TPTP Problem File: SLH0020^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Separation_Logic_Unbounded/0003_FixedPoint/prob_00878_025952__6917924_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1513 ( 524 unt; 234 typ; 0 def)
% Number of atoms : 3913 (1209 equ; 0 cnn)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 10944 ( 501 ~; 35 |; 264 &;8345 @)
% ( 0 <=>;1799 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 3118 (3118 >; 0 *; 0 +; 0 <<)
% Number of symbols : 214 ( 211 usr; 30 con; 0-5 aty)
% Number of variables : 3484 ( 182 ^;3199 !; 103 ?;3484 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:39.034
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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thf(ty_n_t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Filter__Ofilter_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
filter_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_tf__d,type,
d: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (211)
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
condit5292637031048566470_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
condit3378509905675676198_set_a: ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Set__Oset_Itf__a_J,type,
condit6315317455391067509_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Filter_Ocofinite_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
cofinite_c_d_set_a: filter_c_d_set_a ).
thf(sy_c_Filter_Ocofinite_001tf__a,type,
cofinite_a: filter_a ).
thf(sy_c_Finite__Set_OFpow_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
finite1740182815655637662_set_a: set_option_c_d_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_Itf__a_J,type,
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finite457288119118821432_set_a: set_set_c_d_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_FixedPoint_Ologic_ODD_001tf__c_001tf__d_001tf__a,type,
dD_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_FixedPoint_Ologic_OD_001tf__c_001tf__d_001tf__a,type,
d_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_FixedPoint_Ologic_OGFP_001tf__c_001tf__d_001tf__a,type,
gFP_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_OInf_001tf__c_001tf__d_001tf__a,type,
inf_c_d_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_OLFP_001tf__c_001tf__d_001tf__a,type,
lFP_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_OSup_001tf__c_001tf__d_001tf__a,type,
sup_c_d_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oempty__interp_001_062_Itf__c_Mtf__d_J_001tf__a,type,
empty_interp_c_d_a: ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Ofull__interp_001tf__c_001tf__d_001tf__a,type,
full_interp_c_d_a: ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oless_001tf__c_001tf__d_001tf__a,type,
less_c_d_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Omonotonic_001tf__c_001tf__d_001tf__a,type,
monotonic_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Onon__increasing_001tf__c_001tf__d_001tf__a,type,
non_increasing_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Osmaller__interp_001tf__c_001tf__d_001tf__a,type,
smaller_interp_c_d_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Ofun__upd_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
fun_up1723300674898315325_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Fun_Ofun__upd_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001tf__a,type,
fun_upd_c_d_set_a_a: ( ( ( c > d ) > set_a ) > a ) > ( ( c > d ) > set_a ) > a > ( ( c > d ) > set_a ) > a ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
fun_up4138888537510562013_set_a: ( set_c_d_set_a > ( c > d ) > set_a ) > set_c_d_set_a > ( ( c > d ) > set_a ) > set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001tf__a,type,
fun_up4087578694834399916et_a_a: ( set_c_d_set_a > a ) > set_c_d_set_a > a > set_c_d_set_a > a ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_Itf__a_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
fun_up4868139172692436302_set_a: ( set_a > ( c > d ) > set_a ) > set_a > ( ( c > d ) > set_a ) > set_a > ( c > d ) > set_a ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
fun_upd_set_a_set_a: ( set_a > set_a ) > set_a > set_a > set_a > set_a ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_Itf__a_J_001tf__a,type,
fun_upd_set_a_a: ( set_a > a ) > set_a > a > set_a > a ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
fun_upd_a_c_d_set_a: ( a > ( c > d ) > set_a ) > a > ( ( c > d ) > set_a ) > a > ( c > d ) > set_a ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
fun_upd_a_a: ( a > a ) > a > a > a > a ).
thf(sy_c_Fun_Oinj__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
inj_on2268522623953733425_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001tf__a,type,
inj_on_c_d_set_a_a: ( ( ( c > d ) > set_a ) > a ) > set_c_d_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
inj_on_a_c_d_set_a: ( a > ( c > d ) > set_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
minus_926187851963594727et_a_o: ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
minus_6165026464846083862_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
minus_1665977719694084726_set_a: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
minus_3753830358241515990_set_a: set_set_c_d_set_a > set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
uminus6307618635820417879et_a_o: ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
uminus3002763893361803174_set_a: ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_M_Eo_J,type,
uminus_uminus_a_o: ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
uminus8771976365291672326_set_a: set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
uminus8902946929875755622_set_a: set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
uminus6103902357914783669_set_a: set_set_a > set_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
inf_inf_c_d_set_a_o: ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
inf_inf_c_d_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
inf_in754637537901350525_set_a: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
inf_in650668748222022109_set_a: set_set_c_d_set_a > set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
sup_sup_c_d_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
sup_su3175602471750379875_set_a: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
lattic5191877960299797817_set_a: ( ( ( c > d ) > set_a ) > set_c_d_set_a ) > set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
lattic2349170783384439560_set_a: ( ( ( c > d ) > set_a ) > set_a ) > set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
lattic1032982531657200042_set_a: ( a > set_c_d_set_a ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
lattic2425909027367666425_set_a: ( a > set_a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic3893622604919961804_set_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
lattic8453104748687127596_set_a: set_set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic1995125144389820681_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
lattic5078705180708912344_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic5899416730317975049_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001tf__a,type,
lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic8365952737566729574_set_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Option_Ooption_ONone_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
none_c_d_set_a: option_c_d_set_a ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
thf(sy_c_Option_Ooption_OSome_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
some_c_d_set_a: ( ( c > d ) > set_a ) > option_c_d_set_a ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
thf(sy_c_Option_Othese_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
these_c_d_set_a: set_option_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Option_Othese_001tf__a,type,
these_a: set_option_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
bot_bot_c_d_set_a_o: ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
bot_bot_c_d_set_a: ( c > d ) > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_M_Eo_J,type,
bot_bo3591254198091563330et_a_o: set_c_d_set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
bot_bo1170640541930586401_set_a: filter_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_Itf__a_J,type,
bot_bot_filter_a: filter_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
bot_bo738396921950161403_set_a: set_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
bot_bo6666349697208826049_set_a: set_option_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
bot_bot_set_option_a: set_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
bot_bo58555506362910043_set_a: set_set_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
ord_less_c_d_set_a_o: ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_c_d_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le3685282097655362107_set_a: set_c_d_set_a > set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
ord_le7529600783926193563_set_a: set_set_c_d_set_a > set_set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
ord_le961293222253252206et_a_o: ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le8464990428230162895_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le252514701126353884_set_a: ( $o > ( c > d ) > set_a ) > ( $o > ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
ord_le6704328240068426556_set_a: ( $o > set_c_d_set_a ) > ( $o > set_c_d_set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le5982164083705284911_set_a: set_c_d_set_a > set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
ord_le7272806397018272911_set_a: set_set_c_d_set_a > set_set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
order_551701534984366216_set_a: ( ( ( c > d ) > set_a ) > $o ) > ( c > d ) > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
order_7154941061327320040_set_a: ( set_c_d_set_a > $o ) > set_c_d_set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oordering_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
ordering_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
orderi3582402084717419303_set_a: ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_Itf__a_J,type,
ordering_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
orderi3930633105564526365et_a_o: ( ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ) > ( ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
orderi5785346111247480928_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_Itf__a_M_Eo_J,type,
ordering_top_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > ( ( a > $o ) > ( a > $o ) > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
orderi13773357969974208_set_a: ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
partia701112543150332005_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
partia1270112395057131461_set_a: ( set_c_d_set_a > set_c_d_set_a > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_Itf__a_J,type,
partia6602192050731689876_set_a: ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
top_top_c_d_set_a_o: ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
top_top_c_d_set_a: ( c > d ) > set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_M_Eo_J,type,
top_to6119605859643668830et_a_o: set_c_d_set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
top_top_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
top_to4267977599310771935_set_a: set_c_d_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to1333438998097461157_set_a: set_option_c_d_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to3895570120271872023_set_a: set_Pr2676350728994116295_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_Mtf__a_J_J,type,
top_to9215656001106096358et_a_a: set_Pr2440150136174226326et_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to6342870235713707528_set_a: set_Pr8790736407636613304_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
top_to8063371432257647191od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to5717711934741766719_set_a: set_set_c_d_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to279427854467338187_set_a: set_Su1130066786674581787_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_Mtf__a_J_J,type,
top_to5142853716682987162et_a_a: set_Su6007176542983585770et_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to2270067951290598332_set_a: set_Su3134390777591196940_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
top_to8848906000605539851um_a_a: set_Sum_sum_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
collect_c_d_set_a: ( ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
collec3354561713582630522_set_a: ( set_c_d_set_a > $o ) > set_set_c_d_set_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_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_5710119992958135237_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_1393004410334431755_set_a: ( ( ( c > d ) > set_a ) > option_c_d_set_a ) > set_c_d_set_a > set_option_c_d_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_1181342993027318565_set_a: ( ( ( c > d ) > set_a ) > set_c_d_set_a ) > set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_5050625251388476148_set_a: ( ( ( c > d ) > set_a ) > set_a ) > set_c_d_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001tf__a,type,
image_c_d_set_a_a: ( ( ( c > d ) > set_a ) > a ) > set_c_d_set_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_212549500329102437_set_a: ( set_c_d_set_a > ( c > d ) > set_a ) > set_set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_5418612861375423429_set_a: ( set_c_d_set_a > set_c_d_set_a ) > set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001t__Set__Oset_Itf__a_J,type,
image_4522397567451716500_set_a: ( set_c_d_set_a > set_a ) > set_set_c_d_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001tf__a,type,
image_702032380087044660et_a_a: ( set_c_d_set_a > a ) > set_set_c_d_set_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
image_7012603752887491525_set_a: ( set_set_c_d_set_a > set_set_c_d_set_a ) > set_se3584202636623819855_set_a > set_se3584202636623819855_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_1482592857945081046_set_a: ( set_a > ( c > d ) > set_a ) > set_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_3326132959979657270_set_a: ( set_a > set_c_d_set_a ) > set_set_a > set_set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_a_c_d_set_a: ( a > ( c > d ) > set_a ) > set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_3734436999661236630_set_a: ( a > set_c_d_set_a ) > set_a > set_set_c_d_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_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
insert_c_d_set_a: ( ( c > d ) > set_a ) > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
insert1935891768494221125_set_a: option_c_d_set_a > set_option_c_d_set_a > set_option_c_d_set_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
insert_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
insert_set_c_d_set_a: set_c_d_set_a > set_set_c_d_set_a > set_set_c_d_set_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_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
is_empty_c_d_set_a: set_c_d_set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
is_sin6979784932356128547_set_a: set_c_d_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
is_sin5290792544168550019_set_a: set_set_c_d_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Opairwise_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
pairwise_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
pairwi5502267298322432890_set_a: ( set_c_d_set_a > set_c_d_set_a > $o ) > set_set_c_d_set_a > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_Itf__a_J,type,
pairwise_set_a: ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Set_Opairwise_001tf__a,type,
pairwise_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Set_Oremove_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
remove_c_d_set_a: ( ( c > d ) > set_a ) > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oremove_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
remove_set_c_d_set_a: set_c_d_set_a > set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Set_Oremove_001t__Set__Oset_Itf__a_J,type,
remove_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
the_elem_c_d_set_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
member_c_d_set_a: ( ( c > d ) > set_a ) > set_c_d_set_a > $o ).
thf(sy_c_member_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
member4306893881663408030_set_a: option_c_d_set_a > set_option_c_d_set_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
member_set_c_d_set_a: set_c_d_set_a > set_set_c_d_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
member6574826897039512728_set_a: set_set_c_d_set_a > set_se3584202636623819855_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_x,type,
x: ( c > d ) > set_a ).
% Relevant facts (1278)
thf(fact_0_smaller__interpI,axiom,
! [Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a] :
( ! [S: c > d,X: a] :
( ( member_a @ X @ ( Delta @ S ) )
=> ( member_a @ X @ ( Delta2 @ S ) ) )
=> ( smaller_interp_c_d_a @ Delta @ Delta2 ) ) ).
% smaller_interpI
thf(fact_1_smaller__interp__antisym,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ A )
=> ( A = B ) ) ) ).
% smaller_interp_antisym
thf(fact_2_smaller__interp__refl,axiom,
! [Delta: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ Delta @ Delta ) ).
% smaller_interp_refl
thf(fact_3_smaller__interp__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ C )
=> ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% smaller_interp_trans
thf(fact_4_less__def,axiom,
( less_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% less_def
thf(fact_5_smaller__empty,axiom,
! [X2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ empty_interp_c_d_a @ X2 ) ).
% smaller_empty
thf(fact_6_smaller__full,axiom,
! [X2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ X2 @ full_interp_c_d_a ) ).
% smaller_full
thf(fact_7_test__axiom__inf,axiom,
! [A3: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( smaller_interp_c_d_a @ Z @ X ) )
=> ( smaller_interp_c_d_a @ Z @ ( inf_c_d_a @ A3 ) ) ) ).
% test_axiom_inf
thf(fact_8_test__axiom__sup,axiom,
! [A3: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( smaller_interp_c_d_a @ X @ Z ) )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a @ A3 ) @ Z ) ) ).
% test_axiom_sup
thf(fact_9_smaller__interp__def,axiom,
( smaller_interp_c_d_a
= ( ^ [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
! [S2: c > d] : ( ord_less_eq_set_a @ ( Delta3 @ S2 ) @ ( Delta4 @ S2 ) ) ) ) ).
% smaller_interp_def
thf(fact_10_monotonicI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F @ Delta5 ) @ ( F @ Delta6 ) ) )
=> ( monotonic_c_d_a @ F ) ) ).
% monotonicI
thf(fact_11_monotonic__def,axiom,
( monotonic_c_d_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F2 @ Delta3 ) @ ( F2 @ Delta4 ) ) ) ) ) ).
% monotonic_def
thf(fact_12_non__increasingI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F @ Delta6 ) @ ( F @ Delta5 ) ) )
=> ( non_increasing_c_d_a @ F ) ) ).
% non_increasingI
thf(fact_13_non__increasing__def,axiom,
( non_increasing_c_d_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F2 @ Delta4 ) @ ( F2 @ Delta3 ) ) ) ) ) ).
% non_increasing_def
thf(fact_14_GFP__lub,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( d_c_d_a @ F ) )
=> ( smaller_interp_c_d_a @ X @ Y ) )
=> ( smaller_interp_c_d_a @ ( gFP_c_d_a @ F ) @ Y ) ) ).
% GFP_lub
thf(fact_15_LFP__glb,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( dD_c_d_a @ F ) )
=> ( smaller_interp_c_d_a @ Y @ X ) )
=> ( smaller_interp_c_d_a @ Y @ ( lFP_c_d_a @ F ) ) ) ).
% LFP_glb
thf(fact_16_smaller__interp__D,axiom,
! [X2: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ ( d_c_d_a @ F ) )
=> ( smaller_interp_c_d_a @ X2 @ ( gFP_c_d_a @ F ) ) ) ).
% smaller_interp_D
thf(fact_17_smaller__interp__DD,axiom,
! [X2: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ ( dD_c_d_a @ F ) )
=> ( smaller_interp_c_d_a @ ( lFP_c_d_a @ F ) @ X2 ) ) ).
% smaller_interp_DD
thf(fact_18_mem__Collect__eq,axiom,
! [A: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( member_set_c_d_set_a @ A @ ( collec3354561713582630522_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_19_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_20_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_21_mem__Collect__eq,axiom,
! [A: ( c > d ) > set_a,P: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a @ A @ ( collect_c_d_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_22_Collect__mem__eq,axiom,
! [A3: set_set_c_d_set_a] :
( ( collec3354561713582630522_set_a
@ ^ [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_23_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_24_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_25_Collect__mem__eq,axiom,
! [A3: set_c_d_set_a] :
( ( collect_c_d_set_a
@ ^ [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_26_Collect__cong,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ! [X: ( c > d ) > set_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_c_d_set_a @ P )
= ( collect_c_d_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_27_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_28_sup__empty,axiom,
( ( sup_c_d_a @ bot_bo738396921950161403_set_a )
= empty_interp_c_d_a ) ).
% sup_empty
thf(fact_29_inf__empty,axiom,
( ( inf_c_d_a @ bot_bo738396921950161403_set_a )
= full_interp_c_d_a ) ).
