TPTP Problem File: SLH0053^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0017_Preliminaries/prob_00171_006307__16363918_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1504 ( 613 unt; 223 typ; 0 def)
% Number of atoms : 3610 (1139 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 10414 ( 343 ~; 47 |; 213 &;8232 @)
% ( 0 <=>;1579 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 490 ( 490 >; 0 *; 0 +; 0 <<)
% Number of symbols : 207 ( 204 usr; 18 con; 0-4 aty)
% Number of variables : 3240 ( 198 ^;2958 !; 84 ?;3240 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:24:29.341
%------------------------------------------------------------------------------
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collec10116633892588882od_a_a: ( produc3498347346309940967od_a_a > $o ) > set_Pr8600417178894128327od_a_a ).
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collec3859266087184377170od_a_a: ( produc7983190928304305895od_a_a > $o ) > set_Pr5931134355633282247od_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
collec8259436133773553042_set_a: ( produc1703568184450464039_set_a > $o ) > set_Pr5845495582615845127_set_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec1673347964119250290od_a_a: ( set_Product_prod_a_a > $o ) > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Odisjnt_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
disjnt6673878776331334894od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Set_Odisjnt_001tf__a,type,
disjnt_a: set_a > set_a > $o ).
thf(sy_c_Set_Oimage_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_7999903765548794282od_a_a: ( ( product_prod_a_a > $o ) > set_Product_prod_a_a ) > set_Pr974637816708178892_a_a_o > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_4636654165204879301od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_6690255128444368805od_a_a: ( product_prod_a_a > set_Product_prod_a_a ) > set_Product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_4506799131697958853od_a_a: ( set_Product_prod_a_a > set_Product_prod_a_a ) > set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_3052423715192255772_set_a: ( set_Product_prod_a_a > set_a ) > set_se5735800977113168103od_a_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
insert8933621553020740791od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
insert914553114930139863od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
is_emp2923524183434141113od_a_a: set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
is_emp2937470224744679417od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
is_emp3465682670623713241od_a_a: set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_Itf__a_J,type,
is_empty_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
is_sin3171834905898671131od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
remove8198300757409973004od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
the_el8589169208993665564od_a_a: set_Product_prod_a_a > product_prod_a_a ).
thf(sy_c_Transitive__Closure_Oacyclic_001tf__a,type,
transitive_acyclic_a: set_Product_prod_a_a > $o ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
transi7229704755246292010od_a_a: set_Pr8600417178894128327od_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
transi1172912939347638282od_a_a: set_Pr5931134355633282247od_a_a > set_Pr5931134355633282247od_a_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Set__Oset_Itf__a_J,type,
transi3782967724755053153_set_a: set_Pr5845495582615845127_set_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
transi3369178822797952200od_a_a: set_Pr8600417178894128327od_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
transi728970796719974568od_a_a: set_Pr5931134355633282247od_a_a > set_Pr5931134355633282247od_a_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Set__Oset_Itf__a_J,type,
transi6946170248087431423_set_a: set_Pr5845495582615845127_set_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Transitive__Closure_Otrancl_001tf__a,type,
transitive_trancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
max_ex7229726517752475106od_a_a: set_Pr8600417178894128327od_a_a > set_Pr5931134355633282247od_a_a ).
thf(sy_c_Wellfounded_Omax__ext_001tf__a,type,
max_ext_a: set_Product_prod_a_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
wf_Product_prod_a_a: set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_Wellfounded_Owf_001t__Set__Oset_Itf__a_J,type,
wf_set_a: set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_Wellfounded_Owf_001tf__a,type,
wf_a: set_Product_prod_a_a > $o ).
thf(sy_c_Zorn_OChains_001tf__a,type,
chains_a: set_Product_prod_a_a > set_set_a ).
thf(sy_c_Zorn_Ochain__subset_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
chain_7119315380238734464od_a_a: set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member4903802553882211088od_a_a: produc7983190928304305895od_a_a > set_Pr5931134355633282247od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member92994164477555952od_a_a: set_Pr8600417178894128327od_a_a > set_se7936120678642572967od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
member7393593894965162800_set_a: set_Pr5845495582615845127_set_a > set_se6363330096766809319_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member2792204094846457616od_a_a: set_se5735800977113168103od_a_a > set_se5775600271191333063od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_r,type,
r: set_Product_prod_a_a ).
% Relevant facts (1277)
thf(fact_0_Field__trancl,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( field_1126092520709947252od_a_a @ ( transi3369178822797952200od_a_a @ R ) )
= ( field_1126092520709947252od_a_a @ R ) ) ).
% Field_trancl
thf(fact_1_Field__trancl,axiom,
! [R: set_Product_prod_a_a] :
( ( field_a @ ( transitive_trancl_a @ R ) )
= ( field_a @ R ) ) ).
% Field_trancl
thf(fact_2_Field__symcl,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( field_1126092520709947252od_a_a @ ( symcl_8029934039450908030od_a_a @ R ) )
= ( field_1126092520709947252od_a_a @ R ) ) ).
% Field_symcl
thf(fact_3_Field__symcl,axiom,
! [R: set_Product_prod_a_a] :
( ( field_a @ ( symcl_a @ R ) )
= ( field_a @ R ) ) ).
% Field_symcl
thf(fact_4_transymcl__def,axiom,
( transymcl_a
= ( ^ [R2: set_Product_prod_a_a] : ( transitive_trancl_a @ ( symcl_a @ R2 ) ) ) ) ).
% transymcl_def
thf(fact_5_r__into__trancl_H,axiom,
! [P: produc7983190928304305895od_a_a,R: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ P @ R )
=> ( member4903802553882211088od_a_a @ P @ ( transi728970796719974568od_a_a @ R ) ) ) ).
% r_into_trancl'
thf(fact_6_r__into__trancl_H,axiom,
! [P: produc1703568184450464039_set_a,R: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ P @ R )
=> ( member7983343339038529360_set_a @ P @ ( transi6946170248087431423_set_a @ R ) ) ) ).
% r_into_trancl'
thf(fact_7_r__into__trancl_H,axiom,
! [P: product_prod_a_a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ R )
=> ( member1426531477525435216od_a_a @ P @ ( transitive_trancl_a @ R ) ) ) ).
% r_into_trancl'
thf(fact_8_Domain__symcl,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( domain990338261037515724od_a_a @ ( symcl_8029934039450908030od_a_a @ R ) )
= ( field_1126092520709947252od_a_a @ R ) ) ).
% Domain_symcl
thf(fact_9_Domain__symcl,axiom,
! [R: set_Product_prod_a_a] :
( ( domain_a_a @ ( symcl_a @ R ) )
= ( field_a @ R ) ) ).
% Domain_symcl
thf(fact_10_trancl__range,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( range_2510312826226007605od_a_a @ ( transi3369178822797952200od_a_a @ R ) )
= ( range_2510312826226007605od_a_a @ R ) ) ).
% trancl_range
thf(fact_11_trancl__range,axiom,
! [R: set_Product_prod_a_a] :
( ( range_a_a @ ( transitive_trancl_a @ R ) )
= ( range_a_a @ R ) ) ).
% trancl_range
thf(fact_12_trancl__domain,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( domain990338261037515724od_a_a @ ( transi3369178822797952200od_a_a @ R ) )
= ( domain990338261037515724od_a_a @ R ) ) ).
% trancl_domain
thf(fact_13_trancl__domain,axiom,
! [R: set_Product_prod_a_a] :
( ( domain_a_a @ ( transitive_trancl_a @ R ) )
= ( domain_a_a @ R ) ) ).
% trancl_domain
thf(fact_14_trancl__empty,axiom,
( ( transi3369178822797952200od_a_a @ bot_bo510284599550014259od_a_a )
= bot_bo510284599550014259od_a_a ) ).
% trancl_empty
thf(fact_15_trancl__empty,axiom,
( ( transitive_trancl_a @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% trancl_empty
thf(fact_16_classes__def,axiom,
( classe2514596149140685704od_a_a
= ( ^ [Qeq: set_Pr8600417178894128327od_a_a] : ( equiv_6838630135511005422od_a_a @ ( field_1126092520709947252od_a_a @ Qeq ) @ ( transy2380204104860363337od_a_a @ Qeq ) ) ) ) ).
% classes_def
thf(fact_17_classes__def,axiom,
( classes_a
= ( ^ [Qeq: set_Product_prod_a_a] : ( equiv_quotient_a @ ( field_a @ Qeq ) @ ( transymcl_a @ Qeq ) ) ) ) ).
% classes_def
thf(fact_18_finite__trancl,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ ( transi3369178822797952200od_a_a @ R ) )
= ( finite20598220287489552od_a_a @ R ) ) ).
% finite_trancl
thf(fact_19_finite__trancl,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( transitive_trancl_a @ R ) )
= ( finite6544458595007987280od_a_a @ R ) ) ).
% finite_trancl
thf(fact_20_total__on__trancl,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( total_3048927264068743339od_a_a @ A @ R )
=> ( total_3048927264068743339od_a_a @ A @ ( transi3369178822797952200od_a_a @ R ) ) ) ).
% total_on_trancl
thf(fact_21_total__on__trancl,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( total_on_a @ A @ R )
=> ( total_on_a @ A @ ( transitive_trancl_a @ R ) ) ) ).
% total_on_trancl
thf(fact_22_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr8600417178894128327od_a_a] :
( ( transi3369178822797952200od_a_a @ ( transi7229704755246292010od_a_a @ R3 ) )
= ( transi7229704755246292010od_a_a @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_23_rtrancl__trancl__absorb,axiom,
! [R3: set_Product_prod_a_a] :
( ( transitive_trancl_a @ ( transitive_rtrancl_a @ R3 ) )
= ( transitive_rtrancl_a @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_24_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr8600417178894128327od_a_a] :
( ( transi7229704755246292010od_a_a @ ( transi3369178822797952200od_a_a @ R3 ) )
= ( transi7229704755246292010od_a_a @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_25_trancl__rtrancl__absorb,axiom,
! [R3: set_Product_prod_a_a] :
( ( transitive_rtrancl_a @ ( transitive_trancl_a @ R3 ) )
= ( transitive_rtrancl_a @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_26_trancl__mono,axiom,
! [P: produc7983190928304305895od_a_a,R: set_Pr5931134355633282247od_a_a,S: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ P @ ( transi728970796719974568od_a_a @ R ) )
=> ( ( ord_le6923081475299677287od_a_a @ R @ S )
=> ( member4903802553882211088od_a_a @ P @ ( transi728970796719974568od_a_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_27_trancl__mono,axiom,
! [P: produc1703568184450464039_set_a,R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ P @ ( transi6946170248087431423_set_a @ R ) )
=> ( ( ord_le8376522849517564071_set_a @ R @ S )
=> ( member7983343339038529360_set_a @ P @ ( transi6946170248087431423_set_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_28_trancl__mono,axiom,
! [P: produc3498347346309940967od_a_a,R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ P @ ( transi3369178822797952200od_a_a @ R ) )
=> ( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( member6330455413206600464od_a_a @ P @ ( transi3369178822797952200od_a_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_29_trancl__mono,axiom,
! [P: product_prod_a_a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ ( transitive_trancl_a @ R ) )
=> ( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( member1426531477525435216od_a_a @ P @ ( transitive_trancl_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_30_rtrancl__idemp,axiom,
! [R: set_Product_prod_a_a] :
( ( transitive_rtrancl_a @ ( transitive_rtrancl_a @ R ) )
= ( transitive_rtrancl_a @ R ) ) ).
% rtrancl_idemp
thf(fact_31_rtrancl__idemp,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( transi7229704755246292010od_a_a @ ( transi7229704755246292010od_a_a @ R ) )
= ( transi7229704755246292010od_a_a @ R ) ) ).
% rtrancl_idemp
thf(fact_32_r__into__rtrancl,axiom,
! [P: produc7983190928304305895od_a_a,R: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ P @ R )
=> ( member4903802553882211088od_a_a @ P @ ( transi1172912939347638282od_a_a @ R ) ) ) ).
% r_into_rtrancl
thf(fact_33_r__into__rtrancl,axiom,
! [P: produc1703568184450464039_set_a,R: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ P @ R )
=> ( member7983343339038529360_set_a @ P @ ( transi3782967724755053153_set_a @ R ) ) ) ).
% r_into_rtrancl
thf(fact_34_r__into__rtrancl,axiom,
! [P: produc3498347346309940967od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ P @ R )
=> ( member6330455413206600464od_a_a @ P @ ( transi7229704755246292010od_a_a @ R ) ) ) ).
% r_into_rtrancl
thf(fact_35_r__into__rtrancl,axiom,
! [P: product_prod_a_a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ R )
=> ( member1426531477525435216od_a_a @ P @ ( transitive_rtrancl_a @ R ) ) ) ).
% r_into_rtrancl
thf(fact_36_rtrancl__mono,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ R ) @ ( transitive_rtrancl_a @ S ) ) ) ).
% rtrancl_mono
thf(fact_37_rtrancl__mono,axiom,
! [R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ R @ S )
=> ( ord_le8376522849517564071_set_a @ ( transi3782967724755053153_set_a @ R ) @ ( transi3782967724755053153_set_a @ S ) ) ) ).
% rtrancl_mono
thf(fact_38_rtrancl__mono,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( ord_le3469131294019144807od_a_a @ ( transi7229704755246292010od_a_a @ R ) @ ( transi7229704755246292010od_a_a @ S ) ) ) ).
% rtrancl_mono
thf(fact_39_rtrancl__subset,axiom,
! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3 @ S2 )
=> ( ( ord_le746702958409616551od_a_a @ S2 @ ( transitive_rtrancl_a @ R3 ) )
=> ( ( transitive_rtrancl_a @ S2 )
= ( transitive_rtrancl_a @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_40_rtrancl__subset,axiom,
! [R3: set_Pr5845495582615845127_set_a,S2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ R3 @ S2 )
=> ( ( ord_le8376522849517564071_set_a @ S2 @ ( transi3782967724755053153_set_a @ R3 ) )
=> ( ( transi3782967724755053153_set_a @ S2 )
= ( transi3782967724755053153_set_a @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_41_rtrancl__subset,axiom,
! [R3: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R3 @ S2 )
=> ( ( ord_le3469131294019144807od_a_a @ S2 @ ( transi7229704755246292010od_a_a @ R3 ) )
=> ( ( transi7229704755246292010od_a_a @ S2 )
= ( transi7229704755246292010od_a_a @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_42_rtrancl__subset__rtrancl,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ ( transitive_rtrancl_a @ S ) )
=> ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ R ) @ ( transitive_rtrancl_a @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_43_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ R @ ( transi3782967724755053153_set_a @ S ) )
=> ( ord_le8376522849517564071_set_a @ ( transi3782967724755053153_set_a @ R ) @ ( transi3782967724755053153_set_a @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_44_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ ( transi7229704755246292010od_a_a @ S ) )
=> ( ord_le3469131294019144807od_a_a @ ( transi7229704755246292010od_a_a @ R ) @ ( transi7229704755246292010od_a_a @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_45_Range__empty,axiom,
( ( range_2510312826226007605od_a_a @ bot_bo510284599550014259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Range_empty
thf(fact_46_Range__empty,axiom,
( ( range_a_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_a ) ).
% Range_empty
thf(fact_47_Domain__empty,axiom,
( ( domain990338261037515724od_a_a @ bot_bo510284599550014259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Domain_empty
thf(fact_48_Domain__empty,axiom,
( ( domain_a_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_a ) ).
% Domain_empty
thf(fact_49_Field__empty,axiom,
( ( field_1595709187305778996od_a_a @ bot_bo514369784327480115od_a_a )
= bot_bo510284599550014259od_a_a ) ).
% Field_empty
thf(fact_50_Field__empty,axiom,
( ( field_set_a @ bot_bo5799363139946352499_set_a )
= bot_bot_set_set_a ) ).
% Field_empty
thf(fact_51_Field__empty,axiom,
( ( field_8083607959440978772od_a_a @ bot_bo8313862440737889075od_a_a )
= bot_bo777872063958040403od_a_a ) ).
% Field_empty
thf(fact_52_Field__empty,axiom,
( ( field_1126092520709947252od_a_a @ bot_bo510284599550014259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Field_empty
thf(fact_53_Field__empty,axiom,
( ( field_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_a ) ).
% Field_empty
thf(fact_54_quotient__empty,axiom,
! [R: set_Pr3617678592830649543od_a_a] :
( ( equiv_5809209031780029614od_a_a @ bot_bo510284599550014259od_a_a @ R )
= bot_bo2005289091711683859od_a_a ) ).
% quotient_empty
thf(fact_55_quotient__empty,axiom,
! [R: set_Pr5845495582615845127_set_a] :
( ( equiv_quotient_set_a @ bot_bot_set_set_a @ R )
= bot_bo3380559777022489994_set_a ) ).
% quotient_empty
thf(fact_56_quotient__empty,axiom,
! [R: set_Pr5931134355633282247od_a_a] :
( ( equiv_2247238863381808334od_a_a @ bot_bo777872063958040403od_a_a @ R )
= bot_bo5729467828521840435od_a_a ) ).
% quotient_empty
thf(fact_57_quotient__empty,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( equiv_6838630135511005422od_a_a @ bot_bo3357376287454694259od_a_a @ R )
= bot_bo777872063958040403od_a_a ) ).
% quotient_empty
thf(fact_58_quotient__empty,axiom,
! [R: set_Product_prod_a_a] :
( ( equiv_quotient_a @ bot_bot_set_a @ R )
= bot_bot_set_set_a ) ).
% quotient_empty
thf(fact_59_quotient__is__empty,axiom,
! [A: set_Pr8600417178894128327od_a_a,R: set_Pr3617678592830649543od_a_a] :
( ( ( equiv_5809209031780029614od_a_a @ A @ R )
= bot_bo2005289091711683859od_a_a )
= ( A = bot_bo510284599550014259od_a_a ) ) ).
% quotient_is_empty
thf(fact_60_quotient__is__empty,axiom,
! [A: set_set_a,R: set_Pr5845495582615845127_set_a] :
( ( ( equiv_quotient_set_a @ A @ R )
= bot_bo3380559777022489994_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% quotient_is_empty
thf(fact_61_quotient__is__empty,axiom,
! [A: set_se5735800977113168103od_a_a,R: set_Pr5931134355633282247od_a_a] :
( ( ( equiv_2247238863381808334od_a_a @ A @ R )
= bot_bo5729467828521840435od_a_a )
= ( A = bot_bo777872063958040403od_a_a ) ) ).
% quotient_is_empty
thf(fact_62_quotient__is__empty,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( ( equiv_6838630135511005422od_a_a @ A @ R )
= bot_bo777872063958040403od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% quotient_is_empty
thf(fact_63_quotient__is__empty,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( ( equiv_quotient_a @ A @ R )
= bot_bot_set_set_a )
= ( A = bot_bot_set_a ) ) ).
% quotient_is_empty
thf(fact_64_quotient__is__empty2,axiom,
! [A: set_Pr8600417178894128327od_a_a,R: set_Pr3617678592830649543od_a_a] :
( ( bot_bo2005289091711683859od_a_a
= ( equiv_5809209031780029614od_a_a @ A @ R ) )
= ( A = bot_bo510284599550014259od_a_a ) ) ).
% quotient_is_empty2
thf(fact_65_quotient__is__empty2,axiom,
! [A: set_set_a,R: set_Pr5845495582615845127_set_a] :
( ( bot_bo3380559777022489994_set_a
= ( equiv_quotient_set_a @ A @ R ) )
= ( A = bot_bot_set_set_a ) ) ).
% quotient_is_empty2
thf(fact_66_quotient__is__empty2,axiom,
! [A: set_se5735800977113168103od_a_a,R: set_Pr5931134355633282247od_a_a] :
( ( bot_bo5729467828521840435od_a_a
= ( equiv_2247238863381808334od_a_a @ A @ R ) )
= ( A = bot_bo777872063958040403od_a_a ) ) ).
% quotient_is_empty2
thf(fact_67_quotient__is__empty2,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( bot_bo777872063958040403od_a_a
= ( equiv_6838630135511005422od_a_a @ A @ R ) )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% quotient_is_empty2
thf(fact_68_quotient__is__empty2,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( bot_bot_set_set_a
= ( equiv_quotient_a @ A @ R ) )
= ( A = bot_bot_set_a ) ) ).
% quotient_is_empty2
thf(fact_69_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_70_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_71_subset__empty,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ bot_bo777872063958040403od_a_a )
= ( A = bot_bo777872063958040403od_a_a ) ) ).
% subset_empty
thf(fact_72_subset__empty,axiom,
! [A: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ bot_bo5799363139946352499_set_a )
= ( A = bot_bo5799363139946352499_set_a ) ) ).
% subset_empty
thf(fact_73_subset__empty,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ bot_bo510284599550014259od_a_a )
= ( A = bot_bo510284599550014259od_a_a ) ) ).
% subset_empty
thf(fact_74_subset__empty,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_75_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_76_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_77_empty__subsetI,axiom,
! [A: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ bot_bo777872063958040403od_a_a @ A ) ).
% empty_subsetI
thf(fact_78_empty__subsetI,axiom,
! [A: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ bot_bo5799363139946352499_set_a @ A ) ).
% empty_subsetI
thf(fact_79_empty__subsetI,axiom,
! [A: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ bot_bo510284599550014259od_a_a @ A ) ).
% empty_subsetI
thf(fact_80_empty__subsetI,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% empty_subsetI
thf(fact_81_finite__Range,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ R )
=> ( finite6544458595007987280od_a_a @ ( range_2510312826226007605od_a_a @ R ) ) ) ).
% finite_Range
thf(fact_82_finite__Range,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( finite_finite_a @ ( range_a_a @ R ) ) ) ).
% finite_Range
thf(fact_83_finite__Domain,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ R )
=> ( finite6544458595007987280od_a_a @ ( domain990338261037515724od_a_a @ R ) ) ) ).
% finite_Domain
thf(fact_84_finite__Domain,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( finite_finite_a @ ( domain_a_a @ R ) ) ) ).
% finite_Domain
thf(fact_85_Range__empty__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( ( range_2510312826226007605od_a_a @ R )
= bot_bo3357376287454694259od_a_a )
= ( R = bot_bo510284599550014259od_a_a ) ) ).
% Range_empty_iff
thf(fact_86_Range__empty__iff,axiom,
! [R: set_Product_prod_a_a] :
( ( ( range_a_a @ R )
= bot_bot_set_a )
= ( R = bot_bo3357376287454694259od_a_a ) ) ).
% Range_empty_iff
thf(fact_87_Domain__empty__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( ( domain990338261037515724od_a_a @ R )
= bot_bo3357376287454694259od_a_a )
= ( R = bot_bo510284599550014259od_a_a ) ) ).
% Domain_empty_iff
thf(fact_88_Domain__empty__iff,axiom,
! [R: set_Product_prod_a_a] :
( ( ( domain_a_a @ R )
= bot_bot_set_a )
= ( R = bot_bo3357376287454694259od_a_a ) ) ).
% Domain_empty_iff
thf(fact_89_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_90_empty__Collect__eq,axiom,
! [P2: produc3498347346309940967od_a_a > $o] :
( ( bot_bo510284599550014259od_a_a
= ( collec10116633892588882od_a_a @ P2 ) )
= ( ! [X: produc3498347346309940967od_a_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_91_empty__Collect__eq,axiom,
! [P2: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P2 ) )
= ( ! [X: set_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_92_empty__Collect__eq,axiom,
! [P2: set_Product_prod_a_a > $o] :
( ( bot_bo777872063958040403od_a_a
= ( collec1673347964119250290od_a_a @ P2 ) )
= ( ! [X: set_Product_prod_a_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_93_empty__Collect__eq,axiom,
! [P2: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P2 ) )
= ( ! [X: product_prod_a_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_94_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_95_Collect__empty__eq,axiom,
! [P2: produc3498347346309940967od_a_a > $o] :
( ( ( collec10116633892588882od_a_a @ P2 )
= bot_bo510284599550014259od_a_a )
= ( ! [X: produc3498347346309940967od_a_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_96_Collect__empty__eq,axiom,
! [P2: set_a > $o] :
( ( ( collect_set_a @ P2 )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_97_Collect__empty__eq,axiom,
! [P2: set_Product_prod_a_a > $o] :
( ( ( collec1673347964119250290od_a_a @ P2 )
= bot_bo777872063958040403od_a_a )
= ( ! [X: set_Product_prod_a_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_98_Collect__empty__eq,axiom,
! [P2: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: product_prod_a_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_99_all__not__in__conv,axiom,
! [A: set_Pr5931134355633282247od_a_a] :
( ( ! [X: produc7983190928304305895od_a_a] :
~ ( member4903802553882211088od_a_a @ X @ A ) )
= ( A = bot_bo8313862440737889075od_a_a ) ) ).
% all_not_in_conv
thf(fact_100_all__not__in__conv,axiom,
! [A: set_Pr5845495582615845127_set_a] :
( ( ! [X: produc1703568184450464039_set_a] :
~ ( member7983343339038529360_set_a @ X @ A ) )
= ( A = bot_bo5799363139946352499_set_a ) ) ).
% all_not_in_conv
thf(fact_101_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_102_all__not__in__conv,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ( ! [X: produc3498347346309940967od_a_a] :
~ ( member6330455413206600464od_a_a @ X @ A ) )
= ( A = bot_bo510284599550014259od_a_a ) ) ).
% all_not_in_conv
thf(fact_103_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_104_all__not__in__conv,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( ! [X: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ X @ A ) )
= ( A = bot_bo777872063958040403od_a_a ) ) ).
% all_not_in_conv
thf(fact_105_all__not__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ! [X: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X @ A ) )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_106_empty__iff,axiom,
! [C: produc7983190928304305895od_a_a] :
~ ( member4903802553882211088od_a_a @ C @ bot_bo8313862440737889075od_a_a ) ).
% empty_iff
thf(fact_107_empty__iff,axiom,
! [C: produc1703568184450464039_set_a] :
~ ( member7983343339038529360_set_a @ C @ bot_bo5799363139946352499_set_a ) ).
% empty_iff
thf(fact_108_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_109_empty__iff,axiom,
! [C: produc3498347346309940967od_a_a] :
~ ( member6330455413206600464od_a_a @ C @ bot_bo510284599550014259od_a_a ) ).
% empty_iff
thf(fact_110_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_111_empty__iff,axiom,
! [C: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ C @ bot_bo777872063958040403od_a_a ) ).
