,Y4: set_o] :
( ( ( image_7870716977482210326_set_o @ F @ top_to7721136755696657239od_o_o )
= top_top_set_set_o )
=> ~ ! [X: product_prod_o_o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_945_surjE,axiom,
! [F: set_o > $o,Y4: $o] :
( ( ( image_set_o_o2 @ F @ top_top_set_set_o )
= top_top_set_o )
=> ~ ! [X: set_o] :
( Y4
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_946_surjE,axiom,
! [F: set_o > product_prod_o_o,Y4: product_prod_o_o] :
( ( ( image_6524031425630931892od_o_o @ F @ top_top_set_set_o )
= top_to7721136755696657239od_o_o )
=> ~ ! [X: set_o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_947_surjE,axiom,
! [F: set_o > set_o,Y4: set_o] :
( ( ( image_set_o_set_o2 @ F @ top_top_set_set_o )
= top_top_set_set_o )
=> ~ ! [X: set_o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_948_surjE,axiom,
! [F: $o > $o,Y4: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X: $o] :
( Y4
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_949_surjD,axiom,
! [F: $o > set_set_o,Y4: set_set_o] :
( ( ( image_o_set_set_o @ F @ top_top_set_o )
= top_to7699215511130388872_set_o )
=> ? [X: $o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_950_surjD,axiom,
! [F: $o > product_prod_o_o,Y4: product_prod_o_o] :
( ( ( image_4057150146340385428od_o_o @ F @ top_top_set_o )
= top_to7721136755696657239od_o_o )
=> ? [X: $o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_951_surjD,axiom,
! [F: $o > set_o,Y4: set_o] :
( ( ( image_o_set_o2 @ F @ top_top_set_o )
= top_top_set_set_o )
=> ? [X: $o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_952_surjD,axiom,
! [F: product_prod_o_o > $o,Y4: $o] :
( ( ( image_7896445794123959606_o_o_o @ F @ top_to7721136755696657239od_o_o )
= top_top_set_o )
=> ? [X: product_prod_o_o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_953_surjD,axiom,
! [F: product_prod_o_o > product_prod_o_o,Y4: product_prod_o_o] :
( ( ( image_9131363867636255685od_o_o @ F @ top_to7721136755696657239od_o_o )
= top_to7721136755696657239od_o_o )
=> ? [X: product_prod_o_o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_954_surjD,axiom,
! [F: product_prod_o_o > set_o,Y4: set_o] :
( ( ( image_7870716977482210326_set_o @ F @ top_to7721136755696657239od_o_o )
= top_top_set_set_o )
=> ? [X: product_prod_o_o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_955_surjD,axiom,
! [F: set_o > $o,Y4: $o] :
( ( ( image_set_o_o2 @ F @ top_top_set_set_o )
= top_top_set_o )
=> ? [X: set_o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_956_surjD,axiom,
! [F: set_o > product_prod_o_o,Y4: product_prod_o_o] :
( ( ( image_6524031425630931892od_o_o @ F @ top_top_set_set_o )
= top_to7721136755696657239od_o_o )
=> ? [X: set_o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_957_surjD,axiom,
! [F: set_o > set_o,Y4: set_o] :
( ( ( image_set_o_set_o2 @ F @ top_top_set_set_o )
= top_top_set_set_o )
=> ? [X: set_o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_958_surjD,axiom,
! [F: $o > $o,Y4: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X: $o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_959_range__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_960_range__eqI,axiom,
! [B: set_o,F: $o > set_o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_o_set_o2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_961_range__eqI,axiom,
! [B: $o,F: set_o > $o,X2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_set_o_o2 @ F @ top_top_set_set_o ) ) ) ).
% range_eqI
thf(fact_962_range__eqI,axiom,
! [B: set_set_o,F: $o > set_set_o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_set_set_o @ B @ ( image_o_set_set_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_963_range__eqI,axiom,
! [B: b > prat,F: $o > b > prat,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_b_prat @ B @ ( image_o_b_prat2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_964_range__eqI,axiom,
! [B: product_prod_o_o,F: $o > product_prod_o_o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member7466972457876170832od_o_o @ B @ ( image_4057150146340385428od_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_965_range__eqI,axiom,
! [B: $o,F: product_prod_o_o > $o,X2: product_prod_o_o] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_7896445794123959606_o_o_o @ F @ top_to7721136755696657239od_o_o ) ) ) ).
% range_eqI
thf(fact_966_range__eqI,axiom,
! [B: set_o,F: set_o > set_o,X2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_set_o_set_o2 @ F @ top_top_set_set_o ) ) ) ).
% range_eqI
thf(fact_967_range__eqI,axiom,
! [B: set_o,F: product_prod_o_o > set_o,X2: product_prod_o_o] :
( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_7870716977482210326_set_o @ F @ top_to7721136755696657239od_o_o ) ) ) ).
% range_eqI
thf(fact_968_range__eqI,axiom,
! [B: b > prat,F: set_o > b > prat,X2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( member_b_prat @ B @ ( image_set_o_b_prat2 @ F @ top_top_set_set_o ) ) ) ).
% range_eqI
thf(fact_969_rangeI,axiom,
! [F: $o > $o,X2: $o] : ( member_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_970_rangeI,axiom,
! [F: $o > set_o,X2: $o] : ( member_set_o @ ( F @ X2 ) @ ( image_o_set_o2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_971_rangeI,axiom,
! [F: set_o > $o,X2: set_o] : ( member_o @ ( F @ X2 ) @ ( image_set_o_o2 @ F @ top_top_set_set_o ) ) ).
% rangeI
thf(fact_972_rangeI,axiom,
! [F: $o > set_set_o,X2: $o] : ( member_set_set_o @ ( F @ X2 ) @ ( image_o_set_set_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_973_rangeI,axiom,
! [F: $o > b > prat,X2: $o] : ( member_b_prat @ ( F @ X2 ) @ ( image_o_b_prat2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_974_rangeI,axiom,
! [F: $o > product_prod_o_o,X2: $o] : ( member7466972457876170832od_o_o @ ( F @ X2 ) @ ( image_4057150146340385428od_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_975_rangeI,axiom,
! [F: product_prod_o_o > $o,X2: product_prod_o_o] : ( member_o @ ( F @ X2 ) @ ( image_7896445794123959606_o_o_o @ F @ top_to7721136755696657239od_o_o ) ) ).
% rangeI
thf(fact_976_rangeI,axiom,
! [F: set_o > set_o,X2: set_o] : ( member_set_o @ ( F @ X2 ) @ ( image_set_o_set_o2 @ F @ top_top_set_set_o ) ) ).
% rangeI
thf(fact_977_rangeI,axiom,
! [F: product_prod_o_o > set_o,X2: product_prod_o_o] : ( member_set_o @ ( F @ X2 ) @ ( image_7870716977482210326_set_o @ F @ top_to7721136755696657239od_o_o ) ) ).
% rangeI
thf(fact_978_rangeI,axiom,
! [F: set_o > b > prat,X2: set_o] : ( member_b_prat @ ( F @ X2 ) @ ( image_set_o_b_prat2 @ F @ top_top_set_set_o ) ) ).
% rangeI
thf(fact_979_empty__not__UNIV,axiom,
bot_bot_set_b_prat != top_top_set_b_prat ).
% empty_not_UNIV
thf(fact_980_empty__not__UNIV,axiom,
bot_bo7073875226086086771od_o_o != top_to7721136755696657239od_o_o ).
% empty_not_UNIV
thf(fact_981_empty__not__UNIV,axiom,
bot_bot_set_set_o != top_top_set_set_o ).
% empty_not_UNIV
thf(fact_982_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_983_insert__UNIV,axiom,
! [X2: product_prod_o_o] :
( ( insert6201435330877294327od_o_o @ X2 @ top_to7721136755696657239od_o_o )
= top_to7721136755696657239od_o_o ) ).
% insert_UNIV
thf(fact_984_insert__UNIV,axiom,
! [X2: set_o] :
( ( insert_set_o @ X2 @ top_top_set_set_o )
= top_top_set_set_o ) ).