% inf_empty
thf(fact_30_subsetI,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ! [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
=> ( member_set_c_d_set_a @ X @ B3 ) )
=> ( ord_le7272806397018272911_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_31_subsetI,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_32_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_33_subsetI,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( member_c_d_set_a @ X @ B3 ) )
=> ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_34_subset__antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_35_subset__antisym,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_36_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_37_order__refl,axiom,
! [X2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_38_order__refl,axiom,
! [X2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_39_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_40_dual__order_Orefl,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_41_dual__order_Orefl,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_42_full__interp__def,axiom,
( full_interp_c_d_a
= ( ^ [S2: c > d] : top_top_set_a ) ) ).
% full_interp_def
thf(fact_43_empty__interp__def,axiom,
( empty_interp_c_d_a
= ( ^ [S2: c > d] : bot_bot_set_a ) ) ).
% empty_interp_def
thf(fact_44_bot__apply,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_45_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_46_top__apply,axiom,
( top_top_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : top_top_o ) ) ).
% top_apply
thf(fact_47_top__apply,axiom,
( top_top_a_o
= ( ^ [X3: a] : top_top_o ) ) ).
% top_apply
thf(fact_48_UNIV__I,axiom,
! [X2: set_c_d_set_a] : ( member_set_c_d_set_a @ X2 @ top_to5717711934741766719_set_a ) ).
% UNIV_I
thf(fact_49_UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_50_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_51_UNIV__I,axiom,
! [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ top_to4267977599310771935_set_a ) ).
% UNIV_I
thf(fact_52_empty__Collect__eq,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( bot_bo738396921950161403_set_a
= ( collect_c_d_set_a @ P ) )
= ( ! [X3: ( c > d ) > set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_53_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_54_Collect__empty__eq,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( ( collect_c_d_set_a @ P )
= bot_bo738396921950161403_set_a )
= ( ! [X3: ( c > d ) > set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_55_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_56_all__not__in__conv,axiom,
! [A3: set_set_c_d_set_a] :
( ( ! [X3: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ X3 @ A3 ) )
= ( A3 = bot_bo58555506362910043_set_a ) ) ).
% all_not_in_conv
thf(fact_57_all__not__in__conv,axiom,
! [A3: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_58_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_59_all__not__in__conv,axiom,
! [A3: set_c_d_set_a] :
( ( ! [X3: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ X3 @ A3 ) )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% all_not_in_conv
thf(fact_60_empty__iff,axiom,
! [C: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ C @ bot_bo58555506362910043_set_a ) ).
% empty_iff
thf(fact_61_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_62_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_63_empty__iff,axiom,
! [C: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ C @ bot_bo738396921950161403_set_a ) ).
% empty_iff
thf(fact_64_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_65_empty__subsetI,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A3 ) ).
% empty_subsetI
thf(fact_66_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_67_subset__empty,axiom,
! [A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ bot_bo738396921950161403_set_a )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% subset_empty
thf(fact_68_bot__fun__def,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_69_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_70_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_71_empty__not__UNIV,axiom,
bot_bo738396921950161403_set_a != top_to4267977599310771935_set_a ).
% empty_not_UNIV
thf(fact_72_UNIV__witness,axiom,
? [X: set_c_d_set_a] : ( member_set_c_d_set_a @ X @ top_to5717711934741766719_set_a ) ).
% UNIV_witness
thf(fact_73_UNIV__witness,axiom,
? [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_74_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_75_UNIV__witness,axiom,
? [X: ( c > d ) > set_a] : ( member_c_d_set_a @ X @ top_to4267977599310771935_set_a ) ).
% UNIV_witness
thf(fact_76_ex__in__conv,axiom,
! [A3: set_set_c_d_set_a] :
( ( ? [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ A3 ) )
= ( A3 != bot_bo58555506362910043_set_a ) ) ).
% ex_in_conv
thf(fact_77_ex__in__conv,axiom,
! [A3: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_78_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_79_ex__in__conv,axiom,
! [A3: set_c_d_set_a] :
( ( ? [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ A3 ) )
= ( A3 != bot_bo738396921950161403_set_a ) ) ).
% ex_in_conv
thf(fact_80_UNIV__eq__I,axiom,
! [A3: set_set_c_d_set_a] :
( ! [X: set_c_d_set_a] : ( member_set_c_d_set_a @ X @ A3 )
=> ( top_to5717711934741766719_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_81_UNIV__eq__I,axiom,
! [A3: set_set_a] :
( ! [X: set_a] : ( member_set_a @ X @ A3 )
=> ( top_top_set_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_82_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X: a] : ( member_a @ X @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_83_UNIV__eq__I,axiom,
! [A3: set_c_d_set_a] :
( ! [X: ( c > d ) > set_a] : ( member_c_d_set_a @ X @ A3 )
=> ( top_to4267977599310771935_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_84_equals0I,axiom,
! [A3: set_set_c_d_set_a] :
( ! [Y2: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ Y2 @ A3 )
=> ( A3 = bot_bo58555506362910043_set_a ) ) ).
% equals0I
thf(fact_85_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_86_equals0I,axiom,
! [A3: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_87_equals0I,axiom,
! [A3: set_c_d_set_a] :
( ! [Y2: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ Y2 @ A3 )
=> ( A3 = bot_bo738396921950161403_set_a ) ) ).
% equals0I
thf(fact_88_equals0D,axiom,
! [A3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( A3 = bot_bo58555506362910043_set_a )
=> ~ ( member_set_c_d_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_89_equals0D,axiom,
! [A3: set_set_a,A: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_90_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_91_equals0D,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( A3 = bot_bo738396921950161403_set_a )
=> ~ ( member_c_d_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_92_emptyE,axiom,
! [A: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) ).
% emptyE
thf(fact_93_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_94_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_95_emptyE,axiom,
! [A: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% emptyE
thf(fact_96_bot_Oextremum__uniqueI,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ A @ bot_bot_c_d_set_a_o )
=> ( A = bot_bot_c_d_set_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_97_bot_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
=> ( A = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_98_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_99_bot_Oextremum__uniqueI,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ bot_bo738396921950161403_set_a )
=> ( A = bot_bo738396921950161403_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_100_bot_Oextremum__uniqueI,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ bot_bot_c_d_set_a )
=> ( A = bot_bot_c_d_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_101_bot_Oextremum__unique,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ A @ bot_bot_c_d_set_a_o )
= ( A = bot_bot_c_d_set_a_o ) ) ).
% bot.extremum_unique
thf(fact_102_bot_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
= ( A = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_103_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_104_bot_Oextremum__unique,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ bot_bo738396921950161403_set_a )
= ( A = bot_bo738396921950161403_set_a ) ) ).
% bot.extremum_unique
thf(fact_105_bot_Oextremum__unique,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ bot_bot_c_d_set_a )
= ( A = bot_bot_c_d_set_a ) ) ).
% bot.extremum_unique
thf(fact_106_bot_Oextremum,axiom,
! [A: ( ( c > d ) > set_a ) > $o] : ( ord_le961293222253252206et_a_o @ bot_bot_c_d_set_a_o @ A ) ).
% bot.extremum
thf(fact_107_bot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% bot.extremum
thf(fact_108_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_109_bot_Oextremum,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A ) ).
% bot.extremum
thf(fact_110_bot_Oextremum,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ bot_bot_c_d_set_a @ A ) ).
% bot.extremum
thf(fact_111_top_Oextremum__uniqueI,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ top_top_c_d_set_a_o @ A )
=> ( A = top_top_c_d_set_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_112_top_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
=> ( A = top_top_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_113_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_114_top_Oextremum__uniqueI,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ top_to4267977599310771935_set_a @ A )
=> ( A = top_to4267977599310771935_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_115_top_Oextremum__uniqueI,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ top_top_c_d_set_a @ A )
=> ( A = top_top_c_d_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_116_top_Oextremum__unique,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ top_top_c_d_set_a_o @ A )
= ( A = top_top_c_d_set_a_o ) ) ).
% top.extremum_unique
thf(fact_117_top_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
= ( A = top_top_a_o ) ) ).
% top.extremum_unique
thf(fact_118_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_119_top_Oextremum__unique,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ top_to4267977599310771935_set_a @ A )
= ( A = top_to4267977599310771935_set_a ) ) ).
% top.extremum_unique
thf(fact_120_top_Oextremum__unique,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ top_top_c_d_set_a @ A )
= ( A = top_top_c_d_set_a ) ) ).
% top.extremum_unique
thf(fact_121_top__greatest,axiom,
! [A: ( ( c > d ) > set_a ) > $o] : ( ord_le961293222253252206et_a_o @ A @ top_top_c_d_set_a_o ) ).
% top_greatest
thf(fact_122_top__greatest,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ top_top_a_o ) ).
% top_greatest
thf(fact_123_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_124_top__greatest,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ top_to4267977599310771935_set_a ) ).
% top_greatest
thf(fact_125_top__greatest,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ top_top_c_d_set_a ) ).
% top_greatest
thf(fact_126_subset__UNIV,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_127_subset__UNIV,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ top_to4267977599310771935_set_a ) ).
% subset_UNIV
thf(fact_128_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_129_order__antisym__conv,axiom,
! [Y: set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y @ X2 )
=> ( ( ord_le5982164083705284911_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_130_order__antisym__conv,axiom,
! [Y: ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y @ X2 )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_131_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_132_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_133_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_134_ord__le__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_135_ord__le__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_136_ord__le__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_137_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_138_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_139_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_140_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_142_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_143_ord__eq__le__subst,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_144_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_145_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: set_c_d_set_a > ( c > d ) > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_146_ord__eq__le__subst,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_147_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_148_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_150_order__eq__refl,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( X2 = Y )
=> ( ord_le5982164083705284911_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_151_order__eq__refl,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( X2 = Y )
=> ( ord_le8464990428230162895_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_152_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_160_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_161_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_162_order__subst1,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_163_order__subst1,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_164_order__subst1,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_165_order__subst1,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_166_order__subst1,axiom,
! [A: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_167_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_168_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_c_d_set_a > ( c > d ) > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_169_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_170_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_171_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A2 @ B2 )
& ( ord_le5982164083705284911_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_172_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A2 @ B2 )
& ( ord_le8464990428230162895_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_173_le__fun__def,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [F2: ( c > d ) > set_a,G: ( c > d ) > set_a] :
! [X3: c > d] : ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_174_le__funI,axiom,
! [F: ( c > d ) > set_a,G2: ( c > d ) > set_a] :
( ! [X: c > d] : ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_le8464990428230162895_set_a @ F @ G2 ) ) ).
% le_funI
thf(fact_175_le__funE,axiom,
! [F: ( c > d ) > set_a,G2: ( c > d ) > set_a,X2: c > d] :
( ( ord_le8464990428230162895_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_176_le__funD,axiom,
! [F: ( c > d ) > set_a,G2: ( c > d ) > set_a,X2: c > d] :
( ( ord_le8464990428230162895_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_177_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_179_antisym,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_180_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_181_dual__order_Otrans,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ C @ B )
=> ( ord_le5982164083705284911_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_182_dual__order_Otrans,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ C @ B )
=> ( ord_le8464990428230162895_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_183_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_184_dual__order_Oantisym,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_185_dual__order_Oantisym,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_186_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_187_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B2 @ A2 )
& ( ord_le5982164083705284911_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_188_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B2 @ A2 )
& ( ord_le8464990428230162895_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_189_order__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_190_order__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ Z )
=> ( ord_le5982164083705284911_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_191_order__trans,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ Z )
=> ( ord_le8464990428230162895_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_192_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_193_order_Otrans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_194_order_Otrans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_195_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_196_order__antisym,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_197_order__antisym,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_198_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_199_ord__le__eq__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_200_ord__le__eq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_201_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_202_ord__eq__le__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A = B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__eq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A = B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_204_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_205_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
& ( ord_le5982164083705284911_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_206_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y3 = Z2 ) )
= ( ^ [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
& ( ord_le8464990428230162895_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_207_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_208_Collect__mono__iff,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ( ord_le5982164083705284911_set_a @ ( collect_c_d_set_a @ P ) @ ( collect_c_d_set_a @ Q ) )
= ( ! [X3: ( c > d ) > set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_209_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_210_set__eq__subset,axiom,
( ( ^ [Y3: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A4 @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_211_subset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_212_subset__trans,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ C2 )
=> ( ord_le5982164083705284911_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_213_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_214_Collect__mono,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ! [X: ( c > d ) > set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le5982164083705284911_set_a @ ( collect_c_d_set_a @ P ) @ ( collect_c_d_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_215_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_216_subset__refl,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_217_subset__iff,axiom,
( ord_le7272806397018272911_set_a
= ( ^ [A4: set_set_c_d_set_a,B4: set_set_c_d_set_a] :
! [T: set_c_d_set_a] :
( ( member_set_c_d_set_a @ T @ A4 )
=> ( member_set_c_d_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_218_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_219_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_220_subset__iff,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A4: set_c_d_set_a,B4: set_c_d_set_a] :
! [T: ( c > d ) > set_a] :
( ( member_c_d_set_a @ T @ A4 )
=> ( member_c_d_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_221_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_222_equalityD2,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( A3 = B3 )
=> ( ord_le5982164083705284911_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_223_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_224_equalityD1,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( A3 = B3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_225_subset__eq,axiom,
( ord_le7272806397018272911_set_a
= ( ^ [A4: set_set_c_d_set_a,B4: set_set_c_d_set_a] :
! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A4 )
=> ( member_set_c_d_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_226_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_227_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_228_subset__eq,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A4: set_c_d_set_a,B4: set_c_d_set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A4 )
=> ( member_c_d_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_229_equalityE,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_230_equalityE,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( A3 = B3 )
=> ~ ( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ~ ( ord_le5982164083705284911_set_a @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_231_subsetD,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A3 @ B3 )
=> ( ( member_set_c_d_set_a @ C @ A3 )
=> ( member_set_c_d_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_232_subsetD,axiom,
! [A3: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_233_subsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_234_subsetD,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_235_in__mono,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A3 @ B3 )
=> ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( member_set_c_d_set_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_236_in__mono,axiom,
! [A3: set_set_a,B3: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_237_in__mono,axiom,
! [A3: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_238_in__mono,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_c_d_set_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_239_iso__tuple__UNIV__I,axiom,
! [X2: set_c_d_set_a] : ( member_set_c_d_set_a @ X2 @ top_to5717711934741766719_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_240_iso__tuple__UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_241_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_242_iso__tuple__UNIV__I,axiom,
! [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ top_to4267977599310771935_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_243_subset__emptyI,axiom,
! [A3: set_set_c_d_set_a] :
( ! [X: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ X @ A3 )
=> ( ord_le7272806397018272911_set_a @ A3 @ bot_bo58555506362910043_set_a ) ) ).
% subset_emptyI
thf(fact_244_subset__emptyI,axiom,
! [A3: set_set_a] :
( ! [X: set_a] :
~ ( member_set_a @ X @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_245_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_246_subset__emptyI,axiom,
! [A3: set_c_d_set_a] :
( ! [X: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ X @ A3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ bot_bo738396921950161403_set_a ) ) ).
% subset_emptyI
thf(fact_247_Set_Ois__empty__def,axiom,
( is_empty_c_d_set_a
= ( ^ [A4: set_c_d_set_a] : ( A4 = bot_bo738396921950161403_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_248_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_249_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_250_Greatest__equality,axiom,
! [P: set_c_d_set_a > $o,X2: set_c_d_set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_c_d_set_a] :
( ( P @ Y2 )
=> ( ord_le5982164083705284911_set_a @ Y2 @ X2 ) )
=> ( ( order_7154941061327320040_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_251_Greatest__equality,axiom,
! [P: ( ( c > d ) > set_a ) > $o,X2: ( c > d ) > set_a] :
( ( P @ X2 )
=> ( ! [Y2: ( c > d ) > set_a] :
( ( P @ Y2 )
=> ( ord_le8464990428230162895_set_a @ Y2 @ X2 ) )
=> ( ( order_551701534984366216_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_252_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_253_GreatestI2__order,axiom,
! [P: set_c_d_set_a > $o,X2: set_c_d_set_a,Q: set_c_d_set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_c_d_set_a] :
( ( P @ Y2 )
=> ( ord_le5982164083705284911_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_c_d_set_a] :
( ( P @ X )
=> ( ! [Y5: set_c_d_set_a] :
( ( P @ Y5 )
=> ( ord_le5982164083705284911_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_7154941061327320040_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_254_GreatestI2__order,axiom,
! [P: ( ( c > d ) > set_a ) > $o,X2: ( c > d ) > set_a,Q: ( ( c > d ) > set_a ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: ( c > d ) > set_a] :
( ( P @ Y2 )
=> ( ord_le8464990428230162895_set_a @ Y2 @ X2 ) )
=> ( ! [X: ( c > d ) > set_a] :
( ( P @ X )
=> ( ! [Y5: ( c > d ) > set_a] :
( ( P @ Y5 )
=> ( ord_le8464990428230162895_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_551701534984366216_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_255_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X4: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_256_le__rel__bool__arg__iff,axiom,
( ord_le6704328240068426556_set_a
= ( ^ [X4: $o > set_c_d_set_a,Y6: $o > set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le5982164083705284911_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_257_le__rel__bool__arg__iff,axiom,
( ord_le252514701126353884_set_a
= ( ^ [X4: $o > ( c > d ) > set_a,Y6: $o > ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le8464990428230162895_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_258_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_259_verit__comp__simplify1_I2_J,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_260_verit__comp__simplify1_I2_J,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_261_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_262_top__set__def,axiom,
( top_to4267977599310771935_set_a
= ( collect_c_d_set_a @ top_top_c_d_set_a_o ) ) ).