% empty_iff
thf(fact_112_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_113_subset__antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_114_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_115_subset__antisym,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( ord_le1995061765932249223od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_116_subset__antisym,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( ( ord_le8376522849517564071_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_117_subset__antisym,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( ( ord_le3469131294019144807od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_118_subsetI,axiom,
! [A: set_Pr5931134355633282247od_a_a,B: set_Pr5931134355633282247od_a_a] :
( ! [X2: produc7983190928304305895od_a_a] :
( ( member4903802553882211088od_a_a @ X2 @ A )
=> ( member4903802553882211088od_a_a @ X2 @ B ) )
=> ( ord_le6923081475299677287od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_119_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_120_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_121_subsetI,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
=> ( member1816616512716248880od_a_a @ X2 @ B ) )
=> ( ord_le1995061765932249223od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_122_subsetI,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ! [X2: produc1703568184450464039_set_a] :
( ( member7983343339038529360_set_a @ X2 @ A )
=> ( member7983343339038529360_set_a @ X2 @ B ) )
=> ( ord_le8376522849517564071_set_a @ A @ B ) ) ).
% subsetI
thf(fact_123_subsetI,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ! [X2: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X2 @ A )
=> ( member6330455413206600464od_a_a @ X2 @ B ) )
=> ( ord_le3469131294019144807od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_124_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_125_ex__in__conv,axiom,
! [A: set_Pr5931134355633282247od_a_a] :
( ( ? [X: produc7983190928304305895od_a_a] : ( member4903802553882211088od_a_a @ X @ A ) )
= ( A != bot_bo8313862440737889075od_a_a ) ) ).
% ex_in_conv
thf(fact_126_ex__in__conv,axiom,
! [A: set_Pr5845495582615845127_set_a] :
( ( ? [X: produc1703568184450464039_set_a] : ( member7983343339038529360_set_a @ X @ A ) )
= ( A != bot_bo5799363139946352499_set_a ) ) ).
% ex_in_conv
thf(fact_127_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_128_ex__in__conv,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ( ? [X: produc3498347346309940967od_a_a] : ( member6330455413206600464od_a_a @ X @ A ) )
= ( A != bot_bo510284599550014259od_a_a ) ) ).
% ex_in_conv
thf(fact_129_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_130_ex__in__conv,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( ? [X: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X @ A ) )
= ( A != bot_bo777872063958040403od_a_a ) ) ).
% ex_in_conv
thf(fact_131_ex__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ? [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A ) )
= ( A != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_132_equals0I,axiom,
! [A: set_Pr5931134355633282247od_a_a] :
( ! [Y: produc7983190928304305895od_a_a] :
~ ( member4903802553882211088od_a_a @ Y @ A )
=> ( A = bot_bo8313862440737889075od_a_a ) ) ).
% equals0I
thf(fact_133_equals0I,axiom,
! [A: set_Pr5845495582615845127_set_a] :
( ! [Y: produc1703568184450464039_set_a] :
~ ( member7983343339038529360_set_a @ Y @ A )
=> ( A = bot_bo5799363139946352499_set_a ) ) ).
% equals0I
thf(fact_134_equals0I,axiom,
! [A: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_135_equals0I,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ! [Y: produc3498347346309940967od_a_a] :
~ ( member6330455413206600464od_a_a @ Y @ A )
=> ( A = bot_bo510284599550014259od_a_a ) ) ).
% equals0I
thf(fact_136_equals0I,axiom,
! [A: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_137_equals0I,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ! [Y: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ Y @ A )
=> ( A = bot_bo777872063958040403od_a_a ) ) ).
% equals0I
thf(fact_138_equals0I,axiom,
! [A: set_Product_prod_a_a] :
( ! [Y: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y @ A )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_139_equals0D,axiom,
! [A: set_Pr5931134355633282247od_a_a,A2: produc7983190928304305895od_a_a] :
( ( A = bot_bo8313862440737889075od_a_a )
=> ~ ( member4903802553882211088od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_140_equals0D,axiom,
! [A: set_Pr5845495582615845127_set_a,A2: produc1703568184450464039_set_a] :
( ( A = bot_bo5799363139946352499_set_a )
=> ~ ( member7983343339038529360_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_141_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_142_equals0D,axiom,
! [A: set_Pr8600417178894128327od_a_a,A2: produc3498347346309940967od_a_a] :
( ( A = bot_bo510284599550014259od_a_a )
=> ~ ( member6330455413206600464od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_143_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_144_equals0D,axiom,
! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( A = bot_bo777872063958040403od_a_a )
=> ~ ( member1816616512716248880od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_145_equals0D,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( A = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_146_emptyE,axiom,
! [A2: produc7983190928304305895od_a_a] :
~ ( member4903802553882211088od_a_a @ A2 @ bot_bo8313862440737889075od_a_a ) ).
% emptyE
thf(fact_147_emptyE,axiom,
! [A2: produc1703568184450464039_set_a] :
~ ( member7983343339038529360_set_a @ A2 @ bot_bo5799363139946352499_set_a ) ).
% emptyE
thf(fact_148_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_149_emptyE,axiom,
! [A2: produc3498347346309940967od_a_a] :
~ ( member6330455413206600464od_a_a @ A2 @ bot_bo510284599550014259od_a_a ) ).
% emptyE
thf(fact_150_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_151_emptyE,axiom,
! [A2: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ A2 @ bot_bo777872063958040403od_a_a ) ).
% emptyE
thf(fact_152_emptyE,axiom,
! [A2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_153_Collect__mono__iff,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P2 ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X: product_prod_a_a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_154_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_155_Collect__mono__iff,axiom,
! [P2: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ( ord_le1995061765932249223od_a_a @ ( collec1673347964119250290od_a_a @ P2 ) @ ( collec1673347964119250290od_a_a @ Q ) )
= ( ! [X: set_Product_prod_a_a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_156_Collect__mono__iff,axiom,
! [P2: produc1703568184450464039_set_a > $o,Q: produc1703568184450464039_set_a > $o] :
( ( ord_le8376522849517564071_set_a @ ( collec8259436133773553042_set_a @ P2 ) @ ( collec8259436133773553042_set_a @ Q ) )
= ( ! [X: produc1703568184450464039_set_a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_157_Collect__mono__iff,axiom,
! [P2: produc3498347346309940967od_a_a > $o,Q: produc3498347346309940967od_a_a > $o] :
( ( ord_le3469131294019144807od_a_a @ ( collec10116633892588882od_a_a @ P2 ) @ ( collec10116633892588882od_a_a @ Q ) )
= ( ! [X: produc3498347346309940967od_a_a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_158_set__eq__subset,axiom,
( ( ^ [Y2: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y2 = Z ) )
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
& ( ord_le746702958409616551od_a_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_159_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_160_set__eq__subset,axiom,
( ( ^ [Y2: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] : ( Y2 = Z ) )
= ( ^ [A3: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A3 @ B2 )
& ( ord_le1995061765932249223od_a_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_161_set__eq__subset,axiom,
( ( ^ [Y2: set_Pr5845495582615845127_set_a,Z: set_Pr5845495582615845127_set_a] : ( Y2 = Z ) )
= ( ^ [A3: set_Pr5845495582615845127_set_a,B2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A3 @ B2 )
& ( ord_le8376522849517564071_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_162_set__eq__subset,axiom,
( ( ^ [Y2: set_Pr8600417178894128327od_a_a,Z: set_Pr8600417178894128327od_a_a] : ( Y2 = Z ) )
= ( ^ [A3: set_Pr8600417178894128327od_a_a,B2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A3 @ B2 )
& ( ord_le3469131294019144807od_a_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_163_subset__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_164_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_165_subset__trans,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( ord_le1995061765932249223od_a_a @ B @ C2 )
=> ( ord_le1995061765932249223od_a_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_166_subset__trans,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a,C2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( ( ord_le8376522849517564071_set_a @ B @ C2 )
=> ( ord_le8376522849517564071_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_167_subset__trans,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,C2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( ( ord_le3469131294019144807od_a_a @ B @ C2 )
=> ( ord_le3469131294019144807od_a_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_168_Collect__mono,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P2 ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_169_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_170_Collect__mono,axiom,
! [P2: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ! [X2: set_Product_prod_a_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1995061765932249223od_a_a @ ( collec1673347964119250290od_a_a @ P2 ) @ ( collec1673347964119250290od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_171_Collect__mono,axiom,
! [P2: produc1703568184450464039_set_a > $o,Q: produc1703568184450464039_set_a > $o] :
( ! [X2: produc1703568184450464039_set_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le8376522849517564071_set_a @ ( collec8259436133773553042_set_a @ P2 ) @ ( collec8259436133773553042_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_172_Collect__mono,axiom,
! [P2: produc3498347346309940967od_a_a > $o,Q: produc3498347346309940967od_a_a > $o] :
( ! [X2: produc3498347346309940967od_a_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3469131294019144807od_a_a @ ( collec10116633892588882od_a_a @ P2 ) @ ( collec10116633892588882od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_173_mem__Collect__eq,axiom,
! [A2: produc7983190928304305895od_a_a,P2: produc7983190928304305895od_a_a > $o] :
( ( member4903802553882211088od_a_a @ A2 @ ( collec3859266087184377170od_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_174_mem__Collect__eq,axiom,
! [A2: produc1703568184450464039_set_a,P2: produc1703568184450464039_set_a > $o] :
( ( member7983343339038529360_set_a @ A2 @ ( collec8259436133773553042_set_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_175_mem__Collect__eq,axiom,
! [A2: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( member1816616512716248880od_a_a @ A2 @ ( collec1673347964119250290od_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_176_mem__Collect__eq,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_177_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_178_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_179_Collect__mem__eq,axiom,
! [A: set_Pr5931134355633282247od_a_a] :
( ( collec3859266087184377170od_a_a
@ ^ [X: produc7983190928304305895od_a_a] : ( member4903802553882211088od_a_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_180_Collect__mem__eq,axiom,
! [A: set_Pr5845495582615845127_set_a] :
( ( collec8259436133773553042_set_a
@ ^ [X: produc1703568184450464039_set_a] : ( member7983343339038529360_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_181_Collect__mem__eq,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( collec1673347964119250290od_a_a
@ ^ [X: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_182_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_183_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_184_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_185_Collect__cong,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec3336397797384452498od_a_a @ P2 )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_186_subset__refl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% subset_refl
thf(fact_187_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_188_subset__refl,axiom,
! [A: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ A @ A ) ).
% subset_refl
thf(fact_189_subset__refl,axiom,
! [A: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ A @ A ) ).
% subset_refl
thf(fact_190_subset__refl,axiom,
! [A: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ A @ A ) ).
% subset_refl
thf(fact_191_subset__iff,axiom,
( ord_le6923081475299677287od_a_a
= ( ^ [A3: set_Pr5931134355633282247od_a_a,B2: set_Pr5931134355633282247od_a_a] :
! [T: produc7983190928304305895od_a_a] :
( ( member4903802553882211088od_a_a @ T @ A3 )
=> ( member4903802553882211088od_a_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_192_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_193_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_194_subset__iff,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
! [T: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ T @ A3 )
=> ( member1816616512716248880od_a_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_195_subset__iff,axiom,
( ord_le8376522849517564071_set_a
= ( ^ [A3: set_Pr5845495582615845127_set_a,B2: set_Pr5845495582615845127_set_a] :
! [T: produc1703568184450464039_set_a] :
( ( member7983343339038529360_set_a @ T @ A3 )
=> ( member7983343339038529360_set_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_196_subset__iff,axiom,
( ord_le3469131294019144807od_a_a
= ( ^ [A3: set_Pr8600417178894128327od_a_a,B2: set_Pr8600417178894128327od_a_a] :
! [T: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ T @ A3 )
=> ( member6330455413206600464od_a_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_197_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_198_equalityD2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ( ord_le746702958409616551od_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_199_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_200_equalityD2,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( A = B )
=> ( ord_le1995061765932249223od_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_201_equalityD2,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( A = B )
=> ( ord_le8376522849517564071_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_202_equalityD2,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( A = B )
=> ( ord_le3469131294019144807od_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_203_equalityD1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_204_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_205_equalityD1,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( A = B )
=> ( ord_le1995061765932249223od_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_206_equalityD1,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( A = B )
=> ( ord_le8376522849517564071_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_207_equalityD1,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( A = B )
=> ( ord_le3469131294019144807od_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_208_subset__eq,axiom,
( ord_le6923081475299677287od_a_a
= ( ^ [A3: set_Pr5931134355633282247od_a_a,B2: set_Pr5931134355633282247od_a_a] :
! [X: produc7983190928304305895od_a_a] :
( ( member4903802553882211088od_a_a @ X @ A3 )
=> ( member4903802553882211088od_a_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_209_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_210_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_211_subset__eq,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ A3 )
=> ( member1816616512716248880od_a_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_212_subset__eq,axiom,
( ord_le8376522849517564071_set_a
= ( ^ [A3: set_Pr5845495582615845127_set_a,B2: set_Pr5845495582615845127_set_a] :
! [X: produc1703568184450464039_set_a] :
( ( member7983343339038529360_set_a @ X @ A3 )
=> ( member7983343339038529360_set_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_213_subset__eq,axiom,
( ord_le3469131294019144807od_a_a
= ( ^ [A3: set_Pr8600417178894128327od_a_a,B2: set_Pr8600417178894128327od_a_a] :
! [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ A3 )
=> ( member6330455413206600464od_a_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_214_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_215_equalityE,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ~ ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ~ ( ord_le746702958409616551od_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_216_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_217_equalityE,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( A = B )
=> ~ ( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ~ ( ord_le1995061765932249223od_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_218_equalityE,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( A = B )
=> ~ ( ( ord_le8376522849517564071_set_a @ A @ B )
=> ~ ( ord_le8376522849517564071_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_219_equalityE,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( A = B )
=> ~ ( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ~ ( ord_le3469131294019144807od_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_220_subsetD,axiom,
! [A: set_Pr5931134355633282247od_a_a,B: set_Pr5931134355633282247od_a_a,C: produc7983190928304305895od_a_a] :
( ( ord_le6923081475299677287od_a_a @ A @ B )
=> ( ( member4903802553882211088od_a_a @ C @ A )
=> ( member4903802553882211088od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_221_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_222_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_223_subsetD,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( member1816616512716248880od_a_a @ C @ A )
=> ( member1816616512716248880od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_224_subsetD,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a,C: produc1703568184450464039_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( ( member7983343339038529360_set_a @ C @ A )
=> ( member7983343339038529360_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_225_subsetD,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,C: produc3498347346309940967od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( ( member6330455413206600464od_a_a @ C @ A )
=> ( member6330455413206600464od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_226_subsetD,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_227_in__mono,axiom,
! [A: set_Pr5931134355633282247od_a_a,B: set_Pr5931134355633282247od_a_a,X3: produc7983190928304305895od_a_a] :
( ( ord_le6923081475299677287od_a_a @ A @ B )
=> ( ( member4903802553882211088od_a_a @ X3 @ A )
=> ( member4903802553882211088od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_228_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_229_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_230_in__mono,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,X3: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( member1816616512716248880od_a_a @ X3 @ A )
=> ( member1816616512716248880od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_231_in__mono,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a,X3: produc1703568184450464039_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( ( member7983343339038529360_set_a @ X3 @ A )
=> ( member7983343339038529360_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_232_in__mono,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,X3: produc3498347346309940967od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( ( member6330455413206600464od_a_a @ X3 @ A )
=> ( member6330455413206600464od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_233_in__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( member1426531477525435216od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_234_total__on__empty,axiom,
! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% total_on_empty
thf(fact_235_total__on__empty,axiom,
! [R: set_Pr3617678592830649543od_a_a] : ( total_755287270405358187od_a_a @ bot_bo510284599550014259od_a_a @ R ) ).
% total_on_empty
thf(fact_236_total__on__empty,axiom,
! [R: set_Pr5845495582615845127_set_a] : ( total_on_set_a @ bot_bot_set_set_a @ R ) ).
% total_on_empty
thf(fact_237_total__on__empty,axiom,
! [R: set_Pr5931134355633282247od_a_a] : ( total_4423108994320943243od_a_a @ bot_bo777872063958040403od_a_a @ R ) ).
% total_on_empty
thf(fact_238_total__on__empty,axiom,
! [R: set_Pr8600417178894128327od_a_a] : ( total_3048927264068743339od_a_a @ bot_bo3357376287454694259od_a_a @ R ) ).
% total_on_empty
thf(fact_239_total__on__subset,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a] :
( ( total_3048927264068743339od_a_a @ A @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( total_3048927264068743339od_a_a @ B @ R ) ) ) ).
% total_on_subset
thf(fact_240_total__on__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a] :
( ( total_on_a @ A @ R )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( total_on_a @ B @ R ) ) ) ).
% total_on_subset
thf(fact_241_total__on__subset,axiom,
! [A: set_se5735800977113168103od_a_a,R: set_Pr5931134355633282247od_a_a,B: set_se5735800977113168103od_a_a] :
( ( total_4423108994320943243od_a_a @ A @ R )
=> ( ( ord_le1995061765932249223od_a_a @ B @ A )
=> ( total_4423108994320943243od_a_a @ B @ R ) ) ) ).
% total_on_subset
thf(fact_242_total__on__subset,axiom,
! [A: set_Pr5845495582615845127_set_a,R: set_Pr3538216614922818759_set_a,B: set_Pr5845495582615845127_set_a] :
( ( total_1305401759943531691_set_a @ A @ R )
=> ( ( ord_le8376522849517564071_set_a @ B @ A )
=> ( total_1305401759943531691_set_a @ B @ R ) ) ) ).
% total_on_subset
thf(fact_243_total__on__subset,axiom,
! [A: set_Pr8600417178894128327od_a_a,R: set_Pr3617678592830649543od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( total_755287270405358187od_a_a @ A @ R )
=> ( ( ord_le3469131294019144807od_a_a @ B @ A )
=> ( total_755287270405358187od_a_a @ B @ R ) ) ) ).
% total_on_subset
thf(fact_244_mono__Field,axiom,
! [R: set_Pr5931134355633282247od_a_a,S: set_Pr5931134355633282247od_a_a] :
( ( ord_le6923081475299677287od_a_a @ R @ S )
=> ( ord_le1995061765932249223od_a_a @ ( field_8083607959440978772od_a_a @ R ) @ ( field_8083607959440978772od_a_a @ S ) ) ) ).
% mono_Field
thf(fact_245_mono__Field,axiom,
! [R: set_Pr3538216614922818759_set_a,S: set_Pr3538216614922818759_set_a] :
( ( ord_le6153589704918764647_set_a @ R @ S )
=> ( ord_le8376522849517564071_set_a @ ( field_2314949223707911028_set_a @ R ) @ ( field_2314949223707911028_set_a @ S ) ) ) ).
% mono_Field
thf(fact_246_mono__Field,axiom,
! [R: set_Pr3617678592830649543od_a_a,S: set_Pr3617678592830649543od_a_a] :
( ( ord_le726539776577973351od_a_a @ R @ S )
=> ( ord_le3469131294019144807od_a_a @ ( field_1595709187305778996od_a_a @ R ) @ ( field_1595709187305778996od_a_a @ S ) ) ) ).
% mono_Field
thf(fact_247_mono__Field,axiom,
! [R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ R @ S )
=> ( ord_le3724670747650509150_set_a @ ( field_set_a @ R ) @ ( field_set_a @ S ) ) ) ).
% mono_Field
thf(fact_248_mono__Field,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( ord_le746702958409616551od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ ( field_1126092520709947252od_a_a @ S ) ) ) ).
% mono_Field
thf(fact_249_mono__Field,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_less_eq_set_a @ ( field_a @ R ) @ ( field_a @ S ) ) ) ).
% mono_Field
thf(fact_250_finite__Field,axiom,
! [R: set_Pr3617678592830649543od_a_a] :
( ( finite4202127094207432208od_a_a @ R )
=> ( finite20598220287489552od_a_a @ ( field_1595709187305778996od_a_a @ R ) ) ) ).
% finite_Field
thf(fact_251_finite__Field,axiom,
! [R: set_Pr5931134355633282247od_a_a] :
( ( finite296458618915675664od_a_a @ R )
=> ( finite8717734299975451184od_a_a @ ( field_8083607959440978772od_a_a @ R ) ) ) ).
% finite_Field
thf(fact_252_finite__Field,axiom,
! [R: set_Pr5845495582615845127_set_a] :
( ( finite5416632273030758992_set_a @ R )
=> ( finite_finite_set_a @ ( field_set_a @ R ) ) ) ).
% finite_Field
thf(fact_253_finite__Field,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ R )
=> ( finite6544458595007987280od_a_a @ ( field_1126092520709947252od_a_a @ R ) ) ) ).
% finite_Field
thf(fact_254_finite__Field,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( finite_finite_a @ ( field_a @ R ) ) ) ).
% finite_Field
thf(fact_255_Domain__mono,axiom,
! [R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ R @ S )
=> ( ord_le3724670747650509150_set_a @ ( domain_set_a_set_a @ R ) @ ( domain_set_a_set_a @ S ) ) ) ).
% Domain_mono
thf(fact_256_Domain__mono,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( ord_le746702958409616551od_a_a @ ( domain990338261037515724od_a_a @ R ) @ ( domain990338261037515724od_a_a @ S ) ) ) ).
% Domain_mono
thf(fact_257_Domain__mono,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_less_eq_set_a @ ( domain_a_a @ R ) @ ( domain_a_a @ S ) ) ) ).
% Domain_mono
thf(fact_258_Range__mono,axiom,
! [R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ R @ S )
=> ( ord_le3724670747650509150_set_a @ ( range_set_a_set_a @ R ) @ ( range_set_a_set_a @ S ) ) ) ).
% Range_mono
thf(fact_259_Range__mono,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( ord_le746702958409616551od_a_a @ ( range_2510312826226007605od_a_a @ R ) @ ( range_2510312826226007605od_a_a @ S ) ) ) ).
% Range_mono
thf(fact_260_Range__mono,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_less_eq_set_a @ ( range_a_a @ R ) @ ( range_a_a @ S ) ) ) ).
% Range_mono
thf(fact_261_finite__has__minimal,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( A != bot_bo777872063958040403od_a_a )
=> ? [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_262_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_263_finite__has__minimal,axiom,
! [A: set_se5775600271191333063od_a_a] :
( ( finite225493028838687248od_a_a @ A )
=> ( ( A != bot_bo5729467828521840435od_a_a )
=> ? [X2: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ X2 @ A )
& ! [Xa: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ Xa @ A )
=> ( ( ord_le1995061765932249223od_a_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_264_finite__has__minimal,axiom,
! [A: set_se6363330096766809319_set_a] :
( ( finite7892596592296074800_set_a @ A )
=> ( ( A != bot_bo5258460409194027859_set_a )
=> ? [X2: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ X2 @ A )
& ! [Xa: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ Xa @ A )
=> ( ( ord_le8376522849517564071_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_265_finite__has__minimal,axiom,
! [A: set_se7936120678642572967od_a_a] :
( ( finite5992499909718138864od_a_a @ A )
=> ( ( A != bot_bo2005289091711683859od_a_a )
=> ? [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ A )
& ! [Xa: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ Xa @ A )
=> ( ( ord_le3469131294019144807od_a_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_266_finite__has__maximal,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( A != bot_bo777872063958040403od_a_a )
=> ? [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_267_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_268_finite__has__maximal,axiom,
! [A: set_se5775600271191333063od_a_a] :
( ( finite225493028838687248od_a_a @ A )
=> ( ( A != bot_bo5729467828521840435od_a_a )
=> ? [X2: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ X2 @ A )
& ! [Xa: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ Xa @ A )
=> ( ( ord_le1995061765932249223od_a_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_269_finite__has__maximal,axiom,
! [A: set_se6363330096766809319_set_a] :
( ( finite7892596592296074800_set_a @ A )
=> ( ( A != bot_bo5258460409194027859_set_a )
=> ? [X2: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ X2 @ A )
& ! [Xa: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ Xa @ A )
=> ( ( ord_le8376522849517564071_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_270_finite__has__maximal,axiom,
! [A: set_se7936120678642572967od_a_a] :
( ( finite5992499909718138864od_a_a @ A )
=> ( ( A != bot_bo2005289091711683859od_a_a )
=> ? [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ A )
& ! [Xa: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ Xa @ A )
=> ( ( ord_le3469131294019144807od_a_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_271_bot__apply,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X: product_prod_a_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_272_dual__order_Orefl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_273_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_274_dual__order_Orefl,axiom,
! [A2: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_275_dual__order_Orefl,axiom,
! [A2: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_276_dual__order_Orefl,axiom,
! [A2: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_277_order__refl,axiom,
! [X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_278_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_279_order__refl,axiom,
! [X3: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_280_order__refl,axiom,
! [X3: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_281_order__refl,axiom,
! [X3: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_282_less__by__empty,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( A = bot_bo5799363139946352499_set_a )
=> ( ord_le8376522849517564071_set_a @ A @ B ) ) ).
% less_by_empty
thf(fact_283_less__by__empty,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( A = bot_bo510284599550014259od_a_a )
=> ( ord_le3469131294019144807od_a_a @ A @ B ) ) ).
% less_by_empty
thf(fact_284_less__by__empty,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = bot_bo3357376287454694259od_a_a )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% less_by_empty
thf(fact_285_rev__finite__subset,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_286_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_287_rev__finite__subset,axiom,
! [B: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ B )
=> ( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( finite8717734299975451184od_a_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_288_rev__finite__subset,axiom,
! [B: set_Pr5845495582615845127_set_a,A: set_Pr5845495582615845127_set_a] :
( ( finite5416632273030758992_set_a @ B )
=> ( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( finite5416632273030758992_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_289_rev__finite__subset,axiom,
! [B: set_Pr8600417178894128327od_a_a,A: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ B )
=> ( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( finite20598220287489552od_a_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_290_rev__finite__subset,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( finite6544458595007987280od_a_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_291_infinite__super,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_292_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_293_infinite__super,axiom,
! [S2: set_se5735800977113168103od_a_a,T2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ S2 @ T2 )
=> ( ~ ( finite8717734299975451184od_a_a @ S2 )
=> ~ ( finite8717734299975451184od_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_294_infinite__super,axiom,
! [S2: set_Pr5845495582615845127_set_a,T2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ S2 @ T2 )
=> ( ~ ( finite5416632273030758992_set_a @ S2 )
=> ~ ( finite5416632273030758992_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_295_infinite__super,axiom,
! [S2: set_Pr8600417178894128327od_a_a,T2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ S2 @ T2 )
=> ( ~ ( finite20598220287489552od_a_a @ S2 )
=> ~ ( finite20598220287489552od_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_296_infinite__super,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ S2 @ T2 )
=> ( ~ ( finite6544458595007987280od_a_a @ S2 )
=> ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_297_finite__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_298_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_299_finite__subset,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( finite8717734299975451184od_a_a @ B )
=> ( finite8717734299975451184od_a_a @ A ) ) ) ).