% insert_UNIV
thf(fact_985_insert__UNIV,axiom,
! [X2: $o] :
( ( insert_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_986_SUP__eq__const,axiom,
! [I2: set_o,F: $o > $o,X2: $o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_987_SUP__eq__const,axiom,
! [I2: set_o,F: $o > set_o,X2: set_o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_988_SUP__eq__const,axiom,
! [I2: set_set_o,F: set_o > $o,X2: $o] :
( ( I2 != bot_bot_set_set_o )
=> ( ! [I3: set_o] :
( ( member_set_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_o_o2 @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_989_SUP__eq__const,axiom,
! [I2: set_set_o,F: set_o > set_o,X2: set_o] :
( ( I2 != bot_bot_set_set_o )
=> ( ! [I3: set_o] :
( ( member_set_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_990_SUP__eq__const,axiom,
! [I2: set_b_prat,F: ( b > prat ) > $o,X2: $o] :
( ( I2 != bot_bot_set_b_prat )
=> ( ! [I3: b > prat] :
( ( member_b_prat @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_b_prat_o2 @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_991_SUP__eq__const,axiom,
! [I2: set_Product_prod_o_o,F: product_prod_o_o > $o,X2: $o] :
( ( I2 != bot_bo7073875226086086771od_o_o )
=> ( ! [I3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_7896445794123959606_o_o_o @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_992_SUP__eq__const,axiom,
! [I2: set_o,F: $o > set_set_o,X2: set_set_o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_993_SUP__eq__const,axiom,
! [I2: set_b_prat,F: ( b > prat ) > set_o,X2: set_o] :
( ( I2 != bot_bot_set_b_prat )
=> ( ! [I3: b > prat] :
( ( member_b_prat @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_994_SUP__eq__const,axiom,
! [I2: set_Product_prod_o_o,F: product_prod_o_o > set_o,X2: set_o] :
( ( I2 != bot_bo7073875226086086771od_o_o )
=> ( ! [I3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_995_SUP__eq__const,axiom,
! [I2: set_set_o,F: set_o > set_set_o,X2: set_set_o] :
( ( I2 != bot_bot_set_set_o )
=> ( ! [I3: set_o] :
( ( member_set_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_996_Domain_Ocases,axiom,
! [A: b > prat,R3: set_Pr2868754470712218413prat_b] :
( ( member_b_prat @ A @ ( domain4185636366779699890prat_b @ R3 ) )
=> ~ ! [B5: produc7326829233880046182prat_b] :
~ ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A @ B5 ) @ R3 ) ) ).
% Domain.cases
thf(fact_997_Domain_Ocases,axiom,
! [A: b > prat,R3: set_Pr8957522257986161222prat_b] :
( ( member_b_prat @ A @ ( domain_b_prat_b @ R3 ) )
=> ~ ! [B5: b] :
~ ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B5 ) @ R3 ) ) ).
% Domain.cases
thf(fact_998_Domain_Ocases,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( domain_o_o @ R3 ) )
=> ~ ! [B5: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B5 ) @ R3 ) ) ).
% Domain.cases
thf(fact_999_Domain_Osimps,axiom,
! [A: b > prat,R3: set_Pr2868754470712218413prat_b] :
( ( member_b_prat @ A @ ( domain4185636366779699890prat_b @ R3 ) )
= ( ? [A2: b > prat,B2: produc7326829233880046182prat_b] :
( ( A = A2 )
& ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A2 @ B2 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_1000_Domain_Osimps,axiom,
! [A: b > prat,R3: set_Pr8957522257986161222prat_b] :
( ( member_b_prat @ A @ ( domain_b_prat_b @ R3 ) )
= ( ? [A2: b > prat,B2: b] :
( ( A = A2 )
& ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A2 @ B2 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_1001_Domain_Osimps,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( domain_o_o @ R3 ) )
= ( ? [A2: $o,B2: $o] :
( ( A = A2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_1002_Domain_ODomainI,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A @ B ) @ R3 )
=> ( member_b_prat @ A @ ( domain4185636366779699890prat_b @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_1003_Domain_ODomainI,axiom,
! [A: b > prat,B: b,R3: set_Pr8957522257986161222prat_b] :
( ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 )
=> ( member_b_prat @ A @ ( domain_b_prat_b @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_1004_Domain_ODomainI,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member_o @ A @ ( domain_o_o @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_1005_DomainE,axiom,
! [A: b > prat,R3: set_Pr2868754470712218413prat_b] :
( ( member_b_prat @ A @ ( domain4185636366779699890prat_b @ R3 ) )
=> ~ ! [B5: produc7326829233880046182prat_b] :
~ ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A @ B5 ) @ R3 ) ) ).
% DomainE
thf(fact_1006_DomainE,axiom,
! [A: b > prat,R3: set_Pr8957522257986161222prat_b] :
( ( member_b_prat @ A @ ( domain_b_prat_b @ R3 ) )
=> ~ ! [B5: b] :
~ ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B5 ) @ R3 ) ) ).
% DomainE
thf(fact_1007_DomainE,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( domain_o_o @ R3 ) )
=> ~ ! [B5: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B5 ) @ R3 ) ) ).
% DomainE
thf(fact_1008_Domain__iff,axiom,
! [A: b > prat,R3: set_Pr2868754470712218413prat_b] :
( ( member_b_prat @ A @ ( domain4185636366779699890prat_b @ R3 ) )
= ( ? [Y: produc7326829233880046182prat_b] : ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A @ Y ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_1009_Domain__iff,axiom,
! [A: b > prat,R3: set_Pr8957522257986161222prat_b] :
( ( member_b_prat @ A @ ( domain_b_prat_b @ R3 ) )
= ( ? [Y: b] : ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ Y ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_1010_Domain__iff,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( domain_o_o @ R3 ) )
= ( ? [Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ Y ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_1011_Range_Ocases,axiom,
! [A: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( member5302574828040144399prat_b @ A @ ( range_3221829133846087963prat_b @ R3 ) )
=> ~ ! [A6: b > prat] :
~ ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A6 @ A ) @ R3 ) ) ).
% Range.cases
thf(fact_1012_Range_Ocases,axiom,
! [A: b,R3: set_Pr8957522257986161222prat_b] :
( ( member_b @ A @ ( range_b_prat_b @ R3 ) )
=> ~ ! [A6: b > prat] :
~ ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A6 @ A ) @ R3 ) ) ).
% Range.cases
thf(fact_1013_Range_Ocases,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( range_o_o @ R3 ) )
=> ~ ! [A6: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A6 @ A ) @ R3 ) ) ).
% Range.cases
thf(fact_1014_Range_Osimps,axiom,
! [A: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( member5302574828040144399prat_b @ A @ ( range_3221829133846087963prat_b @ R3 ) )
= ( ? [A2: b > prat,B2: produc7326829233880046182prat_b] :
( ( A = B2 )
& ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A2 @ B2 ) @ R3 ) ) ) ) ).
% Range.simps
thf(fact_1015_Range_Osimps,axiom,
! [A: b,R3: set_Pr8957522257986161222prat_b] :
( ( member_b @ A @ ( range_b_prat_b @ R3 ) )
= ( ? [A2: b > prat,B2: b] :
( ( A = B2 )
& ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A2 @ B2 ) @ R3 ) ) ) ) ).
% Range.simps
thf(fact_1016_Range_Osimps,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( range_o_o @ R3 ) )
= ( ? [A2: $o,B2: $o] :
( ( A = B2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R3 ) ) ) ) ).
% Range.simps
thf(fact_1017_Range_Ointros,axiom,
! [A: b > prat,B: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A @ B ) @ R3 )
=> ( member5302574828040144399prat_b @ B @ ( range_3221829133846087963prat_b @ R3 ) ) ) ).
% Range.intros
thf(fact_1018_Range_Ointros,axiom,
! [A: b > prat,B: b,R3: set_Pr8957522257986161222prat_b] :
( ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A @ B ) @ R3 )
=> ( member_b @ B @ ( range_b_prat_b @ R3 ) ) ) ).
% Range.intros
thf(fact_1019_Range_Ointros,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member_o @ B @ ( range_o_o @ R3 ) ) ) ).
% Range.intros
thf(fact_1020_RangeE,axiom,
! [B: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( member5302574828040144399prat_b @ B @ ( range_3221829133846087963prat_b @ R3 ) )
=> ~ ! [A6: b > prat] :
~ ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ A6 @ B ) @ R3 ) ) ).
% RangeE
thf(fact_1021_RangeE,axiom,
! [B: b,R3: set_Pr8957522257986161222prat_b] :
( ( member_b @ B @ ( range_b_prat_b @ R3 ) )
=> ~ ! [A6: b > prat] :
~ ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ A6 @ B ) @ R3 ) ) ).
% RangeE
thf(fact_1022_RangeE,axiom,
! [B: $o,R3: set_Product_prod_o_o] :
( ( member_o @ B @ ( range_o_o @ R3 ) )
=> ~ ! [A6: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A6 @ B ) @ R3 ) ) ).
% RangeE
thf(fact_1023_Range__iff,axiom,
! [A: produc7326829233880046182prat_b,R3: set_Pr2868754470712218413prat_b] :
( ( member5302574828040144399prat_b @ A @ ( range_3221829133846087963prat_b @ R3 ) )
= ( ? [Y: b > prat] : ( member412304827439910006prat_b @ ( produc4863932588836083261prat_b @ Y @ A ) @ R3 ) ) ) ).