% top_set_def
thf(fact_263_bot__set__def,axiom,
( bot_bo738396921950161403_set_a
= ( collect_c_d_set_a @ bot_bot_c_d_set_a_o ) ) ).
% bot_set_def
thf(fact_264_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_265_top1I,axiom,
! [X2: ( c > d ) > set_a] : ( top_top_c_d_set_a_o @ X2 ) ).
% top1I
thf(fact_266_top1I,axiom,
! [X2: a] : ( top_top_a_o @ X2 ) ).
% top1I
thf(fact_267_top__empty__eq,axiom,
( top_to6119605859643668830et_a_o
= ( ^ [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ top_to5717711934741766719_set_a ) ) ) ).
% top_empty_eq
thf(fact_268_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_269_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_270_top__empty__eq,axiom,
( top_top_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ top_to4267977599310771935_set_a ) ) ) ).
% top_empty_eq
thf(fact_271_Collect__empty__eq__bot,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( ( collect_c_d_set_a @ P )
= bot_bo738396921950161403_set_a )
= ( P = bot_bot_c_d_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_272_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_273_bot__empty__eq,axiom,
( bot_bo3591254198091563330et_a_o
= ( ^ [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ bot_bo58555506362910043_set_a ) ) ) ).
% bot_empty_eq
thf(fact_274_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_275_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_276_bot__empty__eq,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ).
% bot_empty_eq
thf(fact_277_top__conj_I1_J,axiom,
! [X2: ( c > d ) > set_a,P: $o] :
( ( ( top_top_c_d_set_a_o @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_278_top__conj_I1_J,axiom,
! [X2: a,P: $o] :
( ( ( top_top_a_o @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_279_top__conj_I2_J,axiom,
! [P: $o,X2: ( c > d ) > set_a] :
( ( P
& ( top_top_c_d_set_a_o @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_280_top__conj_I2_J,axiom,
! [P: $o,X2: a] :
( ( P
& ( top_top_a_o @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_281_is__singletonI_H,axiom,
! [A3: set_set_c_d_set_a] :
( ( A3 != bot_bo58555506362910043_set_a )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
=> ( ( member_set_c_d_set_a @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_sin5290792544168550019_set_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_282_is__singletonI_H,axiom,
! [A3: set_set_a] :
( ( A3 != bot_bot_set_set_a )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( member_set_a @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_singleton_set_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_283_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ A3 )
=> ( ( member_a @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_284_is__singletonI_H,axiom,
! [A3: set_c_d_set_a] :
( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( member_c_d_set_a @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_sin6979784932356128547_set_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_285_Diff__eq__empty__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_286_Diff__eq__empty__iff,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( minus_1665977719694084726_set_a @ A3 @ B3 )
= bot_bo738396921950161403_set_a )
= ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_287_Diff__UNIV,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ top_to4267977599310771935_set_a )
= bot_bo738396921950161403_set_a ) ).
% Diff_UNIV
thf(fact_288_Diff__UNIV,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_289_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_290_singleton__insert__inj__eq_H,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ A3 )
= ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
= ( ( A = B )
& ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_291_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_292_singleton__insert__inj__eq,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a )
= ( insert_c_d_set_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_293_insertCI,axiom,
! [A: set_c_d_set_a,B3: set_set_c_d_set_a,B: set_c_d_set_a] :
( ( ~ ( member_set_c_d_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_set_c_d_set_a @ A @ ( insert_set_c_d_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_294_insertCI,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_295_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_296_insertCI,axiom,
! [A: ( c > d ) > set_a,B3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ~ ( member_c_d_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_297_insert__iff,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ ( insert_set_c_d_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_set_c_d_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_298_insert__iff,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_299_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_300_insert__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_c_d_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_301_insert__absorb2,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ X2 @ ( insert_c_d_set_a @ X2 @ A3 ) )
= ( insert_c_d_set_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_302_insert__absorb2,axiom,
! [X2: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A3 ) )
= ( insert_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_303_DiffI,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ A3 )
=> ( ~ ( member_set_c_d_set_a @ C @ B3 )
=> ( member_set_c_d_set_a @ C @ ( minus_3753830358241515990_set_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_304_DiffI,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( ~ ( member_set_a @ C @ B3 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_305_DiffI,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_306_DiffI,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ A3 )
=> ( ~ ( member_c_d_set_a @ C @ B3 )
=> ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_307_Diff__iff,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ ( minus_3753830358241515990_set_a @ A3 @ B3 ) )
= ( ( member_set_c_d_set_a @ C @ A3 )
& ~ ( member_set_c_d_set_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_308_Diff__iff,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) )
= ( ( member_set_a @ C @ A3 )
& ~ ( member_set_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_309_Diff__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_310_Diff__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) )
= ( ( member_c_d_set_a @ C @ A3 )
& ~ ( member_c_d_set_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_311_Diff__idemp,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) @ B3 )
= ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_312_Diff__idemp,axiom,
! [A3: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_313_singletonI,axiom,
! [A: set_c_d_set_a] : ( member_set_c_d_set_a @ A @ ( insert_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) ) ).
% singletonI
thf(fact_314_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_315_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_316_singletonI,axiom,
! [A: ( c > d ) > set_a] : ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ).
% singletonI
thf(fact_317_insert__subset,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( ( member_set_c_d_set_a @ X2 @ B3 )
& ( ord_le7272806397018272911_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_318_insert__subset,axiom,
! [X2: set_a,A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A3 ) @ B3 )
= ( ( member_set_a @ X2 @ B3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_319_insert__subset,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
= ( ( member_a @ X2 @ B3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_320_insert__subset,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( ( member_c_d_set_a @ X2 @ B3 )
& ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_321_Diff__cancel,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ A3 )
= bot_bo738396921950161403_set_a ) ).
% Diff_cancel
thf(fact_322_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_323_empty__Diff,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ bot_bo738396921950161403_set_a @ A3 )
= bot_bo738396921950161403_set_a ) ).
% empty_Diff
thf(fact_324_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_325_Diff__empty,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ bot_bo738396921950161403_set_a )
= A3 ) ).
% Diff_empty
thf(fact_326_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_327_Diff__insert0,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ B3 ) )
= ( minus_3753830358241515990_set_a @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_328_Diff__insert0,axiom,
! [X2: set_a,A3: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ B3 ) )
= ( minus_5736297505244876581_set_a @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_329_Diff__insert0,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
= ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_330_Diff__insert0,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ B3 ) )
= ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_331_insert__Diff1,axiom,
! [X2: set_c_d_set_a,B3: set_set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ B3 )
=> ( ( minus_3753830358241515990_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( minus_3753830358241515990_set_a @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_332_insert__Diff1,axiom,
! [X2: set_a,B3: set_set_a,A3: set_set_a] :
( ( member_set_a @ X2 @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_333_insert__Diff1,axiom,
! [X2: a,B3: set_a,A3: set_a] :
( ( member_a @ X2 @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
= ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_334_insert__Diff1,axiom,
! [X2: ( c > d ) > set_a,B3: set_c_d_set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X2 @ B3 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_335_insert__Diff__single,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ A @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) )
= ( insert_c_d_set_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_336_insert__Diff__single,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_337_is__singletonI,axiom,
! [X2: ( c > d ) > set_a] : ( is_sin6979784932356128547_set_a @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ).
% is_singletonI
thf(fact_338_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_339_Diff__single__insert,axiom,
! [A3: set_a,X2: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_340_Diff__single__insert,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) @ B3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_341_subset__insert__iff,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ B3 ) )
= ( ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ord_le7272806397018272911_set_a @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) @ B3 ) )
& ( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ord_le7272806397018272911_set_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_342_subset__insert__iff,axiom,
! [A3: set_set_a,X2: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X2 @ B3 ) )
= ( ( ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_343_subset__insert__iff,axiom,
! [A3: set_a,X2: a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
= ( ( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_344_subset__insert__iff,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ B3 ) )
= ( ( ( member_c_d_set_a @ X2 @ A3 )
=> ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) @ B3 ) )
& ( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_345_subset__Diff__insert,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a,X2: set_c_d_set_a,C2: set_set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A3 @ ( minus_3753830358241515990_set_a @ B3 @ ( insert_set_c_d_set_a @ X2 @ C2 ) ) )
= ( ( ord_le7272806397018272911_set_a @ A3 @ ( minus_3753830358241515990_set_a @ B3 @ C2 ) )
& ~ ( member_set_c_d_set_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_346_subset__Diff__insert,axiom,
! [A3: set_set_a,B3: set_set_a,X2: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B3 @ ( insert_set_a @ X2 @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B3 @ C2 ) )
& ~ ( member_set_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_347_subset__Diff__insert,axiom,
! [A3: set_a,B3: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B3 @ C2 ) )
& ~ ( member_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_348_subset__Diff__insert,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,X2: ( c > d ) > set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B3 @ ( insert_c_d_set_a @ X2 @ C2 ) ) )
= ( ( ord_le5982164083705284911_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B3 @ C2 ) )
& ~ ( member_c_d_set_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_349_DiffE,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ ( minus_3753830358241515990_set_a @ A3 @ B3 ) )
=> ~ ( ( member_set_c_d_set_a @ C @ A3 )
=> ( member_set_c_d_set_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_350_DiffE,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) )
=> ~ ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_351_DiffE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_352_DiffE,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) )
=> ~ ( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_353_DiffD1,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ ( minus_3753830358241515990_set_a @ A3 @ B3 ) )
=> ( member_set_c_d_set_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_354_DiffD1,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) )
=> ( member_set_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_355_DiffD1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_356_DiffD1,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) )
=> ( member_c_d_set_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_357_DiffD2,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ ( minus_3753830358241515990_set_a @ A3 @ B3 ) )
=> ~ ( member_set_c_d_set_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_358_DiffD2,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) )
=> ~ ( member_set_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_359_DiffD2,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ~ ( member_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_360_DiffD2,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) )
=> ~ ( member_c_d_set_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_361_insertE,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ ( insert_set_c_d_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_c_d_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_362_insertE,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_363_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_364_insertE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_c_d_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_365_insertI1,axiom,
! [A: set_c_d_set_a,B3: set_set_c_d_set_a] : ( member_set_c_d_set_a @ A @ ( insert_set_c_d_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_366_insertI1,axiom,
! [A: set_a,B3: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_367_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_368_insertI1,axiom,
! [A: ( c > d ) > set_a,B3: set_c_d_set_a] : ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_369_insertI2,axiom,
! [A: set_c_d_set_a,B3: set_set_c_d_set_a,B: set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ B3 )
=> ( member_set_c_d_set_a @ A @ ( insert_set_c_d_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_370_insertI2,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( member_set_a @ A @ B3 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_371_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_372_insertI2,axiom,
! [A: ( c > d ) > set_a,B3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A @ B3 )
=> ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_373_Set_Oset__insert,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A3 )
=> ~ ! [B5: set_set_c_d_set_a] :
( ( A3
= ( insert_set_c_d_set_a @ X2 @ B5 ) )
=> ( member_set_c_d_set_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_374_Set_Oset__insert,axiom,
! [X2: set_a,A3: set_set_a] :
( ( member_set_a @ X2 @ A3 )
=> ~ ! [B5: set_set_a] :
( ( A3
= ( insert_set_a @ X2 @ B5 ) )
=> ( member_set_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_375_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B5: set_a] :
( ( A3
= ( insert_a @ X2 @ B5 ) )
=> ( member_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_376_Set_Oset__insert,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ~ ! [B5: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ X2 @ B5 ) )
=> ( member_c_d_set_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_377_insert__ident,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ~ ( member_set_c_d_set_a @ X2 @ B3 )
=> ( ( ( insert_set_c_d_set_a @ X2 @ A3 )
= ( insert_set_c_d_set_a @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_378_insert__ident,axiom,
! [X2: set_a,A3: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ~ ( member_set_a @ X2 @ B3 )
=> ( ( ( insert_set_a @ X2 @ A3 )
= ( insert_set_a @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_379_insert__ident,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ~ ( member_a @ X2 @ B3 )
=> ( ( ( insert_a @ X2 @ A3 )
= ( insert_a @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_380_insert__ident,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ~ ( member_c_d_set_a @ X2 @ B3 )
=> ( ( ( insert_c_d_set_a @ X2 @ A3 )
= ( insert_c_d_set_a @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_381_insert__absorb,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ A3 )
=> ( ( insert_set_c_d_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_382_insert__absorb,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( insert_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_383_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_384_insert__absorb,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ( ( insert_c_d_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_385_insert__eq__iff,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a,B: set_c_d_set_a,B3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ A @ A3 )
=> ( ~ ( member_set_c_d_set_a @ B @ B3 )
=> ( ( ( insert_set_c_d_set_a @ A @ A3 )
= ( insert_set_c_d_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_set_c_d_set_a] :
( ( A3
= ( insert_set_c_d_set_a @ B @ C3 ) )
& ~ ( member_set_c_d_set_a @ B @ C3 )
& ( B3
= ( insert_set_c_d_set_a @ A @ C3 ) )
& ~ ( member_set_c_d_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_386_insert__eq__iff,axiom,
! [A: set_a,A3: set_set_a,B: set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ~ ( member_set_a @ B @ B3 )
=> ( ( ( insert_set_a @ A @ A3 )
= ( insert_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_set_a] :
( ( A3
= ( insert_set_a @ B @ C3 ) )
& ~ ( member_set_a @ B @ C3 )
& ( B3
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_387_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B3
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_388_insert__eq__iff,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ A3 )
=> ( ~ ( member_c_d_set_a @ B @ B3 )
=> ( ( ( insert_c_d_set_a @ A @ A3 )
= ( insert_c_d_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ B @ C3 ) )
& ~ ( member_c_d_set_a @ B @ C3 )
& ( B3
= ( insert_c_d_set_a @ A @ C3 ) )
& ~ ( member_c_d_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_389_insert__Diff__if,axiom,
! [X2: set_c_d_set_a,B3: set_set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( ( member_set_c_d_set_a @ X2 @ B3 )
=> ( ( minus_3753830358241515990_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( minus_3753830358241515990_set_a @ A3 @ B3 ) ) )
& ( ~ ( member_set_c_d_set_a @ X2 @ B3 )
=> ( ( minus_3753830358241515990_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( insert_set_c_d_set_a @ X2 @ ( minus_3753830358241515990_set_a @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_390_insert__Diff__if,axiom,
! [X2: set_a,B3: set_set_a,A3: set_set_a] :
( ( ( member_set_a @ X2 @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A3 @ B3 ) ) )
& ( ~ ( member_set_a @ X2 @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ B3 )
= ( insert_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_391_insert__Diff__if,axiom,
! [X2: a,B3: set_a,A3: set_a] :
( ( ( member_a @ X2 @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
= ( minus_minus_set_a @ A3 @ B3 ) ) )
& ( ~ ( member_a @ X2 @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_392_insert__Diff__if,axiom,
! [X2: ( c > d ) > set_a,B3: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ( member_c_d_set_a @ X2 @ B3 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) )
& ( ~ ( member_c_d_set_a @ X2 @ B3 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) @ B3 )
= ( insert_c_d_set_a @ X2 @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_393_insert__commute,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ X2 @ ( insert_c_d_set_a @ Y @ A3 ) )
= ( insert_c_d_set_a @ Y @ ( insert_c_d_set_a @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_394_insert__commute,axiom,
! [X2: a,Y: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_395_mk__disjoint__insert,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ A3 )
=> ? [B5: set_set_c_d_set_a] :
( ( A3
= ( insert_set_c_d_set_a @ A @ B5 ) )
& ~ ( member_set_c_d_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_396_mk__disjoint__insert,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ? [B5: set_set_a] :
( ( A3
= ( insert_set_a @ A @ B5 ) )
& ~ ( member_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_397_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B5: set_a] :
( ( A3
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_398_mk__disjoint__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ? [B5: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ A @ B5 ) )
& ~ ( member_c_d_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_399_Diff__insert__absorb,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( minus_3753830358241515990_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_400_Diff__insert__absorb,axiom,
! [X2: set_a,A3: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_401_Diff__insert__absorb,axiom,
! [X2: a,A3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_402_Diff__insert__absorb,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_403_Diff__insert2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ B3 ) )
= ( minus_1665977719694084726_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_404_Diff__insert2,axiom,
! [A3: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_405_insert__Diff,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ A3 )
=> ( ( insert_set_c_d_set_a @ A @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_406_insert__Diff,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_407_insert__Diff,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_408_insert__Diff,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ( ( insert_c_d_set_a @ A @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_409_Diff__insert,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ B3 ) )
= ( minus_1665977719694084726_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ).