% finite_subset
thf(fact_300_finite__subset,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( ( finite5416632273030758992_set_a @ B )
=> ( finite5416632273030758992_set_a @ A ) ) ) ).
% finite_subset
thf(fact_301_finite__subset,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( ( finite20598220287489552od_a_a @ B )
=> ( finite20598220287489552od_a_a @ A ) ) ) ).
% finite_subset
thf(fact_302_finite__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( finite6544458595007987280od_a_a @ B )
=> ( finite6544458595007987280od_a_a @ A ) ) ) ).
% finite_subset
thf(fact_303_subset__emptyI,axiom,
! [A: set_Pr5931134355633282247od_a_a] :
( ! [X2: produc7983190928304305895od_a_a] :
~ ( member4903802553882211088od_a_a @ X2 @ A )
=> ( ord_le6923081475299677287od_a_a @ A @ bot_bo8313862440737889075od_a_a ) ) ).
% subset_emptyI
thf(fact_304_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_305_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_306_subset__emptyI,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ! [X2: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ X2 @ A )
=> ( ord_le1995061765932249223od_a_a @ A @ bot_bo777872063958040403od_a_a ) ) ).
% subset_emptyI
thf(fact_307_subset__emptyI,axiom,
! [A: set_Pr5845495582615845127_set_a] :
( ! [X2: produc1703568184450464039_set_a] :
~ ( member7983343339038529360_set_a @ X2 @ A )
=> ( ord_le8376522849517564071_set_a @ A @ bot_bo5799363139946352499_set_a ) ) ).
% subset_emptyI
thf(fact_308_subset__emptyI,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ! [X2: produc3498347346309940967od_a_a] :
~ ( member6330455413206600464od_a_a @ X2 @ A )
=> ( ord_le3469131294019144807od_a_a @ A @ bot_bo510284599550014259od_a_a ) ) ).
% subset_emptyI
thf(fact_309_subset__emptyI,axiom,
! [A: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_310_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_311_bot__set__def,axiom,
( bot_bo510284599550014259od_a_a
= ( collec10116633892588882od_a_a @ bot_bo7912853196105230186_a_a_o ) ) ).
% bot_set_def
thf(fact_312_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_313_bot__set__def,axiom,
( bot_bo777872063958040403od_a_a
= ( collec1673347964119250290od_a_a @ bot_bo3434591380065676106_a_a_o ) ) ).
% bot_set_def
thf(fact_314_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_315_bot__empty__eq,axiom,
( bot_bo428180006308831082_a_a_o
= ( ^ [X: produc7983190928304305895od_a_a] : ( member4903802553882211088od_a_a @ X @ bot_bo8313862440737889075od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_316_bot__empty__eq,axiom,
( bot_bo4307338768736490282et_a_o
= ( ^ [X: produc1703568184450464039_set_a] : ( member7983343339038529360_set_a @ X @ bot_bo5799363139946352499_set_a ) ) ) ).
% bot_empty_eq
thf(fact_317_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_318_bot__empty__eq,axiom,
( bot_bo7912853196105230186_a_a_o
= ( ^ [X: produc3498347346309940967od_a_a] : ( member6330455413206600464od_a_a @ X @ bot_bo510284599550014259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_319_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X: set_a] : ( member_set_a @ X @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_320_bot__empty__eq,axiom,
( bot_bo3434591380065676106_a_a_o
= ( ^ [X: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X @ bot_bo777872063958040403od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_321_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_322_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y2 = Z ) )
= ( ^ [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y3 )
& ( ord_le746702958409616551od_a_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_323_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_324_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] : ( Y2 = Z ) )
= ( ^ [X: set_se5735800977113168103od_a_a,Y3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X @ Y3 )
& ( ord_le1995061765932249223od_a_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_325_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Pr5845495582615845127_set_a,Z: set_Pr5845495582615845127_set_a] : ( Y2 = Z ) )
= ( ^ [X: set_Pr5845495582615845127_set_a,Y3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X @ Y3 )
& ( ord_le8376522849517564071_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_326_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Pr8600417178894128327od_a_a,Z: set_Pr8600417178894128327od_a_a] : ( Y2 = Z ) )
= ( ^ [X: set_Pr8600417178894128327od_a_a,Y3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ X @ Y3 )
& ( ord_le3469131294019144807od_a_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_327_ord__eq__le__trans,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2 = B3 )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_328_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_329_ord__eq__le__trans,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( A2 = B3 )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ C )
=> ( ord_le1995061765932249223od_a_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_330_ord__eq__le__trans,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( A2 = B3 )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ C )
=> ( ord_le8376522849517564071_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_331_ord__eq__le__trans,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a] :
( ( A2 = B3 )
=> ( ( ord_le3469131294019144807od_a_a @ B3 @ C )
=> ( ord_le3469131294019144807od_a_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_332_ord__le__eq__trans,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_333_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_334_ord__le__eq__trans,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le1995061765932249223od_a_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_335_ord__le__eq__trans,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le8376522849517564071_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_336_ord__le__eq__trans,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le3469131294019144807od_a_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_337_order__antisym,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y4 )
=> ( ( ord_le746702958409616551od_a_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_338_order__antisym,axiom,
! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_339_order__antisym,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X3 @ Y4 )
=> ( ( ord_le1995061765932249223od_a_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_340_order__antisym,axiom,
! [X3: set_Pr5845495582615845127_set_a,Y4: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X3 @ Y4 )
=> ( ( ord_le8376522849517564071_set_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_341_order__antisym,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ X3 @ Y4 )
=> ( ( ord_le3469131294019144807od_a_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_342_order_Otrans,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_343_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_344_order_Otrans,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ C )
=> ( ord_le1995061765932249223od_a_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_345_order_Otrans,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ B3 )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ C )
=> ( ord_le8376522849517564071_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_346_order_Otrans,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ B3 )
=> ( ( ord_le3469131294019144807od_a_a @ B3 @ C )
=> ( ord_le3469131294019144807od_a_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_347_order__trans,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y4 )
=> ( ( ord_le746702958409616551od_a_a @ Y4 @ Z2 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_348_order__trans,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_349_order__trans,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a,Z2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X3 @ Y4 )
=> ( ( ord_le1995061765932249223od_a_a @ Y4 @ Z2 )
=> ( ord_le1995061765932249223od_a_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_350_order__trans,axiom,
! [X3: set_Pr5845495582615845127_set_a,Y4: set_Pr5845495582615845127_set_a,Z2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X3 @ Y4 )
=> ( ( ord_le8376522849517564071_set_a @ Y4 @ Z2 )
=> ( ord_le8376522849517564071_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_351_order__trans,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ X3 @ Y4 )
=> ( ( ord_le3469131294019144807od_a_a @ Y4 @ Z2 )
=> ( ord_le3469131294019144807od_a_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_352_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ A4 )
& ( ord_le746702958409616551od_a_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_353_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_354_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_se5735800977113168103od_a_a,B4: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B4 @ A4 )
& ( ord_le1995061765932249223od_a_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_355_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr5845495582615845127_set_a,Z: set_Pr5845495582615845127_set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_Pr5845495582615845127_set_a,B4: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ B4 @ A4 )
& ( ord_le8376522849517564071_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_356_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr8600417178894128327od_a_a,Z: set_Pr8600417178894128327od_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_Pr8600417178894128327od_a_a,B4: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ B4 @ A4 )
& ( ord_le3469131294019144807od_a_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_357_dual__order_Oantisym,axiom,
! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B3 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_358_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_359_dual__order_Oantisym,axiom,
! [B3: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B3 @ A2 )
=> ( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_360_dual__order_Oantisym,axiom,
! [B3: set_Pr5845495582615845127_set_a,A2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ B3 @ A2 )
=> ( ( ord_le8376522849517564071_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_361_dual__order_Oantisym,axiom,
! [B3: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ B3 @ A2 )
=> ( ( ord_le3469131294019144807od_a_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_362_dual__order_Otrans,axiom,
! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B3 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C @ B3 )
=> ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_363_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_364_dual__order_Otrans,axiom,
! [B3: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B3 @ A2 )
=> ( ( ord_le1995061765932249223od_a_a @ C @ B3 )
=> ( ord_le1995061765932249223od_a_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_365_dual__order_Otrans,axiom,
! [B3: set_Pr5845495582615845127_set_a,A2: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ B3 @ A2 )
=> ( ( ord_le8376522849517564071_set_a @ C @ B3 )
=> ( ord_le8376522849517564071_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_366_dual__order_Otrans,axiom,
! [B3: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ B3 @ A2 )
=> ( ( ord_le3469131294019144807od_a_a @ C @ B3 )
=> ( ord_le3469131294019144807od_a_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_367_antisym,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_368_antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_369_antisym,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_370_antisym,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ B3 )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_371_antisym,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ B3 )
=> ( ( ord_le3469131294019144807od_a_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_372_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B4 )
& ( ord_le746702958409616551od_a_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_373_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_374_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_se5735800977113168103od_a_a,B4: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A4 @ B4 )
& ( ord_le1995061765932249223od_a_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_375_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Pr5845495582615845127_set_a,Z: set_Pr5845495582615845127_set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_Pr5845495582615845127_set_a,B4: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A4 @ B4 )
& ( ord_le8376522849517564071_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_376_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Pr8600417178894128327od_a_a,Z: set_Pr8600417178894128327od_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_Pr8600417178894128327od_a_a,B4: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A4 @ B4 )
& ( ord_le3469131294019144807od_a_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_377_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_378_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: set_a > set_Product_prod_a_a,B3: set_a,C: set_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_379_order__subst1,axiom,
! [A2: set_a,F: set_Product_prod_a_a > set_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_380_order__subst1,axiom,
! [A2: set_a,F: set_se5735800977113168103od_a_a > set_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_381_order__subst1,axiom,
! [A2: set_se5735800977113168103od_a_a,F: set_a > set_se5735800977113168103od_a_a,B3: set_a,C: set_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_382_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_383_order__subst1,axiom,
! [A2: set_a,F: set_Pr5845495582615845127_set_a > set_a,B3: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ C )
=> ( ! [X2: set_Pr5845495582615845127_set_a,Y: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_384_order__subst1,axiom,
! [A2: set_Pr5845495582615845127_set_a,F: set_a > set_Pr5845495582615845127_set_a,B3: set_a,C: set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le8376522849517564071_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le8376522849517564071_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_385_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: set_se5735800977113168103od_a_a > set_Product_prod_a_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_386_order__subst1,axiom,
! [A2: set_se5735800977113168103od_a_a,F: set_Product_prod_a_a > set_se5735800977113168103od_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_387_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_388_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > set_a,C: set_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_389_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_390_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_le1995061765932249223od_a_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_391_order__subst2,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,F: set_se5735800977113168103od_a_a > set_a,C: set_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_392_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_393_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_le8376522849517564071_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le8376522849517564071_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le8376522849517564071_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_394_order__subst2,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,F: set_Pr5845495582615845127_set_a > set_a,C: set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_Pr5845495582615845127_set_a,Y: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_395_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ord_le1995061765932249223od_a_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_396_order__subst2,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,F: set_se5735800977113168103od_a_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_397_order__eq__refl,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( X3 = Y4 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_398_order__eq__refl,axiom,
! [X3: set_a,Y4: set_a] :
( ( X3 = Y4 )
=> ( ord_less_eq_set_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_399_order__eq__refl,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] :
( ( X3 = Y4 )
=> ( ord_le1995061765932249223od_a_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_400_order__eq__refl,axiom,
! [X3: set_Pr5845495582615845127_set_a,Y4: set_Pr5845495582615845127_set_a] :
( ( X3 = Y4 )
=> ( ord_le8376522849517564071_set_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_401_order__eq__refl,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] :
( ( X3 = Y4 )
=> ( ord_le3469131294019144807od_a_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_402_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_403_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_Product_prod_a_a > set_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_404_ord__eq__le__subst,axiom,
! [A2: set_Product_prod_a_a,F: set_a > set_Product_prod_a_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_405_ord__eq__le__subst,axiom,
! [A2: set_se5735800977113168103od_a_a,F: set_a > set_se5735800977113168103od_a_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_406_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_se5735800977113168103od_a_a > set_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_407_ord__eq__le__subst,axiom,
! [A2: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_408_ord__eq__le__subst,axiom,
! [A2: set_Pr5845495582615845127_set_a,F: set_a > set_Pr5845495582615845127_set_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le8376522849517564071_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le8376522849517564071_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_409_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_Pr5845495582615845127_set_a > set_a,B3: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ C )
=> ( ! [X2: set_Pr5845495582615845127_set_a,Y: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_410_ord__eq__le__subst,axiom,
! [A2: set_se5735800977113168103od_a_a,F: set_Product_prod_a_a > set_se5735800977113168103od_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_411_ord__eq__le__subst,axiom,
! [A2: set_Product_prod_a_a,F: set_se5735800977113168103od_a_a > set_Product_prod_a_a,B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_412_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_413_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > set_a,C: set_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_414_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_415_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_416_ord__le__eq__subst,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,F: set_se5735800977113168103od_a_a > set_a,C: set_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_417_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_418_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le8376522849517564071_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le8376522849517564071_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_419_ord__le__eq__subst,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,F: set_Pr5845495582615845127_set_a > set_a,C: set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_Pr5845495582615845127_set_a,Y: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_420_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le1995061765932249223od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_421_ord__le__eq__subst,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,F: set_se5735800977113168103od_a_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_422_order__antisym__conv,axiom,
! [Y4: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y4 @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_423_order__antisym__conv,axiom,
! [Y4: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_424_order__antisym__conv,axiom,
! [Y4: set_se5735800977113168103od_a_a,X3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ Y4 @ X3 )
=> ( ( ord_le1995061765932249223od_a_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_425_order__antisym__conv,axiom,
! [Y4: set_Pr5845495582615845127_set_a,X3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ Y4 @ X3 )
=> ( ( ord_le8376522849517564071_set_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_426_order__antisym__conv,axiom,
! [Y4: set_Pr8600417178894128327od_a_a,X3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ Y4 @ X3 )
=> ( ( ord_le3469131294019144807od_a_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_427_bot__fun__def,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X: product_prod_a_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_428_bot_Oextremum,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% bot.extremum
thf(fact_429_bot_Oextremum,axiom,
! [A2: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ bot_bo4160289986317612842_a_a_o @ A2 ) ).
% bot.extremum
thf(fact_430_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_431_bot_Oextremum,axiom,
! [A2: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ bot_bo777872063958040403od_a_a @ A2 ) ).
% bot.extremum
thf(fact_432_bot_Oextremum,axiom,
! [A2: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ bot_bo5799363139946352499_set_a @ A2 ) ).
% bot.extremum
thf(fact_433_bot_Oextremum,axiom,
! [A2: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ bot_bo510284599550014259od_a_a @ A2 ) ).
% bot.extremum
thf(fact_434_bot_Oextremum,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% bot.extremum
thf(fact_435_bot_Oextremum__unique,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_436_bot_Oextremum__unique,axiom,
! [A2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ bot_bo4160289986317612842_a_a_o )
= ( A2 = bot_bo4160289986317612842_a_a_o ) ) ).
% bot.extremum_unique
thf(fact_437_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_438_bot_Oextremum__unique,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ bot_bo777872063958040403od_a_a )
= ( A2 = bot_bo777872063958040403od_a_a ) ) ).
% bot.extremum_unique
thf(fact_439_bot_Oextremum__unique,axiom,
! [A2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ bot_bo5799363139946352499_set_a )
= ( A2 = bot_bo5799363139946352499_set_a ) ) ).
% bot.extremum_unique
thf(fact_440_bot_Oextremum__unique,axiom,
! [A2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ bot_bo510284599550014259od_a_a )
= ( A2 = bot_bo510284599550014259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_441_bot_Oextremum__unique,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_442_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
=> ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_443_bot_Oextremum__uniqueI,axiom,
! [A2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ bot_bo4160289986317612842_a_a_o )
=> ( A2 = bot_bo4160289986317612842_a_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_444_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_445_bot_Oextremum__uniqueI,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ bot_bo777872063958040403od_a_a )
=> ( A2 = bot_bo777872063958040403od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_446_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ bot_bo5799363139946352499_set_a )
=> ( A2 = bot_bo5799363139946352499_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_447_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ bot_bo510284599550014259od_a_a )
=> ( A2 = bot_bo510284599550014259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_448_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_449_finite__has__maximal2,axiom,
! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( member1816616512716248880od_a_a @ A2 @ A )
=> ? [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
& ( ord_le746702958409616551od_a_a @ A2 @ X2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_450_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_451_finite__has__maximal2,axiom,
! [A: set_se5775600271191333063od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( finite225493028838687248od_a_a @ A )
=> ( ( member2792204094846457616od_a_a @ A2 @ A )
=> ? [X2: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ X2 @ A )
& ( ord_le1995061765932249223od_a_a @ A2 @ X2 )
& ! [Xa: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ Xa @ A )
=> ( ( ord_le1995061765932249223od_a_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_452_finite__has__maximal2,axiom,
! [A: set_se6363330096766809319_set_a,A2: set_Pr5845495582615845127_set_a] :
( ( finite7892596592296074800_set_a @ A )
=> ( ( member7393593894965162800_set_a @ A2 @ A )
=> ? [X2: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ X2 @ A )
& ( ord_le8376522849517564071_set_a @ A2 @ X2 )
& ! [Xa: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ Xa @ A )
=> ( ( ord_le8376522849517564071_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_453_finite__has__maximal2,axiom,
! [A: set_se7936120678642572967od_a_a,A2: set_Pr8600417178894128327od_a_a] :
( ( finite5992499909718138864od_a_a @ A )
=> ( ( member92994164477555952od_a_a @ A2 @ A )
=> ? [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ A )
& ( ord_le3469131294019144807od_a_a @ A2 @ X2 )
& ! [Xa: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ Xa @ A )
=> ( ( ord_le3469131294019144807od_a_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_454_finite__has__minimal2,axiom,
! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( member1816616512716248880od_a_a @ A2 @ A )
=> ? [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
& ( ord_le746702958409616551od_a_a @ X2 @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_455_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_456_finite__has__minimal2,axiom,
! [A: set_se5775600271191333063od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( finite225493028838687248od_a_a @ A )
=> ( ( member2792204094846457616od_a_a @ A2 @ A )
=> ? [X2: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ X2 @ A )
& ( ord_le1995061765932249223od_a_a @ X2 @ A2 )
& ! [Xa: set_se5735800977113168103od_a_a] :
( ( member2792204094846457616od_a_a @ Xa @ A )
=> ( ( ord_le1995061765932249223od_a_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_457_finite__has__minimal2,axiom,
! [A: set_se6363330096766809319_set_a,A2: set_Pr5845495582615845127_set_a] :
( ( finite7892596592296074800_set_a @ A )
=> ( ( member7393593894965162800_set_a @ A2 @ A )
=> ? [X2: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ X2 @ A )
& ( ord_le8376522849517564071_set_a @ X2 @ A2 )
& ! [Xa: set_Pr5845495582615845127_set_a] :
( ( member7393593894965162800_set_a @ Xa @ A )
=> ( ( ord_le8376522849517564071_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_458_finite__has__minimal2,axiom,
! [A: set_se7936120678642572967od_a_a,A2: set_Pr8600417178894128327od_a_a] :
( ( finite5992499909718138864od_a_a @ A )
=> ( ( member92994164477555952od_a_a @ A2 @ A )
=> ? [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ A )
& ( ord_le3469131294019144807od_a_a @ X2 @ A2 )
& ! [Xa: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ Xa @ A )
=> ( ( ord_le3469131294019144807od_a_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_459_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_460_finite_OemptyI,axiom,
finite20598220287489552od_a_a @ bot_bo510284599550014259od_a_a ).
% finite.emptyI
thf(fact_461_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_462_finite_OemptyI,axiom,
finite8717734299975451184od_a_a @ bot_bo777872063958040403od_a_a ).
% finite.emptyI
thf(fact_463_finite_OemptyI,axiom,
finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).
% finite.emptyI
thf(fact_464_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_465_infinite__imp__nonempty,axiom,
! [S2: set_Pr8600417178894128327od_a_a] :
( ~ ( finite20598220287489552od_a_a @ S2 )
=> ( S2 != bot_bo510284599550014259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_466_infinite__imp__nonempty,axiom,
! [S2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ( S2 != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_467_infinite__imp__nonempty,axiom,
! [S2: set_se5735800977113168103od_a_a] :
( ~ ( finite8717734299975451184od_a_a @ S2 )
=> ( S2 != bot_bo777872063958040403od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_468_infinite__imp__nonempty,axiom,
! [S2: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S2 )
=> ( S2 != bot_bo3357376287454694259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_469_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_470_Set_Ois__empty__def,axiom,
( is_emp2923524183434141113od_a_a
= ( ^ [A3: set_Pr8600417178894128327od_a_a] : ( A3 = bot_bo510284599550014259od_a_a ) ) ) ).
% Set.is_empty_def
thf(fact_471_Set_Ois__empty__def,axiom,
( is_empty_set_a
= ( ^ [A3: set_set_a] : ( A3 = bot_bot_set_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_472_Set_Ois__empty__def,axiom,
( is_emp3465682670623713241od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a] : ( A3 = bot_bo777872063958040403od_a_a ) ) ) ).
% Set.is_empty_def
thf(fact_473_Set_Ois__empty__def,axiom,
( is_emp2937470224744679417od_a_a
= ( ^ [A3: set_Product_prod_a_a] : ( A3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% Set.is_empty_def
thf(fact_474_Id__on__empty,axiom,
( ( id_on_5942634073268129412od_a_a @ bot_bo510284599550014259od_a_a )
= bot_bo514369784327480115od_a_a ) ).
% Id_on_empty
thf(fact_475_Id__on__empty,axiom,
( ( id_on_set_a @ bot_bot_set_set_a )
= bot_bo5799363139946352499_set_a ) ).
% Id_on_empty
thf(fact_476_Id__on__empty,axiom,
( ( id_on_8064794247841821348od_a_a @ bot_bo777872063958040403od_a_a )
= bot_bo8313862440737889075od_a_a ) ).
% Id_on_empty
thf(fact_477_Id__on__empty,axiom,
( ( id_on_a @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% Id_on_empty
thf(fact_478_Id__on__empty,axiom,
( ( id_on_7193410965977890500od_a_a @ bot_bo3357376287454694259od_a_a )
= bot_bo510284599550014259od_a_a ) ).
% Id_on_empty
thf(fact_479_GreatestI2__order,axiom,
! [P2: set_Product_prod_a_a > $o,X3: set_Product_prod_a_a,Q: set_Product_prod_a_a > $o] :
( ( P2 @ X3 )
=> ( ! [Y: set_Product_prod_a_a] :
( ( P2 @ Y )
=> ( ord_le746702958409616551od_a_a @ Y @ X3 ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( P2 @ X2 )
=> ( ! [Y5: set_Product_prod_a_a] :
( ( P2 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_6315983676290676192od_a_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_480_GreatestI2__order,axiom,
! [P2: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P2 @ X3 )
=> ( ! [Y: set_a] :
( ( P2 @ Y )
=> ( ord_less_eq_set_a @ Y @ X3 ) )
=> ( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( ! [Y5: set_a] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_481_GreatestI2__order,axiom,
! [P2: set_se5735800977113168103od_a_a > $o,X3: set_se5735800977113168103od_a_a,Q: set_se5735800977113168103od_a_a > $o] :
( ( P2 @ X3 )
=> ( ! [Y: set_se5735800977113168103od_a_a] :
( ( P2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ Y @ X3 ) )
=> ( ! [X2: set_se5735800977113168103od_a_a] :
( ( P2 @ X2 )
=> ( ! [Y5: set_se5735800977113168103od_a_a] :
( ( P2 @ Y5 )
=> ( ord_le1995061765932249223od_a_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_6951509314964895168od_a_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_482_GreatestI2__order,axiom,
! [P2: set_Pr5845495582615845127_set_a > $o,X3: set_Pr5845495582615845127_set_a,Q: set_Pr5845495582615845127_set_a > $o] :
( ( P2 @ X3 )
=> ( ! [Y: set_Pr5845495582615845127_set_a] :
( ( P2 @ Y )
=> ( ord_le8376522849517564071_set_a @ Y @ X3 ) )
=> ( ! [X2: set_Pr5845495582615845127_set_a] :
( ( P2 @ X2 )
=> ( ! [Y5: set_Pr5845495582615845127_set_a] :
( ( P2 @ Y5 )
=> ( ord_le8376522849517564071_set_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_7864067821098553824_set_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_483_GreatestI2__order,axiom,
! [P2: set_Pr8600417178894128327od_a_a > $o,X3: set_Pr8600417178894128327od_a_a,Q: set_Pr8600417178894128327od_a_a > $o] :
( ( P2 @ X3 )
=> ( ! [Y: set_Pr8600417178894128327od_a_a] :
( ( P2 @ Y )
=> ( ord_le3469131294019144807od_a_a @ Y @ X3 ) )
=> ( ! [X2: set_Pr8600417178894128327od_a_a] :
( ( P2 @ X2 )
=> ( ! [Y5: set_Pr8600417178894128327od_a_a] :
( ( P2 @ Y5 )
=> ( ord_le3469131294019144807od_a_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_6423662517715940256od_a_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_484_Greatest__equality,axiom,
! [P2: set_Product_prod_a_a > $o,X3: set_Product_prod_a_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_Product_prod_a_a] :
( ( P2 @ Y )
=> ( ord_le746702958409616551od_a_a @ Y @ X3 ) )
=> ( ( order_6315983676290676192od_a_a @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_485_Greatest__equality,axiom,
! [P2: set_a > $o,X3: set_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_a] :
( ( P2 @ Y )
=> ( ord_less_eq_set_a @ Y @ X3 ) )
=> ( ( order_Greatest_set_a @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_486_Greatest__equality,axiom,
! [P2: set_se5735800977113168103od_a_a > $o,X3: set_se5735800977113168103od_a_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_se5735800977113168103od_a_a] :
( ( P2 @ Y )
=> ( ord_le1995061765932249223od_a_a @ Y @ X3 ) )
=> ( ( order_6951509314964895168od_a_a @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_487_Greatest__equality,axiom,
! [P2: set_Pr5845495582615845127_set_a > $o,X3: set_Pr5845495582615845127_set_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_Pr5845495582615845127_set_a] :
( ( P2 @ Y )
=> ( ord_le8376522849517564071_set_a @ Y @ X3 ) )
=> ( ( order_7864067821098553824_set_a @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_488_Greatest__equality,axiom,
! [P2: set_Pr8600417178894128327od_a_a > $o,X3: set_Pr8600417178894128327od_a_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_Pr8600417178894128327od_a_a] :
( ( P2 @ Y )
=> ( ord_le3469131294019144807od_a_a @ Y @ X3 ) )
=> ( ( order_6423662517715940256od_a_a @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_489_Field__def,axiom,
( field_1126092520709947252od_a_a
= ( ^ [R2: set_Pr8600417178894128327od_a_a] : ( sup_su3048258781599657691od_a_a @ ( domain990338261037515724od_a_a @ R2 ) @ ( range_2510312826226007605od_a_a @ R2 ) ) ) ) ).