% Range_iff
thf(fact_1024_Range__iff,axiom,
! [A: b,R3: set_Pr8957522257986161222prat_b] :
( ( member_b @ A @ ( range_b_prat_b @ R3 ) )
= ( ? [Y: b > prat] : ( member5302574828040144399prat_b @ ( produc7508179496679232342prat_b @ Y @ A ) @ R3 ) ) ) ).
% Range_iff
thf(fact_1025_Range__iff,axiom,
! [A: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ ( range_o_o @ R3 ) )
= ( ? [Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ A ) @ R3 ) ) ) ).
% Range_iff
thf(fact_1026_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: $o,B3: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1027_UN__extend__simps_I1_J,axiom,
! [C2: set_set_o,A: $o,B3: set_o > set_o] :
( ( ( C2 = bot_bot_set_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bot_set_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1028_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: set_o,B3: $o > set_set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ C2 ) ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ C2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1029_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: b > prat,B3: $o > set_b_prat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert_b_prat @ A @ ( comple8554269323607207949b_prat @ ( image_o_set_b_prat @ B3 @ C2 ) ) )
= ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert_b_prat @ A @ ( comple8554269323607207949b_prat @ ( image_o_set_b_prat @ B3 @ C2 ) ) )
= ( comple8554269323607207949b_prat
@ ( image_o_set_b_prat
@ ^ [X5: $o] : ( insert_b_prat @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1030_UN__extend__simps_I1_J,axiom,
! [C2: set_b_prat,A: $o,B3: ( b > prat ) > set_o] :
( ( ( C2 = bot_bot_set_b_prat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bot_set_b_prat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_b_prat_set_o2 @ B3 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_b_prat_set_o2
@ ^ [X5: b > prat] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1031_UN__extend__simps_I1_J,axiom,
! [C2: set_Product_prod_o_o,A: $o,B3: product_prod_o_o > set_o] :
( ( ( C2 = bot_bo7073875226086086771od_o_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bo7073875226086086771od_o_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_7870716977482210326_set_o @ B3 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_7870716977482210326_set_o
@ ^ [X5: product_prod_o_o] : ( insert_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1032_UN__extend__simps_I1_J,axiom,
! [C2: set_set_o,A: set_o,B3: set_o > set_set_o] :
( ( ( C2 = bot_bot_set_set_o )
=> ( ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ C2 ) ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
& ( ( C2 != bot_bot_set_set_o )
=> ( ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ C2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1033_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: product_prod_o_o,B3: $o > set_Product_prod_o_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert6201435330877294327od_o_o @ A @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ C2 ) ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert6201435330877294327od_o_o @ A @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ C2 ) ) )
= ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( insert6201435330877294327od_o_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1034_UN__extend__simps_I1_J,axiom,
! [C2: set_set_o,A: b > prat,B3: set_o > set_b_prat] :
( ( ( C2 = bot_bot_set_set_o )
=> ( ( insert_b_prat @ A @ ( comple8554269323607207949b_prat @ ( image_2300625924940167469b_prat @ B3 @ C2 ) ) )
= ( insert_b_prat @ A @ bot_bot_set_b_prat ) ) )
& ( ( C2 != bot_bot_set_set_o )
=> ( ( insert_b_prat @ A @ ( comple8554269323607207949b_prat @ ( image_2300625924940167469b_prat @ B3 @ C2 ) ) )
= ( comple8554269323607207949b_prat
@ ( image_2300625924940167469b_prat
@ ^ [X5: set_o] : ( insert_b_prat @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1035_UN__extend__simps_I1_J,axiom,
! [C2: set_b_prat,A: set_o,B3: ( b > prat ) > set_set_o] :
( ( ( C2 = bot_bot_set_b_prat )
=> ( ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ C2 ) ) )
= ( insert_set_o @ A @ bot_bot_set_set_o ) ) )
& ( ( C2 != bot_bot_set_b_prat )
=> ( ( insert_set_o @ A @ ( comple4436988014476444997_set_o @ ( image_4142778379334362525_set_o @ B3 @ C2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_4142778379334362525_set_o
@ ^ [X5: b > prat] : ( insert_set_o @ A @ ( B3 @ X5 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1036_Image__UN,axiom,
! [R3: set_Product_prod_o_o,B3: $o > set_o,A3: set_o] :
( ( image_o_o @ R3 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( image_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1037_Image__UN,axiom,
! [R3: set_Product_prod_o_o,B3: set_o > set_o,A3: set_set_o] :
( ( image_o_o @ R3 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( image_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1038_Image__UN,axiom,
! [R3: set_Pr3508996721749391591_set_o,B3: $o > set_o,A3: set_o] :
( ( image_o_set_o @ R3 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( image_o_set_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1039_Image__UN,axiom,
! [R3: set_Pr5232024646447722279et_o_o,B3: $o > set_set_o,A3: set_o] :
( ( image_set_o_o @ R3 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( image_set_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1040_Image__UN,axiom,
! [R3: set_Pr3508996721749391591_set_o,B3: set_o > set_o,A3: set_set_o] :
( ( image_o_set_o @ R3 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_5023573440332574309_set_o
@ ^ [X5: set_o] : ( image_o_set_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1041_Image__UN,axiom,
! [R3: set_Pr5237325540968292374od_o_o,B3: $o > set_o,A3: set_o] :
( ( image_5252864891027546466od_o_o @ R3 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ A3 ) ) )
= ( comple2567785453016644468od_o_o
@ ( image_5520585345982086004od_o_o
@ ^ [X5: $o] : ( image_5252864891027546466od_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1042_Image__UN,axiom,
! [R3: set_Pr5232024646447722279et_o_o,B3: set_o > set_set_o,A3: set_set_o] :
( ( image_set_o_o @ R3 @ ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_o_set_o2
@ ^ [X5: set_o] : ( image_set_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1043_Image__UN,axiom,
! [R3: set_Pr4577759397028426247_set_o,B3: $o > set_set_o,A3: set_o] :
( ( image_set_o_set_o @ R3 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B3 @ A3 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( image_set_o_set_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1044_Image__UN,axiom,
! [R3: set_Pr2246928807395521336_o_o_o,B3: $o > set_Product_prod_o_o,A3: set_o] :
( ( image_9092160538811120644_o_o_o @ R3 @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ B3 @ A3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( image_9092160538811120644_o_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1045_Image__UN,axiom,
! [R3: set_Pr5237325540968292374od_o_o,B3: set_o > set_o,A3: set_set_o] :
( ( image_5252864891027546466od_o_o @ R3 @ ( comple90263536869209701_set_o @ ( image_set_o_set_o2 @ B3 @ A3 ) ) )
= ( comple2567785453016644468od_o_o
@ ( image_4627112110306186388od_o_o
@ ^ [X5: set_o] : ( image_5252864891027546466od_o_o @ R3 @ ( B3 @ X5 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_1046_range__composition,axiom,
! [F: $o > $o,G: $o > $o] :
( ( image_o_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_o_o2 @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1047_range__composition,axiom,
! [F: set_o > $o,G: $o > set_o] :
( ( image_o_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_set_o_o2 @ F @ ( image_o_set_o2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1048_range__composition,axiom,
! [F: $o > set_o,G: $o > $o] :
( ( image_o_set_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_o_set_o2 @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1049_range__composition,axiom,
! [F: $o > $o,G: set_o > $o] :
( ( image_set_o_o2
@ ^ [X5: set_o] : ( F @ ( G @ X5 ) )
@ top_top_set_set_o )
= ( image_o_o2 @ F @ ( image_set_o_o2 @ G @ top_top_set_set_o ) ) ) ).
% range_composition
thf(fact_1050_range__composition,axiom,
! [F: set_set_o > $o,G: $o > set_set_o] :
( ( image_o_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_set_set_o_o @ F @ ( image_o_set_set_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1051_range__composition,axiom,
! [F: set_o > set_o,G: $o > set_o] :
( ( image_o_set_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_set_o_set_o2 @ F @ ( image_o_set_o2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1052_range__composition,axiom,
! [F: $o > set_set_o,G: $o > $o] :
( ( image_o_set_set_o
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_o_set_set_o @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1053_range__composition,axiom,
! [F: $o > $o,G: product_prod_o_o > $o] :
( ( image_7896445794123959606_o_o_o
@ ^ [X5: product_prod_o_o] : ( F @ ( G @ X5 ) )
@ top_to7721136755696657239od_o_o )
= ( image_o_o2 @ F @ ( image_7896445794123959606_o_o_o @ G @ top_to7721136755696657239od_o_o ) ) ) ).