% Diff_insert
thf(fact_410_Diff__insert,axiom,
! [A3: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_411_is__singletonE,axiom,
! [A3: set_c_d_set_a] :
( ( is_sin6979784932356128547_set_a @ A3 )
=> ~ ! [X: ( c > d ) > set_a] :
( A3
!= ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) ).
% is_singletonE
thf(fact_412_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X: a] :
( A3
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_413_is__singleton__def,axiom,
( is_sin6979784932356128547_set_a
= ( ^ [A4: set_c_d_set_a] :
? [X3: ( c > d ) > set_a] :
( A4
= ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_414_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X3: a] :
( A4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_415_insert__UNIV,axiom,
! [X2: a] :
( ( insert_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_416_insert__UNIV,axiom,
! [X2: ( c > d ) > set_a] :
( ( insert_c_d_set_a @ X2 @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a ) ).
% insert_UNIV
thf(fact_417_singleton__inject,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a )
= ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_418_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_419_insert__not__empty,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ A @ A3 )
!= bot_bo738396921950161403_set_a ) ).
% insert_not_empty
thf(fact_420_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_421_doubleton__eq__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a,D: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
= ( insert_c_d_set_a @ C @ ( insert_c_d_set_a @ D @ bot_bo738396921950161403_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_422_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_423_singleton__iff,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( member_set_c_d_set_a @ B @ ( insert_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_424_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_425_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_426_singleton__iff,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_427_singletonD,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( member_set_c_d_set_a @ B @ ( insert_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_428_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_429_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_430_singletonD,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_431_subset__insertI2,axiom,
! [A3: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_432_subset__insertI2,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_433_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_434_subset__insertI,axiom,
! [B3: set_c_d_set_a,A: ( c > d ) > set_a] : ( ord_le5982164083705284911_set_a @ B3 @ ( insert_c_d_set_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_435_subset__insert,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( ord_le7272806397018272911_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ B3 ) )
= ( ord_le7272806397018272911_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_436_subset__insert,axiom,
! [X2: set_a,A3: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X2 @ B3 ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_437_subset__insert,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_438_subset__insert,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ B3 ) )
= ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_439_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_440_insert__mono,axiom,
! [C2: set_c_d_set_a,D2: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ C2 @ D2 )
=> ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ A @ C2 ) @ ( insert_c_d_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_441_insert__subsetI,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,X5: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( ord_le7272806397018272911_set_a @ X5 @ A3 )
=> ( ord_le7272806397018272911_set_a @ ( insert_set_c_d_set_a @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_442_insert__subsetI,axiom,
! [X2: set_a,A3: set_set_a,X5: set_set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_443_insert__subsetI,axiom,
! [X2: a,A3: set_a,X5: set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ X5 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_444_insert__subsetI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,X5: set_c_d_set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ X5 @ A3 )
=> ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_445_double__diff,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_446_double__diff,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ C2 )
=> ( ( minus_1665977719694084726_set_a @ B3 @ ( minus_1665977719694084726_set_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_447_Diff__subset,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_448_Diff__subset,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_449_Diff__mono,axiom,
! [A3: set_a,C2: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_450_Diff__mono,axiom,
! [A3: set_c_d_set_a,C2: set_c_d_set_a,D2: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ C2 )
=> ( ( ord_le5982164083705284911_set_a @ D2 @ B3 )
=> ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) @ ( minus_1665977719694084726_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_451_subset__singletonD,axiom,
! [A3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_452_subset__singletonD,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
=> ( ( A3 = bot_bo738396921950161403_set_a )
| ( A3
= ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_453_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_454_subset__singleton__iff,axiom,
! [X5: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ X5 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
= ( ( X5 = bot_bo738396921950161403_set_a )
| ( X5
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_455_diff__shunt__var,axiom,
! [X2: ( ( c > d ) > set_a ) > $o,Y: ( ( c > d ) > set_a ) > $o] :
( ( ( minus_926187851963594727et_a_o @ X2 @ Y )
= bot_bot_c_d_set_a_o )
= ( ord_le961293222253252206et_a_o @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_456_diff__shunt__var,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ( minus_minus_a_o @ X2 @ Y )
= bot_bot_a_o )
= ( ord_less_eq_a_o @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_457_diff__shunt__var,axiom,
! [X2: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_458_diff__shunt__var,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ( minus_1665977719694084726_set_a @ X2 @ Y )
= bot_bo738396921950161403_set_a )
= ( ord_le5982164083705284911_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_459_diff__shunt__var,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( minus_6165026464846083862_set_a @ X2 @ Y )
= bot_bot_c_d_set_a )
= ( ord_le8464990428230162895_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_460_is__singleton__the__elem,axiom,
( is_sin6979784932356128547_set_a
= ( ^ [A4: set_c_d_set_a] :
( A4
= ( insert_c_d_set_a @ ( the_elem_c_d_set_a @ A4 ) @ bot_bo738396921950161403_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_461_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_462_the__elem__eq,axiom,
! [X2: ( c > d ) > set_a] :
( ( the_elem_c_d_set_a @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_463_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_464_remove__def,axiom,
( remove_c_d_set_a
= ( ^ [X3: ( c > d ) > set_a,A4: set_c_d_set_a] : ( minus_1665977719694084726_set_a @ A4 @ ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% remove_def
thf(fact_465_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_466_pairwise__alt,axiom,
( pairwise_c_d_set_a
= ( ^ [R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S3: set_c_d_set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ S3 )
=> ! [Y4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y4 @ ( minus_1665977719694084726_set_a @ S3 @ ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) )
=> ( R @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_467_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R: a > a > $o,S3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S3 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( minus_minus_set_a @ S3 @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
=> ( R @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_468_psubset__insert__iff,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( ord_le7529600783926193563_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ B3 ) )
= ( ( ( member_set_c_d_set_a @ X2 @ B3 )
=> ( ord_le7529600783926193563_set_a @ A3 @ B3 ) )
& ( ~ ( member_set_c_d_set_a @ X2 @ B3 )
=> ( ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ord_le7529600783926193563_set_a @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) @ B3 ) )
& ( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ord_le7272806397018272911_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_469_psubset__insert__iff,axiom,
! [A3: set_set_a,X2: set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ ( insert_set_a @ X2 @ B3 ) )
= ( ( ( member_set_a @ X2 @ B3 )
=> ( ord_less_set_set_a @ A3 @ B3 ) )
& ( ~ ( member_set_a @ X2 @ B3 )
=> ( ( ( member_set_a @ X2 @ A3 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_470_psubset__insert__iff,axiom,
! [A3: set_a,X2: a,B3: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
= ( ( ( member_a @ X2 @ B3 )
=> ( ord_less_set_a @ A3 @ B3 ) )
& ( ~ ( member_a @ X2 @ B3 )
=> ( ( ( member_a @ X2 @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_471_psubset__insert__iff,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ B3 ) )
= ( ( ( member_c_d_set_a @ X2 @ B3 )
=> ( ord_le3685282097655362107_set_a @ A3 @ B3 ) )
& ( ~ ( member_c_d_set_a @ X2 @ B3 )
=> ( ( ( member_c_d_set_a @ X2 @ A3 )
=> ( ord_le3685282097655362107_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) @ B3 ) )
& ( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_472_subset__Compl__singleton,axiom,
! [A3: set_set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A3 @ ( uminus8902946929875755622_set_a @ ( insert_set_c_d_set_a @ B @ bot_bo58555506362910043_set_a ) ) )
= ( ~ ( member_set_c_d_set_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_473_subset__Compl__singleton,axiom,
! [A3: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_474_subset__Compl__singleton,axiom,
! [A3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_475_subset__Compl__singleton,axiom,
! [A3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ ( uminus8771976365291672326_set_a @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) )
= ( ~ ( member_c_d_set_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_476_Compl__insert,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( uminus8771976365291672326_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) )
= ( minus_1665977719694084726_set_a @ ( uminus8771976365291672326_set_a @ A3 ) @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ).
% Compl_insert
thf(fact_477_Compl__insert,axiom,
! [X2: a,A3: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X2 @ A3 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_478_ComplI,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ C @ A3 )
=> ( member_set_c_d_set_a @ C @ ( uminus8902946929875755622_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_479_ComplI,axiom,
! [C: set_a,A3: set_set_a] :
( ~ ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_480_ComplI,axiom,
! [C: a,A3: set_a] :
( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_481_ComplI,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ ( uminus8771976365291672326_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_482_Compl__iff,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ ( uminus8902946929875755622_set_a @ A3 ) )
= ( ~ ( member_set_c_d_set_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_483_Compl__iff,axiom,
! [C: set_a,A3: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A3 ) )
= ( ~ ( member_set_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_484_Compl__iff,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
= ( ~ ( member_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_485_Compl__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( uminus8771976365291672326_set_a @ A3 ) )
= ( ~ ( member_c_d_set_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_486_Compl__eq__Compl__iff,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( uminus8771976365291672326_set_a @ A3 )
= ( uminus8771976365291672326_set_a @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_487_Compl__eq__Compl__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( uminus_uminus_set_a @ A3 )
= ( uminus_uminus_set_a @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_488_member__remove,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ ( remove_set_c_d_set_a @ Y @ A3 ) )
= ( ( member_set_c_d_set_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_489_member__remove,axiom,
! [X2: set_a,Y: set_a,A3: set_set_a] :
( ( member_set_a @ X2 @ ( remove_set_a @ Y @ A3 ) )
= ( ( member_set_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_490_member__remove,axiom,
! [X2: a,Y: a,A3: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y @ A3 ) )
= ( ( member_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_491_member__remove,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X2 @ ( remove_c_d_set_a @ Y @ A3 ) )
= ( ( member_c_d_set_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_492_compl__le__compl__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_493_compl__le__compl__iff,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ ( uminus8771976365291672326_set_a @ Y ) )
= ( ord_le5982164083705284911_set_a @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_494_compl__le__compl__iff,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ ( uminus3002763893361803174_set_a @ Y ) )
= ( ord_le8464990428230162895_set_a @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_495_psubsetI,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_496_psubsetI,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_le3685282097655362107_set_a @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_497_Compl__subset__Compl__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B3 ) )
= ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_498_Compl__subset__Compl__iff,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ A3 ) @ ( uminus8771976365291672326_set_a @ B3 ) )
= ( ord_le5982164083705284911_set_a @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_499_Compl__anti__mono,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_500_Compl__anti__mono,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ B3 ) @ ( uminus8771976365291672326_set_a @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_501_boolean__algebra_Ocompl__zero,axiom,
( ( uminus6307618635820417879et_a_o @ bot_bot_c_d_set_a_o )
= top_top_c_d_set_a_o ) ).
% boolean_algebra.compl_zero
thf(fact_502_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_a_o @ bot_bot_a_o )
= top_top_a_o ) ).
% boolean_algebra.compl_zero
thf(fact_503_boolean__algebra_Ocompl__zero,axiom,
( ( uminus8771976365291672326_set_a @ bot_bo738396921950161403_set_a )
= top_to4267977599310771935_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_504_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_505_boolean__algebra_Ocompl__one,axiom,
( ( uminus6307618635820417879et_a_o @ top_top_c_d_set_a_o )
= bot_bot_c_d_set_a_o ) ).
% boolean_algebra.compl_one
thf(fact_506_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_a_o @ top_top_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.compl_one
thf(fact_507_boolean__algebra_Ocompl__one,axiom,
( ( uminus8771976365291672326_set_a @ top_to4267977599310771935_set_a )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.compl_one
thf(fact_508_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_509_compl__le__swap2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_510_compl__le__swap2,axiom,
! [Y: set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ Y ) @ X2 )
=> ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_511_compl__le__swap2,axiom,
! [Y: ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( uminus3002763893361803174_set_a @ Y ) @ X2 )
=> ( ord_le8464990428230162895_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_512_compl__le__swap1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X2 ) )
=> ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_513_compl__le__swap1,axiom,
! [Y: set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y @ ( uminus8771976365291672326_set_a @ X2 ) )
=> ( ord_le5982164083705284911_set_a @ X2 @ ( uminus8771976365291672326_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_514_compl__le__swap1,axiom,
! [Y: ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y @ ( uminus3002763893361803174_set_a @ X2 ) )
=> ( ord_le8464990428230162895_set_a @ X2 @ ( uminus3002763893361803174_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_515_compl__mono,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X2 ) ) ) ).
% compl_mono
thf(fact_516_compl__mono,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ Y ) @ ( uminus8771976365291672326_set_a @ X2 ) ) ) ).
% compl_mono
thf(fact_517_compl__mono,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ord_le8464990428230162895_set_a @ ( uminus3002763893361803174_set_a @ Y ) @ ( uminus3002763893361803174_set_a @ X2 ) ) ) ).
% compl_mono
thf(fact_518_bot_Oextremum__strict,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
~ ( ord_less_c_d_set_a_o @ A @ bot_bot_c_d_set_a_o ) ).
% bot.extremum_strict
thf(fact_519_bot_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_520_bot_Oextremum__strict,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% bot.extremum_strict
thf(fact_521_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_522_bot_Onot__eq__extremum,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( A != bot_bot_c_d_set_a_o )
= ( ord_less_c_d_set_a_o @ bot_bot_c_d_set_a_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_523_bot_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_524_bot_Onot__eq__extremum,axiom,
! [A: set_c_d_set_a] :
( ( A != bot_bo738396921950161403_set_a )
= ( ord_le3685282097655362107_set_a @ bot_bo738396921950161403_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_525_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_526_top_Oextremum__strict,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
~ ( ord_less_c_d_set_a_o @ top_top_c_d_set_a_o @ A ) ).
% top.extremum_strict
thf(fact_527_top_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ top_top_a_o @ A ) ).
% top.extremum_strict
thf(fact_528_top_Oextremum__strict,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ top_to4267977599310771935_set_a @ A ) ).
% top.extremum_strict
thf(fact_529_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_530_top_Onot__eq__extremum,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( A != top_top_c_d_set_a_o )
= ( ord_less_c_d_set_a_o @ A @ top_top_c_d_set_a_o ) ) ).
% top.not_eq_extremum
thf(fact_531_top_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != top_top_a_o )
= ( ord_less_a_o @ A @ top_top_a_o ) ) ).
% top.not_eq_extremum
thf(fact_532_top_Onot__eq__extremum,axiom,
! [A: set_c_d_set_a] :
( ( A != top_to4267977599310771935_set_a )
= ( ord_le3685282097655362107_set_a @ A @ top_to4267977599310771935_set_a ) ) ).
% top.not_eq_extremum
thf(fact_533_top_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != top_top_set_a )
= ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_534_not__psubset__empty,axiom,
! [A3: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A3 @ bot_bo738396921950161403_set_a ) ).
% not_psubset_empty
thf(fact_535_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_536_psubsetE,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_537_psubsetE,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ~ ( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ord_le5982164083705284911_set_a @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_538_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_539_psubset__eq,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [A4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_540_psubset__imp__subset,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_541_psubset__imp__subset,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_542_psubset__subset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_543_psubset__subset__trans,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ C2 )
=> ( ord_le3685282097655362107_set_a @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_544_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_545_subset__not__subset__eq,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [A4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A4 @ B4 )
& ~ ( ord_le5982164083705284911_set_a @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_546_subset__psubset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_547_subset__psubset__trans,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( ord_le3685282097655362107_set_a @ B3 @ C2 )
=> ( ord_le3685282097655362107_set_a @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_548_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_549_subset__iff__psubset__eq,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_550_psubset__imp__ex__mem,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( ord_le7529600783926193563_set_a @ A3 @ B3 )
=> ? [B6: set_c_d_set_a] : ( member_set_c_d_set_a @ B6 @ ( minus_3753830358241515990_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_551_psubset__imp__ex__mem,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
=> ? [B6: set_a] : ( member_set_a @ B6 @ ( minus_5736297505244876581_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_552_psubset__imp__ex__mem,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_553_psubset__imp__ex__mem,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ? [B6: ( c > d ) > set_a] : ( member_c_d_set_a @ B6 @ ( minus_1665977719694084726_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_554_verit__comp__simplify1_I1_J,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_555_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_556_less__imp__neq,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_557_less__imp__neq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_558_order_Oasym,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ~ ( ord_le3685282097655362107_set_a @ B @ A ) ) ).