% Field_def
thf(fact_490_Field__def,axiom,
( field_1595709187305778996od_a_a
= ( ^ [R2: set_Pr3617678592830649543od_a_a] : ( sup_su8193676293155882651od_a_a @ ( domain6325608428002865100od_a_a @ R2 ) @ ( range_5058862847143773749od_a_a @ R2 ) ) ) ) ).
% Field_def
thf(fact_491_Field__def,axiom,
( field_a
= ( ^ [R2: set_Product_prod_a_a] : ( sup_sup_set_a @ ( domain_a_a @ R2 ) @ ( range_a_a @ R2 ) ) ) ) ).
% Field_def
thf(fact_492_Collect__empty__eq__bot,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( P2 = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_493_Collect__empty__eq__bot,axiom,
! [P2: produc3498347346309940967od_a_a > $o] :
( ( ( collec10116633892588882od_a_a @ P2 )
= bot_bo510284599550014259od_a_a )
= ( P2 = bot_bo7912853196105230186_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_494_Collect__empty__eq__bot,axiom,
! [P2: set_a > $o] :
( ( ( collect_set_a @ P2 )
= bot_bot_set_set_a )
= ( P2 = bot_bot_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_495_Collect__empty__eq__bot,axiom,
! [P2: set_Product_prod_a_a > $o] :
( ( ( collec1673347964119250290od_a_a @ P2 )
= bot_bo777872063958040403od_a_a )
= ( P2 = bot_bo3434591380065676106_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_496_Collect__empty__eq__bot,axiom,
! [P2: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P2 )
= bot_bo3357376287454694259od_a_a )
= ( P2 = bot_bo4160289986317612842_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_497_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_498_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_499_verit__comp__simplify1_I2_J,axiom,
! [A2: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_500_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_501_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_502_UnCI,axiom,
! [C: produc7983190928304305895od_a_a,B: set_Pr5931134355633282247od_a_a,A: set_Pr5931134355633282247od_a_a] :
( ( ~ ( member4903802553882211088od_a_a @ C @ B )
=> ( member4903802553882211088od_a_a @ C @ A ) )
=> ( member4903802553882211088od_a_a @ C @ ( sup_su6534410975031996059od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_503_UnCI,axiom,
! [C: produc1703568184450464039_set_a,B: set_Pr5845495582615845127_set_a,A: set_Pr5845495582615845127_set_a] :
( ( ~ ( member7983343339038529360_set_a @ C @ B )
=> ( member7983343339038529360_set_a @ C @ A ) )
=> ( member7983343339038529360_set_a @ C @ ( sup_su5624169484884619995_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_504_UnCI,axiom,
! [C: set_Product_prod_a_a,B: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a] :
( ( ~ ( member1816616512716248880od_a_a @ C @ B )
=> ( member1816616512716248880od_a_a @ C @ A ) )
=> ( member1816616512716248880od_a_a @ C @ ( sup_su6839823885543768763od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_505_UnCI,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_506_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_507_UnCI,axiom,
! [C: produc3498347346309940967od_a_a,B: set_Pr8600417178894128327od_a_a,A: set_Pr8600417178894128327od_a_a] :
( ( ~ ( member6330455413206600464od_a_a @ C @ B )
=> ( member6330455413206600464od_a_a @ C @ A ) )
=> ( member6330455413206600464od_a_a @ C @ ( sup_su8193676293155882651od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_508_UnCI,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ A ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_509_Un__iff,axiom,
! [C: produc7983190928304305895od_a_a,A: set_Pr5931134355633282247od_a_a,B: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ C @ ( sup_su6534410975031996059od_a_a @ A @ B ) )
= ( ( member4903802553882211088od_a_a @ C @ A )
| ( member4903802553882211088od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_510_Un__iff,axiom,
! [C: produc1703568184450464039_set_a,A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ C @ ( sup_su5624169484884619995_set_a @ A @ B ) )
= ( ( member7983343339038529360_set_a @ C @ A )
| ( member7983343339038529360_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_511_Un__iff,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( sup_su6839823885543768763od_a_a @ A @ B ) )
= ( ( member1816616512716248880od_a_a @ C @ A )
| ( member1816616512716248880od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_512_Un__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_513_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_514_Un__iff,axiom,
! [C: produc3498347346309940967od_a_a,A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ C @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
= ( ( member6330455413206600464od_a_a @ C @ A )
| ( member6330455413206600464od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_515_Un__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
| ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_516_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_517_Un__empty,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( ( sup_su8193676293155882651od_a_a @ A @ B )
= bot_bo510284599550014259od_a_a )
= ( ( A = bot_bo510284599550014259od_a_a )
& ( B = bot_bo510284599550014259od_a_a ) ) ) ).
% Un_empty
thf(fact_518_Un__empty,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_519_Un__empty,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ( sup_su6839823885543768763od_a_a @ A @ B )
= bot_bo777872063958040403od_a_a )
= ( ( A = bot_bo777872063958040403od_a_a )
& ( B = bot_bo777872063958040403od_a_a ) ) ) ).
% Un_empty
thf(fact_520_Un__empty,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% Un_empty
thf(fact_521_finite__Un,axiom,
! [F2: set_se5735800977113168103od_a_a,G: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ ( sup_su6839823885543768763od_a_a @ F2 @ G ) )
= ( ( finite8717734299975451184od_a_a @ F2 )
& ( finite8717734299975451184od_a_a @ G ) ) ) ).
% finite_Un
thf(fact_522_finite__Un,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) )
= ( ( finite_finite_set_a @ F2 )
& ( finite_finite_set_a @ G ) ) ) ).
% finite_Un
thf(fact_523_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_524_finite__Un,axiom,
! [F2: set_Pr8600417178894128327od_a_a,G: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ ( sup_su8193676293155882651od_a_a @ F2 @ G ) )
= ( ( finite20598220287489552od_a_a @ F2 )
& ( finite20598220287489552od_a_a @ G ) ) ) ).
% finite_Un
thf(fact_525_finite__Un,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F2 @ G ) )
= ( ( finite6544458595007987280od_a_a @ F2 )
& ( finite6544458595007987280od_a_a @ G ) ) ) ).
% finite_Un
thf(fact_526_Un__subset__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C2 )
= ( ( ord_le746702958409616551od_a_a @ A @ C2 )
& ( ord_le746702958409616551od_a_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_527_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_528_Un__subset__iff,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ A @ B ) @ C2 )
= ( ( ord_le1995061765932249223od_a_a @ A @ C2 )
& ( ord_le1995061765932249223od_a_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_529_Un__subset__iff,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a,C2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ A @ B ) @ C2 )
= ( ( ord_le8376522849517564071_set_a @ A @ C2 )
& ( ord_le8376522849517564071_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_530_Un__subset__iff,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,C2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ A @ B ) @ C2 )
= ( ( ord_le3469131294019144807od_a_a @ A @ C2 )
& ( ord_le3469131294019144807od_a_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_531_Field__Un,axiom,
! [R: set_Pr3617678592830649543od_a_a,S: set_Pr3617678592830649543od_a_a] :
( ( field_1595709187305778996od_a_a @ ( sup_su964460112276788891od_a_a @ R @ S ) )
= ( sup_su8193676293155882651od_a_a @ ( field_1595709187305778996od_a_a @ R ) @ ( field_1595709187305778996od_a_a @ S ) ) ) ).
% Field_Un
thf(fact_532_Field__Un,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( field_1126092520709947252od_a_a @ ( sup_su8193676293155882651od_a_a @ R @ S ) )
= ( sup_su3048258781599657691od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ ( field_1126092520709947252od_a_a @ S ) ) ) ).
% Field_Un
thf(fact_533_Field__Un,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( field_a @ ( sup_su3048258781599657691od_a_a @ R @ S ) )
= ( sup_sup_set_a @ ( field_a @ R ) @ ( field_a @ S ) ) ) ).
% Field_Un
thf(fact_534_Domain__Id__on,axiom,
! [A: set_set_a] :
( ( domain_set_a_set_a @ ( id_on_set_a @ A ) )
= A ) ).
% Domain_Id_on
thf(fact_535_Domain__Id__on,axiom,
! [A: set_Product_prod_a_a] :
( ( domain990338261037515724od_a_a @ ( id_on_7193410965977890500od_a_a @ A ) )
= A ) ).
% Domain_Id_on
thf(fact_536_Domain__Id__on,axiom,
! [A: set_a] :
( ( domain_a_a @ ( id_on_a @ A ) )
= A ) ).
% Domain_Id_on
thf(fact_537_Range__Id__on,axiom,
! [A: set_set_a] :
( ( range_set_a_set_a @ ( id_on_set_a @ A ) )
= A ) ).
% Range_Id_on
thf(fact_538_Range__Id__on,axiom,
! [A: set_Product_prod_a_a] :
( ( range_2510312826226007605od_a_a @ ( id_on_7193410965977890500od_a_a @ A ) )
= A ) ).
% Range_Id_on
thf(fact_539_Range__Id__on,axiom,
! [A: set_a] :
( ( range_a_a @ ( id_on_a @ A ) )
= A ) ).
% Range_Id_on
thf(fact_540_UnE,axiom,
! [C: produc7983190928304305895od_a_a,A: set_Pr5931134355633282247od_a_a,B: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ C @ ( sup_su6534410975031996059od_a_a @ A @ B ) )
=> ( ~ ( member4903802553882211088od_a_a @ C @ A )
=> ( member4903802553882211088od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_541_UnE,axiom,
! [C: produc1703568184450464039_set_a,A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ C @ ( sup_su5624169484884619995_set_a @ A @ B ) )
=> ( ~ ( member7983343339038529360_set_a @ C @ A )
=> ( member7983343339038529360_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_542_UnE,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( sup_su6839823885543768763od_a_a @ A @ B ) )
=> ( ~ ( member1816616512716248880od_a_a @ C @ A )
=> ( member1816616512716248880od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_543_UnE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_544_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_545_UnE,axiom,
! [C: produc3498347346309940967od_a_a,A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ C @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
=> ( ~ ( member6330455413206600464od_a_a @ C @ A )
=> ( member6330455413206600464od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_546_UnE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_547_UnI1,axiom,
! [C: produc7983190928304305895od_a_a,A: set_Pr5931134355633282247od_a_a,B: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ C @ A )
=> ( member4903802553882211088od_a_a @ C @ ( sup_su6534410975031996059od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_548_UnI1,axiom,
! [C: produc1703568184450464039_set_a,A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ C @ A )
=> ( member7983343339038529360_set_a @ C @ ( sup_su5624169484884619995_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_549_UnI1,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ A )
=> ( member1816616512716248880od_a_a @ C @ ( sup_su6839823885543768763od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_550_UnI1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_551_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_552_UnI1,axiom,
! [C: produc3498347346309940967od_a_a,A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ C @ A )
=> ( member6330455413206600464od_a_a @ C @ ( sup_su8193676293155882651od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_553_UnI1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_554_UnI2,axiom,
! [C: produc7983190928304305895od_a_a,B: set_Pr5931134355633282247od_a_a,A: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ C @ B )
=> ( member4903802553882211088od_a_a @ C @ ( sup_su6534410975031996059od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_555_UnI2,axiom,
! [C: produc1703568184450464039_set_a,B: set_Pr5845495582615845127_set_a,A: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ C @ B )
=> ( member7983343339038529360_set_a @ C @ ( sup_su5624169484884619995_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_556_UnI2,axiom,
! [C: set_Product_prod_a_a,B: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ B )
=> ( member1816616512716248880od_a_a @ C @ ( sup_su6839823885543768763od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_557_UnI2,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_558_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_559_UnI2,axiom,
! [C: produc3498347346309940967od_a_a,B: set_Pr8600417178894128327od_a_a,A: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ C @ B )
=> ( member6330455413206600464od_a_a @ C @ ( sup_su8193676293155882651od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_560_UnI2,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_561_bex__Un,axiom,
! [A: set_a,B: set_a,P2: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A @ B ) )
& ( P2 @ X ) ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( P2 @ X ) )
| ? [X: a] :
( ( member_a @ X @ B )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_562_bex__Un,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ( ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
& ( P2 @ X ) ) )
= ( ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
& ( P2 @ X ) )
| ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ B )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_563_bex__Un,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,P2: produc3498347346309940967od_a_a > $o] :
( ( ? [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
& ( P2 @ X ) ) )
= ( ? [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ A )
& ( P2 @ X ) )
| ? [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ B )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_564_ball__Un,axiom,
! [A: set_a,B: set_a,P2: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A @ B ) )
=> ( P2 @ X ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( P2 @ X ) )
& ! [X: a] :
( ( member_a @ X @ B )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_565_ball__Un,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ( P2 @ X ) ) )
= ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( P2 @ X ) )
& ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ B )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_566_ball__Un,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,P2: produc3498347346309940967od_a_a > $o] :
( ( ! [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
=> ( P2 @ X ) ) )
= ( ! [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ A )
=> ( P2 @ X ) )
& ! [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ B )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_567_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_568_Un__assoc,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_569_Un__assoc,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,C2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ ( sup_su8193676293155882651od_a_a @ A @ B ) @ C2 )
= ( sup_su8193676293155882651od_a_a @ A @ ( sup_su8193676293155882651od_a_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_570_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_571_Un__absorb,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_572_Un__absorb,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_573_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B2: set_a] : ( sup_sup_set_a @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_574_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_575_Un__commute,axiom,
( sup_su8193676293155882651od_a_a
= ( ^ [A3: set_Pr8600417178894128327od_a_a,B2: set_Pr8600417178894128327od_a_a] : ( sup_su8193676293155882651od_a_a @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_576_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_577_Un__left__absorb,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_578_Un__left__absorb,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
= ( sup_su8193676293155882651od_a_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_579_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_580_Un__left__commute,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_581_Un__left__commute,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,C2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A @ ( sup_su8193676293155882651od_a_a @ B @ C2 ) )
= ( sup_su8193676293155882651od_a_a @ B @ ( sup_su8193676293155882651od_a_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_582_Domain__Un__eq,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( domain990338261037515724od_a_a @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
= ( sup_su3048258781599657691od_a_a @ ( domain990338261037515724od_a_a @ A ) @ ( domain990338261037515724od_a_a @ B ) ) ) ).
% Domain_Un_eq
thf(fact_583_Domain__Un__eq,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( domain_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( sup_sup_set_a @ ( domain_a_a @ A ) @ ( domain_a_a @ B ) ) ) ).
% Domain_Un_eq
thf(fact_584_Range__Un__eq,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( range_2510312826226007605od_a_a @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
= ( sup_su3048258781599657691od_a_a @ ( range_2510312826226007605od_a_a @ A ) @ ( range_2510312826226007605od_a_a @ B ) ) ) ).
% Range_Un_eq
thf(fact_585_Range__Un__eq,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( range_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( sup_sup_set_a @ ( range_a_a @ A ) @ ( range_a_a @ B ) ) ) ).
% Range_Un_eq
thf(fact_586_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_587_Un__empty__right,axiom,
! [A: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A @ bot_bo510284599550014259od_a_a )
= A ) ).
% Un_empty_right
thf(fact_588_Un__empty__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Un_empty_right
thf(fact_589_Un__empty__right,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ A @ bot_bo777872063958040403od_a_a )
= A ) ).
% Un_empty_right
thf(fact_590_Un__empty__right,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= A ) ).
% Un_empty_right
thf(fact_591_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_592_Un__empty__left,axiom,
! [B: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ bot_bo510284599550014259od_a_a @ B )
= B ) ).
% Un_empty_left
thf(fact_593_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_594_Un__empty__left,axiom,
! [B: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ bot_bo777872063958040403od_a_a @ B )
= B ) ).
% Un_empty_left
thf(fact_595_Un__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= B ) ).
% Un_empty_left
thf(fact_596_infinite__Un,axiom,
! [S2: set_se5735800977113168103od_a_a,T2: set_se5735800977113168103od_a_a] :
( ( ~ ( finite8717734299975451184od_a_a @ ( sup_su6839823885543768763od_a_a @ S2 @ T2 ) ) )
= ( ~ ( finite8717734299975451184od_a_a @ S2 )
| ~ ( finite8717734299975451184od_a_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_597_infinite__Un,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_set_a @ S2 )
| ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_598_infinite__Un,axiom,
! [S2: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_a @ S2 )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_599_infinite__Un,axiom,
! [S2: set_Pr8600417178894128327od_a_a,T2: set_Pr8600417178894128327od_a_a] :
( ( ~ ( finite20598220287489552od_a_a @ ( sup_su8193676293155882651od_a_a @ S2 @ T2 ) ) )
= ( ~ ( finite20598220287489552od_a_a @ S2 )
| ~ ( finite20598220287489552od_a_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_600_infinite__Un,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S2 @ T2 ) ) )
= ( ~ ( finite6544458595007987280od_a_a @ S2 )
| ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_601_Un__infinite,axiom,
! [S2: set_se5735800977113168103od_a_a,T2: set_se5735800977113168103od_a_a] :
( ~ ( finite8717734299975451184od_a_a @ S2 )
=> ~ ( finite8717734299975451184od_a_a @ ( sup_su6839823885543768763od_a_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_602_Un__infinite,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_603_Un__infinite,axiom,
! [S2: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_604_Un__infinite,axiom,
! [S2: set_Pr8600417178894128327od_a_a,T2: set_Pr8600417178894128327od_a_a] :
( ~ ( finite20598220287489552od_a_a @ S2 )
=> ~ ( finite20598220287489552od_a_a @ ( sup_su8193676293155882651od_a_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_605_Un__infinite,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S2 )
=> ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_606_finite__UnI,axiom,
! [F2: set_se5735800977113168103od_a_a,G: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ F2 )
=> ( ( finite8717734299975451184od_a_a @ G )
=> ( finite8717734299975451184od_a_a @ ( sup_su6839823885543768763od_a_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_607_finite__UnI,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_608_finite__UnI,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_609_finite__UnI,axiom,
! [F2: set_Pr8600417178894128327od_a_a,G: set_Pr8600417178894128327od_a_a] :
( ( finite20598220287489552od_a_a @ F2 )
=> ( ( finite20598220287489552od_a_a @ G )
=> ( finite20598220287489552od_a_a @ ( sup_su8193676293155882651od_a_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_610_finite__UnI,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( finite6544458595007987280od_a_a @ G )
=> ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_611_subset__Un__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_612_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_613_subset__Un__eq,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_614_subset__Un__eq,axiom,
( ord_le8376522849517564071_set_a
= ( ^ [A3: set_Pr5845495582615845127_set_a,B2: set_Pr5845495582615845127_set_a] :
( ( sup_su5624169484884619995_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_615_subset__Un__eq,axiom,
( ord_le3469131294019144807od_a_a
= ( ^ [A3: set_Pr8600417178894128327od_a_a,B2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_616_subset__UnE,axiom,
! [C2: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ~ ! [A5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ A )
=> ! [B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ B )
=> ( C2
!= ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_617_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_618_subset__UnE,axiom,
! [C2: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ C2 @ ( sup_su6839823885543768763od_a_a @ A @ B ) )
=> ~ ! [A5: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A5 @ A )
=> ! [B5: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B5 @ B )
=> ( C2
!= ( sup_su6839823885543768763od_a_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_619_subset__UnE,axiom,
! [C2: set_Pr5845495582615845127_set_a,A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ C2 @ ( sup_su5624169484884619995_set_a @ A @ B ) )
=> ~ ! [A5: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A5 @ A )
=> ! [B5: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ B5 @ B )
=> ( C2
!= ( sup_su5624169484884619995_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_620_subset__UnE,axiom,
! [C2: set_Pr8600417178894128327od_a_a,A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ C2 @ ( sup_su8193676293155882651od_a_a @ A @ B ) )
=> ~ ! [A5: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A5 @ A )
=> ! [B5: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ B5 @ B )
=> ( C2
!= ( sup_su8193676293155882651od_a_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_621_Un__absorb2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_622_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_623_Un__absorb2,axiom,
! [B: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B @ A )
=> ( ( sup_su6839823885543768763od_a_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_624_Un__absorb2,axiom,
! [B: set_Pr5845495582615845127_set_a,A: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ B @ A )
=> ( ( sup_su5624169484884619995_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_625_Un__absorb2,axiom,
! [B: set_Pr8600417178894128327od_a_a,A: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ B @ A )
=> ( ( sup_su8193676293155882651od_a_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_626_Un__absorb1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_627_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_628_Un__absorb1,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( sup_su6839823885543768763od_a_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_629_Un__absorb1,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ B )
=> ( ( sup_su5624169484884619995_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_630_Un__absorb1,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ B )
=> ( ( sup_su8193676293155882651od_a_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_631_Un__upper2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% Un_upper2
thf(fact_632_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_633_Un__upper2,axiom,
! [B: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ B @ ( sup_su6839823885543768763od_a_a @ A @ B ) ) ).
% Un_upper2
thf(fact_634_Un__upper2,axiom,
! [B: set_Pr5845495582615845127_set_a,A: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ B @ ( sup_su5624169484884619995_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_635_Un__upper2,axiom,
! [B: set_Pr8600417178894128327od_a_a,A: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ B @ ( sup_su8193676293155882651od_a_a @ A @ B ) ) ).
% Un_upper2
thf(fact_636_Un__upper1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% Un_upper1
thf(fact_637_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_638_Un__upper1,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ A @ ( sup_su6839823885543768763od_a_a @ A @ B ) ) ).
% Un_upper1
thf(fact_639_Un__upper1,axiom,
! [A: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ A @ ( sup_su5624169484884619995_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_640_Un__upper1,axiom,
! [A: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ A @ ( sup_su8193676293155882651od_a_a @ A @ B ) ) ).
% Un_upper1
thf(fact_641_Un__least,axiom,
! [A: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_642_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_643_Un__least,axiom,
! [A: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ C2 )
=> ( ( ord_le1995061765932249223od_a_a @ B @ C2 )
=> ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_644_Un__least,axiom,
! [A: set_Pr5845495582615845127_set_a,C2: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ C2 )
=> ( ( ord_le8376522849517564071_set_a @ B @ C2 )
=> ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_645_Un__least,axiom,
! [A: set_Pr8600417178894128327od_a_a,C2: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ C2 )
=> ( ( ord_le3469131294019144807od_a_a @ B @ C2 )
=> ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_646_Un__mono,axiom,
! [A: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B @ D )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ ( sup_su3048258781599657691od_a_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_647_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_648_Un__mono,axiom,
! [A: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,D: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ C2 )
=> ( ( ord_le1995061765932249223od_a_a @ B @ D )
=> ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ A @ B ) @ ( sup_su6839823885543768763od_a_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_649_Un__mono,axiom,
! [A: set_Pr5845495582615845127_set_a,C2: set_Pr5845495582615845127_set_a,B: set_Pr5845495582615845127_set_a,D: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A @ C2 )
=> ( ( ord_le8376522849517564071_set_a @ B @ D )
=> ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ A @ B ) @ ( sup_su5624169484884619995_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_650_Un__mono,axiom,
! [A: set_Pr8600417178894128327od_a_a,C2: set_Pr8600417178894128327od_a_a,B: set_Pr8600417178894128327od_a_a,D: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A @ C2 )
=> ( ( ord_le3469131294019144807od_a_a @ B @ D )
=> ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ A @ B ) @ ( sup_su8193676293155882651od_a_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_651_rtrancl__Un__rtrancl,axiom,
! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( transitive_rtrancl_a @ ( sup_su3048258781599657691od_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( transitive_rtrancl_a @ S2 ) ) )
= ( transitive_rtrancl_a @ ( sup_su3048258781599657691od_a_a @ R3 @ S2 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_652_rtrancl__Un__rtrancl,axiom,
! [R3: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
( ( transi7229704755246292010od_a_a @ ( sup_su8193676293155882651od_a_a @ ( transi7229704755246292010od_a_a @ R3 ) @ ( transi7229704755246292010od_a_a @ S2 ) ) )
= ( transi7229704755246292010od_a_a @ ( sup_su8193676293155882651od_a_a @ R3 @ S2 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_653_in__rtrancl__UnI,axiom,
! [X3: produc7983190928304305895od_a_a,R3: set_Pr5931134355633282247od_a_a,S2: set_Pr5931134355633282247od_a_a] :
( ( ( member4903802553882211088od_a_a @ X3 @ ( transi1172912939347638282od_a_a @ R3 ) )
| ( member4903802553882211088od_a_a @ X3 @ ( transi1172912939347638282od_a_a @ S2 ) ) )
=> ( member4903802553882211088od_a_a @ X3 @ ( transi1172912939347638282od_a_a @ ( sup_su6534410975031996059od_a_a @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_654_in__rtrancl__UnI,axiom,
! [X3: produc1703568184450464039_set_a,R3: set_Pr5845495582615845127_set_a,S2: set_Pr5845495582615845127_set_a] :
( ( ( member7983343339038529360_set_a @ X3 @ ( transi3782967724755053153_set_a @ R3 ) )
| ( member7983343339038529360_set_a @ X3 @ ( transi3782967724755053153_set_a @ S2 ) ) )
=> ( member7983343339038529360_set_a @ X3 @ ( transi3782967724755053153_set_a @ ( sup_su5624169484884619995_set_a @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_655_in__rtrancl__UnI,axiom,
! [X3: produc3498347346309940967od_a_a,R3: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
( ( ( member6330455413206600464od_a_a @ X3 @ ( transi7229704755246292010od_a_a @ R3 ) )
| ( member6330455413206600464od_a_a @ X3 @ ( transi7229704755246292010od_a_a @ S2 ) ) )
=> ( member6330455413206600464od_a_a @ X3 @ ( transi7229704755246292010od_a_a @ ( sup_su8193676293155882651od_a_a @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_656_in__rtrancl__UnI,axiom,
! [X3: product_prod_a_a,R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ ( transitive_rtrancl_a @ R3 ) )
| ( member1426531477525435216od_a_a @ X3 @ ( transitive_rtrancl_a @ S2 ) ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( transitive_rtrancl_a @ ( sup_su3048258781599657691od_a_a @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_657_rtrancl__Un__subset,axiom,
! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( transitive_rtrancl_a @ S2 ) ) @ ( transitive_rtrancl_a @ ( sup_su3048258781599657691od_a_a @ R3 @ S2 ) ) ) ).