% range_composition
thf(fact_1054_range__composition,axiom,
! [F: $o > set_o,G: set_o > $o] :
( ( image_set_o_set_o2
@ ^ [X5: set_o] : ( F @ ( G @ X5 ) )
@ top_top_set_set_o )
= ( image_o_set_o2 @ F @ ( image_set_o_o2 @ G @ top_top_set_set_o ) ) ) ).
% range_composition
thf(fact_1055_range__composition,axiom,
! [F: set_set_o > set_o,G: $o > set_set_o] :
( ( image_o_set_o2
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_1148209357520620709_set_o @ F @ ( image_o_set_set_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1056_rangeE,axiom,
! [B: product_prod_o_o,F: set_o > product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( image_6524031425630931892od_o_o @ F @ top_top_set_set_o ) )
=> ~ ! [X: set_o] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1057_rangeE,axiom,
! [B: set_o,F: set_o > set_o] :
( ( member_set_o @ B @ ( image_set_o_set_o2 @ F @ top_top_set_set_o ) )
=> ~ ! [X: set_o] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1058_SUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_1059_SUP__constant,axiom,
! [A3: set_o,C: set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= bot_bot_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1060_reflD,axiom,
! [R3: set_Product_prod_o_o,A: $o] :
( ( refl_on_o @ top_top_set_o @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ R3 ) ) ).
% reflD
thf(fact_1061_reflI,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ X ) @ R3 )
=> ( refl_on_o @ top_top_set_o @ R3 ) ) ).
% reflI
thf(fact_1062_range__eq__singletonD,axiom,
! [F: $o > $o,A: $o,X2: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1063_ccpo__Sup__singleton,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% ccpo_Sup_singleton
thf(fact_1064_cSup__singleton,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% cSup_singleton
thf(fact_1065_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_1066_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_1067_bdd__above__empty,axiom,
condit5488710616941104124bove_o @ bot_bot_set_o ).
% bdd_above_empty
thf(fact_1068_Field__empty,axiom,
( ( field_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Field_empty
thf(fact_1069_Times__empty,axiom,
! [A3: set_o,B3: set_o] :
( ( ( product_Sigma_o_o @ A3
@ ^ [Uu: $o] : B3 )
= bot_bo7073875226086086771od_o_o )
= ( ( A3 = bot_bot_set_o )
| ( B3 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_1070_UNIV__Times__UNIV,axiom,
( ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : top_top_set_o )
= top_to7721136755696657239od_o_o ) ).
% UNIV_Times_UNIV
thf(fact_1071_times__eq__iff,axiom,
! [A3: set_o,B3: set_o,C2: set_o,D2: set_o] :
( ( ( product_Sigma_o_o @ A3
@ ^ [Uu: $o] : B3 )
= ( product_Sigma_o_o @ C2
@ ^ [Uu: $o] : D2 ) )
= ( ( ( A3 = C2 )
& ( B3 = D2 ) )
| ( ( ( A3 = bot_bot_set_o )
| ( B3 = bot_bot_set_o ) )
& ( ( C2 = bot_bot_set_o )
| ( D2 = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_1072_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_o_o,A: $o,B: $o] :
( ( refl_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( antisym_on_o @ top_top_set_o @ R3 )
=> ( ( member_o @ A @ ( field_o @ R3 ) )
=> ( ( member_o @ B @ ( field_o @ R3 ) )
=> ( ( ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( image_o_o @ R3 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( A = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1073_Field__insert,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) )
= ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R3 ) ) ) ).
% Field_insert
thf(fact_1074_Partial__order__eq__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_o_o,A: $o,B: $o] :
( ( order_8538205470752950541r_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( member_o @ A @ ( field_o @ R3 ) )
=> ( ( member_o @ B @ ( field_o @ R3 ) )
=> ( ( ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( image_o_o @ R3 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1075_insert__Times__insert,axiom,
! [A: $o,A3: set_o,B: $o,B3: set_o] :
( ( product_Sigma_o_o @ ( insert_o @ A @ A3 )
@ ^ [Uu: $o] : ( insert_o @ B @ B3 ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
@ ( sup_su5769328420594410459od_o_o
@ ( product_Sigma_o_o @ A3
@ ^ [Uu: $o] : ( insert_o @ B @ B3 ) )
@ ( product_Sigma_o_o @ ( insert_o @ A @ A3 )
@ ^ [Uu: $o] : B3 ) ) ) ) ).
% insert_Times_insert
thf(fact_1076_Un__empty,axiom,
! [A3: set_o,B3: set_o] :
( ( ( sup_sup_set_o @ A3 @ B3 )
= bot_bot_set_o )
= ( ( A3 = bot_bot_set_o )
& ( B3 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_1077_Un__insert__left,axiom,
! [A: $o,B3: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B3 ) @ C2 )
= ( insert_o @ A @ ( sup_sup_set_o @ B3 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1078_Un__insert__right,axiom,
! [A3: set_o,A: $o,B3: set_o] :
( ( sup_sup_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_1079_UN__simps_I3_J,axiom,
! [C2: set_o,A3: set_o,B3: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( sup_sup_set_o @ A3 @ ( B3 @ X5 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( sup_sup_set_o @ A3 @ ( B3 @ X5 ) )
@ C2 ) )
= ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1080_UN__simps_I2_J,axiom,
! [C2: set_o,A3: $o > set_o,B3: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( sup_sup_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( sup_sup_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ A3 @ C2 ) ) @ B3 ) ) ) ) ).
% UN_simps(2)
thf(fact_1081_Un__UNIV__left,axiom,
! [B3: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B3 )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_1082_Un__UNIV__right,axiom,
! [A3: set_o] :
( ( sup_sup_set_o @ A3 @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_1083_Un__empty__right,axiom,
! [A3: set_o] :
( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
= A3 ) ).
% Un_empty_right
thf(fact_1084_Un__empty__left,axiom,
! [B3: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_1085_cSup__union__distrib,axiom,
! [A3: set_o,B3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A3 )
=> ( ( B3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B3 )
=> ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A3 @ B3 ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1086_insert__def,axiom,
( insert_o
= ( ^ [A2: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X5: $o] : ( X5 = A2 ) ) ) ) ) ).
% insert_def
thf(fact_1087_insert__is__Un,axiom,
( insert_o
= ( ^ [A2: $o] : ( sup_sup_set_o @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_1088_Un__singleton__iff,axiom,
! [A3: set_o,B3: set_o,X2: $o] :
( ( ( sup_sup_set_o @ A3 @ B3 )
= ( insert_o @ X2 @ bot_bot_set_o ) )
= ( ( ( A3 = bot_bot_set_o )
& ( B3
= ( insert_o @ X2 @ bot_bot_set_o ) ) )
| ( ( A3
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B3 = bot_bot_set_o ) )
| ( ( A3
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B3
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1089_singleton__Un__iff,axiom,
! [X2: $o,A3: set_o,B3: set_o] :
( ( ( insert_o @ X2 @ bot_bot_set_o )
= ( sup_sup_set_o @ A3 @ B3 ) )
= ( ( ( A3 = bot_bot_set_o )
& ( B3
= ( insert_o @ X2 @ bot_bot_set_o ) ) )
| ( ( A3
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B3 = bot_bot_set_o ) )
| ( ( A3
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B3
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1090_antisymI,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X: $o,Y3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X ) @ R3 )
=> ( X = Y3 ) ) )
=> ( antisym_on_o @ top_top_set_o @ R3 ) ) ).
% antisymI
thf(fact_1091_antisymD,axiom,
! [R3: set_Product_prod_o_o,X2: $o,Y4: $o] :
( ( antisym_on_o @ top_top_set_o @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X2 ) @ R3 )
=> ( X2 = Y4 ) ) ) ) ).
% antisymD
thf(fact_1092_antisym__empty,axiom,
antisym_on_o @ top_top_set_o @ bot_bo7073875226086086771od_o_o ).
% antisym_empty
thf(fact_1093_antisym__Id__on,axiom,
! [A3: set_o] : ( antisym_on_o @ top_top_set_o @ ( id_on_o @ A3 ) ) ).
% antisym_Id_on
thf(fact_1094_partial__order__on__empty,axiom,
order_8538205470752950541r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% partial_order_on_empty
thf(fact_1095_cSup__insert__If,axiom,
! [X7: set_o,A: $o] :
( ( condit5488710616941104124bove_o @ X7 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X7 ) )
= ( ( ( X7 = bot_bot_set_o )
=> A )
& ( ( X7 != bot_bot_set_o )
=> ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X7 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1096_cSup__insert,axiom,
! [X7: set_o,A: $o] :
( ( X7 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ X7 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X7 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X7 ) ) ) ) ) ).
% cSup_insert
thf(fact_1097_antisym__singleton,axiom,
! [X2: product_prod_o_o] : ( antisym_on_o @ top_top_set_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) ) ).