% order.asym
thf(fact_559_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_560_ord__eq__less__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A = B )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ord_le3685282097655362107_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_561_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_562_ord__less__eq__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3685282097655362107_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_563_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_564_dual__order_Oasym,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ B @ A )
=> ~ ( ord_le3685282097655362107_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_565_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_566_dual__order_Oirrefl,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_567_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_568_order_Ostrict__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ord_le3685282097655362107_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_569_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_570_dual__order_Ostrict__trans,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ B @ A )
=> ( ( ord_le3685282097655362107_set_a @ C @ B )
=> ( ord_le3685282097655362107_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_571_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_572_order_Ostrict__implies__not__eq,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_573_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_574_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_575_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_576_ComplD,axiom,
! [C: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ C @ ( uminus8902946929875755622_set_a @ A3 ) )
=> ~ ( member_set_c_d_set_a @ C @ A3 ) ) ).
% ComplD
thf(fact_577_ComplD,axiom,
! [C: set_a,A3: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A3 ) )
=> ~ ( member_set_a @ C @ A3 ) ) ).
% ComplD
thf(fact_578_ComplD,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
=> ~ ( member_a @ C @ A3 ) ) ).
% ComplD
thf(fact_579_ComplD,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( uminus8771976365291672326_set_a @ A3 ) )
=> ~ ( member_c_d_set_a @ C @ A3 ) ) ).
% ComplD
thf(fact_580_psubsetD,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le7529600783926193563_set_a @ A3 @ B3 )
=> ( ( member_set_c_d_set_a @ C @ A3 )
=> ( member_set_c_d_set_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_581_psubsetD,axiom,
! [A3: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_582_psubsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_583_psubsetD,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C: ( c > d ) > set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_584_pairwiseD,axiom,
! [R2: set_c_d_set_a > set_c_d_set_a > $o,S4: set_set_c_d_set_a,X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( pairwi5502267298322432890_set_a @ R2 @ S4 )
=> ( ( member_set_c_d_set_a @ X2 @ S4 )
=> ( ( member_set_c_d_set_a @ Y @ S4 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_585_pairwiseD,axiom,
! [R2: set_a > set_a > $o,S4: set_set_a,X2: set_a,Y: set_a] :
( ( pairwise_set_a @ R2 @ S4 )
=> ( ( member_set_a @ X2 @ S4 )
=> ( ( member_set_a @ Y @ S4 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_586_pairwiseD,axiom,
! [R2: a > a > $o,S4: set_a,X2: a,Y: a] :
( ( pairwise_a @ R2 @ S4 )
=> ( ( member_a @ X2 @ S4 )
=> ( ( member_a @ Y @ S4 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_587_pairwiseD,axiom,
! [R2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S4: set_c_d_set_a,X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( pairwise_c_d_set_a @ R2 @ S4 )
=> ( ( member_c_d_set_a @ X2 @ S4 )
=> ( ( member_c_d_set_a @ Y @ S4 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_588_pairwiseI,axiom,
! [S4: set_set_c_d_set_a,R2: set_c_d_set_a > set_c_d_set_a > $o] :
( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ S4 )
=> ( ( member_set_c_d_set_a @ Y2 @ S4 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwi5502267298322432890_set_a @ R2 @ S4 ) ) ).
% pairwiseI
thf(fact_589_pairwiseI,axiom,
! [S4: set_set_a,R2: set_a > set_a > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ S4 )
=> ( ( member_set_a @ Y2 @ S4 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_set_a @ R2 @ S4 ) ) ).
% pairwiseI
thf(fact_590_pairwiseI,axiom,
! [S4: set_a,R2: a > a > $o] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ S4 )
=> ( ( member_a @ Y2 @ S4 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_a @ R2 @ S4 ) ) ).
% pairwiseI
thf(fact_591_pairwiseI,axiom,
! [S4: set_c_d_set_a,R2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ S4 )
=> ( ( member_c_d_set_a @ Y2 @ S4 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_c_d_set_a @ R2 @ S4 ) ) ).
% pairwiseI
thf(fact_592_pairwise__def,axiom,
( pairwise_c_d_set_a
= ( ^ [R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S3: set_c_d_set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ S3 )
=> ! [Y4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y4 @ S3 )
=> ( ( X3 != Y4 )
=> ( R @ X3 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_593_pairwise__def,axiom,
( pairwise_a
= ( ^ [R: a > a > $o,S3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S3 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ S3 )
=> ( ( X3 != Y4 )
=> ( R @ X3 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_594_psubset__trans,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ( ( ord_le3685282097655362107_set_a @ B3 @ C2 )
=> ( ord_le3685282097655362107_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_595_psubset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_596_double__complement,axiom,
! [A3: set_c_d_set_a] :
( ( uminus8771976365291672326_set_a @ ( uminus8771976365291672326_set_a @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_597_double__complement,axiom,
! [A3: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_598_order__less__asym,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ~ ( ord_le3685282097655362107_set_a @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_599_order__less__asym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_600_order__less__asym_H,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ~ ( ord_le3685282097655362107_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_601_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_602_order__less__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ( ord_le3685282097655362107_set_a @ Y @ Z )
=> ( ord_le3685282097655362107_set_a @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_603_order__less__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_604_ord__eq__less__subst,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_605_ord__eq__less__subst,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_606_ord__eq__less__subst,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_607_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_608_ord__less__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_609_ord__less__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_610_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_611_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_612_order__less__irrefl,axiom,
! [X2: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_613_order__less__irrefl,axiom,
! [X2: set_a] :
~ ( ord_less_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_614_order__less__subst1,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_615_order__less__subst1,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( ord_le3685282097655362107_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_616_order__less__subst1,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_617_order__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_618_order__less__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_619_order__less__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_620_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_621_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_622_order__less__not__sym,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ~ ( ord_le3685282097655362107_set_a @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_623_order__less__not__sym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_624_order__less__imp__triv,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,P: $o] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ( ord_le3685282097655362107_set_a @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_625_order__less__imp__triv,axiom,
! [X2: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_626_order__less__imp__not__eq,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_627_order__less__imp__not__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_628_order__less__imp__not__eq2,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_629_order__less__imp__not__eq2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_630_order__less__imp__not__less,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ~ ( ord_le3685282097655362107_set_a @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_631_order__less__imp__not__less,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_632_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_633_leD,axiom,
! [Y: set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y @ X2 )
=> ~ ( ord_le3685282097655362107_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_634_leD,axiom,
! [Y: ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y @ X2 )
=> ~ ( ord_less_c_d_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_635_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_636_nless__le,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ~ ( ord_le3685282097655362107_set_a @ A @ B ) )
= ( ~ ( ord_le5982164083705284911_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_637_nless__le,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ~ ( ord_less_c_d_set_a @ A @ B ) )
= ( ~ ( ord_le8464990428230162895_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_638_antisym__conv1,axiom,
! [X2: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_639_antisym__conv1,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ~ ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_640_antisym__conv1,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ~ ( ord_less_c_d_set_a @ X2 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_641_antisym__conv2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_642_antisym__conv2,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ~ ( ord_le3685282097655362107_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_643_antisym__conv2,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_c_d_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_644_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_645_less__le__not__le,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
& ~ ( ord_le5982164083705284911_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_646_less__le__not__le,axiom,
( ord_less_c_d_set_a
= ( ^ [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
& ~ ( ord_le8464990428230162895_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_647_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_648_order_Oorder__iff__strict,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_649_order_Oorder__iff__strict,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_650_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_651_order_Ostrict__iff__order,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_652_order_Ostrict__iff__order,axiom,
( ord_less_c_d_set_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_653_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_654_order_Ostrict__trans1,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ord_le3685282097655362107_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_655_order_Ostrict__trans1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ B @ C )
=> ( ord_less_c_d_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_656_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_657_order_Ostrict__trans2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ord_le3685282097655362107_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_658_order_Ostrict__trans2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ord_less_c_d_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_659_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_660_order_Ostrict__iff__not,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A2 @ B2 )
& ~ ( ord_le5982164083705284911_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_661_order_Ostrict__iff__not,axiom,
( ord_less_c_d_set_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A2 @ B2 )
& ~ ( ord_le8464990428230162895_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_662_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_663_dual__order_Oorder__iff__strict,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [B2: set_c_d_set_a,A2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_664_dual__order_Oorder__iff__strict,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_665_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_666_dual__order_Ostrict__iff__order,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [B2: set_c_d_set_a,A2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_667_dual__order_Ostrict__iff__order,axiom,
( ord_less_c_d_set_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_668_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_669_dual__order_Ostrict__trans1,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le3685282097655362107_set_a @ C @ B )
=> ( ord_le3685282097655362107_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_670_dual__order_Ostrict__trans1,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_less_c_d_set_a @ C @ B )
=> ( ord_less_c_d_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_671_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_672_dual__order_Ostrict__trans2,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ C @ B )
=> ( ord_le3685282097655362107_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_673_dual__order_Ostrict__trans2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ C @ B )
=> ( ord_less_c_d_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_674_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_675_dual__order_Ostrict__iff__not,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [B2: set_c_d_set_a,A2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B2 @ A2 )
& ~ ( ord_le5982164083705284911_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_676_dual__order_Ostrict__iff__not,axiom,
( ord_less_c_d_set_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B2 @ A2 )
& ~ ( ord_le8464990428230162895_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_677_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_678_order_Ostrict__implies__order,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ord_le5982164083705284911_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_679_order_Ostrict__implies__order,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A @ B )
=> ( ord_le8464990428230162895_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_680_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_681_dual__order_Ostrict__implies__order,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ B @ A )
=> ( ord_le5982164083705284911_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_682_dual__order_Ostrict__implies__order,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ B @ A )
=> ( ord_le8464990428230162895_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_683_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_684_order__le__less,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_685_order__le__less,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_686_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_687_order__less__le,axiom,
( ord_le3685282097655362107_set_a
= ( ^ [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_688_order__less__le,axiom,
( ord_less_c_d_set_a
= ( ^ [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_689_order__less__imp__le,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_690_order__less__imp__le,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ord_le5982164083705284911_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_691_order__less__imp__le,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X2 @ Y )
=> ( ord_le8464990428230162895_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_692_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_693_order__le__neq__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_le3685282097655362107_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_694_order__le__neq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_c_d_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_695_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_696_order__neq__le__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( A != B )
=> ( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ord_le3685282097655362107_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_697_order__neq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A != B )
=> ( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ord_less_c_d_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_698_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_699_order__le__less__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ord_le3685282097655362107_set_a @ Y @ Z )
=> ( ord_le3685282097655362107_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_700_order__le__less__trans,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ord_less_c_d_set_a @ Y @ Z )
=> ( ord_less_c_d_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_701_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_702_order__less__le__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ Z )
=> ( ord_le3685282097655362107_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_703_order__less__le__trans,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X2 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ Z )
=> ( ord_less_c_d_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_704_order__le__less__subst1,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_705_order__le__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_706_order__le__less__subst1,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_707_order__le__less__subst1,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_708_order__le__less__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_c_d_set_a > ( c > d ) > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3685282097655362107_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_c_d_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_709_order__le__less__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_c_d_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_710_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_711_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_712_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_713_order__le__less__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_714_order__le__less__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_715_order__le__less__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_716_order__le__less__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_717_order__le__less__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le3685282097655362107_set_a @ ( F @ B ) @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_718_order__le__less__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ ( F @ B ) @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_719_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_720_order__less__le__subst1,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( ord_le3685282097655362107_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_721_order__less__le__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C: set_a] :
( ( ord_less_c_d_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_722_order__less__le__subst1,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_723_order__less__le__subst1,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_724_order__less__le__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_c_d_set_a > ( c > d ) > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_c_d_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_725_order__less__le__subst1,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_726_order__less__le__subst1,axiom,
! [A: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le3685282097655362107_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_727_order__less__le__subst1,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_728_order__less__le__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_729_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_730_order__less__le__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_731_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_732_order__less__le__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_c_d_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_733_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_c_d_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_734_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_735_order__le__imp__less__or__eq,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ord_le3685282097655362107_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_736_order__le__imp__less__or__eq,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ord_less_c_d_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_737_pairwise__empty,axiom,
! [P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] : ( pairwise_c_d_set_a @ P @ bot_bo738396921950161403_set_a ) ).
% pairwise_empty
thf(fact_738_pairwise__empty,axiom,
! [P: a > a > $o] : ( pairwise_a @ P @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_739_pairwise__subset,axiom,
! [P: a > a > $o,S4: set_a,T2: set_a] :
( ( pairwise_a @ P @ S4 )
=> ( ( ord_less_eq_set_a @ T2 @ S4 )
=> ( pairwise_a @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_740_pairwise__subset,axiom,
! [P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( pairwise_c_d_set_a @ P @ S4 )
=> ( ( ord_le5982164083705284911_set_a @ T2 @ S4 )
=> ( pairwise_c_d_set_a @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_741_pairwise__mono,axiom,
! [P: a > a > $o,A3: set_a,Q: a > a > $o,B3: set_a] :
( ( pairwise_a @ P @ A3 )
=> ( ! [X: a,Y2: a] :
( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( pairwise_a @ Q @ B3 ) ) ) ) ).
% pairwise_mono
thf(fact_742_pairwise__mono,axiom,
! [P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A3: set_c_d_set_a,Q: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,B3: set_c_d_set_a] :
( ( pairwise_c_d_set_a @ P @ A3 )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ A3 )
=> ( pairwise_c_d_set_a @ Q @ B3 ) ) ) ) ).
% pairwise_mono
thf(fact_743_pairwise__insert,axiom,
! [R3: set_c_d_set_a > set_c_d_set_a > $o,X2: set_c_d_set_a,S5: set_set_c_d_set_a] :
( ( pairwi5502267298322432890_set_a @ R3 @ ( insert_set_c_d_set_a @ X2 @ S5 ) )
= ( ! [Y4: set_c_d_set_a] :
( ( ( member_set_c_d_set_a @ Y4 @ S5 )
& ( Y4 != X2 ) )
=> ( ( R3 @ X2 @ Y4 )
& ( R3 @ Y4 @ X2 ) ) )
& ( pairwi5502267298322432890_set_a @ R3 @ S5 ) ) ) ).
% pairwise_insert
thf(fact_744_pairwise__insert,axiom,
! [R3: set_a > set_a > $o,X2: set_a,S5: set_set_a] :
( ( pairwise_set_a @ R3 @ ( insert_set_a @ X2 @ S5 ) )
= ( ! [Y4: set_a] :
( ( ( member_set_a @ Y4 @ S5 )
& ( Y4 != X2 ) )
=> ( ( R3 @ X2 @ Y4 )
& ( R3 @ Y4 @ X2 ) ) )
& ( pairwise_set_a @ R3 @ S5 ) ) ) ).
% pairwise_insert
thf(fact_745_pairwise__insert,axiom,
! [R3: a > a > $o,X2: a,S5: set_a] :
( ( pairwise_a @ R3 @ ( insert_a @ X2 @ S5 ) )
= ( ! [Y4: a] :
( ( ( member_a @ Y4 @ S5 )
& ( Y4 != X2 ) )
=> ( ( R3 @ X2 @ Y4 )
& ( R3 @ Y4 @ X2 ) ) )
& ( pairwise_a @ R3 @ S5 ) ) ) ).
% pairwise_insert
thf(fact_746_pairwise__insert,axiom,
! [R3: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,X2: ( c > d ) > set_a,S5: set_c_d_set_a] :
( ( pairwise_c_d_set_a @ R3 @ ( insert_c_d_set_a @ X2 @ S5 ) )
= ( ! [Y4: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ Y4 @ S5 )
& ( Y4 != X2 ) )
=> ( ( R3 @ X2 @ Y4 )
& ( R3 @ Y4 @ X2 ) ) )
& ( pairwise_c_d_set_a @ R3 @ S5 ) ) ) ).
% pairwise_insert
thf(fact_747_Compl__empty__eq,axiom,
( ( uminus8771976365291672326_set_a @ bot_bo738396921950161403_set_a )
= top_to4267977599310771935_set_a ) ).
% Compl_empty_eq
thf(fact_748_Compl__empty__eq,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% Compl_empty_eq
thf(fact_749_Compl__UNIV__eq,axiom,
( ( uminus8771976365291672326_set_a @ top_to4267977599310771935_set_a )
= bot_bo738396921950161403_set_a ) ).