% rtrancl_Un_subset
thf(fact_658_rtrancl__Un__subset,axiom,
! [R3: set_Pr5845495582615845127_set_a,S2: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ ( transi3782967724755053153_set_a @ R3 ) @ ( transi3782967724755053153_set_a @ S2 ) ) @ ( transi3782967724755053153_set_a @ ( sup_su5624169484884619995_set_a @ R3 @ S2 ) ) ) ).
% rtrancl_Un_subset
thf(fact_659_rtrancl__Un__subset,axiom,
! [R3: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ ( transi7229704755246292010od_a_a @ R3 ) @ ( transi7229704755246292010od_a_a @ S2 ) ) @ ( transi7229704755246292010od_a_a @ ( sup_su8193676293155882651od_a_a @ R3 @ S2 ) ) ) ).
% rtrancl_Un_subset
thf(fact_660_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_661_sup__bot_Oright__neutral,axiom,
! [A2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A2 @ bot_bo510284599550014259od_a_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_662_sup__bot_Oright__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_663_sup__bot_Oright__neutral,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ A2 @ bot_bo777872063958040403od_a_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_664_sup__bot_Oright__neutral,axiom,
! [A2: product_prod_a_a > $o] :
( ( sup_su1640154476453168578_a_a_o @ A2 @ bot_bo4160289986317612842_a_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_665_sup__bot_Oright__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_666_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_667_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( bot_bo510284599550014259od_a_a
= ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) )
= ( ( A2 = bot_bo510284599550014259od_a_a )
& ( B3 = bot_bo510284599550014259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_668_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B3 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_669_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( bot_bo777872063958040403od_a_a
= ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) )
= ( ( A2 = bot_bo777872063958040403od_a_a )
& ( B3 = bot_bo777872063958040403od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_670_sup__bot_Oneutr__eq__iff,axiom,
! [A2: product_prod_a_a > $o,B3: product_prod_a_a > $o] :
( ( bot_bo4160289986317612842_a_a_o
= ( sup_su1640154476453168578_a_a_o @ A2 @ B3 ) )
= ( ( A2 = bot_bo4160289986317612842_a_a_o )
& ( B3 = bot_bo4160289986317612842_a_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_671_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_672_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_673_sup__bot_Oleft__neutral,axiom,
! [A2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ bot_bo510284599550014259od_a_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_674_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_675_sup__bot_Oleft__neutral,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ bot_bo777872063958040403od_a_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_676_sup__bot_Oleft__neutral,axiom,
! [A2: product_prod_a_a > $o] :
( ( sup_su1640154476453168578_a_a_o @ bot_bo4160289986317612842_a_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_677_sup__bot_Oleft__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_678_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_679_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( ( sup_su8193676293155882651od_a_a @ A2 @ B3 )
= bot_bo510284599550014259od_a_a )
= ( ( A2 = bot_bo510284599550014259od_a_a )
& ( B3 = bot_bo510284599550014259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_680_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B3 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_681_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( ( sup_su6839823885543768763od_a_a @ A2 @ B3 )
= bot_bo777872063958040403od_a_a )
= ( ( A2 = bot_bo777872063958040403od_a_a )
& ( B3 = bot_bo777872063958040403od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_682_sup__bot_Oeq__neutr__iff,axiom,
! [A2: product_prod_a_a > $o,B3: product_prod_a_a > $o] :
( ( ( sup_su1640154476453168578_a_a_o @ A2 @ B3 )
= bot_bo4160289986317612842_a_a_o )
= ( ( A2 = bot_bo4160289986317612842_a_a_o )
& ( B3 = bot_bo4160289986317612842_a_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_683_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A2 @ B3 )
= bot_bo3357376287454694259od_a_a )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_684_sup__eq__bot__iff,axiom,
! [X3: set_a,Y4: set_a] :
( ( ( sup_sup_set_a @ X3 @ Y4 )
= bot_bot_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_685_sup__eq__bot__iff,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] :
( ( ( sup_su8193676293155882651od_a_a @ X3 @ Y4 )
= bot_bo510284599550014259od_a_a )
= ( ( X3 = bot_bo510284599550014259od_a_a )
& ( Y4 = bot_bo510284599550014259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_686_sup__eq__bot__iff,axiom,
! [X3: set_set_a,Y4: set_set_a] :
( ( ( sup_sup_set_set_a @ X3 @ Y4 )
= bot_bot_set_set_a )
= ( ( X3 = bot_bot_set_set_a )
& ( Y4 = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_687_sup__eq__bot__iff,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] :
( ( ( sup_su6839823885543768763od_a_a @ X3 @ Y4 )
= bot_bo777872063958040403od_a_a )
= ( ( X3 = bot_bo777872063958040403od_a_a )
& ( Y4 = bot_bo777872063958040403od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_688_sup__eq__bot__iff,axiom,
! [X3: product_prod_a_a > $o,Y4: product_prod_a_a > $o] :
( ( ( sup_su1640154476453168578_a_a_o @ X3 @ Y4 )
= bot_bo4160289986317612842_a_a_o )
= ( ( X3 = bot_bo4160289986317612842_a_a_o )
& ( Y4 = bot_bo4160289986317612842_a_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_689_sup__eq__bot__iff,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X3 @ Y4 )
= bot_bo3357376287454694259od_a_a )
= ( ( X3 = bot_bo3357376287454694259od_a_a )
& ( Y4 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_690_bot__eq__sup__iff,axiom,
! [X3: set_a,Y4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X3 @ Y4 ) )
= ( ( X3 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_691_bot__eq__sup__iff,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] :
( ( bot_bo510284599550014259od_a_a
= ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) )
= ( ( X3 = bot_bo510284599550014259od_a_a )
& ( Y4 = bot_bo510284599550014259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_692_bot__eq__sup__iff,axiom,
! [X3: set_set_a,Y4: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X3 @ Y4 ) )
= ( ( X3 = bot_bot_set_set_a )
& ( Y4 = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_693_bot__eq__sup__iff,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] :
( ( bot_bo777872063958040403od_a_a
= ( sup_su6839823885543768763od_a_a @ X3 @ Y4 ) )
= ( ( X3 = bot_bo777872063958040403od_a_a )
& ( Y4 = bot_bo777872063958040403od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_694_bot__eq__sup__iff,axiom,
! [X3: product_prod_a_a > $o,Y4: product_prod_a_a > $o] :
( ( bot_bo4160289986317612842_a_a_o
= ( sup_su1640154476453168578_a_a_o @ X3 @ Y4 ) )
= ( ( X3 = bot_bo4160289986317612842_a_a_o )
& ( Y4 = bot_bo4160289986317612842_a_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_695_bot__eq__sup__iff,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) )
= ( ( X3 = bot_bo3357376287454694259od_a_a )
& ( Y4 = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_696_sup__bot__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_697_sup__bot__right,axiom,
! [X3: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X3 @ bot_bo510284599550014259od_a_a )
= X3 ) ).
% sup_bot_right
thf(fact_698_sup__bot__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ bot_bot_set_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_699_sup__bot__right,axiom,
! [X3: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ X3 @ bot_bo777872063958040403od_a_a )
= X3 ) ).
% sup_bot_right
thf(fact_700_sup__bot__right,axiom,
! [X3: product_prod_a_a > $o] :
( ( sup_su1640154476453168578_a_a_o @ X3 @ bot_bo4160289986317612842_a_a_o )
= X3 ) ).
% sup_bot_right
thf(fact_701_sup__bot__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= X3 ) ).
% sup_bot_right
thf(fact_702_sup__bot__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_703_sup__bot__left,axiom,
! [X3: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ bot_bo510284599550014259od_a_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_704_sup__bot__left,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_705_sup__bot__left,axiom,
! [X3: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ bot_bo777872063958040403od_a_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_706_sup__bot__left,axiom,
! [X3: product_prod_a_a > $o] :
( ( sup_su1640154476453168578_a_a_o @ bot_bo4160289986317612842_a_a_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_707_sup__bot__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_708_sup_Obounded__iff,axiom,
! [B3: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B3 @ C ) @ A2 )
= ( ( ord_le746702958409616551od_a_a @ B3 @ A2 )
& ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_709_sup_Obounded__iff,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B3 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_710_sup_Obounded__iff,axiom,
! [B3: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ B3 @ C ) @ A2 )
= ( ( ord_le1995061765932249223od_a_a @ B3 @ A2 )
& ( ord_le1995061765932249223od_a_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_711_sup_Obounded__iff,axiom,
! [B3: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a,A2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ B3 @ C ) @ A2 )
= ( ( ord_le8376522849517564071_set_a @ B3 @ A2 )
& ( ord_le8376522849517564071_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_712_sup_Obounded__iff,axiom,
! [B3: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ B3 @ C ) @ A2 )
= ( ( ord_le3469131294019144807od_a_a @ B3 @ A2 )
& ( ord_le3469131294019144807od_a_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_713_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_714_sup_Oidem,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_715_sup_Oidem,axiom,
! [A2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_716_sup__idem,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_717_sup__idem,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_718_sup__idem,axiom,
! [X3: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_719_sup_Oleft__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B3 ) )
= ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_720_sup_Oleft__idem,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_721_sup_Oleft__idem,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ A2 @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) )
= ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_722_sup__left__idem,axiom,
! [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y4 ) )
= ( sup_sup_set_a @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_723_sup__left__idem,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) )
= ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_724_sup__left__idem,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) )
= ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_725_sup_Oright__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ B3 )
= ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_726_sup_Oright__idem,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) @ B3 )
= ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_727_sup_Oright__idem,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) @ B3 )
= ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_728_le__sup__iff,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le746702958409616551od_a_a @ X3 @ Z2 )
& ( ord_le746702958409616551od_a_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_729_le__sup__iff,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X3 @ Z2 )
& ( ord_less_eq_set_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_730_le__sup__iff,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a,Z2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le1995061765932249223od_a_a @ X3 @ Z2 )
& ( ord_le1995061765932249223od_a_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_731_le__sup__iff,axiom,
! [X3: set_Pr5845495582615845127_set_a,Y4: set_Pr5845495582615845127_set_a,Z2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le8376522849517564071_set_a @ X3 @ Z2 )
& ( ord_le8376522849517564071_set_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_732_le__sup__iff,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le3469131294019144807od_a_a @ X3 @ Z2 )
& ( ord_le3469131294019144807od_a_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_733_inf__sup__aci_I8_J,axiom,
! [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y4 ) )
= ( sup_sup_set_a @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_734_inf__sup__aci_I8_J,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) )
= ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_735_inf__sup__aci_I8_J,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) )
= ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_736_inf__sup__aci_I7_J,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_a @ Y4 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_737_inf__sup__aci_I7_J,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_738_inf__sup__aci_I7_J,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ Y4 @ Z2 ) )
= ( sup_su8193676293155882651od_a_a @ Y4 @ ( sup_su8193676293155882651od_a_a @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_739_inf__sup__aci_I6_J,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_740_inf__sup__aci_I6_J,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_741_inf__sup__aci_I6_J,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) @ Z2 )
= ( sup_su8193676293155882651od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_742_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_743_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_744_inf__sup__aci_I5_J,axiom,
( sup_su8193676293155882651od_a_a
= ( ^ [X: set_Pr8600417178894128327od_a_a,Y3: set_Pr8600417178894128327od_a_a] : ( sup_su8193676293155882651od_a_a @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_745_sup_Oassoc,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_746_sup_Oassoc,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) @ C )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_747_sup_Oassoc,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) @ C )
= ( sup_su8193676293155882651od_a_a @ A2 @ ( sup_su8193676293155882651od_a_a @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_748_sup__assoc,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_749_sup__assoc,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_750_sup__assoc,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) @ Z2 )
= ( sup_su8193676293155882651od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_751_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_752_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_753_sup_Ocommute,axiom,
( sup_su8193676293155882651od_a_a
= ( ^ [A4: set_Pr8600417178894128327od_a_a,B4: set_Pr8600417178894128327od_a_a] : ( sup_su8193676293155882651od_a_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_754_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_755_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_756_sup__commute,axiom,
( sup_su8193676293155882651od_a_a
= ( ^ [X: set_Pr8600417178894128327od_a_a,Y3: set_Pr8600417178894128327od_a_a] : ( sup_su8193676293155882651od_a_a @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_757_sup_Oleft__commute,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_758_sup_Oleft__commute,axiom,
! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B3 @ ( sup_su3048258781599657691od_a_a @ A2 @ C ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_759_sup_Oleft__commute,axiom,
! [B3: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ B3 @ ( sup_su8193676293155882651od_a_a @ A2 @ C ) )
= ( sup_su8193676293155882651od_a_a @ A2 @ ( sup_su8193676293155882651od_a_a @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_760_sup__left__commute,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_a @ Y4 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_761_sup__left__commute,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_762_sup__left__commute,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ Y4 @ Z2 ) )
= ( sup_su8193676293155882651od_a_a @ Y4 @ ( sup_su8193676293155882651od_a_a @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_763_inf__sup__ord_I4_J,axiom,
! [Y4: set_Product_prod_a_a,X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_764_inf__sup__ord_I4_J,axiom,
! [Y4: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_765_inf__sup__ord_I4_J,axiom,
! [Y4: set_se5735800977113168103od_a_a,X3: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ Y4 @ ( sup_su6839823885543768763od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_766_inf__sup__ord_I4_J,axiom,
! [Y4: set_Pr5845495582615845127_set_a,X3: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ Y4 @ ( sup_su5624169484884619995_set_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_767_inf__sup__ord_I4_J,axiom,
! [Y4: set_Pr8600417178894128327od_a_a,X3: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ Y4 @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_768_inf__sup__ord_I3_J,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_769_inf__sup__ord_I3_J,axiom,
! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_770_inf__sup__ord_I3_J,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ X3 @ ( sup_su6839823885543768763od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_771_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr5845495582615845127_set_a,Y4: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ X3 @ ( sup_su5624169484884619995_set_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_772_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_773_le__supE,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ X3 )
=> ~ ( ord_le746702958409616551od_a_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_774_le__supE,axiom,
! [A2: set_a,B3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X3 )
=> ~ ( ord_less_eq_set_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_775_le__supE,axiom,
! [A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,X3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_le1995061765932249223od_a_a @ A2 @ X3 )
=> ~ ( ord_le1995061765932249223od_a_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_776_le__supE,axiom,
! [A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,X3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_le8376522849517564071_set_a @ A2 @ X3 )
=> ~ ( ord_le8376522849517564071_set_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_777_le__supE,axiom,
! [A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,X3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_le3469131294019144807od_a_a @ A2 @ X3 )
=> ~ ( ord_le3469131294019144807od_a_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_778_le__supI,axiom,
! [A2: set_Product_prod_a_a,X3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ X3 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_779_le__supI,axiom,
! [A2: set_a,X3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X3 )
=> ( ( ord_less_eq_set_a @ B3 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_780_le__supI,axiom,
! [A2: set_se5735800977113168103od_a_a,X3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ X3 )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ X3 )
=> ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_781_le__supI,axiom,
! [A2: set_Pr5845495582615845127_set_a,X3: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ X3 )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ X3 )
=> ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_782_le__supI,axiom,
! [A2: set_Pr8600417178894128327od_a_a,X3: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ X3 )
=> ( ( ord_le3469131294019144807od_a_a @ B3 @ X3 )
=> ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_783_sup__ge1,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_784_sup__ge1,axiom,
! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_785_sup__ge1,axiom,
! [X3: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ X3 @ ( sup_su6839823885543768763od_a_a @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_786_sup__ge1,axiom,
! [X3: set_Pr5845495582615845127_set_a,Y4: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ X3 @ ( sup_su5624169484884619995_set_a @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_787_sup__ge1,axiom,
! [X3: set_Pr8600417178894128327od_a_a,Y4: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_788_sup__ge2,axiom,
! [Y4: set_Product_prod_a_a,X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_789_sup__ge2,axiom,
! [Y4: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_790_sup__ge2,axiom,
! [Y4: set_se5735800977113168103od_a_a,X3: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ Y4 @ ( sup_su6839823885543768763od_a_a @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_791_sup__ge2,axiom,
! [Y4: set_Pr5845495582615845127_set_a,X3: set_Pr5845495582615845127_set_a] : ( ord_le8376522849517564071_set_a @ Y4 @ ( sup_su5624169484884619995_set_a @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_792_sup__ge2,axiom,
! [Y4: set_Pr8600417178894128327od_a_a,X3: set_Pr8600417178894128327od_a_a] : ( ord_le3469131294019144807od_a_a @ Y4 @ ( sup_su8193676293155882651od_a_a @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_793_le__supI1,axiom,
! [X3: set_Product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_794_le__supI1,axiom,
! [X3: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_795_le__supI1,axiom,
! [X3: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X3 @ A2 )
=> ( ord_le1995061765932249223od_a_a @ X3 @ ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_796_le__supI1,axiom,
! [X3: set_Pr5845495582615845127_set_a,A2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X3 @ A2 )
=> ( ord_le8376522849517564071_set_a @ X3 @ ( sup_su5624169484884619995_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_797_le__supI1,axiom,
! [X3: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ X3 @ A2 )
=> ( ord_le3469131294019144807od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_798_le__supI2,axiom,
! [X3: set_Product_prod_a_a,B3: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ B3 )
=> ( ord_le746702958409616551od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_799_le__supI2,axiom,
! [X3: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X3 @ B3 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_800_le__supI2,axiom,
! [X3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X3 @ B3 )
=> ( ord_le1995061765932249223od_a_a @ X3 @ ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_801_le__supI2,axiom,
! [X3: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,A2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ X3 @ B3 )
=> ( ord_le8376522849517564071_set_a @ X3 @ ( sup_su5624169484884619995_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_802_le__supI2,axiom,
! [X3: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ X3 @ B3 )
=> ( ord_le3469131294019144807od_a_a @ X3 @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_803_sup_Omono,axiom,
! [C: set_Product_prod_a_a,A2: set_Product_prod_a_a,D2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ D2 @ B3 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ C @ D2 ) @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_804_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_805_sup_Omono,axiom,
! [C: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a,D2: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ C @ A2 )
=> ( ( ord_le1995061765932249223od_a_a @ D2 @ B3 )
=> ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ C @ D2 ) @ ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_806_sup_Omono,axiom,
! [C: set_Pr5845495582615845127_set_a,A2: set_Pr5845495582615845127_set_a,D2: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ C @ A2 )
=> ( ( ord_le8376522849517564071_set_a @ D2 @ B3 )
=> ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ C @ D2 ) @ ( sup_su5624169484884619995_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_807_sup_Omono,axiom,
! [C: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a,D2: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ C @ A2 )
=> ( ( ord_le3469131294019144807od_a_a @ D2 @ B3 )
=> ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ C @ D2 ) @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_808_sup__mono,axiom,
! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B3: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) @ ( sup_su3048258781599657691od_a_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_809_sup__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_810_sup__mono,axiom,
! [A2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a,D2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ C )
=> ( ( ord_le1995061765932249223od_a_a @ B3 @ D2 )
=> ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ A2 @ B3 ) @ ( sup_su6839823885543768763od_a_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_811_sup__mono,axiom,
! [A2: set_Pr5845495582615845127_set_a,C: set_Pr5845495582615845127_set_a,B3: set_Pr5845495582615845127_set_a,D2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ A2 @ C )
=> ( ( ord_le8376522849517564071_set_a @ B3 @ D2 )
=> ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ A2 @ B3 ) @ ( sup_su5624169484884619995_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_812_sup__mono,axiom,
! [A2: set_Pr8600417178894128327od_a_a,C: set_Pr8600417178894128327od_a_a,B3: set_Pr8600417178894128327od_a_a,D2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A2 @ C )
=> ( ( ord_le3469131294019144807od_a_a @ B3 @ D2 )
=> ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) @ ( sup_su8193676293155882651od_a_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_813_sup__least,axiom,
! [Y4: set_Product_prod_a_a,X3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y4 @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ Z2 @ X3 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_814_sup__least,axiom,
! [Y4: set_a,X3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( ( ord_less_eq_set_a @ Z2 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_815_sup__least,axiom,
! [Y4: set_se5735800977113168103od_a_a,X3: set_se5735800977113168103od_a_a,Z2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ Y4 @ X3 )
=> ( ( ord_le1995061765932249223od_a_a @ Z2 @ X3 )
=> ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_816_sup__least,axiom,
! [Y4: set_Pr5845495582615845127_set_a,X3: set_Pr5845495582615845127_set_a,Z2: set_Pr5845495582615845127_set_a] :
( ( ord_le8376522849517564071_set_a @ Y4 @ X3 )
=> ( ( ord_le8376522849517564071_set_a @ Z2 @ X3 )
=> ( ord_le8376522849517564071_set_a @ ( sup_su5624169484884619995_set_a @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_817_sup__least,axiom,
! [Y4: set_Pr8600417178894128327od_a_a,X3: set_Pr8600417178894128327od_a_a,Z2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ Y4 @ X3 )
=> ( ( ord_le3469131294019144807od_a_a @ Z2 @ X3 )
=> ( ord_le3469131294019144807od_a_a @ ( sup_su8193676293155882651od_a_a @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_818_le__iff__sup,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_819_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_820_le__iff__sup,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [X: set_se5735800977113168103od_a_a,Y3: set_se5735800977113168103od_a_a] :
( ( sup_su6839823885543768763od_a_a @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_821_le__iff__sup,axiom,
( ord_le8376522849517564071_set_a
= ( ^ [X: set_Pr5845495582615845127_set_a,Y3: set_Pr5845495582615845127_set_a] :
( ( sup_su5624169484884619995_set_a @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_822_le__iff__sup,axiom,
( ord_le3469131294019144807od_a_a
= ( ^ [X: set_Pr8600417178894128327od_a_a,Y3: set_Pr8600417178894128327od_a_a] :
( ( sup_su8193676293155882651od_a_a @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_823_sup_OorderE,axiom,
! [B3: set_Pr8600417178894128327od_a_a,A2: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ B3 @ A2 )
=> ( A2
= ( sup_su8193676293155882651od_a_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_824_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_825_trancl__reflcl,axiom,
! [R: set_Product_prod_a_a] :
( ( transitive_trancl_a @ ( sup_su3048258781599657691od_a_a @ R @ id_a ) )
= ( transitive_rtrancl_a @ R ) ) ).
% trancl_reflcl
thf(fact_826_rtrancl__empty,axiom,
( ( transitive_rtrancl_a @ bot_bo3357376287454694259od_a_a )
= id_a ) ).
% rtrancl_empty
thf(fact_827_rtrancl__trancl__reflcl,axiom,
( transitive_rtrancl_a
= ( ^ [R2: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ ( transitive_trancl_a @ R2 ) @ id_a ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_828_finite__Int,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ F2 )
| ( finite6544458595007987280od_a_a @ G ) )
=> ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_829_Int__insert__right__if1,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_830_Int__insert__right__if0,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_831_Int__insert__left__if1,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_832_Int__insert__left__if0,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_833_insert__iff,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_834_insertCI,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_835_inf__bot__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_836_inf__bot__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_837_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_838_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_839_disjoint__insert_I2_J,axiom,
! [A: set_Product_prod_a_a,B3: product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ B3 @ B ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B3 @ A )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_840_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( ( inf_in8905007599844390133od_a_a @ B @ A )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_841_insert__disjoint_I2_J,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B ) )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_842_insert__disjoint_I1_J,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_843_singletonI,axiom,
! [A2: product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_844_finite__insert,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
= ( finite6544458595007987280od_a_a @ A ) ) ).
% finite_insert
thf(fact_845_insert__subset,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) @ B )
= ( ( member1426531477525435216od_a_a @ X3 @ B )
& ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_846_singleton__insert__inj__eq,axiom,
! [B3: product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_847_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ A )
= ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) )
= ( ( A2 = B3 )
& ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_848_mk__disjoint__insert,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ? [B6: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ A2 @ B6 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_849_Int__insert__right,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_850_Int__insert__left,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_851_insert__eq__iff,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ B3 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ A2 @ A )
= ( insert4534936382041156343od_a_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ B3 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ B3 @ C3 )
& ( B
= ( insert4534936382041156343od_a_a @ A2 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_852_insert__absorb,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( insert4534936382041156343od_a_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_853_insert__ident,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ X3 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ X3 @ A )
= ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_854_Set_Oset__insert,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ~ ! [B6: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ X3 @ B6 ) )
=> ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_855_insertI2,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ B )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_856_insertI1,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ B ) ) ).
% insertI1
thf(fact_857_insertE,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_858_Domain__Int__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_less_eq_set_a @ ( domain_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) @ ( inf_inf_set_a @ ( domain_a_a @ A ) @ ( domain_a_a @ B ) ) ) ).
% Domain_Int_subset
thf(fact_859_Range__Int__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_less_eq_set_a @ ( range_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) @ ( inf_inf_set_a @ ( range_a_a @ A ) @ ( range_a_a @ B ) ) ) ).
% Range_Int_subset
thf(fact_860_singletonD,axiom,
! [B3: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_861_singleton__iff,axiom,
! [B3: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_862_doubleton__eq__iff,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,C: product_prod_a_a,D2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A2 = C )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_863_insert__not__empty,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A2 @ A )
!= bot_bo3357376287454694259od_a_a ) ).