% antisym_singleton
thf(fact_1098_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_1099_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B ) )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1100_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1101_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1102_sup__eq__bot__iff,axiom,
! [X2: set_o,Y4: set_o] :
( ( ( sup_sup_set_o @ X2 @ Y4 )
= bot_bot_set_o )
= ( ( X2 = bot_bot_set_o )
& ( Y4 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_1103_sup__bot__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_1104_sup__bot__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% sup_bot_right
thf(fact_1105_bot__eq__sup__iff,axiom,
! [X2: set_o,Y4: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_o )
& ( Y4 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_1106_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1107_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1108_UNION__fun__upd,axiom,
! [A3: $o > set_o,I4: $o,B3: set_o,J2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o2 @ ( fun_upd_o_set_o @ A3 @ I4 @ B3 ) @ J2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ A3 @ ( minus_minus_set_o @ J2 @ ( insert_o @ I4 @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I4 @ J2 ) @ B3 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_1109_quotient__def,axiom,
( equiv_quotient_o
= ( ^ [A4: set_o,R: set_Product_prod_o_o] :
( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X5: $o] : ( insert_set_o @ ( image_o_o @ R @ ( insert_o @ X5 @ bot_bot_set_o ) ) @ bot_bot_set_set_o )
@ A4 ) ) ) ) ).
% quotient_def
thf(fact_1110_Diff__cancel,axiom,
! [A3: set_o] :
( ( minus_minus_set_o @ A3 @ A3 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_1111_empty__Diff,axiom,
! [A3: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_1112_Diff__empty,axiom,
! [A3: set_o] :
( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
= A3 ) ).
% Diff_empty
thf(fact_1113_Diff__insert0,axiom,
! [X2: $o,A3: set_o,B3: set_o] :
( ~ ( member_o @ X2 @ A3 )
=> ( ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ B3 ) )
= ( minus_minus_set_o @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1114_insert__Diff1,axiom,
! [X2: $o,B3: set_o,A3: set_o] :
( ( member_o @ X2 @ B3 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B3 )
= ( minus_minus_set_o @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1115_Inf__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inf_UNIV
thf(fact_1116_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_1117_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_1118_Inf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inf_empty
thf(fact_1119_Diff__UNIV,axiom,
! [A3: set_o] :
( ( minus_minus_set_o @ A3 @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_1120_cInf__singleton,axiom,
! [X2: $o] :
( ( complete_Inf_Inf_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% cInf_singleton
thf(fact_1121_insert__Diff__single,axiom,
! [A: $o,A3: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_1122_quotient__empty,axiom,
! [R3: set_Product_prod_o_o] :
( ( equiv_quotient_o @ bot_bot_set_o @ R3 )
= bot_bot_set_set_o ) ).
% quotient_empty
thf(fact_1123_quotient__is__empty,axiom,
! [A3: set_o,R3: set_Product_prod_o_o] :
( ( ( equiv_quotient_o @ A3 @ R3 )
= bot_bot_set_set_o )
= ( A3 = bot_bot_set_o ) ) ).
% quotient_is_empty
thf(fact_1124_quotient__is__empty2,axiom,
! [A3: set_o,R3: set_Product_prod_o_o] :
( ( bot_bot_set_set_o
= ( equiv_quotient_o @ A3 @ R3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% quotient_is_empty2
thf(fact_1125_INT__constant,axiom,
! [A3: set_o,C: set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= top_top_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1126_INT__simps_I3_J,axiom,
! [C2: set_o,A3: $o > set_o,B3: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( minus_minus_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( minus_minus_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ A3 @ C2 ) ) @ B3 ) ) ) ) ).
% INT_simps(3)
thf(fact_1127_INT__simps_I4_J,axiom,
! [C2: set_o,A3: set_o,B3: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( minus_minus_set_o @ A3 @ ( B3 @ X5 ) )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( minus_minus_set_o @ A3 @ ( B3 @ X5 ) )
@ C2 ) )
= ( minus_minus_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o2 @ B3 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1128_Inter__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inter_UNIV
thf(fact_1129_insert__Diff__if,axiom,
! [X2: $o,B3: set_o,A3: set_o] :
( ( ( member_o @ X2 @ B3 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B3 )
= ( minus_minus_set_o @ A3 @ B3 ) ) )
& ( ~ ( member_o @ X2 @ B3 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B3 )
= ( insert_o @ X2 @ ( minus_minus_set_o @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1130_INT__extend__simps_I3_J,axiom,
! [C2: set_o,A3: $o > set_o,B3: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ A3 @ C2 ) ) @ B3 )
= ( minus_minus_set_o @ top_top_set_o @ B3 ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ A3 @ C2 ) ) @ B3 )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( minus_minus_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1131_Inf__sup__eq__top__iff,axiom,
! [B3: set_set_o,A: set_o] :
( ( ( sup_sup_set_o @ ( comple3063163877087187839_set_o @ B3 ) @ A )
= top_top_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ B3 )
=> ( ( sup_sup_set_o @ X5 @ A )
= top_top_set_o ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1132_Diff__insert,axiom,
! [A3: set_o,A: $o,B3: set_o] :
( ( minus_minus_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ B3 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_1133_insert__Diff,axiom,
! [A: $o,A3: set_o] :
( ( member_o @ A @ A3 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1134_Diff__insert2,axiom,
! [A3: set_o,A: $o,B3: set_o] :
( ( minus_minus_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1135_Diff__insert__absorb,axiom,
! [X2: $o,A3: set_o] :
( ~ ( member_o @ X2 @ A3 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1136_INF__constant,axiom,
! [A3: set_o,C: set_o] :
( ( ( A3 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= top_top_set_o ) )
& ( ( A3 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [Y: $o] : C
@ A3 ) )
= C ) ) ) ).
% INF_constant
thf(fact_1137_INF__empty,axiom,
! [F: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ F @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_1138_in__image__insert__iff,axiom,
! [B3: set_set_o,X2: $o,A3: set_o] :
( ! [C5: set_o] :
( ( member_set_o @ C5 @ B3 )
=> ~ ( member_o @ X2 @ C5 ) )
=> ( ( member_set_o @ A3 @ ( image_set_o_set_o2 @ ( insert_o @ X2 ) @ B3 ) )
= ( ( member_o @ X2 @ A3 )
& ( member_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1139_INT__empty,axiom,
! [B3: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ B3 @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_1140_quotientI,axiom,
! [X2: $o,A3: set_o,R3: set_Product_prod_o_o] :
( ( member_o @ X2 @ A3 )
=> ( member_set_o @ ( image_o_o @ R3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ ( equiv_quotient_o @ A3 @ R3 ) ) ) ).
% quotientI
thf(fact_1141_quotientE,axiom,
! [X7: set_o,A3: set_o,R3: set_Product_prod_o_o] :
( ( member_set_o @ X7 @ ( equiv_quotient_o @ A3 @ R3 ) )
=> ~ ! [X: $o] :
( ( X7
= ( image_o_o @ R3 @ ( insert_o @ X @ bot_bot_set_o ) ) )
=> ~ ( member_o @ X @ A3 ) ) ) ).
% quotientE
thf(fact_1142_fun__upd__image,axiom,
! [X2: $o,A3: set_o,F: $o > $o,Y4: $o] :
( ( ( member_o @ X2 @ A3 )
=> ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X2 @ Y4 ) @ A3 )
= ( insert_o @ Y4 @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A3 )
=> ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X2 @ Y4 ) @ A3 )
= ( image_o_o2 @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1143_singleton__quotient,axiom,
! [X2: $o,R3: set_Product_prod_o_o] :
( ( equiv_quotient_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ R3 )
= ( insert_set_o @ ( image_o_o @ R3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ bot_bot_set_set_o ) ) ).
% singleton_quotient
thf(fact_1144_remove__def,axiom,
( remove_o
= ( ^ [X5: $o,A4: set_o] : ( minus_minus_set_o @ A4 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_1145_INT__simps_I1_J,axiom,
! [C2: set_o,A3: $o > set_o,B3: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( inf_inf_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( inf_inf_set_o @ ( A3 @ X5 ) @ B3 )
@ C2 ) )
= ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ A3 @ C2 ) ) @ B3 ) ) ) ) ).
% INT_simps(1)
thf(fact_1146_INT__simps_I2_J,axiom,
! [C2: set_o,A3: set_o,B3: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( inf_inf_set_o @ A3 @ ( B3 @ X5 ) )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o2
@ ^ [X5: $o] : ( inf_inf_set_o @ A3 @ ( B3 @ X5 ) )
@ C2 ) )
= ( inf_inf_set_o @ A3 @ ( comple3063163877087187839_set_o @ ( image_o_set_o2 @ B3 @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1147_inf__bot__left,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_1148_inf__bot__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_1149_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1150_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1151_Int__UNIV,axiom,
! [A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ A3 @ B3 )
= top_top_set_o )
= ( ( A3 = top_top_set_o )
& ( B3 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_1152_Compl__disjoint2,axiom,
! [A3: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A3 ) @ A3 )
= bot_bot_set_o ) ).