% Compl_UNIV_eq
thf(fact_750_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_751_subset__Compl__self__eq,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_752_subset__Compl__self__eq,axiom,
! [A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ ( uminus8771976365291672326_set_a @ A3 ) )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_753_Compl__eq__Diff__UNIV,axiom,
( uminus8771976365291672326_set_a
= ( minus_1665977719694084726_set_a @ top_to4267977599310771935_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_754_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_a
= ( minus_minus_set_a @ top_top_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_755_pairwise__singleton,axiom,
! [P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A3: ( c > d ) > set_a] : ( pairwise_c_d_set_a @ P @ ( insert_c_d_set_a @ A3 @ bot_bo738396921950161403_set_a ) ) ).
% pairwise_singleton
thf(fact_756_pairwise__singleton,axiom,
! [P: a > a > $o,A3: a] : ( pairwise_a @ P @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_757_less__fun__def,axiom,
( ord_less_c_d_set_a
= ( ^ [F2: ( c > d ) > set_a,G: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ F2 @ G )
& ~ ( ord_le8464990428230162895_set_a @ G @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_758_top_Oordering__top__axioms,axiom,
orderi3930633105564526365et_a_o @ ord_le961293222253252206et_a_o @ ord_less_c_d_set_a_o @ top_top_c_d_set_a_o ).
% top.ordering_top_axioms
thf(fact_759_top_Oordering__top__axioms,axiom,
ordering_top_a_o @ ord_less_eq_a_o @ ord_less_a_o @ top_top_a_o ).
% top.ordering_top_axioms
thf(fact_760_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_761_top_Oordering__top__axioms,axiom,
orderi13773357969974208_set_a @ ord_le5982164083705284911_set_a @ ord_le3685282097655362107_set_a @ top_to4267977599310771935_set_a ).
% top.ordering_top_axioms
thf(fact_762_top_Oordering__top__axioms,axiom,
orderi5785346111247480928_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a @ top_top_c_d_set_a ).
% top.ordering_top_axioms
thf(fact_763_order_Oordering__axioms,axiom,
ordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% order.ordering_axioms
thf(fact_764_order_Oordering__axioms,axiom,
orderi3582402084717419303_set_a @ ord_le5982164083705284911_set_a @ ord_le3685282097655362107_set_a ).
% order.ordering_axioms
thf(fact_765_order_Oordering__axioms,axiom,
ordering_c_d_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% order.ordering_axioms
thf(fact_766_bdd__above_Opreordering__bdd__axioms,axiom,
condit6315317455391067509_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_767_bdd__above_Opreordering__bdd__axioms,axiom,
condit3378509905675676198_set_a @ ord_le5982164083705284911_set_a @ ord_le3685282097655362107_set_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_768_bdd__above_Opreordering__bdd__axioms,axiom,
condit5292637031048566470_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_769_finite__compl,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( finite3330819693523053784_set_a @ ( uminus8771976365291672326_set_a @ A3 ) )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_compl
thf(fact_770_finite__compl,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( finite_finite_a @ ( uminus_uminus_set_a @ A3 ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_compl
thf(fact_771_finite__induct__select,axiom,
! [S4: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ S4 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [T3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ T3 @ S4 )
=> ( ( P @ T3 )
=> ? [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ ( minus_1665977719694084726_set_a @ S4 @ T3 ) )
& ( P @ ( insert_c_d_set_a @ X6 @ T3 ) ) ) ) )
=> ( P @ S4 ) ) ) ) ).
% finite_induct_select
thf(fact_772_finite__induct__select,axiom,
! [S4: set_a,P: set_a > $o] :
( ( finite_finite_a @ S4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S4 )
=> ( ( P @ T3 )
=> ? [X6: a] :
( ( member_a @ X6 @ ( minus_minus_set_a @ S4 @ T3 ) )
& ( P @ ( insert_a @ X6 @ T3 ) ) ) ) )
=> ( P @ S4 ) ) ) ) ).
% finite_induct_select
thf(fact_773_remove__induct,axiom,
! [P: set_set_c_d_set_a > $o,B3: set_set_c_d_set_a] :
( ( P @ bot_bo58555506362910043_set_a )
=> ( ( ~ ( finite457288119118821432_set_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A5 )
=> ( ( A5 != bot_bo58555506362910043_set_a )
=> ( ( ord_le7272806397018272911_set_a @ A5 @ B3 )
=> ( ! [X6: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X6 @ A5 )
=> ( P @ ( minus_3753830358241515990_set_a @ A5 @ ( insert_set_c_d_set_a @ X6 @ bot_bo58555506362910043_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_774_remove__induct,axiom,
! [P: set_set_a > $o,B3: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ! [X6: set_a] :
( ( member_set_a @ X6 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_775_remove__induct,axiom,
! [P: set_a > $o,B3: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_776_remove__induct,axiom,
! [P: set_c_d_set_a > $o,B3: set_c_d_set_a] :
( ( P @ bot_bo738396921950161403_set_a )
=> ( ( ~ ( finite3330819693523053784_set_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A5 )
=> ( ( A5 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ A5 @ B3 )
=> ( ! [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ A5 )
=> ( P @ ( minus_1665977719694084726_set_a @ A5 @ ( insert_c_d_set_a @ X6 @ bot_bo738396921950161403_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_777_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_778_finite__Plus__UNIV__iff,axiom,
( ( finite793227202826479597_set_a @ top_to2270067951290598332_set_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_779_finite__Plus__UNIV__iff,axiom,
( ( finite5017241612818336203et_a_a @ top_to5142853716682987162et_a_a )
= ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_780_finite__Plus__UNIV__iff,axiom,
( ( finite5989733633321134460_set_a @ top_to279427854467338187_set_a )
= ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
& ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_781_finite__has__maximal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X: set_a] :
( ( member_set_a @ X @ A3 )
& ( ord_less_eq_set_a @ A @ X )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_782_finite__has__maximal2,axiom,
! [A3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( member_set_c_d_set_a @ A @ A3 )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
& ( ord_le5982164083705284911_set_a @ A @ X )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_783_finite__has__maximal2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
& ( ord_le8464990428230162895_set_a @ A @ X )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_784_finite__has__minimal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X: set_a] :
( ( member_set_a @ X @ A3 )
& ( ord_less_eq_set_a @ X @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_785_finite__has__minimal2,axiom,
! [A3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( member_set_c_d_set_a @ A @ A3 )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
& ( ord_le5982164083705284911_set_a @ X @ A )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_786_finite__has__minimal2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
& ( ord_le8464990428230162895_set_a @ X @ A )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_787_ex__new__if__finite,axiom,
! [A3: set_set_c_d_set_a] :
( ~ ( finite457288119118821432_set_a @ top_to5717711934741766719_set_a )
=> ( ( finite457288119118821432_set_a @ A3 )
=> ? [A6: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_788_ex__new__if__finite,axiom,
! [A3: set_set_a] :
( ~ ( finite_finite_set_a @ top_top_set_set_a )
=> ( ( finite_finite_set_a @ A3 )
=> ? [A6: set_a] :
~ ( member_set_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_789_ex__new__if__finite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A3 )
=> ? [A6: a] :
~ ( member_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_790_ex__new__if__finite,axiom,
! [A3: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( ( finite3330819693523053784_set_a @ A3 )
=> ? [A6: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_791_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_792_Finite__Set_Ofinite__set,axiom,
( ( finite457288119118821432_set_a @ top_to5717711934741766719_set_a )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_793_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_794_finite__prod,axiom,
( ( finite4492253876758910081_set_a @ top_to6342870235713707528_set_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_prod
thf(fact_795_finite__prod,axiom,
( ( finite8716268286750766687et_a_a @ top_to9215656001106096358et_a_a )
= ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_796_finite__prod,axiom,
( ( finite2397556900044337168_set_a @ top_to3895570120271872023_set_a )
= ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
& ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_prod
thf(fact_797_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_798_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( finite4492253876758910081_set_a @ top_to6342870235713707528_set_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_799_finite__Prod__UNIV,axiom,
( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite8716268286750766687et_a_a @ top_to9215656001106096358et_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_800_finite__Prod__UNIV,axiom,
( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( finite2397556900044337168_set_a @ top_to3895570120271872023_set_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_801_finite__fun__UNIVD2,axiom,
( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( finite_finite_set_a @ top_top_set_set_a ) ) ).
% finite_fun_UNIVD2
thf(fact_802_infinite__imp__nonempty,axiom,
! [S4: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ S4 )
=> ( S4 != bot_bo738396921950161403_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_803_infinite__imp__nonempty,axiom,
! [S4: set_a] :
( ~ ( finite_finite_a @ S4 )
=> ( S4 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_804_finite_OemptyI,axiom,
finite3330819693523053784_set_a @ bot_bo738396921950161403_set_a ).
% finite.emptyI
thf(fact_805_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_806_finite__subset,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( finite_finite_a @ B3 )
=> ( finite_finite_a @ A3 ) ) ) ).
% finite_subset
thf(fact_807_finite__subset,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( finite3330819693523053784_set_a @ B3 )
=> ( finite3330819693523053784_set_a @ A3 ) ) ) ).
% finite_subset
thf(fact_808_infinite__super,axiom,
! [S4: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S4 @ T2 )
=> ( ~ ( finite_finite_a @ S4 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_809_infinite__super,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ S4 @ T2 )
=> ( ~ ( finite3330819693523053784_set_a @ S4 )
=> ~ ( finite3330819693523053784_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_810_rev__finite__subset,axiom,
! [B3: set_a,A3: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( finite_finite_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_811_rev__finite__subset,axiom,
! [B3: set_c_d_set_a,A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( finite3330819693523053784_set_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_812_finite__has__minimal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a @ X @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_813_finite__has__minimal,axiom,
! [A3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_814_finite__has__minimal,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_815_finite__has__maximal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a @ X @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_816_finite__has__maximal,axiom,
! [A3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_817_finite__has__maximal,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_818_infinite__finite__induct,axiom,
! [P: set_set_c_d_set_a > $o,A3: set_set_c_d_set_a] :
( ! [A5: set_set_c_d_set_a] :
( ~ ( finite457288119118821432_set_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo58555506362910043_set_a )
=> ( ! [X: set_c_d_set_a,F3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ F3 )
=> ( ~ ( member_set_c_d_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_c_d_set_a @ X @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_819_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A3: set_set_a] :
( ! [A5: set_set_a] :
( ~ ( finite_finite_set_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_820_infinite__finite__induct,axiom,
! [P: set_a > $o,A3: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_821_infinite__finite__induct,axiom,
! [P: set_c_d_set_a > $o,A3: set_c_d_set_a] :
( ! [A5: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ~ ( member_c_d_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ X @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_822_finite__ne__induct,axiom,
! [F4: set_set_c_d_set_a,P: set_set_c_d_set_a > $o] :
( ( finite457288119118821432_set_a @ F4 )
=> ( ( F4 != bot_bo58555506362910043_set_a )
=> ( ! [X: set_c_d_set_a] : ( P @ ( insert_set_c_d_set_a @ X @ bot_bo58555506362910043_set_a ) )
=> ( ! [X: set_c_d_set_a,F3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ F3 )
=> ( ( F3 != bot_bo58555506362910043_set_a )
=> ( ~ ( member_set_c_d_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_c_d_set_a @ X @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_823_finite__ne__induct,axiom,
! [F4: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( F4 != bot_bot_set_set_a )
=> ( ! [X: set_a] : ( P @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ! [X: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_824_finite__ne__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_825_finite__ne__induct,axiom,
! [F4: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( F4 != bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a] : ( P @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
=> ( ! [X: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( F3 != bot_bo738396921950161403_set_a )
=> ( ~ ( member_c_d_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ X @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_826_finite__induct,axiom,
! [F4: set_set_c_d_set_a,P: set_set_c_d_set_a > $o] :
( ( finite457288119118821432_set_a @ F4 )
=> ( ( P @ bot_bo58555506362910043_set_a )
=> ( ! [X: set_c_d_set_a,F3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ F3 )
=> ( ~ ( member_set_c_d_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_c_d_set_a @ X @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_827_finite__induct,axiom,
! [F4: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_828_finite__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_829_finite__induct,axiom,
! [F4: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ~ ( member_c_d_set_a @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ X @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_830_finite_Osimps,axiom,
( finite3330819693523053784_set_a
= ( ^ [A2: set_c_d_set_a] :
( ( A2 = bot_bo738396921950161403_set_a )
| ? [A4: set_c_d_set_a,B2: ( c > d ) > set_a] :
( ( A2
= ( insert_c_d_set_a @ B2 @ A4 ) )
& ( finite3330819693523053784_set_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_831_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A2: set_a] :
( ( A2 = bot_bot_set_a )
| ? [A4: set_a,B2: a] :
( ( A2
= ( insert_a @ B2 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_832_finite_Ocases,axiom,
! [A: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A )
=> ( ( A != bot_bo738396921950161403_set_a )
=> ~ ! [A5: set_c_d_set_a] :
( ? [A6: ( c > d ) > set_a] :
( A
= ( insert_c_d_set_a @ A6 @ A5 ) )
=> ~ ( finite3330819693523053784_set_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_833_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A6: a] :
( A
= ( insert_a @ A6 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_834_finite__subset__induct_H,axiom,
! [F4: set_set_c_d_set_a,A3: set_set_c_d_set_a,P: set_set_c_d_set_a > $o] :
( ( finite457288119118821432_set_a @ F4 )
=> ( ( ord_le7272806397018272911_set_a @ F4 @ A3 )
=> ( ( P @ bot_bo58555506362910043_set_a )
=> ( ! [A6: set_c_d_set_a,F3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ F3 )
=> ( ( member_set_c_d_set_a @ A6 @ A3 )
=> ( ( ord_le7272806397018272911_set_a @ F3 @ A3 )
=> ( ~ ( member_set_c_d_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_c_d_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_835_finite__subset__induct_H,axiom,
! [F4: set_set_a,A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A3 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_836_finite__subset__induct_H,axiom,
! [F4: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_837_finite__subset__induct_H,axiom,
! [F4: set_c_d_set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( ord_le5982164083705284911_set_a @ F4 @ A3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A6: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( member_c_d_set_a @ A6 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ F3 @ A3 )
=> ( ~ ( member_c_d_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_838_finite__subset__induct,axiom,
! [F4: set_set_c_d_set_a,A3: set_set_c_d_set_a,P: set_set_c_d_set_a > $o] :
( ( finite457288119118821432_set_a @ F4 )
=> ( ( ord_le7272806397018272911_set_a @ F4 @ A3 )
=> ( ( P @ bot_bo58555506362910043_set_a )
=> ( ! [A6: set_c_d_set_a,F3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ F3 )
=> ( ( member_set_c_d_set_a @ A6 @ A3 )
=> ( ~ ( member_set_c_d_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_c_d_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_839_finite__subset__induct,axiom,
! [F4: set_set_a,A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A3 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_840_finite__subset__induct,axiom,
! [F4: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A3 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_841_finite__subset__induct,axiom,
! [F4: set_c_d_set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( ord_le5982164083705284911_set_a @ F4 @ A3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A6: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( member_c_d_set_a @ A6 @ A3 )
=> ( ~ ( member_c_d_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_842_infinite__remove,axiom,
! [S4: set_c_d_set_a,A: ( c > d ) > set_a] :
( ~ ( finite3330819693523053784_set_a @ S4 )
=> ~ ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ S4 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ) ).