% insert_not_empty
thf(fact_864_singleton__inject,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_865_Int__emptyI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ~ ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_866_disjoint__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ~ ( member1426531477525435216od_a_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_867_Int__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_868_Int__empty__right,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_869_disjoint__iff__not__equal,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ B )
=> ( X != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_870_finite_OinsertI,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_871_insert__subsetI,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,X4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( ord_le746702958409616551od_a_a @ X4 @ A )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_872_subset__insert,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_873_Int__Collect__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) ) @ ( inf_in8905007599844390133od_a_a @ B @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_874_finite_Ocases,axiom,
! [A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( ( A2 != bot_bo3357376287454694259od_a_a )
=> ~ ! [A6: set_Product_prod_a_a] :
( ? [A7: product_prod_a_a] :
( A2
= ( insert4534936382041156343od_a_a @ A7 @ A6 ) )
=> ~ ( finite6544458595007987280od_a_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_875_finite_Osimps,axiom,
( finite6544458595007987280od_a_a
= ( ^ [A4: set_Product_prod_a_a] :
( ( A4 = bot_bo3357376287454694259od_a_a )
| ? [A3: set_Product_prod_a_a,B4: product_prod_a_a] :
( ( A4
= ( insert4534936382041156343od_a_a @ B4 @ A3 ) )
& ( finite6544458595007987280od_a_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_876_finite__induct,axiom,
! [F2: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( P2 @ bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert4534936382041156343od_a_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_877_finite__ne__induct,axiom,
! [F2: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( F2 != bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a] : ( P2 @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
=> ( ! [X2: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( F3 != bot_bo3357376287454694259od_a_a )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert4534936382041156343od_a_a @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_878_infinite__finite__induct,axiom,
! [P2: set_Product_prod_a_a > $o,A: set_Product_prod_a_a] :
( ! [A6: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert4534936382041156343od_a_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_879_subset__singletonD,axiom,
! [A: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A = bot_bo3357376287454694259od_a_a )
| ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_880_subset__singleton__iff,axiom,
! [X4: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( X4 = bot_bo3357376287454694259od_a_a )
| ( X4
= ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_881_insert__is__Un,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A4: product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A4 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% insert_is_Un
thf(fact_882_Un__singleton__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= ( ( ( A = bot_bo3357376287454694259od_a_a )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B = bot_bo3357376287454694259od_a_a ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_883_singleton__Un__iff,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( ( A = bot_bo3357376287454694259od_a_a )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B = bot_bo3357376287454694259od_a_a ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_884_total__on__singleton,axiom,
! [X3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a] : ( total_3048927264068743339od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) @ R ) ).
% total_on_singleton
thf(fact_885_finite__subset__induct,axiom,
! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A )
=> ( ( P2 @ bot_bo3357376287454694259od_a_a )
=> ( ! [A7: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A7 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert4534936382041156343od_a_a @ A7 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_886_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A )
=> ( ( P2 @ bot_bo3357376287454694259od_a_a )
=> ( ! [A7: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A7 @ A )
=> ( ( ord_le746702958409616551od_a_a @ F3 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert4534936382041156343od_a_a @ A7 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_887_the__elem__eq,axiom,
! [X3: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_888_is__singletonI,axiom,
! [X3: product_prod_a_a] : ( is_sin3171834905898671131od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ).
% is_singletonI
thf(fact_889_Field__insert,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) )
= ( sup_sup_set_a @ ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) @ ( field_a @ R ) ) ) ).
% Field_insert
thf(fact_890_Field__insert,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( field_1126092520709947252od_a_a @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) )
= ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) @ ( field_1126092520709947252od_a_a @ R ) ) ) ).
% Field_insert
thf(fact_891_Int__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
& ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_892_IntI,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_893_DiffI,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( ~ ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_894_Diff__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
& ~ ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_895_Diff__cancel,axiom,
! [A: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ A @ A )
= bot_bo3357376287454694259od_a_a ) ).
% Diff_cancel
thf(fact_896_empty__Diff,axiom,
! [A: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ bot_bo3357376287454694259od_a_a @ A )
= bot_bo3357376287454694259od_a_a ) ).
% empty_Diff
thf(fact_897_Diff__empty,axiom,
! [A: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= A ) ).
% Diff_empty
thf(fact_898_finite__Diff2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B ) )
= ( finite6544458595007987280od_a_a @ A ) ) ) ).
% finite_Diff2
thf(fact_899_finite__Diff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_900_Diff__insert0,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_901_insert__Diff1,axiom,
! [X3: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) @ B )
= ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_902_IdI,axiom,
! [A2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ id_a ) ).
% IdI
thf(fact_903_pair__in__Id__conv,axiom,
! [A2: a,B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ id_a )
= ( A2 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_904_Id__onI,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( id_on_a @ A ) ) ) ).
% Id_onI
thf(fact_905_Id__onI,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ A2 ) @ ( id_on_7193410965977890500od_a_a @ A ) ) ) ).
% Id_onI
thf(fact_906_Diff__eq__empty__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( minus_6817036919807184750od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_907_insert__Diff__single,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A2 @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( insert4534936382041156343od_a_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_908_finite__Diff__insert,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) ) )
= ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_909_Diff__disjoint,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ ( minus_6817036919807184750od_a_a @ B @ A ) )
= bot_bo3357376287454694259od_a_a ) ).
% Diff_disjoint
thf(fact_910_Domain__insert,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( domain_a_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) )
= ( insert_a @ A2 @ ( domain_a_a @ R ) ) ) ).
% Domain_insert
thf(fact_911_Range__insert,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( range_a_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) )
= ( insert_a @ B3 @ ( range_a_a @ R ) ) ) ).
% Range_insert
thf(fact_912_IntD2,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ).
% IntD2
thf(fact_913_IntD1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ A ) ) ).
% IntD1
thf(fact_914_IntE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A )
=> ~ ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% IntE
thf(fact_915_ssubst__Pair__rhs,axiom,
! [R: a,S: a,R3: set_Product_prod_a_a,S3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S ) @ R3 )
=> ( ( S3 = S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_916_DiffE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% DiffE
thf(fact_917_DiffD1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ A ) ) ).
% DiffD1
thf(fact_918_DiffD2,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A @ B ) )
=> ~ ( member1426531477525435216od_a_a @ C @ B ) ) ).
% DiffD2
thf(fact_919_Domain__Diff__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( domain_a_a @ A ) @ ( domain_a_a @ B ) ) @ ( domain_a_a @ ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ).
% Domain_Diff_subset
thf(fact_920_Range__Diff__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( range_a_a @ A ) @ ( range_a_a @ B ) ) @ ( range_a_a @ ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ).
% Range_Diff_subset
thf(fact_921_Diff__triv,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( minus_6817036919807184750od_a_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_922_Int__Diff__disjoint,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ ( minus_6817036919807184750od_a_a @ A @ B ) )
= bot_bo3357376287454694259od_a_a ) ).
% Int_Diff_disjoint
thf(fact_923_Diff__infinite__finite,axiom,
! [T2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ T2 )
=> ( ~ ( finite6544458595007987280od_a_a @ S2 )
=> ~ ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_924_insert__Diff__if,axiom,
! [X3: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ B )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) @ B )
= ( minus_6817036919807184750od_a_a @ A @ B ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ B )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) @ B )
= ( insert4534936382041156343od_a_a @ X3 @ ( minus_6817036919807184750od_a_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_925_FieldI1,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R3 )
=> ( member1426531477525435216od_a_a @ I @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% FieldI1
thf(fact_926_FieldI1,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ I @ ( field_a @ R3 ) ) ) ).
% FieldI1
thf(fact_927_FieldI2,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R3 )
=> ( member1426531477525435216od_a_a @ J @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% FieldI2
thf(fact_928_FieldI2,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ J @ ( field_a @ R3 ) ) ) ).
% FieldI2
thf(fact_929_converse__rtrancl__into__rtrancl,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ ( transitive_rtrancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_930_converse__rtrancl__induct,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( P2 @ B3 )
=> ( ! [Y: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z3 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_931_converse__rtranclE,axiom,
! [X3: a,Z2: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_932_rtrancl__induct,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( P2 @ A2 )
=> ( ! [Y: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ Y ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ R )
=> ( ( P2 @ Y )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B3 ) ) ) ) ).
% rtrancl_induct
thf(fact_933_rtrancl__trans,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a,Z2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ ( transitive_rtrancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_934_rtranclE,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( A2 != B3 )
=> ~ ! [Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ Y ) @ ( transitive_rtrancl_a @ R ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ B3 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_935_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_936_rtrancl_Ortrancl__refl,axiom,
! [A2: a,R: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( transitive_rtrancl_a @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_937_rtrancl_Osimps,axiom,
! [A1: a,A22: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_rtrancl_a @ R ) )
= ( ? [A4: a] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: a,B4: a,C4: a] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ ( transitive_rtrancl_a @ R ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_938_rtrancl_Ocases,axiom,
! [A1: a,A22: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B7: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B7 ) @ ( transitive_rtrancl_a @ R ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B7 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_939_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X2: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_940_trancl_Ocases,axiom,
! [A1: a,A22: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_trancl_a @ R ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ R )
=> ~ ! [B7: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B7 ) @ ( transitive_trancl_a @ R ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B7 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_941_trancl_Osimps,axiom,
! [A1: a,A22: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_trancl_a @ R ) )
= ( ? [A4: a,B4: a] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R ) )
| ? [A4: a,B4: a,C4: a] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ ( transitive_trancl_a @ R ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_942_trancl_Or__into__trancl,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_943_tranclE,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ~ ! [C5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C5 ) @ ( transitive_trancl_a @ R ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C5 @ B3 ) @ R ) ) ) ) ).
% tranclE
thf(fact_944_trancl__trans,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a,Z2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ ( transitive_trancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% trancl_trans
thf(fact_945_trancl__induct,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( ! [Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ Y ) @ R )
=> ( P2 @ Y ) )
=> ( ! [Y: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ Y ) @ ( transitive_trancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ R )
=> ( ( P2 @ Y )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B3 ) ) ) ) ).
% trancl_induct
thf(fact_946_r__r__into__trancl,axiom,
! [A2: a,B3: a,R3: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_trancl_a @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_947_converse__tranclE,axiom,
! [X3: a,Z2: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ ( transitive_trancl_a @ R ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ R )
=> ~ ! [Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_trancl_a @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_948_irrefl__trancl__rD,axiom,
! [R: set_Product_prod_a_a,X3: a,Y4: a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ ( transitive_trancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ( X3 != Y4 ) ) ) ).
% irrefl_trancl_rD
thf(fact_949_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_950_trancl__into__trancl2,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ ( transitive_trancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_951_trancl__trans__induct,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a,P2: a > a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ R ) )
=> ( ! [X2: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( P2 @ X2 @ Y ) )
=> ( ! [X2: a,Y: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_trancl_a @ R ) )
=> ( ( P2 @ X2 @ Y )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ ( transitive_trancl_a @ R ) )
=> ( ( P2 @ Y @ Z3 )
=> ( P2 @ X2 @ Z3 ) ) ) ) )
=> ( P2 @ X3 @ Y4 ) ) ) ) ).
% trancl_trans_induct
thf(fact_952_converse__trancl__induct,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( ! [Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ B3 ) @ R )
=> ( P2 @ Y ) )
=> ( ! [Y: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z3 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_953_IdE,axiom,
! [P: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ id_a )
=> ~ ! [X2: a] :
( P
!= ( product_Pair_a_a @ X2 @ X2 ) ) ) ).
% IdE
thf(fact_954_total__on__def,axiom,
( total_on_a
= ( ^ [A3: set_a,R2: set_Product_prod_a_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ( ( X != Y3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_955_total__onI,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ! [X2: a,Y: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y @ A )
=> ( ( X2 != Y )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ R ) ) ) ) )
=> ( total_on_a @ A @ R ) ) ).
% total_onI
thf(fact_956_total__onI,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ! [X2: product_prod_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ( member1426531477525435216od_a_a @ Y @ A )
=> ( ( X2 != Y )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Y ) @ R )
| ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ X2 ) @ R ) ) ) ) )
=> ( total_3048927264068743339od_a_a @ A @ R ) ) ).
% total_onI
thf(fact_957_Domain__iff,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( domain_a_a @ R ) )
= ( ? [Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_958_DomainE,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( domain_a_a @ R ) )
=> ~ ! [B7: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B7 ) @ R ) ) ).
% DomainE
thf(fact_959_Domain_ODomainI,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( member_a @ A2 @ ( domain_a_a @ R ) ) ) ).
% Domain.DomainI
thf(fact_960_Domain_Osimps,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( domain_a_a @ R ) )
= ( ? [A4: a,B4: a] :
( ( A2 = A4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_961_Domain_Ocases,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( domain_a_a @ R ) )
=> ~ ! [B7: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B7 ) @ R ) ) ).
% Domain.cases
thf(fact_962_Range__iff,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( range_a_a @ R ) )
= ( ? [Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_963_RangeE,axiom,
! [B3: a,R: set_Product_prod_a_a] :
( ( member_a @ B3 @ ( range_a_a @ R ) )
=> ~ ! [A7: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A7 @ B3 ) @ R ) ) ).
% RangeE
thf(fact_964_Range_Ointros,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( member_a @ B3 @ ( range_a_a @ R ) ) ) ).
% Range.intros
thf(fact_965_Range_Osimps,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( range_a_a @ R ) )
= ( ? [A4: a,B4: a] :
( ( A2 = B4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_966_Range_Ocases,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ ( range_a_a @ R ) )
=> ~ ! [A7: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A7 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_967_Id__onE,axiom,
! [C: produc3498347346309940967od_a_a,A: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ C @ ( id_on_7193410965977890500od_a_a @ A ) )
=> ~ ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( C
!= ( produc7886510207707329367od_a_a @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_968_Id__onE,axiom,
! [C: product_prod_a_a,A: set_a] :
( ( member1426531477525435216od_a_a @ C @ ( id_on_a @ A ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( C
!= ( product_Pair_a_a @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_969_Id__on__eqI,axiom,
! [A2: a,B3: a,A: set_a] :
( ( A2 = B3 )
=> ( ( member_a @ A2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( id_on_a @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_970_Id__on__eqI,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( A2 = B3 )
=> ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ ( id_on_7193410965977890500od_a_a @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_971_Id__on__iff,axiom,
! [X3: product_prod_a_a,Y4: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ ( id_on_7193410965977890500od_a_a @ A ) )
= ( ( X3 = Y4 )
& ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% Id_on_iff
thf(fact_972_Id__on__iff,axiom,
! [X3: a,Y4: a,A: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( id_on_a @ A ) )
= ( ( X3 = Y4 )
& ( member_a @ X3 @ A ) ) ) ).
% Id_on_iff
thf(fact_973_diff__shunt__var,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ( minus_6817036919807184750od_a_a @ X3 @ Y4 )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_974_Diff__insert,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( minus_6817036919807184750od_a_a @ ( minus_6817036919807184750od_a_a @ A @ B ) @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% Diff_insert
thf(fact_975_insert__Diff,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( insert4534936382041156343od_a_a @ A2 @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_976_Diff__insert2,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( minus_6817036919807184750od_a_a @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_977_Diff__insert__absorb,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_978_insert__remove__id,axiom,
! [X3: product_prod_a_a,X4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ X4 )
=> ( X4
= ( insert4534936382041156343od_a_a @ X3 @ ( minus_6817036919807184750od_a_a @ X4 @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% insert_remove_id
thf(fact_979_subset__Diff__insert,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( minus_6817036919807184750od_a_a @ B @ ( insert4534936382041156343od_a_a @ X3 @ C2 ) ) )
= ( ( ord_le746702958409616551od_a_a @ A @ ( minus_6817036919807184750od_a_a @ B @ C2 ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_980_trancl__rtrancl__trancl,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ ( transitive_rtrancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_981_rtrancl__trancl__trancl,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a,Z2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ ( transitive_trancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_982_rtrancl__into__trancl2,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ ( transitive_rtrancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_983_rtrancl__into__trancl1,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( transitive_trancl_a @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_984_rtrancl__eq__or__trancl,axiom,
! [X3: a,Y4: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R3 ) )
= ( ( X3 = Y4 )
| ( ( X3 != Y4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_985_trancl__into__rtrancl,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_986_tranclD2,axiom,
! [X3: a,Y4: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ R3 ) )
=> ? [Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ ( transitive_rtrancl_a @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z3 @ Y4 ) @ R3 ) ) ) ).
% tranclD2
thf(fact_987_rtranclD,axiom,
! [A2: a,B3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ R3 ) )
=> ( ( A2 = B3 )
| ( ( A2 != B3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_trancl_a @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_988_tranclD,axiom,
! [X3: a,Y4: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ R3 ) )
=> ? [Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z3 @ Y4 ) @ ( transitive_rtrancl_a @ R3 ) ) ) ) ).
% tranclD
thf(fact_989_rtrancl__Un__separator__converseE,axiom,
! [A2: a,B3: a,P2: set_Product_prod_a_a,Q: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ ( sup_su3048258781599657691od_a_a @ P2 @ Q ) ) )
=> ( ! [X2: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ B3 ) @ ( transitive_rtrancl_a @ P2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Q )
=> ( Y = X2 ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_990_rtrancl__Un__separatorE,axiom,
! [A2: a,B3: a,P2: set_Product_prod_a_a,Q: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ ( sup_su3048258781599657691od_a_a @ P2 @ Q ) ) )
=> ( ! [X2: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ X2 ) @ ( transitive_rtrancl_a @ P2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Q )
=> ( X2 = Y ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( transitive_rtrancl_a @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_991_Not__Domain__rtrancl,axiom,
! [X3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,Y4: product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( domain990338261037515724od_a_a @ R3 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ ( transi7229704755246292010od_a_a @ R3 ) )
= ( X3 = Y4 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_992_Not__Domain__rtrancl,axiom,
! [X3: a,R3: set_Product_prod_a_a,Y4: a] :
( ~ ( member_a @ X3 @ ( domain_a_a @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R3 ) )
= ( X3 = Y4 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_993_is__singleton__the__elem,axiom,
( is_sin3171834905898671131od_a_a
= ( ^ [A3: set_Product_prod_a_a] :
( A3
= ( insert4534936382041156343od_a_a @ ( the_el8589169208993665564od_a_a @ A3 ) @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_994_is__singletonI_H,axiom,
! [A: set_Product_prod_a_a] :
( ( A != bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ( member1426531477525435216od_a_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_sin3171834905898671131od_a_a @ A ) ) ) ).
% is_singletonI'
thf(fact_995_finite__empty__induct,axiom,
! [A: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( P2 @ A )
=> ( ! [A7: product_prod_a_a,A6: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A6 )
=> ( ( member1426531477525435216od_a_a @ A7 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_6817036919807184750od_a_a @ A6 @ ( insert4534936382041156343od_a_a @ A7 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) )
=> ( P2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% finite_empty_induct
thf(fact_996_infinite__coinduct,axiom,
! [X4: set_Product_prod_a_a > $o,A: set_Product_prod_a_a] :
( ( X4 @ A )
=> ( ! [A6: set_Product_prod_a_a] :
( ( X4 @ A6 )
=> ? [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A6 )
& ( ( X4 @ ( minus_6817036919807184750od_a_a @ A6 @ ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) )
| ~ ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ A6 @ ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) )
=> ~ ( finite6544458595007987280od_a_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_997_infinite__remove,axiom,
! [S2: set_Product_prod_a_a,A2: product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S2 )
=> ~ ( finite6544458595007987280od_a_a @ ( minus_6817036919807184750od_a_a @ S2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% infinite_remove
thf(fact_998_Diff__single__insert,axiom,
! [A: set_Product_prod_a_a,X3: product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) @ B )
=> ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_999_subset__insert__iff,axiom,
! [A: set_Product_prod_a_a,X3: product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( ( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) @ B ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1000_finite__remove__induct,axiom,
! [B: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( P2 @ bot_bo3357376287454694259od_a_a )
=> ( ! [A6: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A6 )
=> ( ( A6 != bot_bo3357376287454694259od_a_a )
=> ( ( ord_le746702958409616551od_a_a @ A6 @ B )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A6 )
=> ( P2 @ ( minus_6817036919807184750od_a_a @ A6 @ ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1001_remove__induct,axiom,
! [P2: set_Product_prod_a_a > $o,B: set_Product_prod_a_a] :
( ( P2 @ bot_bo3357376287454694259od_a_a )
=> ( ( ~ ( finite6544458595007987280od_a_a @ B )
=> ( P2 @ B ) )
=> ( ! [A6: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A6 )
=> ( ( A6 != bot_bo3357376287454694259od_a_a )
=> ( ( ord_le746702958409616551od_a_a @ A6 @ B )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A6 )
=> ( P2 @ ( minus_6817036919807184750od_a_a @ A6 @ ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1002_is__singleton__def,axiom,
( is_sin3171834905898671131od_a_a
= ( ^ [A3: set_Product_prod_a_a] :
? [X: product_prod_a_a] :
( A3
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% is_singleton_def
thf(fact_1003_is__singletonE,axiom,
! [A: set_Product_prod_a_a] :
( ( is_sin3171834905898671131od_a_a @ A )
=> ~ ! [X2: product_prod_a_a] :
( A
!= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% is_singletonE
thf(fact_1004_Total__subset__Id,axiom,
! [R: set_Product_prod_a_a] :
( ( total_on_a @ ( field_a @ R ) @ R )
=> ( ( ord_le746702958409616551od_a_a @ R @ id_a )
=> ( ( R = bot_bo3357376287454694259od_a_a )
| ? [A7: a] :
( R
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A7 @ A7 ) @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1005_Total__Id__Field,axiom,
! [R: set_Product_prod_a_a] :
( ( total_on_a @ ( field_a @ R ) @ R )
=> ( ~ ( ord_le746702958409616551od_a_a @ R @ id_a )
=> ( ( field_a @ R )
= ( field_a @ ( minus_6817036919807184750od_a_a @ R @ id_a ) ) ) ) ) ).
% Total_Id_Field
thf(fact_1006_linear__order__on__singleton,axiom,
! [X3: a] : ( order_8768733634509060147r_on_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% linear_order_on_singleton
thf(fact_1007_linear__order__on__singleton,axiom,
! [X3: product_prod_a_a] : ( order_435519263512085852od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ X3 ) @ bot_bo510284599550014259od_a_a ) ) ).
% linear_order_on_singleton
thf(fact_1008_remove__def,axiom,
( remove8198300757409973004od_a_a
= ( ^ [X: product_prod_a_a,A3: set_Product_prod_a_a] : ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% remove_def
thf(fact_1009_Linear__order__in__diff__Id,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( order_435519263512085852od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R )
= ( ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ ( minus_2483622581487243566od_a_a @ R @ id_Product_prod_a_a ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1010_Linear__order__in__diff__Id,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
= ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ ( minus_6817036919807184750od_a_a @ R @ id_a ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1011_member__remove,axiom,
! [X3: product_prod_a_a,Y4: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( remove8198300757409973004od_a_a @ Y4 @ A ) )
= ( ( member1426531477525435216od_a_a @ X3 @ A )
& ( X3 != Y4 ) ) ) ).
% member_remove
thf(fact_1012_lnear__order__on__empty,axiom,
order_8768733634509060147r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% lnear_order_on_empty
thf(fact_1013_lnear__order__on__empty,axiom,
order_435519263512085852od_a_a @ bot_bo3357376287454694259od_a_a @ bot_bo510284599550014259od_a_a ).
% lnear_order_on_empty
thf(fact_1014_finite__Linear__order__induct,axiom,
! [R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( order_435519263512085852od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( finite20598220287489552od_a_a @ R )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( order_4352681651421660434od_a_a @ R @ X2 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X2 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_1015_finite__Linear__order__induct,axiom,
! [R: set_Product_prod_a_a,X3: a,P2: a > $o] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ X3 @ ( field_a @ R ) )
=> ( ( finite6544458595007987280od_a_a @ R )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( field_a @ R ) )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ ( order_aboveS_a @ R @ X2 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X2 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_1016_Linear__order__wf__diff__Id,axiom,
! [R: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( wf_a @ ( minus_6817036919807184750od_a_a @ R @ id_a ) )
= ( ! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( field_a @ R ) )
=> ( ( A3 != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1017_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( order_435519263512085852od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( wf_Product_prod_a_a @ ( minus_2483622581487243566od_a_a @ R @ id_Product_prod_a_a ) )
= ( ! [A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( A3 != bot_bo3357376287454694259od_a_a )
=> ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
& ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ A3 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1018_acyclic__insert,axiom,
! [Y4: a,X3: a,R: set_Product_prod_a_a] :
( ( transitive_acyclic_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R ) )
= ( ( transitive_acyclic_a @ R )
& ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).
% acyclic_insert
thf(fact_1019_acyclic__def,axiom,
( transitive_acyclic_a
= ( ^ [R2: set_Product_prod_a_a] :
! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% acyclic_def
thf(fact_1020_acyclicI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ ( transitive_trancl_a @ R ) )
=> ( transitive_acyclic_a @ R ) ) ).
% acyclicI
thf(fact_1021_wf__eq__minimal2,axiom,
( wf_a
= ( ^ [R2: set_Product_prod_a_a] :
! [A3: set_a] :
( ( ( ord_less_eq_set_a @ A3 @ ( field_a @ R2 ) )
& ( A3 != bot_bot_set_a ) )
=> ? [X: a] :
( ( member_a @ X @ A3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R2 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1022_wf__eq__minimal2,axiom,
( wf_Product_prod_a_a
= ( ^ [R2: set_Pr8600417178894128327od_a_a] :
! [A3: set_Product_prod_a_a] :
( ( ( ord_le746702958409616551od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ R2 ) )
& ( A3 != bot_bo3357376287454694259od_a_a ) )
=> ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
& ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ A3 )
=> ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X ) @ R2 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1023_wf__insert,axiom,
! [Y4: a,X3: a,R: set_Product_prod_a_a] :
( ( wf_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R ) )
= ( ( wf_a @ R )
& ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).
% wf_insert
thf(fact_1024_wf__Un,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( wf_a @ R )
=> ( ( wf_a @ S )
=> ( ( ( inf_inf_set_a @ ( domain_a_a @ R ) @ ( range_a_a @ S ) )
= bot_bot_set_a )
=> ( wf_a @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ) ).
% wf_Un
thf(fact_1025_wf__Un,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( wf_Product_prod_a_a @ R )
=> ( ( wf_Product_prod_a_a @ S )
=> ( ( ( inf_in8905007599844390133od_a_a @ ( domain990338261037515724od_a_a @ R ) @ ( range_2510312826226007605od_a_a @ S ) )
= bot_bo3357376287454694259od_a_a )
=> ( wf_Product_prod_a_a @ ( sup_su8193676293155882651od_a_a @ R @ S ) ) ) ) ) ).
% wf_Un
thf(fact_1026_wf__empty,axiom,
wf_a @ bot_bo3357376287454694259od_a_a ).
% wf_empty
thf(fact_1027_wf__trancl,axiom,
! [R: set_Product_prod_a_a] :
( ( wf_a @ R )
=> ( wf_a @ ( transitive_trancl_a @ R ) ) ) ).