% Compl_disjoint2
thf(fact_1153_Compl__disjoint,axiom,
! [A3: set_o] :
( ( inf_inf_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
= bot_bot_set_o ) ).
% Compl_disjoint
thf(fact_1154_Int__insert__right__if1,axiom,
! [A: $o,A3: set_o,B3: set_o] :
( ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1155_Int__insert__right__if0,axiom,
! [A: $o,A3: set_o,B3: set_o] :
( ~ ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( inf_inf_set_o @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1156_insert__inter__insert,axiom,
! [A: $o,A3: set_o,B3: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ ( insert_o @ A @ B3 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1157_Int__insert__left__if1,axiom,
! [A: $o,C2: set_o,B3: set_o] :
( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1158_Int__insert__left__if0,axiom,
! [A: $o,C2: set_o,B3: set_o] :
( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C2 )
= ( inf_inf_set_o @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1159_bdd__below__empty,axiom,
condit5413489452508810728elow_o @ bot_bot_set_o ).
% bdd_below_empty
thf(fact_1160_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% boolean_algebra.compl_zero
thf(fact_1161_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.compl_one
thf(fact_1162_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ ( uminus_uminus_set_o @ X2 ) )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1163_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ X2 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1164_inf__compl__bot__right,axiom,
! [X2: set_o,Y4: set_o] :
( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ Y4 @ ( uminus_uminus_set_o @ X2 ) ) )
= bot_bot_set_o ) ).
% inf_compl_bot_right
thf(fact_1165_inf__compl__bot__left2,axiom,
! [X2: set_o,Y4: set_o] :
( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ Y4 ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left2
thf(fact_1166_inf__compl__bot__left1,axiom,
! [X2: set_o,Y4: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ ( inf_inf_set_o @ X2 @ Y4 ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left1
thf(fact_1167_insert__disjoint_I1_J,axiom,
! [A: $o,A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B3 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B3 )
& ( ( inf_inf_set_o @ A3 @ B3 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_1168_insert__disjoint_I2_J,axiom,
! [A: $o,A3: set_o,B3: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B3 ) )
= ( ~ ( member_o @ A @ B3 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1169_disjoint__insert_I1_J,axiom,
! [B3: set_o,A: $o,A3: set_o] :
( ( ( inf_inf_set_o @ B3 @ ( insert_o @ A @ A3 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B3 )
& ( ( inf_inf_set_o @ B3 @ A3 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_1170_disjoint__insert_I2_J,axiom,
! [A3: set_o,B: $o,B3: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A3 @ ( insert_o @ B @ B3 ) ) )
= ( ~ ( member_o @ B @ A3 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1171_Diff__disjoint,axiom,
! [A3: set_o,B3: set_o] :
( ( inf_inf_set_o @ A3 @ ( minus_minus_set_o @ B3 @ A3 ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_1172_Int__UNIV__right,axiom,
! [A3: set_o] :
( ( inf_inf_set_o @ A3 @ top_top_set_o )
= A3 ) ).
% Int_UNIV_right
thf(fact_1173_Int__UNIV__left,axiom,
! [B3: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_1174_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X2: set_o,Y4: set_o] :
( ( ( inf_inf_set_o @ X2 @ Y4 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ X2 @ Y4 )
= top_top_set_o )
=> ( ( uminus_uminus_set_o @ X2 )
= Y4 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1175_cInf__insert__If,axiom,
! [X7: set_o,A: $o] :
( ( condit5413489452508810728elow_o @ X7 )
=> ( ( complete_Inf_Inf_o @ ( insert_o @ A @ X7 ) )
= ( ( ( X7 = bot_bot_set_o )
=> A )
& ( ( X7 != bot_bot_set_o )
=> ( inf_inf_o @ A @ ( complete_Inf_Inf_o @ X7 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_1176_cInf__insert,axiom,
! [X7: set_o,A: $o] :
( ( X7 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ X7 )
=> ( ( complete_Inf_Inf_o @ ( insert_o @ A @ X7 ) )
= ( inf_inf_o @ A @ ( complete_Inf_Inf_o @ X7 ) ) ) ) ) ).
% cInf_insert
thf(fact_1177_cInf__union__distrib,axiom,
! [A3: set_o,B3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A3 )
=> ( ( B3 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B3 )
=> ( ( complete_Inf_Inf_o @ ( sup_sup_set_o @ A3 @ B3 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ A3 ) @ ( complete_Inf_Inf_o @ B3 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_1178_inf__cancel__left2,axiom,
! [X2: set_o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ A ) @ ( inf_inf_set_o @ X2 @ B ) )
= bot_bot_set_o ) ).
% inf_cancel_left2
thf(fact_1179_inf__cancel__left1,axiom,
! [X2: set_o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X2 @ A ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ B ) )
= bot_bot_set_o ) ).
% inf_cancel_left1
thf(fact_1180_Int__insert__left,axiom,
! [A: $o,C2: set_o,B3: set_o] :
( ( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B3 @ C2 ) ) ) )
& ( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C2 )
= ( inf_inf_set_o @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1181_Int__insert__right,axiom,
! [A: $o,A3: set_o,B3: set_o] :
( ( ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B3 ) ) ) )
& ( ~ ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
= ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1182_disjoint__iff__not__equal,axiom,
! [A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ A3 @ B3 )
= bot_bot_set_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ! [Y: $o] :
( ( member_o @ Y @ B3 )
=> ( X5 = (~ Y) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1183_Int__empty__right,axiom,
! [A3: set_o] :
( ( inf_inf_set_o @ A3 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_1184_Int__empty__left,axiom,
! [B3: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B3 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_1185_disjoint__iff,axiom,
! [A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ A3 @ B3 )
= bot_bot_set_o )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ~ ( member_o @ X5 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1186_Int__emptyI,axiom,
! [A3: set_o,B3: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A3 )
=> ~ ( member_o @ X @ B3 ) )
=> ( ( inf_inf_set_o @ A3 @ B3 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1187_Sup__inf__eq__bot__iff,axiom,
! [B3: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B3 ) @ A )
= bot_bot_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ B3 )
=> ( ( inf_inf_set_o @ X5 @ A )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1188_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_1189_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_1190_Int__Diff__disjoint,axiom,
! [A3: set_o,B3: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A3 @ B3 ) @ ( minus_minus_set_o @ A3 @ B3 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_1191_Diff__triv,axiom,
! [A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ A3 @ B3 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
thf(fact_1192_Union__disjoint,axiom,
! [C2: set_set_o,A3: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A3 )
= bot_bot_set_o )
= ( ! [X5: set_o] :
( ( member_set_o @ X5 @ C2 )
=> ( ( inf_inf_set_o @ X5 @ A3 )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_1193_Compl__partition2,axiom,
! [A3: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A3 ) @ A3 )
= top_top_set_o ) ).
% Compl_partition2
thf(fact_1194_Compl__partition,axiom,
! [A3: set_o] :
( ( sup_sup_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
= top_top_set_o ) ).
% Compl_partition
thf(fact_1195_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1196_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_o,X2: set_o,Y4: set_o] :
( ( ( inf_inf_set_o @ A @ X2 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ X2 )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A @ Y4 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ Y4 )
= top_top_set_o )
=> ( X2 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1197_Compl__insert,axiom,
! [X2: $o,A3: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X2 @ A3 ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_1198_insert__partition,axiom,
! [X2: set_o,F3: set_set_o] :
( ~ ( member_set_o @ X2 @ F3 )
=> ( ! [X: set_o] :
( ( member_set_o @ X @ ( insert_set_o @ X2 @ F3 ) )
=> ! [Xa2: set_o] :
( ( member_set_o @ Xa2 @ ( insert_set_o @ X2 @ F3 ) )
=> ( ( X != Xa2 )
=> ( ( inf_inf_set_o @ X @ Xa2 )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X2 @ ( comple90263536869209701_set_o @ F3 ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_1199_antisym__Restr,axiom,
! [R3: set_Product_prod_o_o,A3: set_o] :
( ( antisym_on_o @ top_top_set_o @ R3 )
=> ( antisym_on_o @ top_top_set_o
@ ( inf_in4898592226082374645od_o_o @ R3
@ ( product_Sigma_o_o @ A3
@ ^ [Uu: $o] : A3 ) ) ) ) ).