% infinite_remove
thf(fact_843_infinite__remove,axiom,
! [S4: set_a,A: a] :
( ~ ( finite_finite_a @ S4 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S4 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_844_infinite__coinduct,axiom,
! [X5: set_c_d_set_a > $o,A3: set_c_d_set_a] :
( ( X5 @ A3 )
=> ( ! [A5: set_c_d_set_a] :
( ( X5 @ A5 )
=> ? [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ A5 )
& ( ( X5 @ ( minus_1665977719694084726_set_a @ A5 @ ( insert_c_d_set_a @ X6 @ bot_bo738396921950161403_set_a ) ) )
| ~ ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ A5 @ ( insert_c_d_set_a @ X6 @ bot_bo738396921950161403_set_a ) ) ) ) ) )
=> ~ ( finite3330819693523053784_set_a @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_845_infinite__coinduct,axiom,
! [X5: set_a > $o,A3: set_a] :
( ( X5 @ A3 )
=> ( ! [A5: set_a] :
( ( X5 @ A5 )
=> ? [X6: a] :
( ( member_a @ X6 @ A5 )
& ( ( X5 @ ( minus_minus_set_a @ A5 @ ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_846_finite__empty__induct,axiom,
! [A3: set_set_c_d_set_a,P: set_set_c_d_set_a > $o] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: set_c_d_set_a,A5: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A5 )
=> ( ( member_set_c_d_set_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_3753830358241515990_set_a @ A5 @ ( insert_set_c_d_set_a @ A6 @ bot_bo58555506362910043_set_a ) ) ) ) ) )
=> ( P @ bot_bo58555506362910043_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_847_finite__empty__induct,axiom,
! [A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: set_a,A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( member_set_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A6 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_848_finite__empty__induct,axiom,
! [A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_849_finite__empty__induct,axiom,
! [A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: ( c > d ) > set_a,A5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A5 )
=> ( ( member_c_d_set_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_1665977719694084726_set_a @ A5 @ ( insert_c_d_set_a @ A6 @ bot_bo738396921950161403_set_a ) ) ) ) ) )
=> ( P @ bot_bo738396921950161403_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_850_finite__remove__induct,axiom,
! [B3: set_set_c_d_set_a,P: set_set_c_d_set_a > $o] :
( ( finite457288119118821432_set_a @ B3 )
=> ( ( P @ bot_bo58555506362910043_set_a )
=> ( ! [A5: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A5 )
=> ( ( A5 != bot_bo58555506362910043_set_a )
=> ( ( ord_le7272806397018272911_set_a @ A5 @ B3 )
=> ( ! [X6: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X6 @ A5 )
=> ( P @ ( minus_3753830358241515990_set_a @ A5 @ ( insert_set_c_d_set_a @ X6 @ bot_bo58555506362910043_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_851_finite__remove__induct,axiom,
! [B3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ! [X6: set_a] :
( ( member_set_a @ X6 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_852_finite__remove__induct,axiom,
! [B3: set_a,P: set_a > $o] :
( ( finite_finite_a @ B3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_853_finite__remove__induct,axiom,
! [B3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ B3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A5 )
=> ( ( A5 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ A5 @ B3 )
=> ( ! [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ A5 )
=> ( P @ ( minus_1665977719694084726_set_a @ A5 @ ( insert_c_d_set_a @ X6 @ bot_bo738396921950161403_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_854_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_855_finite__option__UNIV,axiom,
( ( finite1740182815655637662_set_a @ top_to1333438998097461157_set_a )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ).
% finite_option_UNIV
thf(fact_856_ex__min__if__finite,axiom,
! [S4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ S4 )
=> ( ( S4 != bot_bo738396921950161403_set_a )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ S4 )
& ~ ? [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ S4 )
& ( ord_less_c_d_set_a @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_857_ex__min__if__finite,axiom,
! [S4: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ S4 )
=> ( ( S4 != bot_bo58555506362910043_set_a )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ S4 )
& ~ ? [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ S4 )
& ( ord_le3685282097655362107_set_a @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_858_ex__min__if__finite,axiom,
! [S4: set_set_a] :
( ( finite_finite_set_a @ S4 )
=> ( ( S4 != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a @ X @ S4 )
& ~ ? [Xa: set_a] :
( ( member_set_a @ Xa @ S4 )
& ( ord_less_set_a @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_859_cofinite__bot,axiom,
( ( cofinite_a = bot_bot_filter_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% cofinite_bot
thf(fact_860_cofinite__bot,axiom,
( ( cofinite_c_d_set_a = bot_bo1170640541930586401_set_a )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ).
% cofinite_bot
thf(fact_861_arg__min__if__finite_I2_J,axiom,
! [S4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ S4 )
=> ( ( S4 != bot_bo738396921950161403_set_a )
=> ~ ? [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ S4 )
& ( ord_le3685282097655362107_set_a @ ( F @ X6 ) @ ( F @ ( lattic5191877960299797817_set_a @ F @ S4 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_862_arg__min__if__finite_I2_J,axiom,
! [S4: set_a,F: a > set_c_d_set_a] :
( ( finite_finite_a @ S4 )
=> ( ( S4 != bot_bot_set_a )
=> ~ ? [X6: a] :
( ( member_a @ X6 @ S4 )
& ( ord_le3685282097655362107_set_a @ ( F @ X6 ) @ ( F @ ( lattic1032982531657200042_set_a @ F @ S4 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_863_arg__min__if__finite_I2_J,axiom,
! [S4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ( finite3330819693523053784_set_a @ S4 )
=> ( ( S4 != bot_bo738396921950161403_set_a )
=> ~ ? [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ S4 )
& ( ord_less_set_a @ ( F @ X6 ) @ ( F @ ( lattic2349170783384439560_set_a @ F @ S4 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_864_arg__min__if__finite_I2_J,axiom,
! [S4: set_a,F: a > set_a] :
( ( finite_finite_a @ S4 )
=> ( ( S4 != bot_bot_set_a )
=> ~ ? [X6: a] :
( ( member_a @ X6 @ S4 )
& ( ord_less_set_a @ ( F @ X6 ) @ ( F @ ( lattic2425909027367666425_set_a @ F @ S4 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_865_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A3: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_866_image__eqI,axiom,
! [B: ( c > d ) > set_a,F: a > ( c > d ) > set_a,X2: a,A3: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A3 )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_867_image__eqI,axiom,
! [B: a,F: ( ( c > d ) > set_a ) > a,X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_868_image__eqI,axiom,
! [B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_869_image__eqI,axiom,
! [B: set_a,F: a > set_a,X2: a,A3: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A3 )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_870_image__eqI,axiom,
! [B: a,F: set_a > a,X2: set_a,A3: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a @ X2 @ A3 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_871_image__eqI,axiom,
! [B: set_a,F: set_a > set_a,X2: set_a,A3: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_872_image__eqI,axiom,
! [B: set_c_d_set_a,F: a > set_c_d_set_a,X2: a,A3: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A3 )
=> ( member_set_c_d_set_a @ B @ ( image_3734436999661236630_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_873_image__eqI,axiom,
! [B: set_a,F: ( ( c > d ) > set_a ) > set_a,X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_set_a @ B @ ( image_5050625251388476148_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_874_image__eqI,axiom,
! [B: a,F: set_c_d_set_a > a,X2: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( member_a @ B @ ( image_702032380087044660et_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_875_image__empty,axiom,
! [F: set_a > set_a] :
( ( image_set_a_set_a @ F @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_876_image__empty,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( image_5710119992958135237_set_a @ F @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% image_empty
thf(fact_877_image__empty,axiom,
! [F: ( ( c > d ) > set_a ) > a] :
( ( image_c_d_set_a_a @ F @ bot_bo738396921950161403_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_878_image__empty,axiom,
! [F: a > ( c > d ) > set_a] :
( ( image_a_c_d_set_a @ F @ bot_bot_set_a )
= bot_bo738396921950161403_set_a ) ).
% image_empty
thf(fact_879_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_880_empty__is__image,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( bot_bot_set_set_a
= ( image_set_a_set_a @ F @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_881_empty__is__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% empty_is_image
thf(fact_882_empty__is__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a] :
( ( bot_bo738396921950161403_set_a
= ( image_a_c_d_set_a @ F @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_883_empty__is__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( bot_bot_set_a
= ( image_c_d_set_a_a @ F @ A3 ) )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% empty_is_image
thf(fact_884_empty__is__image,axiom,
! [F: a > a,A3: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_885_image__is__empty,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( ( image_set_a_set_a @ F @ A3 )
= bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_886_image__is__empty,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( image_5710119992958135237_set_a @ F @ A3 )
= bot_bo738396921950161403_set_a )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% image_is_empty
thf(fact_887_image__is__empty,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a] :
( ( ( image_a_c_d_set_a @ F @ A3 )
= bot_bo738396921950161403_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_888_image__is__empty,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( ( image_c_d_set_a_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% image_is_empty
thf(fact_889_image__is__empty,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_890_image__insert,axiom,
! [F: set_a > set_a,A: set_a,B3: set_set_a] :
( ( image_set_a_set_a @ F @ ( insert_set_a @ A @ B3 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_set_a_set_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_891_image__insert,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( image_5710119992958135237_set_a @ F @ ( insert_c_d_set_a @ A @ B3 ) )
= ( insert_c_d_set_a @ ( F @ A ) @ ( image_5710119992958135237_set_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_892_image__insert,axiom,
! [F: ( ( c > d ) > set_a ) > a,A: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( image_c_d_set_a_a @ F @ ( insert_c_d_set_a @ A @ B3 ) )
= ( insert_a @ ( F @ A ) @ ( image_c_d_set_a_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_893_image__insert,axiom,
! [F: a > ( c > d ) > set_a,A: a,B3: set_a] :
( ( image_a_c_d_set_a @ F @ ( insert_a @ A @ B3 ) )
= ( insert_c_d_set_a @ ( F @ A ) @ ( image_a_c_d_set_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_894_image__insert,axiom,
! [F: a > a,A: a,B3: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B3 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_895_insert__image,axiom,
! [X2: a,A3: set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( insert_c_d_set_a @ ( F @ X2 ) @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_896_insert__image,axiom,
! [X2: a,A3: set_a,F: a > a] :
( ( member_a @ X2 @ A3 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_a_a @ F @ A3 ) )
= ( image_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_897_insert__image,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( insert_c_d_set_a @ ( F @ X2 ) @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_898_insert__image,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( image_c_d_set_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_899_insert__image,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a] :
( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( insert_c_d_set_a @ ( F @ X2 ) @ ( image_212549500329102437_set_a @ F @ A3 ) )
= ( image_212549500329102437_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_900_insert__image,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,F: set_c_d_set_a > a] :
( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_702032380087044660et_a_a @ F @ A3 ) )
= ( image_702032380087044660et_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_901_insert__image,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( insert_set_a @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ A3 ) )
= ( image_set_a_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_902_insert__image,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > ( c > d ) > set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( insert_c_d_set_a @ ( F @ X2 ) @ ( image_1482592857945081046_set_a @ F @ A3 ) )
= ( image_1482592857945081046_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_903_insert__image,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_set_a_a @ F @ A3 ) )
= ( image_set_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_904_all__subset__image,axiom,
! [F: set_a > set_a,A3: set_set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( P @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_905_all__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_906_all__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( P @ ( image_c_d_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_907_all__subset__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: set_c_d_set_a > $o] :
( ( ! [B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( P @ ( image_a_c_d_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_908_all__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( ! [B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( P @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_909_rev__image__eqI,axiom,
! [X2: a,A3: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_910_rev__image__eqI,axiom,
! [X2: a,A3: set_a,B: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_911_rev__image__eqI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B: a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_912_rev__image__eqI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_913_rev__image__eqI,axiom,
! [X2: a,A3: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_914_rev__image__eqI,axiom,
! [X2: set_a,A3: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_915_rev__image__eqI,axiom,
! [X2: set_a,A3: set_set_a,B: set_a,F: set_a > set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_916_rev__image__eqI,axiom,
! [X2: a,A3: set_a,B: set_c_d_set_a,F: a > set_c_d_set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_c_d_set_a @ B @ ( image_3734436999661236630_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_917_rev__image__eqI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,B: set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_5050625251388476148_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_918_rev__image__eqI,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,B: a,F: set_c_d_set_a > a] :
( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_702032380087044660et_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_919_ball__imageD,axiom,
! [F: set_a > set_a,A3: set_set_a,P: set_a > $o] :
( ! [X: set_a] :
( ( member_set_a @ X @ ( image_set_a_set_a @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: set_a] :
( ( member_set_a @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_920_ball__imageD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_921_ball__imageD,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_922_ball__imageD,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: ( ( c > d ) > set_a ) > $o] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_923_ball__imageD,axiom,
! [F: a > a,A3: set_a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A3 ) )
=> ( P @ X ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_924_image__cong,axiom,
! [M: set_a,N: set_a,F: a > ( c > d ) > set_a,G2: a > ( c > d ) > set_a] :
( ( M = N )
=> ( ! [X: a] :
( ( member_a @ X @ N )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_a_c_d_set_a @ F @ M )
= ( image_a_c_d_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_925_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G2: a > a] :
( ( M = N )
=> ( ! [X: a] :
( ( member_a @ X @ N )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_926_image__cong,axiom,
! [M: set_c_d_set_a,N: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( M = N )
=> ( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ N )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_5710119992958135237_set_a @ F @ M )
= ( image_5710119992958135237_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_927_image__cong,axiom,
! [M: set_c_d_set_a,N: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,G2: ( ( c > d ) > set_a ) > a] :
( ( M = N )
=> ( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ N )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_c_d_set_a_a @ F @ M )
= ( image_c_d_set_a_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_928_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > set_a,G2: set_a > set_a] :
( ( M = N )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ N )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_set_a_set_a @ F @ M )
= ( image_set_a_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_929_bex__imageD,axiom,
! [F: set_a > set_a,A3: set_set_a,P: set_a > $o] :
( ? [X6: set_a] :
( ( member_set_a @ X6 @ ( image_set_a_set_a @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: set_a] :
( ( member_set_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_930_bex__imageD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o] :
( ? [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_931_bex__imageD,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( image_c_d_set_a_a @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_932_bex__imageD,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: ( ( c > d ) > set_a ) > $o] :
( ? [X6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X6 @ ( image_a_c_d_set_a @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: a] :
( ( member_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_933_bex__imageD,axiom,
! [F: a > a,A3: set_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( image_a_a @ F @ A3 ) )
& ( P @ X6 ) )
=> ? [X: a] :
( ( member_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_934_image__iff,axiom,
! [Z: a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( member_a @ Z @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_935_image__iff,axiom,
! [Z: a,F: a > a,A3: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A3 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_936_image__iff,axiom,
! [Z: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ Z @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_937_image__iff,axiom,
! [Z: ( c > d ) > set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( member_c_d_set_a @ Z @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_938_image__iff,axiom,
! [Z: set_a,F: set_a > set_a,A3: set_set_a] :
( ( member_set_a @ Z @ ( image_set_a_set_a @ F @ A3 ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_939_imageI,axiom,
! [X2: a,A3: set_a,F: a > a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_940_imageI,axiom,
! [X2: a,A3: set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X2 @ A3 )
=> ( member_c_d_set_a @ ( F @ X2 ) @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_941_imageI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_942_imageI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_c_d_set_a @ ( F @ X2 ) @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_943_imageI,axiom,
! [X2: a,A3: set_a,F: a > set_a] :
( ( member_a @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ ( image_a_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_944_imageI,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ ( image_set_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_945_imageI,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_946_imageI,axiom,
! [X2: a,A3: set_a,F: a > set_c_d_set_a] :
( ( member_a @ X2 @ A3 )
=> ( member_set_c_d_set_a @ ( F @ X2 ) @ ( image_3734436999661236630_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_947_imageI,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ ( image_5050625251388476148_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_948_imageI,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,F: set_c_d_set_a > a] :
( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ ( image_702032380087044660et_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_949_subset__image__iff,axiom,
! [B3: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B3
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_950_subset__image__iff,axiom,
! [B3: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B3
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_951_subset__image__iff,axiom,
! [B3: set_a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( ? [AA: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ AA @ A3 )
& ( B3
= ( image_c_d_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_952_subset__image__iff,axiom,
! [B3: set_c_d_set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( ord_le5982164083705284911_set_a @ B3 @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B3
= ( image_a_c_d_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_953_subset__image__iff,axiom,
! [B3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B3 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( ? [AA: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ AA @ A3 )
& ( B3
= ( image_5710119992958135237_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_954_image__subset__iff,axiom,
! [F: set_a > set_a,A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ B3 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_955_image__subset__iff,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ B3 )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_956_image__subset__iff,axiom,
! [F: a > a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_957_image__subset__iff,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ B3 )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( member_c_d_set_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_958_image__subset__iff,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_c_d_set_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_959_subset__imageE,axiom,
! [B3: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F @ A3 ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A3 )
=> ( B3
!= ( image_set_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_960_subset__imageE,axiom,
! [B3: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A3 )
=> ( B3
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_961_subset__imageE,axiom,
! [B3: set_a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ~ ! [C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ A3 )
=> ( B3
!= ( image_c_d_set_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_962_subset__imageE,axiom,
! [B3: set_c_d_set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( ord_le5982164083705284911_set_a @ B3 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A3 )
=> ( B3
!= ( image_a_c_d_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_963_subset__imageE,axiom,
! [B3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B3 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ~ ! [C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ A3 )
=> ( B3
!= ( image_5710119992958135237_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_964_image__subsetI,axiom,
! [A3: set_a,F: a > a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_965_image__subsetI,axiom,
! [A3: set_a,F: a > ( c > d ) > set_a,B3: set_c_d_set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_c_d_set_a @ ( F @ X ) @ B3 ) )
=> ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_966_image__subsetI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,B3: set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_967_image__subsetI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B3: set_c_d_set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( member_c_d_set_a @ ( F @ X ) @ B3 ) )
=> ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_968_image__subsetI,axiom,
! [A3: set_a,F: a > set_a,B3: set_set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_set_a @ ( F @ X ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_969_image__subsetI,axiom,
! [A3: set_set_a,F: set_a > a,B3: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_970_image__subsetI,axiom,
! [A3: set_set_a,F: set_a > set_a,B3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ ( F @ X ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_971_image__subsetI,axiom,
! [A3: set_a,F: a > set_c_d_set_a,B3: set_set_c_d_set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_set_c_d_set_a @ ( F @ X ) @ B3 ) )
=> ( ord_le7272806397018272911_set_a @ ( image_3734436999661236630_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_972_image__subsetI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,B3: set_set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( member_set_a @ ( F @ X ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_5050625251388476148_set_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_973_image__subsetI,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > a,B3: set_a] :
( ! [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_702032380087044660et_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_974_image__mono,axiom,
! [A3: set_set_a,B3: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ ( image_set_a_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_975_image__mono,axiom,
! [A3: set_a,B3: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_976_image__mono,axiom,
! [A3: set_a,B3: set_a,F: a > ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ ( image_a_c_d_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_977_image__mono,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ ( image_c_d_set_a_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_978_image__mono,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ ( image_5710119992958135237_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_979_pairwise__imageI,axiom,
! [A3: set_set_a,F: set_a > set_a,P: set_a > set_a > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( member_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_set_a @ P @ ( image_set_a_set_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_980_pairwise__imageI,axiom,
! [A3: set_a,F: a > ( c > d ) > set_a,P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ A3 )
=> ( ( member_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_c_d_set_a @ P @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_981_pairwise__imageI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( member_c_d_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_c_d_set_a @ P @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_982_pairwise__imageI,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
=> ( ( member_set_c_d_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_c_d_set_a @ P @ ( image_212549500329102437_set_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_983_pairwise__imageI,axiom,
! [A3: set_set_a,F: set_a > ( c > d ) > set_a,P: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( member_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_c_d_set_a @ P @ ( image_1482592857945081046_set_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_984_pairwise__imageI,axiom,
! [A3: set_a,F: a > a,P: a > a > $o] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ A3 )
=> ( ( member_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_a @ P @ ( image_a_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_985_pairwise__imageI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,P: a > a > $o] :
( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( member_c_d_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_a @ P @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_986_pairwise__imageI,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > a,P: a > a > $o] :
( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A3 )
=> ( ( member_set_c_d_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_a @ P @ ( image_702032380087044660et_a_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_987_pairwise__imageI,axiom,
! [A3: set_set_a,F: set_a > a,P: a > a > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( member_set_a @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_a @ P @ ( image_set_a_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_988_order_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% order.partial_preordering_axioms
thf(fact_989_order_Opartial__preordering__axioms,axiom,
partia1270112395057131461_set_a @ ord_le5982164083705284911_set_a ).