% wf_trancl
thf(fact_1028_wfE__min_H,axiom,
! [R3: set_Product_prod_a_a,Q: set_a] :
( ( wf_a @ R3 )
=> ( ( Q != bot_bot_set_a )
=> ~ ! [Z3: a] :
( ( member_a @ Z3 @ Q )
=> ~ ! [Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_a @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1029_wfE__min_H,axiom,
! [R3: set_Pr8600417178894128327od_a_a,Q: set_Product_prod_a_a] :
( ( wf_Product_prod_a_a @ R3 )
=> ( ( Q != bot_bo3357376287454694259od_a_a )
=> ~ ! [Z3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z3 @ Q )
=> ~ ! [Y5: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z3 ) @ R3 )
=> ~ ( member1426531477525435216od_a_a @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1030_wf__iff__acyclic__if__finite,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( ( wf_a @ R )
= ( transitive_acyclic_a @ R ) ) ) ).
% wf_iff_acyclic_if_finite
thf(fact_1031_finite__acyclic__wf,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( ( transitive_acyclic_a @ R )
=> ( wf_a @ R ) ) ) ).
% finite_acyclic_wf
thf(fact_1032_partial__order__on__well__order__on,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( ( order_5272072345360262643r_on_a @ A @ R )
=> ( wf_a @ ( minus_6817036919807184750od_a_a @ R @ id_a ) ) ) ) ).
% partial_order_on_well_order_on
thf(fact_1033_refl__on__singleton,axiom,
! [X3: a] : ( refl_on_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_1034_refl__on__singleton,axiom,
! [X3: product_prod_a_a] : ( refl_o7745108929832855590od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ X3 ) @ bot_bo510284599550014259od_a_a ) ) ).
% refl_on_singleton
thf(fact_1035_finite__Partial__order__induct,axiom,
! [R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( order_7408868903334687516od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( finite20598220287489552od_a_a @ R )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( order_4352681651421660434od_a_a @ R @ X2 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X2 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1036_finite__Partial__order__induct,axiom,
! [R: set_Product_prod_a_a,X3: a,P2: a > $o] :
( ( order_5272072345360262643r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ X3 @ ( field_a @ R ) )
=> ( ( finite6544458595007987280od_a_a @ R )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( field_a @ R ) )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ ( order_aboveS_a @ R @ X2 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X2 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1037_refl__onD2,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R )
=> ( member1426531477525435216od_a_a @ Y4 @ A ) ) ) ).
% refl_onD2
thf(fact_1038_refl__onD2,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
( ( refl_on_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ( member_a @ Y4 @ A ) ) ) ).
% refl_onD2
thf(fact_1039_refl__onD1,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R )
=> ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1040_refl__onD1,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
( ( refl_on_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ( member_a @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1041_refl__onD,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ A @ R )
=> ( ( member_a @ A2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1042_refl__onD,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1043_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% refl_on_empty
thf(fact_1044_refl__on__empty,axiom,
refl_o7745108929832855590od_a_a @ bot_bo3357376287454694259od_a_a @ bot_bo510284599550014259od_a_a ).
% refl_on_empty
thf(fact_1045_partial__order__on__empty,axiom,
order_5272072345360262643r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% partial_order_on_empty
thf(fact_1046_partial__order__on__empty,axiom,
order_7408868903334687516od_a_a @ bot_bo3357376287454694259od_a_a @ bot_bo510284599550014259od_a_a ).
% partial_order_on_empty
thf(fact_1047_wo__rel_Ocases__Total3,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) @ ( field_a @ R ) )
=> ( ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( minus_6817036919807184750od_a_a @ R @ id_a ) )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ ( minus_6817036919807184750od_a_a @ R @ id_a ) ) )
=> ( Phi @ A2 @ B3 ) )
=> ( ( ( A2 = B3 )
=> ( Phi @ A2 @ B3 ) )
=> ( Phi @ A2 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1048_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a,Phi: product_prod_a_a > product_prod_a_a > $o] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ ( minus_2483622581487243566od_a_a @ R @ id_Product_prod_a_a ) )
| ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ ( minus_2483622581487243566od_a_a @ R @ id_Product_prod_a_a ) ) )
=> ( Phi @ A2 @ B3 ) )
=> ( ( ( A2 = B3 )
=> ( Phi @ A2 @ B3 ) )
=> ( Phi @ A2 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1049_wf__Union,axiom,
! [R3: set_se5735800977113168103od_a_a] :
( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ R3 )
=> ( wf_a @ X2 ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ R3 )
=> ! [Xa2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa2 @ R3 )
=> ( ( X2 != Xa2 )
=> ( ( inf_inf_set_a @ ( domain_a_a @ X2 ) @ ( range_a_a @ Xa2 ) )
= bot_bot_set_a ) ) ) )
=> ( wf_a @ ( comple8421679170691845492od_a_a @ R3 ) ) ) ) ).
% wf_Union
thf(fact_1050_wf__Union,axiom,
! [R3: set_se7936120678642572967od_a_a] :
( ! [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ R3 )
=> ( wf_Product_prod_a_a @ X2 ) )
=> ( ! [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ R3 )
=> ! [Xa2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ Xa2 @ R3 )
=> ( ( X2 != Xa2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( domain990338261037515724od_a_a @ X2 ) @ ( range_2510312826226007605od_a_a @ Xa2 ) )
= bot_bo3357376287454694259od_a_a ) ) ) )
=> ( wf_Product_prod_a_a @ ( comple6965155415733691188od_a_a @ R3 ) ) ) ) ).
% wf_Union
thf(fact_1051_in__quotient__imp__in__rel,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X4: set_Product_prod_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) @ X4 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1052_in__quotient__imp__in__rel,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ X4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1053_finite__Union,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ! [M: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ M @ A )
=> ( finite6544458595007987280od_a_a @ M ) )
=> ( finite6544458595007987280od_a_a @ ( comple8421679170691845492od_a_a @ A ) ) ) ) ).
% finite_Union
thf(fact_1054_wo__rel_OREFL,axiom,
! [R: set_Product_prod_a_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( refl_on_a @ ( field_a @ R ) @ R ) ) ).
% wo_rel.REFL
thf(fact_1055_wo__rel_Oin__notinI,axiom,
! [R: set_Pr8600417178894128327od_a_a,J: product_prod_a_a,I: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member1426531477525435216od_a_a @ I @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ J @ ( field_1126092520709947252od_a_a @ R ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1056_wo__rel_Oin__notinI,axiom,
! [R: set_Product_prod_a_a,J: a,I: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member_a @ I @ ( field_a @ R ) )
=> ( ( member_a @ J @ ( field_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1057_wo__rel_OTOTALS,axiom,
! [R: set_Product_prod_a_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ! [X5: a] :
( ( member_a @ X5 @ ( field_a @ R ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( field_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ R )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X5 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1058_wo__rel_OTOTAL,axiom,
! [R: set_Product_prod_a_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( total_on_a @ ( field_a @ R ) @ R ) ) ).
% wo_rel.TOTAL
thf(fact_1059_Union__quotient,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( comple2307003609928055243_set_a @ ( equiv_quotient_a @ A @ R ) )
= A ) ) ).
% Union_quotient
thf(fact_1060_finite__UnionD,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( finite6544458595007987280od_a_a @ ( comple8421679170691845492od_a_a @ A ) )
=> ( finite8717734299975451184od_a_a @ A ) ) ).
% finite_UnionD
thf(fact_1061_wo__rel_OLIN,axiom,
! [R: set_Product_prod_a_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R ) ) ).
% wo_rel.LIN
thf(fact_1062_insert__partition,axiom,
! [X3: set_Product_prod_a_a,F2: set_se5735800977113168103od_a_a] :
( ~ ( member1816616512716248880od_a_a @ X3 @ F2 )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ ( insert914553114930139863od_a_a @ X3 @ F2 ) )
=> ! [Xa2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa2 @ ( insert914553114930139863od_a_a @ X3 @ F2 ) )
=> ( ( X2 != Xa2 )
=> ( ( inf_in8905007599844390133od_a_a @ X2 @ Xa2 )
= bot_bo3357376287454694259od_a_a ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X3 @ ( comple8421679170691845492od_a_a @ F2 ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_partition
thf(fact_1063_well__order__induct__imp,axiom,
! [R: set_Pr8600417178894128327od_a_a,P2: product_prod_a_a > $o,A2: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ! [X2: product_prod_a_a] :
( ! [Y5: product_prod_a_a] :
( ( ( Y5 != X2 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ X2 ) @ R ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( P2 @ Y5 ) ) )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( P2 @ X2 ) ) )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( P2 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1064_well__order__induct__imp,axiom,
! [R: set_Product_prod_a_a,P2: a > $o,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ! [X2: a] :
( ! [Y5: a] :
( ( ( Y5 != X2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ X2 ) @ R ) )
=> ( ( member_a @ Y5 @ ( field_a @ R ) )
=> ( P2 @ Y5 ) ) )
=> ( ( member_a @ X2 @ ( field_a @ R ) )
=> ( P2 @ X2 ) ) )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( P2 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1065_in__quotient__imp__closed,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X4: set_Product_prod_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( member1426531477525435216od_a_a @ X3 @ X4 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R )
=> ( member1426531477525435216od_a_a @ Y4 @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_1066_in__quotient__imp__closed,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( member_a @ X3 @ X4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ( member_a @ Y4 @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_1067_quotient__eq__iff,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X4: set_Product_prod_a_a,Y6: set_Product_prod_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( member1816616512716248880od_a_a @ Y6 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( member1426531477525435216od_a_a @ X3 @ X4 )
=> ( ( member1426531477525435216od_a_a @ Y4 @ Y6 )
=> ( ( X4 = Y6 )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_1068_quotient__eq__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a,Y6: set_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( member_a @ X3 @ X4 )
=> ( ( member_a @ Y4 @ Y6 )
=> ( ( X4 = Y6 )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_1069_quotient__eqI,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X4: set_Product_prod_a_a,Y6: set_Product_prod_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( member1816616512716248880od_a_a @ Y6 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( member1426531477525435216od_a_a @ X3 @ X4 )
=> ( ( member1426531477525435216od_a_a @ Y4 @ Y6 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_1070_quotient__eqI,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a,Y6: set_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( member_a @ X3 @ X4 )
=> ( ( member_a @ Y4 @ Y6 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_1071_in__quotient__imp__non__empty,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X4: set_Product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( X4 != bot_bo3357376287454694259od_a_a ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_1072_in__quotient__imp__non__empty,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( X4 != bot_bot_set_a ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_1073_in__quotient__imp__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( ord_less_eq_set_a @ X4 @ A ) ) ) ).
% in_quotient_imp_subset
thf(fact_1074_finite__refines__finite,axiom,
! [A: set_a,R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( finite_finite_set_a @ ( equiv_quotient_a @ A @ R3 ) )
=> ( ( ord_le746702958409616551od_a_a @ R3 @ S2 )
=> ( ( equiv_equiv_a @ A @ R3 )
=> ( ( equiv_equiv_a @ A @ S2 )
=> ( finite_finite_set_a @ ( equiv_quotient_a @ A @ S2 ) ) ) ) ) ) ).
% finite_refines_finite
thf(fact_1075_quotient__disj,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X4: set_Product_prod_a_a,Y6: set_Product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( member1816616512716248880od_a_a @ Y6 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ( ( X4 = Y6 )
| ( ( inf_in8905007599844390133od_a_a @ X4 @ Y6 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% quotient_disj
thf(fact_1076_quotient__disj,axiom,
! [A: set_a,R: set_Product_prod_a_a,X4: set_a,Y6: set_a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A @ R ) )
=> ( ( X4 = Y6 )
| ( ( inf_inf_set_a @ X4 @ Y6 )
= bot_bot_set_a ) ) ) ) ) ).
% quotient_disj
thf(fact_1077_wo__rel_Ocases__Total,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) @ ( field_a @ R ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( Phi @ A2 @ B3 ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R )
=> ( Phi @ A2 @ B3 ) )
=> ( Phi @ A2 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1078_wo__rel_Ocases__Total,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a,Phi: product_prod_a_a > product_prod_a_a > $o] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R )
=> ( Phi @ A2 @ B3 ) )
=> ( ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ R )
=> ( Phi @ A2 @ B3 ) )
=> ( Phi @ A2 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1079_eq__equiv__class__iff2,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( member1426531477525435216od_a_a @ Y4 @ A )
=> ( ( ( equiv_6838630135511005422od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) @ R )
= ( equiv_6838630135511005422od_a_a @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) @ R ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_1080_eq__equiv__class__iff2,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( ( ( equiv_quotient_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ R )
= ( equiv_quotient_a @ ( insert_a @ Y4 @ bot_bot_set_a ) @ R ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_1081_Sup__bot__conv_I1_J,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( ( comple8421679170691845492od_a_a @ A )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ A )
=> ( X = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1082_Sup__bot__conv_I2_J,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ A ) )
= ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ A )
=> ( X = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1083_Sup__empty,axiom,
( ( comple8421679170691845492od_a_a @ bot_bo777872063958040403od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Sup_empty
thf(fact_1084_empty__Union__conv,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ A ) )
= ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ A )
=> ( X = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% empty_Union_conv
thf(fact_1085_Union__empty__conv,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( ( comple8421679170691845492od_a_a @ A )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ A )
=> ( X = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Union_empty_conv
thf(fact_1086_Union__disjoint,axiom,
! [C2: set_se5735800977113168103od_a_a,A: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ C2 ) @ A )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ X @ A )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Union_disjoint
thf(fact_1087_Union__empty,axiom,
( ( comple8421679170691845492od_a_a @ bot_bo777872063958040403od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Union_empty
thf(fact_1088_finite__subset__Union,axiom,
! [A: set_Product_prod_a_a,B8: set_se5735800977113168103od_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ ( comple8421679170691845492od_a_a @ B8 ) )
=> ~ ! [F4: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ F4 )
=> ( ( ord_le1995061765932249223od_a_a @ F4 @ B8 )
=> ~ ( ord_le746702958409616551od_a_a @ A @ ( comple8421679170691845492od_a_a @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1089_Sup__inf__eq__bot__iff,axiom,
! [B: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ B ) @ A2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ B )
=> ( ( inf_in8905007599844390133od_a_a @ X @ A2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1090_proj__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ A )
=> ( ( ( equiv_proj_a_a @ R @ X3 )
= ( equiv_proj_a_a @ R @ Y4 ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_1091_proj__iff,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) @ A )
=> ( ( ( equiv_769305369208038986od_a_a @ R @ X3 )
= ( equiv_769305369208038986od_a_a @ R @ Y4 ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_1092_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Product_prod_a_a,B: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( B != bot_bot_set_a )
=> ? [X_1: a] : ( bNF_We6697304935525757620inim_a @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1093_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( B != bot_bo3357376287454694259od_a_a )
=> ? [X_1: product_prod_a_a] : ( bNF_We207971382126712541od_a_a @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1094_wo__rel_Ominim__in,axiom,
! [R: set_Product_prod_a_a,B: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( B != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1095_wo__rel_Ominim__in,axiom,
! [R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( B != bot_bo3357376287454694259od_a_a )
=> ( member1426531477525435216od_a_a @ ( bNF_We8688793021278315251od_a_a @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1096_proj__preserves,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( member1816616512716248880od_a_a @ ( equiv_769305369208038986od_a_a @ R @ X3 ) @ ( equiv_6838630135511005422od_a_a @ A @ R ) ) ) ).
% proj_preserves
thf(fact_1097_proj__preserves,axiom,
! [X3: a,A: set_a,R: set_Product_prod_a_a] :
( ( member_a @ X3 @ A )
=> ( member_set_a @ ( equiv_proj_a_a @ R @ X3 ) @ ( equiv_quotient_a @ A @ R ) ) ) ).
% proj_preserves
thf(fact_1098_wo__rel_Ominim__isMinim,axiom,
! [R: set_Product_prod_a_a,B: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( B != bot_bot_set_a )
=> ( bNF_We6697304935525757620inim_a @ R @ B @ ( bNF_We5615626441682584778inim_a @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1099_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( B != bot_bo3357376287454694259od_a_a )
=> ( bNF_We207971382126712541od_a_a @ R @ B @ ( bNF_We8688793021278315251od_a_a @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1100_proj__in__iff,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1816616512716248880od_a_a @ ( equiv_769305369208038986od_a_a @ R @ X3 ) @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
= ( member1426531477525435216od_a_a @ X3 @ A ) ) ) ).
% proj_in_iff
thf(fact_1101_proj__in__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_set_a @ ( equiv_proj_a_a @ R @ X3 ) @ ( equiv_quotient_a @ A @ R ) )
= ( member_a @ X3 @ A ) ) ) ).
% proj_in_iff
thf(fact_1102_wo__rel_Ominim__least,axiom,
! [R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ B )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( bNF_We8688793021278315251od_a_a @ R @ B ) @ B3 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1103_wo__rel_Ominim__least,axiom,
! [R: set_Product_prod_a_a,B: set_a,B3: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ B )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R @ B ) @ B3 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1104_wo__rel_Oequals__minim,axiom,
! [R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ A2 @ B )
=> ( ! [B7: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B7 @ B )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B7 ) @ R ) )
=> ( A2
= ( bNF_We8688793021278315251od_a_a @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1105_wo__rel_Oequals__minim,axiom,
! [R: set_Product_prod_a_a,B: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( member_a @ A2 @ B )
=> ( ! [B7: a] :
( ( member_a @ B7 @ B )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B7 ) @ R ) )
=> ( A2
= ( bNF_We5615626441682584778inim_a @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1106_wo__rel_Ominim__inField,axiom,
! [R: set_Product_prod_a_a,B: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( B != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R @ B ) @ ( field_a @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1107_wo__rel_Ominim__inField,axiom,
! [R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( B != bot_bo3357376287454694259od_a_a )
=> ( member1426531477525435216od_a_a @ ( bNF_We8688793021278315251od_a_a @ R @ B ) @ ( field_1126092520709947252od_a_a @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1108_relImage__proj,axiom,
! [A: set_a,R3: set_Product_prod_a_a] :
( ( equiv_equiv_a @ A @ R3 )
=> ( ord_le8376522849517564071_set_a @ ( bNF_Gr2835393858935354180_set_a @ R3 @ ( equiv_proj_a_a @ R3 ) ) @ ( id_on_set_a @ ( equiv_quotient_a @ A @ R3 ) ) ) ) ).
% relImage_proj
thf(fact_1109_wo__rel_Omax2__greater__among,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 ) ) @ R )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 ) ) @ R )
& ( member_a @ ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 ) @ ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1110_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 ) ) @ R )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 ) ) @ R )
& ( member1426531477525435216od_a_a @ ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 ) @ ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1111_wo__rel_Omax2__among,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( member_a @ ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 ) @ ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1112_wo__rel_Omax2__among,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 ) @ ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1113_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 )
= A2 )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1114_wo__rel_Omax2__equals1,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 )
= A2 )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1115_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 )
= B3 )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1116_wo__rel_Omax2__equals2,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 )
= B3 )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1117_wo__rel_Omax2__greater,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( bNF_We2656984769232325907od_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 ) ) @ R )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ ( bNF_We7050438333583422581od_a_a @ R @ A2 @ B3 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1118_wo__rel_Omax2__greater,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 ) ) @ R )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ ( bNF_We3763454674811381836max2_a @ R @ A2 @ B3 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1119_singleton__quotient,axiom,
! [X3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( equiv_6838630135511005422od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) @ R )
= ( insert914553114930139863od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) @ bot_bo777872063958040403od_a_a ) ) ).
% singleton_quotient
thf(fact_1120_singleton__quotient,axiom,
! [X3: a,R: set_Product_prod_a_a] :
( ( equiv_quotient_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ R )
= ( insert_set_a @ ( image_a_a @ R @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ bot_bot_set_set_a ) ) ).
% singleton_quotient
thf(fact_1121_irrefl__tranclI,axiom,
! [R: set_Product_prod_a_a,X3: a] :
( ( ( inf_in8905007599844390133od_a_a @ ( converse_a_a @ R ) @ ( transitive_rtrancl_a @ R ) )
= bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ ( transitive_trancl_a @ R ) ) ) ).
% irrefl_tranclI
thf(fact_1122_ImageI,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R @ A ) ) ) ) ).
% ImageI
thf(fact_1123_ImageI,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a,A: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( ( member_a @ A2 @ A )
=> ( member_a @ B3 @ ( image_a_a @ R @ A ) ) ) ) ).
% ImageI
thf(fact_1124_converse__iff,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( converse_a_a @ R ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_1125_Image__empty2,axiom,
! [R3: set_Pr8600417178894128327od_a_a] :
( ( image_9076584400576816019od_a_a @ R3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Image_empty2
thf(fact_1126_Field__converse,axiom,
! [R: set_Product_prod_a_a] :
( ( field_a @ ( converse_a_a @ R ) )
= ( field_a @ R ) ) ).
% Field_converse
thf(fact_1127_converse__empty,axiom,
( ( converse_a_a @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% converse_empty
thf(fact_1128_finite__converse,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( converse_a_a @ R ) )
= ( finite6544458595007987280od_a_a @ R ) ) ).
% finite_converse
thf(fact_1129_Image__empty1,axiom,
! [X4: set_a] :
( ( image_a_a @ bot_bo3357376287454694259od_a_a @ X4 )
= bot_bot_set_a ) ).
% Image_empty1
thf(fact_1130_Range__converse,axiom,
! [R: set_Product_prod_a_a] :
( ( range_a_a @ ( converse_a_a @ R ) )
= ( domain_a_a @ R ) ) ).
% Range_converse
thf(fact_1131_Domain__converse,axiom,
! [R: set_Product_prod_a_a] :
( ( domain_a_a @ ( converse_a_a @ R ) )
= ( range_a_a @ R ) ) ).
% Domain_converse
thf(fact_1132_Image__singleton__iff,axiom,
! [B3: a,R: set_Product_prod_a_a,A2: a] :
( ( member_a @ B3 @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1133_Image__singleton__iff,axiom,
! [B3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1134_rtrancl__converseD,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ ( converse_a_a @ R ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ ( transitive_rtrancl_a @ R ) ) ) ).
% rtrancl_converseD
thf(fact_1135_rtrancl__converseI,axiom,
! [Y4: a,X3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ ( transitive_rtrancl_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ ( converse_a_a @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_1136_trancl__converse,axiom,
! [R: set_Product_prod_a_a] :
( ( transitive_trancl_a @ ( converse_a_a @ R ) )
= ( converse_a_a @ ( transitive_trancl_a @ R ) ) ) ).
% trancl_converse
thf(fact_1137_rev__ImageI,axiom,
! [A2: a,A: set_a,B3: a,R: set_Product_prod_a_a] :
( ( member_a @ A2 @ A )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( member_a @ B3 @ ( image_a_a @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1138_rev__ImageI,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R @ A ) ) ) ) ).
% rev_ImageI
thf(fact_1139_converseI,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ ( converse_a_a @ R ) ) ) ).
% converseI
thf(fact_1140_converseE,axiom,
! [Yx: product_prod_a_a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Yx @ ( converse_a_a @ R ) )
=> ~ ! [X2: a,Y: a] :
( ( Yx
= ( product_Pair_a_a @ Y @ X2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) ) ) ).
% converseE
thf(fact_1141_converseD,axiom,
! [A2: a,B3: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( converse_a_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R ) ) ).
% converseD
thf(fact_1142_Image__iff,axiom,
! [B3: a,R: set_Product_prod_a_a,A: set_a] :
( ( member_a @ B3 @ ( image_a_a @ R @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ B3 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_1143_ImageE,axiom,
! [B3: a,R: set_Product_prod_a_a,A: set_a] :
( ( member_a @ B3 @ ( image_a_a @ R @ A ) )
=> ~ ! [X2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ B3 ) @ R )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% ImageE
thf(fact_1144_ImageE,axiom,
! [B3: product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R @ A ) )
=> ~ ! [X2: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ B3 ) @ R )
=> ~ ( member1426531477525435216od_a_a @ X2 @ A ) ) ) ).
% ImageE
thf(fact_1145_converse_Osimps,axiom,
! [A1: a,A22: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( converse_a_a @ R ) )
= ( ? [A4: a,B4: a] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_1146_converse_Ocases,axiom,
! [A1: a,A22: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( converse_a_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_1147_trancl__converseI,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( converse_a_a @ ( transitive_trancl_a @ R ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ ( converse_a_a @ R ) ) ) ) ).
% trancl_converseI
thf(fact_1148_trancl__converseD,axiom,
! [X3: a,Y4: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_trancl_a @ ( converse_a_a @ R ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( converse_a_a @ ( transitive_trancl_a @ R ) ) ) ) ).
% trancl_converseD
thf(fact_1149_finite__Image,axiom,
! [R3: set_Product_prod_a_a,A: set_a] :
( ( finite6544458595007987280od_a_a @ R3 )
=> ( finite_finite_a @ ( image_a_a @ R3 @ A ) ) ) ).
% finite_Image
thf(fact_1150_wf__converse__trancl,axiom,
! [R: set_Product_prod_a_a] :
( ( wf_a @ ( converse_a_a @ R ) )
=> ( wf_a @ ( converse_a_a @ ( transitive_trancl_a @ R ) ) ) ) ).
% wf_converse_trancl
thf(fact_1151_wfI__pf,axiom,
! [R3: set_Pr8600417178894128327od_a_a] :
( ! [A6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A6 @ ( image_9076584400576816019od_a_a @ R3 @ A6 ) )
=> ( A6 = bot_bo3357376287454694259od_a_a ) )
=> ( wf_Product_prod_a_a @ R3 ) ) ).
% wfI_pf
thf(fact_1152_wfE__pf,axiom,
! [R3: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( wf_Product_prod_a_a @ R3 )
=> ( ( ord_le746702958409616551od_a_a @ A @ ( image_9076584400576816019od_a_a @ R3 @ A ) )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ) ).
% wfE_pf
thf(fact_1153_equiv__class__self,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( member1426531477525435216od_a_a @ A2 @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% equiv_class_self
thf(fact_1154_quotientE,axiom,
! [X4: set_Product_prod_a_a,A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ ( equiv_6838630135511005422od_a_a @ A @ R ) )
=> ~ ! [X2: product_prod_a_a] :
( ( X4
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) )
=> ~ ( member1426531477525435216od_a_a @ X2 @ A ) ) ) ).