% antisym_Restr
thf(fact_1200_wfE__min_H,axiom,
! [R2: set_Product_prod_o_o,Q: set_o] :
( ( wf_o @ R2 )
=> ( ( Q != bot_bot_set_o )
=> ~ ! [Z4: $o] :
( ( member_o @ Z4 @ Q )
=> ~ ! [Y5: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z4 ) @ R2 )
=> ~ ( member_o @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1201_preorder__on__empty,axiom,
order_preorder_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% preorder_on_empty
thf(fact_1202_wf__Un,axiom,
! [R3: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
( ( wf_o @ R3 )
=> ( ( wf_o @ S2 )
=> ( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ S2 ) )
= bot_bot_set_o )
=> ( wf_o @ ( sup_su5769328420594410459od_o_o @ R3 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_1203_wf__Union,axiom,
! [R2: set_se491199571785101031od_o_o] :
( ! [X: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X @ R2 )
=> ( wf_o @ X ) )
=> ( ! [X: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X @ R2 )
=> ! [Xa2: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ Xa2 @ R2 )
=> ( ( X != Xa2 )
=> ( ( inf_inf_set_o @ ( domain_o_o @ X ) @ ( range_o_o @ Xa2 ) )
= bot_bot_set_o ) ) ) )
=> ( wf_o @ ( comple2567785453016644468od_o_o @ R2 ) ) ) ) ).
% wf_Union
thf(fact_1204_subset__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_o_o,A: $o,B: $o] :
( ( order_preorder_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( member_o @ A @ ( field_o @ R3 ) )
=> ( ( member_o @ B @ ( field_o @ R3 ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R3 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1205_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_1206_empty__subsetI,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).
% empty_subsetI
thf(fact_1207_subset__empty,axiom,
! [A3: set_o] :
( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
= ( A3 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_1208_insert__subset,axiom,
! [X2: $o,A3: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A3 ) @ B3 )
= ( ( member_o @ X2 @ B3 )
& ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_1209_singleton__insert__inj__eq_H,axiom,
! [A: $o,A3: set_o,B: $o] :
( ( ( insert_o @ A @ A3 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1210_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A3: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1211_Diff__eq__empty__iff,axiom,
! [A3: set_o,B3: set_o] :
( ( ( minus_minus_set_o @ A3 @ B3 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_1212_subset__Compl__singleton,axiom,
! [A3: set_o,B: $o] :
( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_1213_diff__shunt__var,axiom,
! [X2: set_o,Y4: set_o] :
( ( ( minus_minus_set_o @ X2 @ Y4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1214_cInf__superset__mono,axiom,
! [A3: set_o,B3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B3 )
=> ( ( ord_less_eq_set_o @ A3 @ B3 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ B3 ) @ ( complete_Inf_Inf_o @ A3 ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1215_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_1216_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_1217_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_1218_subset__emptyI,axiom,
! [A3: set_o] :
( ! [X: $o] :
~ ( member_o @ X @ A3 )
=> ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1219_subset__insertI2,axiom,
! [A3: set_o,B3: set_o,B: $o] :
( ( ord_less_eq_set_o @ A3 @ B3 )
=> ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1220_subset__insertI,axiom,
! [B3: set_o,A: $o] : ( ord_less_eq_set_o @ B3 @ ( insert_o @ A @ B3 ) ) ).
% subset_insertI
thf(fact_1221_subset__insert,axiom,
! [X2: $o,A3: set_o,B3: set_o] :
( ~ ( member_o @ X2 @ A3 )
=> ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B3 ) )
= ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_1222_insert__mono,axiom,
! [C2: set_o,D2: set_o,A: $o] :
( ( ord_less_eq_set_o @ C2 @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1223_times__subset__iff,axiom,
! [A3: set_o,C2: set_o,B3: set_o,D2: set_o] :
( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A3
@ ^ [Uu: $o] : C2 )
@ ( product_Sigma_o_o @ B3
@ ^ [Uu: $o] : D2 ) )
= ( ( A3 = bot_bot_set_o )
| ( C2 = bot_bot_set_o )
| ( ( ord_less_eq_set_o @ A3 @ B3 )
& ( ord_less_eq_set_o @ C2 @ D2 ) ) ) ) ).
% times_subset_iff
thf(fact_1224_cSup__subset__mono,axiom,
! [A3: set_o,B3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B3 )
=> ( ( ord_less_eq_set_o @ A3 @ B3 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1225_subset__UNIV,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ A3 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_1226_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_1227_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_1228_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_1229_less__eq__Sup,axiom,
! [A3: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A3 )
=> ( ord_less_eq_o @ U2 @ V ) )
=> ( ( A3 != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_1230_cSup__least,axiom,
! [X7: set_o,Z2: $o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X7 )
=> ( ord_less_eq_o @ X @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X7 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_1231_cSup__eq__non__empty,axiom,
! [X7: set_o,A: $o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X7 )
=> ( ord_less_eq_o @ X @ A ) )
=> ( ! [Y3: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X7 )
=> ( ord_less_eq_o @ X6 @ Y3 ) )
=> ( ord_less_eq_o @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ X7 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1232_Inf__less__eq,axiom,
! [A3: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A3 )
=> ( ord_less_eq_o @ V @ U2 ) )
=> ( ( A3 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A3 ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_1233_cInf__eq__non__empty,axiom,
! [X7: set_o,A: $o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X7 )
=> ( ord_less_eq_o @ A @ X ) )
=> ( ! [Y3: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X7 )
=> ( ord_less_eq_o @ Y3 @ X6 ) )
=> ( ord_less_eq_o @ Y3 @ A ) )
=> ( ( complete_Inf_Inf_o @ X7 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1234_cInf__greatest,axiom,
! [X7: set_o,Z2: $o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X7 )
=> ( ord_less_eq_o @ Z2 @ X ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_1235_subset__singleton__iff,axiom,
! [X7: set_o,A: $o] :
( ( ord_less_eq_set_o @ X7 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X7 = bot_bot_set_o )
| ( X7
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1236_subset__singletonD,axiom,
! [A3: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) )
=> ( ( A3 = bot_bot_set_o )
| ( A3
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1237_subset__Diff__insert,axiom,
! [A3: set_o,B3: set_o,X2: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B3 @ ( insert_o @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B3 @ C2 ) )
& ~ ( member_o @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1238_subset__Compl__self__eq,axiom,
! [A3: set_o] :
( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_1239_antisym__subset,axiom,
! [R3: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ R3 @ S2 )
=> ( ( antisym_on_o @ top_top_set_o @ S2 )
=> ( antisym_on_o @ top_top_set_o @ R3 ) ) ) ).
% antisym_subset
thf(fact_1240_inf__shunt,axiom,
! [X2: set_o,Y4: set_o] :
( ( ( inf_inf_set_o @ X2 @ Y4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X2 @ ( uminus_uminus_set_o @ Y4 ) ) ) ).
% inf_shunt
thf(fact_1241_Inf__le__Sup,axiom,
! [A3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A3 ) @ ( complete_Sup_Sup_o @ A3 ) ) ) ).
% Inf_le_Sup
thf(fact_1242_Diff__single__insert,axiom,
! [A3: set_o,X2: $o,B3: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B3 )
=> ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1243_subset__insert__iff,axiom,
! [A3: set_o,X2: $o,B3: set_o] :
( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B3 ) )
= ( ( ( member_o @ X2 @ A3 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B3 ) )
& ( ~ ( member_o @ X2 @ A3 )
=> ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1244_surj__Compl__image__subset,axiom,
! [F: $o > $o,A3: set_o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ ( image_o_o2 @ F @ A3 ) ) @ ( image_o_o2 @ F @ ( uminus_uminus_set_o @ A3 ) ) ) ) ).
% surj_Compl_image_subset
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 )
= ( ! [X5: $o] :
( ( member_o @ X5 @ S )
=> ( ord_less_eq_o @ X5 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1246_cSup__mono,axiom,
! [B3: set_o,A3: set_o] :
( ( B3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A3 )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ B3 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A3 )
& ( ord_less_eq_o @ B5 @ X6 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B3 ) @ ( complete_Sup_Sup_o @ A3 ) ) ) ) ) ).
% cSup_mono
thf(fact_1247_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 ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ S )
=> ( ord_less_eq_o @ A @ X5 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1248_cInf__mono,axiom,
! [B3: set_o,A3: set_o] :
( ( B3 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A3 )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ B3 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A3 )
& ( ord_less_eq_o @ X6 @ B5 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A3 ) @ ( complete_Inf_Inf_o @ B3 ) ) ) ) ) ).
% cInf_mono
thf(fact_1249_disjoint__eq__subset__Compl,axiom,
! [A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ A3 @ B3 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1250_wfE__pf,axiom,
! [R2: set_Product_prod_o_o,A3: set_o] :
( ( wf_o @ R2 )
=> ( ( ord_less_eq_set_o @ A3 @ ( image_o_o @ R2 @ A3 ) )
=> ( A3 = bot_bot_set_o ) ) ) ).
% wfE_pf
thf(fact_1251_wfI__pf,axiom,
! [R2: set_Product_prod_o_o] :
( ! [A7: set_o] :
( ( ord_less_eq_set_o @ A7 @ ( image_o_o @ R2 @ A7 ) )
=> ( A7 = bot_bot_set_o ) )
=> ( wf_o @ R2 ) ) ).
% wfI_pf
thf(fact_1252_wf__eq__minimal2,axiom,
( wf_o
= ( ^ [R: set_Product_prod_o_o] :
! [A4: set_o] :
( ( ( ord_less_eq_set_o @ A4 @ ( field_o @ R ) )
& ( A4 != bot_bot_set_o ) )
=> ? [X5: $o] :
( ( member_o @ X5 @ A4 )
& ! [Y: $o] :
( ( member_o @ Y @ A4 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X5 ) @ R ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1253_cSup__inter__less__eq,axiom,
! [A3: set_o,B3: set_o] :
( ( condit5488710616941104124bove_o @ A3 )
=> ( ( condit5488710616941104124bove_o @ B3 )
=> ( ( ( inf_inf_set_o @ A3 @ B3 )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A3 @ B3 ) ) @ ( sup_sup_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1254_less__eq__cInf__inter,axiom,
! [A3: set_o,B3: set_o] :
( ( condit5413489452508810728elow_o @ A3 )
=> ( ( condit5413489452508810728elow_o @ B3 )
=> ( ( ( inf_inf_set_o @ A3 @ B3 )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( inf_inf_o @ ( complete_Inf_Inf_o @ A3 ) @ ( complete_Inf_Inf_o @ B3 ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1255_cInf__le__cSup,axiom,
! [A3: set_o] :
( ( A3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A3 )
=> ( ( condit5413489452508810728elow_o @ A3 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A3 ) @ ( complete_Sup_Sup_o @ A3 ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1256_Linear__order__wf__diff__Id,axiom,
! [R3: set_Product_prod_o_o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R3 @ id_o ) )
= ( ! [A4: set_o] :
( ( ord_less_eq_set_o @ A4 @ ( field_o @ R3 ) )
=> ( ( A4 != bot_bot_set_o )
=> ? [X5: $o] :
( ( member_o @ X5 @ A4 )
& ! [Y: $o] :
( ( member_o @ Y @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1257_Domain__Id,axiom,
( ( domain_o_o @ id_o )
= top_top_set_o ) ).
% Domain_Id
thf(fact_1258_Range__Id,axiom,
( ( range_o_o @ id_o )
= top_top_set_o ) ).
% Range_Id
thf(fact_1259_refl__Id,axiom,
refl_on_o @ top_top_set_o @ id_o ).
% refl_Id
thf(fact_1260_antisym__Id,axiom,
antisym_on_o @ top_top_set_o @ id_o ).
% antisym_Id
thf(fact_1261_totalI,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X: $o,Y3: $o] :
( ( X != Y3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R3 )
| ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X ) @ R3 ) ) )
=> ( total_on_o @ top_top_set_o @ R3 ) ) ).
% totalI
thf(fact_1262_total__on__singleton,axiom,
! [X2: $o,R3: set_Product_prod_o_o] : ( total_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ R3 ) ).
% total_on_singleton
thf(fact_1263_total__on__empty,axiom,
! [R3: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R3 ) ).
% total_on_empty
thf(fact_1264_flat__lub__def,axiom,
( partial_flat_lub_o
= ( ^ [B2: $o,A4: set_o] :
( ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ B2 @ bot_bot_set_o ) )
=> B2 )
& ( ~ ( ord_less_eq_set_o @ A4 @ ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( the_o
@ ^ [X5: $o] : ( member_o @ X5 @ ( minus_minus_set_o @ A4 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1265_Chains__subset_H,axiom,
! [R3: set_Product_prod_o_o] :
( ( refl_on_o @ top_top_set_o @ R3 )
=> ( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ( pred_chain_o @ top_top_set_o
@ ^ [X5: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ R3 ) ) )
@ ( chains_o @ R3 ) ) ) ).
% Chains_subset'
thf(fact_1266_pred__on_Ochain__empty,axiom,
! [A3: set_o,P: $o > $o > $o] : ( pred_chain_o @ A3 @ P @ bot_bot_set_o ) ).
% pred_on.chain_empty
thf(fact_1267_Chains__subset,axiom,
! [R3: set_Product_prod_o_o] :
( ord_le4374716579403074808_set_o @ ( chains_o @ R3 )
@ ( collect_set_o
@ ( pred_chain_o @ top_top_set_o
@ ^ [X5: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ R3 ) ) ) ) ).
% Chains_subset
thf(fact_1268_pred__on_Ochain__extend,axiom,
! [A3: set_o,P: $o > $o > $o,C2: set_o,Z2: $o] :
( ( pred_chain_o @ A3 @ P @ C2 )
=> ( ( member_o @ Z2 @ A3 )
=> ( ! [X: $o] :
( ( member_o @ X @ C2 )
=> ( sup_sup_o_o_o @ P
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ X
@ Z2 ) )
=> ( pred_chain_o @ A3 @ P @ ( sup_sup_set_o @ ( insert_o @ Z2 @ bot_bot_set_o ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1269_Chains__alt__def,axiom,
! [R3: set_Product_prod_o_o] :
( ( refl_on_o @ top_top_set_o @ R3 )
=> ( ( chains_o @ R3 )
= ( collect_set_o
@ ( pred_chain_o @ top_top_set_o
@ ^ [X5: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y ) @ R3 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_1270_vimage__UNIV,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1271_vimage__empty,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1272_Pow__UNIV,axiom,
( ( pow_o @ top_top_set_o )
= top_top_set_set_o ) ).
% Pow_UNIV
thf(fact_1273_Pow__singleton__iff,axiom,
! [X7: set_o,Y7: set_o] :
( ( ( pow_o @ X7 )
= ( insert_set_o @ Y7 @ bot_bot_set_set_o ) )
= ( ( X7 = bot_bot_set_o )
& ( Y7 = bot_bot_set_o ) ) ) ).
% Pow_singleton_iff
thf(fact_1274_Pow__empty,axiom,
( ( pow_o @ bot_bot_set_o )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% Pow_empty
thf(fact_1275_vimage__subsetD,axiom,
! [F: $o > $o,B3: set_o,A3: set_o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ord_less_eq_set_o @ ( vimage_o_o @ F @ B3 ) @ A3 )
=> ( ord_less_eq_set_o @ B3 @ ( image_o_o2 @ F @ A3 ) ) ) ) ).
% vimage_subsetD
thf(fact_1276_Pow__insert,axiom,
! [A: $o,A3: set_o] :
( ( pow_o @ ( insert_o @ A @ A3 ) )
= ( sup_sup_set_set_o @ ( pow_o @ A3 ) @ ( image_set_o_set_o2 @ ( insert_o @ A ) @ ( pow_o @ A3 ) ) ) ) ).
% Pow_insert
% Helper facts (8)
thf(help_fChoice_1_1_fChoice_001_Eo_T,axiom,
! [P: $o > $o] :
( ( P @ ( fChoice_o @ P ) )
= ( ? [X3: $o] : ( P @ X3 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001tf__b_T,axiom,
! [P: b > $o] :
( ( P @ ( fChoice_b @ P ) )
= ( ? [X3: b] : ( P @ X3 ) ) ) ).
thf(help_If_3_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X2: set_o,Y4: set_o] :
( ( if_set_o @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X2: set_o,Y4: set_o] :
( ( if_set_o @ $true @ X2 @ Y4 )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_I_Eo_J_T,axiom,
! [P: set_o > $o] :
( ( P @ ( fChoice_set_o @ P ) )
= ( ? [X3: set_o] : ( P @ X3 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_Itf__b_Mt__PosRat__Oprat_J_T,axiom,
! [P: ( b > prat ) > $o] :
( ( P @ ( fChoice_b_prat @ P ) )
= ( ? [X3: b > prat] : ( P @ X3 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_I_Eo_M_Eo_J_T,axiom,
! [P: product_prod_o_o > $o] :
( ( P @ ( fChoic50381644832149627od_o_o @ P ) )
= ( ? [X3: product_prod_o_o] : ( P @ X3 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( a2
= ( big_add_masks_a_b @ b2
@ ^ [H: a] :
( fChoice_b_prat
@ ^ [R: b > prat] :
( ( a2 @ H )
= ( add_masks_b @ ( b2 @ H ) @ R ) ) ) ) ) ).
%------------------------------------------------------------------------------