% quotientE
thf(fact_1155_quotientE,axiom,
! [X4: set_a,A: set_a,R: set_Product_prod_a_a] :
( ( member_set_a @ X4 @ ( equiv_quotient_a @ A @ R ) )
=> ~ ! [X2: a] :
( ( X4
= ( image_a_a @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% quotientE
thf(fact_1156_quotientI,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( member1816616512716248880od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) @ ( equiv_6838630135511005422od_a_a @ A @ R ) ) ) ).
% quotientI
thf(fact_1157_quotientI,axiom,
! [X3: a,A: set_a,R: set_Product_prod_a_a] :
( ( member_a @ X3 @ A )
=> ( member_set_a @ ( image_a_a @ R @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ ( equiv_quotient_a @ A @ R ) ) ) ).
% quotientI
thf(fact_1158_finite__rtrancl__Image,axiom,
! [R3: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( finite20598220287489552od_a_a @ R3 )
=> ( ( finite6544458595007987280od_a_a @ A )
=> ( finite6544458595007987280od_a_a @ ( image_9076584400576816019od_a_a @ ( transi7229704755246292010od_a_a @ R3 ) @ A ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_1159_finite__rtrancl__Image,axiom,
! [R3: set_Product_prod_a_a,A: set_a] :
( ( finite6544458595007987280od_a_a @ R3 )
=> ( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( image_a_a @ ( transitive_rtrancl_a @ R3 ) @ A ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_1160_finite__acyclic__wf__converse,axiom,
! [R: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( ( transitive_acyclic_a @ R )
=> ( wf_a @ ( converse_a_a @ R ) ) ) ) ).
% finite_acyclic_wf_converse
thf(fact_1161_eq__equiv__class,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a,A: set_a] :
( ( ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
=> ( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ B3 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_1162_eq__equiv__class,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
=> ( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ B3 @ A )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_1163_equiv__class__eq,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a,B3: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1164_equiv__class__eq,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R )
=> ( ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1165_eq__equiv__class__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( ( ( image_a_a @ R @ ( insert_a @ X3 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1166_eq__equiv__class__iff,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( member1426531477525435216od_a_a @ Y4 @ A )
=> ( ( ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1167_equiv__class__eq__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,Y4: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
= ( ( ( image_a_a @ R @ ( insert_a @ X3 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ A )
& ( member_a @ Y4 @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1168_equiv__class__eq__iff,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y4: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y4 ) @ R )
= ( ( ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X3 @ A )
& ( member1426531477525435216od_a_a @ Y4 @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1169_refines__equiv__class__eq,axiom,
! [R3: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le3469131294019144807od_a_a @ R3 @ S2 )
=> ( ( equiv_4910910634973128413od_a_a @ A @ R3 )
=> ( ( equiv_4910910634973128413od_a_a @ A @ S2 )
=> ( ( image_9076584400576816019od_a_a @ R3 @ ( image_9076584400576816019od_a_a @ S2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( image_9076584400576816019od_a_a @ S2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1170_refines__equiv__class__eq2,axiom,
! [R3: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le3469131294019144807od_a_a @ R3 @ S2 )
=> ( ( equiv_4910910634973128413od_a_a @ A @ R3 )
=> ( ( equiv_4910910634973128413od_a_a @ A @ S2 )
=> ( ( image_9076584400576816019od_a_a @ S2 @ ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( image_9076584400576816019od_a_a @ S2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1171_Partial__order__eq__Image1__Image1__iff,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( order_5272072345360262643r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( A2 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1172_Partial__order__eq__Image1__Image1__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( order_7408868903334687516od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( A2 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1173_equiv__class__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a,B3: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1174_equiv__class__subset,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R )
=> ( ord_le746702958409616551od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1175_subset__equiv__class,axiom,
! [A: set_a,R: set_Product_prod_a_a,B3: a,A2: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
=> ( ( member_a @ B3 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_1176_subset__equiv__class,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,B3: product_prod_a_a,A2: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( ord_le746702958409616551od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ A )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_1177_equiv__class__nondisjoint,axiom,
! [A: set_a,R: set_Product_prod_a_a,X3: a,A2: a,B3: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ X3 @ ( inf_inf_set_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1178_equiv__class__nondisjoint,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( inf_in8905007599844390133od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1179_subset__Image1__Image1__iff,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( order_preorder_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1180_subset__Image1__Image1__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( order_3202267349275844158od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ord_le746702958409616551od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1181_chain__subset__Total__Union,axiom,
! [R3: set_se5735800977113168103od_a_a] :
( ( chain_7119315380238734464od_a_a @ R3 )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ R3 )
=> ( total_on_a @ ( field_a @ X2 ) @ X2 ) )
=> ( total_on_a @ ( field_a @ ( comple8421679170691845492od_a_a @ R3 ) ) @ ( comple8421679170691845492od_a_a @ R3 ) ) ) ) ).
% chain_subset_Total_Union
thf(fact_1182_subset__Image__Image__iff,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( order_preorder_on_a @ ( field_a @ R ) @ R )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ A ) @ ( image_a_a @ R @ B ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Y3: a] :
( ( member_a @ Y3 @ B )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1183_preorder__on__empty,axiom,
order_preorder_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% preorder_on_empty
thf(fact_1184_preorder__on__empty,axiom,
order_3202267349275844158od_a_a @ bot_bo3357376287454694259od_a_a @ bot_bo510284599550014259od_a_a ).
% preorder_on_empty
thf(fact_1185_Zorns__po__lemma,axiom,
! [R: set_Product_prod_a_a] :
( ( order_5272072345360262643r_on_a @ ( field_a @ R ) @ R )
=> ( ! [C6: set_a] :
( ( member_set_a @ C6 @ ( chains_a @ R ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( field_a @ R ) )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ C6 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X5 ) @ R ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( field_a @ R ) )
& ! [Xa: a] :
( ( member_a @ Xa @ ( field_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ R )
=> ( Xa = X2 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_1186_disjnt__equiv__class,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a,B3: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( disjnt_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1187_disjnt__equiv__class,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( disjnt6673878776331334894od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1188_disjnt__self__iff__empty,axiom,
! [S2: set_Product_prod_a_a] :
( ( disjnt6673878776331334894od_a_a @ S2 @ S2 )
= ( S2 = bot_bo3357376287454694259od_a_a ) ) ).
% disjnt_self_iff_empty
thf(fact_1189_disjnt__insert2,axiom,
! [Y6: set_Product_prod_a_a,A2: product_prod_a_a,X4: set_Product_prod_a_a] :
( ( disjnt6673878776331334894od_a_a @ Y6 @ ( insert4534936382041156343od_a_a @ A2 @ X4 ) )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ Y6 )
& ( disjnt6673878776331334894od_a_a @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1190_disjnt__insert1,axiom,
! [A2: product_prod_a_a,X4: set_Product_prod_a_a,Y6: set_Product_prod_a_a] :
( ( disjnt6673878776331334894od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ X4 ) @ Y6 )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ Y6 )
& ( disjnt6673878776331334894od_a_a @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1191_disjnt__iff,axiom,
( disjnt6673878776331334894od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [X: product_prod_a_a] :
~ ( ( member1426531477525435216od_a_a @ X @ A3 )
& ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ) ).
% disjnt_iff
thf(fact_1192_disjnt__empty2,axiom,
! [A: set_Product_prod_a_a] : ( disjnt6673878776331334894od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% disjnt_empty2
thf(fact_1193_disjnt__empty1,axiom,
! [A: set_Product_prod_a_a] : ( disjnt6673878776331334894od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% disjnt_empty1
thf(fact_1194_disjnt__insert,axiom,
! [X3: product_prod_a_a,N: set_Product_prod_a_a,M2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ N )
=> ( ( disjnt6673878776331334894od_a_a @ M2 @ N )
=> ( disjnt6673878776331334894od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ M2 ) @ N ) ) ) ).
% disjnt_insert
thf(fact_1195_disjnt__def,axiom,
( disjnt6673878776331334894od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A3 @ B2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjnt_def
thf(fact_1196_single__valued__def,axiom,
( single_valued_a_a
= ( ^ [R2: set_Product_prod_a_a] :
! [X: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ! [Z4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ R2 )
=> ( Y3 = Z4 ) ) ) ) ) ).
% single_valued_def
thf(fact_1197_single__valuedI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a,Y: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z3 ) @ R )
=> ( Y = Z3 ) ) )
=> ( single_valued_a_a @ R ) ) ).
% single_valuedI
thf(fact_1198_single__valuedD,axiom,
! [R: set_Product_prod_a_a,X3: a,Y4: a,Z2: a] :
( ( single_valued_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ R )
=> ( Y4 = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_1199_single__valued__empty,axiom,
single_valued_a_a @ bot_bo3357376287454694259od_a_a ).
% single_valued_empty
thf(fact_1200_single__valued__confluent,axiom,
! [R: set_Product_prod_a_a,X3: a,Y4: a,Z2: a] :
( ( single_valued_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z2 ) @ ( transitive_rtrancl_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ ( transitive_rtrancl_a @ R ) )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z2 @ Y4 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_1201_image__eqI,axiom,
! [B3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( B3
= ( F @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_4636654165204879301od_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1202_image__is__empty,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( image_4636654165204879301od_a_a @ F @ A )
= bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% image_is_empty
thf(fact_1203_empty__is__image,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( image_4636654165204879301od_a_a @ F @ A ) )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% empty_is_image
thf(fact_1204_image__empty,axiom,
! [F: product_prod_a_a > product_prod_a_a] :
( ( image_4636654165204879301od_a_a @ F @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% image_empty
thf(fact_1205_finite__imageI,axiom,
! [F2: set_Product_prod_a_a,H: product_prod_a_a > product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( finite6544458595007987280od_a_a @ ( image_4636654165204879301od_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1206_finite__UN,axiom,
! [A: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( finite6544458595007987280od_a_a @ ( comple8421679170691845492od_a_a @ ( image_6690255128444368805od_a_a @ B @ A ) ) )
= ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( finite6544458595007987280od_a_a @ ( B @ X ) ) ) ) ) ) ).
% finite_UN
thf(fact_1207_Field__Union,axiom,
! [R3: set_se5735800977113168103od_a_a] :
( ( field_a @ ( comple8421679170691845492od_a_a @ R3 ) )
= ( comple2307003609928055243_set_a @ ( image_3052423715192255772_set_a @ field_a @ R3 ) ) ) ).
% Field_Union
thf(fact_1208_finite__surj,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) )
=> ( finite6544458595007987280od_a_a @ B ) ) ) ).
% finite_surj
thf(fact_1209_finite__subset__image,axiom,
! [B: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) )
=> ? [C6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C6 @ A )
& ( finite6544458595007987280od_a_a @ C6 )
& ( B
= ( image_4636654165204879301od_a_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1210_ex__finite__subset__image,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( ? [B2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B2 )
& ( ord_le746702958409616551od_a_a @ B2 @ ( image_4636654165204879301od_a_a @ F @ A ) )
& ( P2 @ B2 ) ) )
= ( ? [B2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B2 )
& ( ord_le746702958409616551od_a_a @ B2 @ A )
& ( P2 @ ( image_4636654165204879301od_a_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1211_all__finite__subset__image,axiom,
! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( ! [B2: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ B2 )
& ( ord_le746702958409616551od_a_a @ B2 @ ( image_4636654165204879301od_a_a @ F @ A ) ) )
=> ( P2 @ B2 ) ) )
= ( ! [B2: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ B2 )
& ( ord_le746702958409616551od_a_a @ B2 @ A ) )
=> ( P2 @ ( image_4636654165204879301od_a_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1212_infinite__surj,axiom,
! [A: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ ( image_4636654165204879301od_a_a @ F @ B ) )
=> ~ ( finite6544458595007987280od_a_a @ B ) ) ) ).
% infinite_surj
thf(fact_1213_image__subsetI,axiom,
! [A: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ ( F @ X2 ) @ B ) )
=> ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1214_rev__image__eqI,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( B3
= ( F @ X3 ) )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_4636654165204879301od_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1215_imageI,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( member1426531477525435216od_a_a @ ( F @ X3 ) @ ( image_4636654165204879301od_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_1216_Sup__SUP__eq,axiom,
( comple9027675562681848937_a_a_o
= ( ^ [S4: set_Pr974637816708178892_a_a_o,X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ ( comple8421679170691845492od_a_a @ ( image_7999903765548794282od_a_a @ collec3336397797384452498od_a_a @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1217_proj__image,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( image_a_set_a @ ( equiv_proj_a_a @ R ) @ A )
= ( equiv_quotient_a @ A @ R ) ) ).
% proj_image
thf(fact_1218_in__image__insert__iff,axiom,
! [B: set_se5735800977113168103od_a_a,X3: product_prod_a_a,A: set_Product_prod_a_a] :
( ! [C6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ C6 @ B )
=> ~ ( member1426531477525435216od_a_a @ X3 @ C6 ) )
=> ( ( member1816616512716248880od_a_a @ A @ ( image_4506799131697958853od_a_a @ ( insert4534936382041156343od_a_a @ X3 ) @ B ) )
= ( ( member1426531477525435216od_a_a @ X3 @ A )
& ( member1816616512716248880od_a_a @ ( minus_6817036919807184750od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1219_UNION__fun__upd,axiom,
! [A: product_prod_a_a > set_Product_prod_a_a,I: product_prod_a_a,B: set_Product_prod_a_a,J2: set_Product_prod_a_a] :
( ( comple8421679170691845492od_a_a @ ( image_6690255128444368805od_a_a @ ( fun_up3141760493783485469od_a_a @ A @ I @ B ) @ J2 ) )
= ( sup_su3048258781599657691od_a_a @ ( comple8421679170691845492od_a_a @ ( image_6690255128444368805od_a_a @ A @ ( minus_6817036919807184750od_a_a @ J2 @ ( insert4534936382041156343od_a_a @ I @ bot_bo3357376287454694259od_a_a ) ) ) ) @ ( if_set6912549639529188301od_a_a @ ( member1426531477525435216od_a_a @ I @ J2 ) @ B @ bot_bo3357376287454694259od_a_a ) ) ) ).
% UNION_fun_upd
thf(fact_1220_empty__in__Fpow,axiom,
! [A: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ ( finite3511610375897707917od_a_a @ A ) ) ).
% empty_in_Fpow
thf(fact_1221_Compl__disjoint,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ ( uminus5530930396987473918od_a_a @ A ) )
= bot_bo3357376287454694259od_a_a ) ).
% Compl_disjoint
thf(fact_1222_Compl__disjoint2,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( uminus5530930396987473918od_a_a @ A ) @ A )
= bot_bo3357376287454694259od_a_a ) ).
% Compl_disjoint2
thf(fact_1223_boolean__algebra_Oconj__cancel__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ ( uminus5530930396987473918od_a_a @ X3 ) )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1224_boolean__algebra_Oconj__cancel__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( uminus5530930396987473918od_a_a @ X3 ) @ X3 )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1225_inf__compl__bot__right,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ ( inf_in8905007599844390133od_a_a @ Y4 @ ( uminus5530930396987473918od_a_a @ X3 ) ) )
= bot_bo3357376287454694259od_a_a ) ).
% inf_compl_bot_right
thf(fact_1226_inf__compl__bot__left2,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ ( inf_in8905007599844390133od_a_a @ ( uminus5530930396987473918od_a_a @ X3 ) @ Y4 ) )
= bot_bo3357376287454694259od_a_a ) ).
% inf_compl_bot_left2
thf(fact_1227_inf__compl__bot__left1,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( uminus5530930396987473918od_a_a @ X3 ) @ ( inf_in8905007599844390133od_a_a @ X3 @ Y4 ) )
= bot_bo3357376287454694259od_a_a ) ).
% inf_compl_bot_left1
thf(fact_1228_subset__Compl__singleton,axiom,
! [A: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( uminus5530930396987473918od_a_a @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B3 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1229_subset__Compl__self__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( uminus5530930396987473918od_a_a @ A ) )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% subset_Compl_self_eq
thf(fact_1230_inf__cancel__left2,axiom,
! [X3: set_Product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ ( uminus5530930396987473918od_a_a @ X3 ) @ A2 ) @ ( inf_in8905007599844390133od_a_a @ X3 @ B3 ) )
= bot_bo3357376287454694259od_a_a ) ).
% inf_cancel_left2
thf(fact_1231_inf__cancel__left1,axiom,
! [X3: set_Product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X3 @ A2 ) @ ( inf_in8905007599844390133od_a_a @ ( uminus5530930396987473918od_a_a @ X3 ) @ B3 ) )
= bot_bo3357376287454694259od_a_a ) ).
% inf_cancel_left1
thf(fact_1232_inf__shunt,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ X3 @ Y4 )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ X3 @ ( uminus5530930396987473918od_a_a @ Y4 ) ) ) ).
% inf_shunt
thf(fact_1233_disjoint__eq__subset__Compl,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ A @ ( uminus5530930396987473918od_a_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1234_Compl__insert,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( uminus5530930396987473918od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) )
= ( minus_6817036919807184750od_a_a @ ( uminus5530930396987473918od_a_a @ A ) @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% Compl_insert
thf(fact_1235_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
= ( ! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( field_a @ R ) )
=> ( ( A3 != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1236_Linear__order__Well__order__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( order_435519263512085852od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( order_1514034925359088069od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
= ( ! [A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( A3 != bot_bo3357376287454694259od_a_a )
=> ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
& ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ A3 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1237_ComplI,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A ) ) ) ).
% ComplI
thf(fact_1238_Compl__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A ) )
= ( ~ ( member1426531477525435216od_a_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1239_ComplD,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A ) )
=> ~ ( member1426531477525435216od_a_a @ C @ A ) ) ).
% ComplD
thf(fact_1240_wo__rel_OWELL,axiom,
! [R: set_Product_prod_a_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R ) ) ).
% wo_rel.WELL
thf(fact_1241_wo__rel_Ointro,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( bNF_We1162827675446709994_rel_a @ R ) ) ).
% wo_rel.intro
thf(fact_1242_wo__rel__def,axiom,
( bNF_We1162827675446709994_rel_a
= ( ^ [R2: set_Product_prod_a_a] : ( order_6972113574731384220r_on_a @ ( field_a @ R2 ) @ R2 ) ) ) ).
% wo_rel_def
thf(fact_1243_linear__order__on__well__order__on,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( finite6544458595007987280od_a_a @ R )
=> ( ( order_8768733634509060147r_on_a @ A @ R )
= ( order_6972113574731384220r_on_a @ A @ R ) ) ) ).
% linear_order_on_well_order_on
thf(fact_1244_well__order__on__Well__order,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ A @ R )
=> ( ( A
= ( field_a @ R ) )
& ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R ) ) ) ).
% well_order_on_Well_order
thf(fact_1245_well__order__on__Field,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ A @ R )
=> ( A
= ( field_a @ R ) ) ) ).
% well_order_on_Field
thf(fact_1246_well__order__on__empty,axiom,
order_6972113574731384220r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% well_order_on_empty
thf(fact_1247_well__order__on__empty,axiom,
order_1514034925359088069od_a_a @ bot_bo3357376287454694259od_a_a @ bot_bo510284599550014259od_a_a ).
% well_order_on_empty
thf(fact_1248_bsqr__max2,axiom,
! [R: set_Product_prod_a_a,A1: a,A22: a,B1: a,B22: a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( product_Pair_a_a @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr_a @ R ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We3763454674811381836max2_a @ R @ A1 @ A22 ) @ ( bNF_We3763454674811381836max2_a @ R @ B1 @ B22 ) ) @ R ) ) ) ).
% bsqr_max2
thf(fact_1249_bsqr__Linear__order,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( order_435519263512085852od_a_a @ ( field_1126092520709947252od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) ) @ ( bNF_Wellorder_bsqr_a @ R ) ) ) ).
% bsqr_Linear_order
thf(fact_1250_bsqr__Well__order,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( order_1514034925359088069od_a_a @ ( field_1126092520709947252od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) ) @ ( bNF_Wellorder_bsqr_a @ R ) ) ) ).
% bsqr_Well_order
thf(fact_1251_bsqr__Total,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( total_3048927264068743339od_a_a @ ( field_1126092520709947252od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) ) @ ( bNF_Wellorder_bsqr_a @ R ) ) ) ).
% bsqr_Total
thf(fact_1252_univ__preserves,axiom,
! [A: set_a,R: set_Product_prod_a_a,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( equiv_4465467934579669695od_a_a @ R @ F )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ ( F @ X2 ) @ B ) )
=> ! [X5: set_a] :
( ( member_set_a @ X5 @ ( equiv_quotient_a @ A @ R ) )
=> ( member1426531477525435216od_a_a @ ( bNF_Gr7423577183739466833od_a_a @ F @ X5 ) @ B ) ) ) ) ) ).
% univ_preserves
thf(fact_1253_wf__ofilterIncl,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( wf_set_a @ ( bNF_We6398142586224878104Incl_a @ R ) ) ) ).
% wf_ofilterIncl
thf(fact_1254_min__bot,axiom,
! [X3: set_Product_prod_a_a] :
( ( ord_mi406616029830906478od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= bot_bo3357376287454694259od_a_a ) ).
% min_bot
thf(fact_1255_min__bot2,axiom,
! [X3: set_Product_prod_a_a] :
( ( ord_mi406616029830906478od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% min_bot2
thf(fact_1256_max__bot2,axiom,
! [X3: set_Product_prod_a_a] :
( ( ord_ma7118944045319509724od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= X3 ) ).
% max_bot2
thf(fact_1257_max__bot,axiom,
! [X3: set_Product_prod_a_a] :
( ( ord_ma7118944045319509724od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= X3 ) ).
% max_bot
thf(fact_1258_underS__incl__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( order_435519263512085852od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ord_le746702958409616551od_a_a @ ( order_8408155380505527533od_a_a @ R @ A2 ) @ ( order_8408155380505527533od_a_a @ R @ B3 ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B3 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1259_underS__incl__iff,axiom,
! [R: set_Product_prod_a_a,A2: a,B3: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B3 @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( order_underS_a @ R @ A2 ) @ ( order_underS_a @ R @ B3 ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1260_max__ext_Ocases,axiom,
! [A1: set_a,A22: set_a,R3: set_Product_prod_a_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R3 ) )
=> ~ ( ( finite_finite_a @ A1 )
=> ( ( finite_finite_a @ A22 )
=> ( ( A22 != bot_bot_set_a )
=> ~ ! [X5: a] :
( ( member_a @ X5 @ A1 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ A22 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_1261_max__ext_Ocases,axiom,
! [A1: set_Product_prod_a_a,A22: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A1 @ A22 ) @ ( max_ex7229726517752475106od_a_a @ R3 ) )
=> ~ ( ( finite6544458595007987280od_a_a @ A1 )
=> ( ( finite6544458595007987280od_a_a @ A22 )
=> ( ( A22 != bot_bo3357376287454694259od_a_a )
=> ~ ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A1 )
=> ? [Xa2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa2 @ A22 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_1262_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,J: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I @ ( order_8408155380505527533od_a_a @ R3 @ J ) )
=> ( member1426531477525435216od_a_a @ I @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1263_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I: a,R3: set_Product_prod_a_a,J: a] :
( ( member_a @ I @ ( order_underS_a @ R3 @ J ) )
=> ( member_a @ I @ ( field_a @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1264_Order__Relation_OunderS__Field,axiom,
! [R: set_Product_prod_a_a,A2: a] : ( ord_less_eq_set_a @ ( order_underS_a @ R @ A2 ) @ ( field_a @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1265_underS__empty,axiom,
! [A2: a,R: set_Product_prod_a_a] :
( ~ ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( order_underS_a @ R @ A2 )
= bot_bot_set_a ) ) ).
% underS_empty
thf(fact_1266_underS__empty,axiom,
! [A2: product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( order_8408155380505527533od_a_a @ R @ A2 )
= bot_bo3357376287454694259od_a_a ) ) ).
% underS_empty
thf(fact_1267_underS__E,axiom,
! [I: a,R3: set_Product_prod_a_a,J: a] :
( ( member_a @ I @ ( order_underS_a @ R3 @ J ) )
=> ( ( I != J )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_1268_underS__E,axiom,
! [I: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,J: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I @ ( order_8408155380505527533od_a_a @ R3 @ J ) )
=> ( ( I != J )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_1269_underS__I,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( I != J )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R3 )
=> ( member1426531477525435216od_a_a @ I @ ( order_8408155380505527533od_a_a @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_1270_underS__I,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( I != J )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ I @ ( order_underS_a @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_1271_max__ext_Omax__extI,axiom,
! [X4: set_a,Y6: set_a,R3: set_Product_prod_a_a] :
( ( finite_finite_a @ X4 )
=> ( ( finite_finite_a @ Y6 )
=> ( ( Y6 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X4 )
=> ? [Xa: a] :
( ( member_a @ Xa @ Y6 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ R3 ) ) )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y6 ) @ ( max_ext_a @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1272_max__ext_Omax__extI,axiom,
! [X4: set_Product_prod_a_a,Y6: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( finite6544458595007987280od_a_a @ X4 )
=> ( ( finite6544458595007987280od_a_a @ Y6 )
=> ( ( Y6 != bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ X4 )
=> ? [Xa: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa @ Y6 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Xa ) @ R3 ) ) )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y6 ) @ ( max_ex7229726517752475106od_a_a @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1273_max__ext_Osimps,axiom,
! [A1: set_a,A22: set_a,R3: set_Product_prod_a_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R3 ) )
= ( ( finite_finite_a @ A1 )
& ( finite_finite_a @ A22 )
& ( A22 != bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ A1 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ A22 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_1274_max__ext_Osimps,axiom,
! [A1: set_Product_prod_a_a,A22: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A1 @ A22 ) @ ( max_ex7229726517752475106od_a_a @ R3 ) )
= ( ( finite6544458595007987280od_a_a @ A1 )
& ( finite6544458595007987280od_a_a @ A22 )
& ( A22 != bot_bo3357376287454694259od_a_a )
& ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A1 )
=> ? [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ A22 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_1275_Refl__under__underS,axiom,
! [R: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( order_under_a @ R @ A2 )
= ( sup_sup_set_a @ ( order_underS_a @ R @ A2 ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1276_Refl__under__underS,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( order_865818352684159372od_a_a @ R @ A2 )
= ( sup_su3048258781599657691od_a_a @ ( order_8408155380505527533od_a_a @ R @ A2 ) @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% Refl_under_underS
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( if_set6912549639529188301od_a_a @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( if_set6912549639529188301od_a_a @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( field_a @ ( transitive_trancl_a @ ( symcl_a @ r ) ) )
= ( field_a @ r ) ) ).
%------------------------------------------------------------------------------