% order.partial_preordering_axioms
thf(fact_990_order_Opartial__preordering__axioms,axiom,
partia701112543150332005_set_a @ ord_le8464990428230162895_set_a ).
% order.partial_preordering_axioms
thf(fact_991_rangeI,axiom,
! [F: set_a > set_a,X2: set_a] : ( member_set_a @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ top_top_set_set_a ) ) ).
% rangeI
thf(fact_992_rangeI,axiom,
! [F: a > a,X2: a] : ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_993_rangeI,axiom,
! [F: a > ( c > d ) > set_a,X2: a] : ( member_c_d_set_a @ ( F @ X2 ) @ ( image_a_c_d_set_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_994_rangeI,axiom,
! [F: a > set_c_d_set_a,X2: a] : ( member_set_c_d_set_a @ ( F @ X2 ) @ ( image_3734436999661236630_set_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_995_rangeI,axiom,
! [F: a > set_a,X2: a] : ( member_set_a @ ( F @ X2 ) @ ( image_a_set_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_996_rangeI,axiom,
! [F: ( ( c > d ) > set_a ) > a,X2: ( c > d ) > set_a] : ( member_a @ ( F @ X2 ) @ ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a ) ) ).
% rangeI
thf(fact_997_rangeI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X2: ( c > d ) > set_a] : ( member_c_d_set_a @ ( F @ X2 ) @ ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a ) ) ).
% rangeI
thf(fact_998_rangeI,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,X2: ( c > d ) > set_a] : ( member_set_c_d_set_a @ ( F @ X2 ) @ ( image_1181342993027318565_set_a @ F @ top_to4267977599310771935_set_a ) ) ).
% rangeI
thf(fact_999_rangeI,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X2: ( c > d ) > set_a] : ( member_set_a @ ( F @ X2 ) @ ( image_5050625251388476148_set_a @ F @ top_to4267977599310771935_set_a ) ) ).
% rangeI
thf(fact_1000_range__eqI,axiom,
! [B: set_a,F: set_a > set_a,X2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ top_top_set_set_a ) ) ) ).
% range_eqI
thf(fact_1001_range__eqI,axiom,
! [B: a,F: a > a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1002_range__eqI,axiom,
! [B: ( c > d ) > set_a,F: a > ( c > d ) > set_a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1003_range__eqI,axiom,
! [B: set_c_d_set_a,F: a > set_c_d_set_a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_c_d_set_a @ B @ ( image_3734436999661236630_set_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1004_range__eqI,axiom,
! [B: set_a,F: a > set_a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1005_range__eqI,axiom,
! [B: a,F: ( ( c > d ) > set_a ) > a,X2: ( c > d ) > set_a] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a ) ) ) ).
% range_eqI
thf(fact_1006_range__eqI,axiom,
! [B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( B
= ( F @ X2 ) )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a ) ) ) ).
% range_eqI
thf(fact_1007_range__eqI,axiom,
! [B: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_c_d_set_a @ B @ ( image_1181342993027318565_set_a @ F @ top_to4267977599310771935_set_a ) ) ) ).
% range_eqI
thf(fact_1008_range__eqI,axiom,
! [B: set_a,F: ( ( c > d ) > set_a ) > set_a,X2: ( c > d ) > set_a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_5050625251388476148_set_a @ F @ top_to4267977599310771935_set_a ) ) ) ).
% range_eqI
thf(fact_1009_range__subsetD,axiom,
! [F: set_a > set_a,B3: set_set_a,I: set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ top_top_set_set_a ) @ B3 )
=> ( member_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1010_range__subsetD,axiom,
! [F: a > set_c_d_set_a,B3: set_set_c_d_set_a,I: a] :
( ( ord_le7272806397018272911_set_a @ ( image_3734436999661236630_set_a @ F @ top_top_set_a ) @ B3 )
=> ( member_set_c_d_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1011_range__subsetD,axiom,
! [F: a > set_a,B3: set_set_a,I: a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ top_top_set_a ) @ B3 )
=> ( member_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1012_range__subsetD,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,B3: set_set_c_d_set_a,I: ( c > d ) > set_a] :
( ( ord_le7272806397018272911_set_a @ ( image_1181342993027318565_set_a @ F @ top_to4267977599310771935_set_a ) @ B3 )
=> ( member_set_c_d_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1013_range__subsetD,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,B3: set_set_a,I: ( c > d ) > set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_5050625251388476148_set_a @ F @ top_to4267977599310771935_set_a ) @ B3 )
=> ( member_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1014_range__subsetD,axiom,
! [F: a > a,B3: set_a,I: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ top_top_set_a ) @ B3 )
=> ( member_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1015_range__subsetD,axiom,
! [F: ( ( c > d ) > set_a ) > a,B3: set_a,I: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a ) @ B3 )
=> ( member_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1016_range__subsetD,axiom,
! [F: a > ( c > d ) > set_a,B3: set_c_d_set_a,I: a] :
( ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ top_top_set_a ) @ B3 )
=> ( member_c_d_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1017_range__subsetD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B3: set_c_d_set_a,I: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a ) @ B3 )
=> ( member_c_d_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1018_finite__surj,axiom,
! [A3: set_set_a,B3: set_set_a,F: set_a > set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F @ A3 ) )
=> ( finite_finite_set_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1019_finite__surj,axiom,
! [A3: set_c_d_set_a,B3: set_a,F: ( ( c > d ) > set_a ) > a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ( finite_finite_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1020_finite__surj,axiom,
! [A3: set_a,B3: set_a,F: a > a] :
( ( finite_finite_a @ A3 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ( finite_finite_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1021_finite__surj,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ( finite3330819693523053784_set_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1022_finite__surj,axiom,
! [A3: set_a,B3: set_c_d_set_a,F: a > ( c > d ) > set_a] :
( ( finite_finite_a @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ( finite3330819693523053784_set_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1023_finite__subset__image,axiom,
! [B3: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F @ A3 ) )
=> ? [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A3 )
& ( finite_finite_set_a @ C4 )
& ( B3
= ( image_set_a_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1024_finite__subset__image,axiom,
! [B3: set_a,F: a > a,A3: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A3 )
& ( finite_finite_a @ C4 )
& ( B3
= ( image_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1025_finite__subset__image,axiom,
! [B3: set_a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ? [C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ A3 )
& ( finite3330819693523053784_set_a @ C4 )
& ( B3
= ( image_c_d_set_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1026_finite__subset__image,axiom,
! [B3: set_c_d_set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( finite3330819693523053784_set_a @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A3 )
& ( finite_finite_a @ C4 )
& ( B3
= ( image_a_c_d_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1027_finite__subset__image,axiom,
! [B3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ B3 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ? [C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ A3 )
& ( finite3330819693523053784_set_a @ C4 )
& ( B3
= ( image_5710119992958135237_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1028_ex__finite__subset__image,axiom,
! [F: set_a > set_a,A3: set_set_a,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A3 )
& ( P @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1029_ex__finite__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A3 )
& ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1030_ex__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ A3 )
& ( P @ ( image_c_d_set_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1031_ex__finite__subset__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: set_c_d_set_a > $o] :
( ( ? [B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ ( image_a_c_d_set_a @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A3 )
& ( P @ ( image_a_c_d_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1032_ex__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( ? [B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ A3 )
& ( P @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1033_all__finite__subset__image,axiom,
! [F: set_a > set_a,A3: set_set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A3 ) )
=> ( P @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1034_all__finite__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A3 ) )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1035_all__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ A3 ) )
=> ( P @ ( image_c_d_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1036_all__finite__subset__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: set_c_d_set_a > $o] :
( ( ! [B4: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ ( image_a_c_d_set_a @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A3 ) )
=> ( P @ ( image_a_c_d_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1037_all__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( ! [B4: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B4 )
& ( ord_le5982164083705284911_set_a @ B4 @ A3 ) )
=> ( P @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1038_image__diff__subset,axiom,
! [F: set_a > set_a,A3: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_set_a_set_a @ F @ A3 ) @ ( image_set_a_set_a @ F @ B3 ) ) @ ( image_set_a_set_a @ F @ ( minus_5736297505244876581_set_a @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_1039_image__diff__subset,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,B3: set_c_d_set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ ( image_c_d_set_a_a @ F @ B3 ) ) @ ( image_c_d_set_a_a @ F @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_1040_image__diff__subset,axiom,
! [F: a > a,A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_1041_image__diff__subset,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ ( image_5710119992958135237_set_a @ F @ B3 ) ) @ ( image_5710119992958135237_set_a @ F @ ( minus_1665977719694084726_set_a @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_1042_image__diff__subset,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,B3: set_a] : ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ ( image_a_c_d_set_a @ F @ B3 ) ) @ ( image_a_c_d_set_a @ F @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_1043_the__elem__image__unique,axiom,
! [A3: set_set_a,F: set_a > set_a,X2: set_a] :
( ( A3 != bot_bot_set_set_a )
=> ( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ A3 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_a @ ( image_set_a_set_a @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1044_the__elem__image__unique,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y2 @ A3 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_c_d_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1045_the__elem__image__unique,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,X2: ( c > d ) > set_a] :
( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y2 @ A3 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_a @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1046_the__elem__image__unique,axiom,
! [A3: set_a,F: a > ( c > d ) > set_a,X2: a] :
( ( A3 != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A3 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_c_d_set_a @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1047_the__elem__image__unique,axiom,
! [A3: set_a,F: a > a,X2: a] :
( ( A3 != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A3 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F @ A3 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1048_range__eq__singletonD,axiom,
! [F: set_a > set_a,A: set_a,X2: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1049_range__eq__singletonD,axiom,
! [F: a > ( c > d ) > set_a,A: ( c > d ) > set_a,X2: a] :
( ( ( image_a_c_d_set_a @ F @ top_top_set_a )
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1050_range__eq__singletonD,axiom,
! [F: a > a,A: a,X2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= ( insert_a @ A @ bot_bot_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1051_range__eq__singletonD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A: ( c > d ) > set_a,X2: ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1052_range__eq__singletonD,axiom,
! [F: ( ( c > d ) > set_a ) > a,A: a,X2: ( c > d ) > set_a] :
( ( ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a )
= ( insert_a @ A @ bot_bot_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1053_notin__range__Some,axiom,
! [X2: option_a] :
( ( ~ ( member_option_a @ X2 @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
= ( X2 = none_a ) ) ).
% notin_range_Some
thf(fact_1054_notin__range__Some,axiom,
! [X2: option_c_d_set_a] :
( ( ~ ( member4306893881663408030_set_a @ X2 @ ( image_1393004410334431755_set_a @ some_c_d_set_a @ top_to4267977599310771935_set_a ) ) )
= ( X2 = none_c_d_set_a ) ) ).
% notin_range_Some
thf(fact_1055_finite__range__Some,axiom,
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ some_a @ top_top_set_a ) )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_range_Some
thf(fact_1056_finite__range__Some,axiom,
( ( finite1740182815655637662_set_a @ ( image_1393004410334431755_set_a @ some_c_d_set_a @ top_to4267977599310771935_set_a ) )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ).
% finite_range_Some
thf(fact_1057_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_1058_UNIV__option__conv,axiom,
( top_to1333438998097461157_set_a
= ( insert1935891768494221125_set_a @ none_c_d_set_a @ ( image_1393004410334431755_set_a @ some_c_d_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_1059_fun__upd__image,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( image_212549500329102437_set_a @ ( fun_up4138888537510562013_set_a @ F @ X2 @ Y ) @ A3 )
= ( insert_c_d_set_a @ Y @ ( image_212549500329102437_set_a @ F @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) ) ) ) )
& ( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( image_212549500329102437_set_a @ ( fun_up4138888537510562013_set_a @ F @ X2 @ Y ) @ A3 )
= ( image_212549500329102437_set_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1060_fun__upd__image,axiom,
! [X2: set_c_d_set_a,A3: set_set_c_d_set_a,F: set_c_d_set_a > a,Y: a] :
( ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( image_702032380087044660et_a_a @ ( fun_up4087578694834399916et_a_a @ F @ X2 @ Y ) @ A3 )
= ( insert_a @ Y @ ( image_702032380087044660et_a_a @ F @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) ) ) ) )
& ( ~ ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( image_702032380087044660et_a_a @ ( fun_up4087578694834399916et_a_a @ F @ X2 @ Y ) @ A3 )
= ( image_702032380087044660et_a_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1061_fun__upd__image,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > set_a,Y: set_a] :
( ( ( member_set_a @ X2 @ A3 )
=> ( ( image_set_a_set_a @ ( fun_upd_set_a_set_a @ F @ X2 @ Y ) @ A3 )
= ( insert_set_a @ Y @ ( image_set_a_set_a @ F @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ( image_set_a_set_a @ ( fun_upd_set_a_set_a @ F @ X2 @ Y ) @ A3 )
= ( image_set_a_set_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1062_fun__upd__image,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( member_set_a @ X2 @ A3 )
=> ( ( image_1482592857945081046_set_a @ ( fun_up4868139172692436302_set_a @ F @ X2 @ Y ) @ A3 )
= ( insert_c_d_set_a @ Y @ ( image_1482592857945081046_set_a @ F @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ( image_1482592857945081046_set_a @ ( fun_up4868139172692436302_set_a @ F @ X2 @ Y ) @ A3 )
= ( image_1482592857945081046_set_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1063_fun__upd__image,axiom,
! [X2: set_a,A3: set_set_a,F: set_a > a,Y: a] :
( ( ( member_set_a @ X2 @ A3 )
=> ( ( image_set_a_a @ ( fun_upd_set_a_a @ F @ X2 @ Y ) @ A3 )
= ( insert_a @ Y @ ( image_set_a_a @ F @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ( image_set_a_a @ ( fun_upd_set_a_a @ F @ X2 @ Y ) @ A3 )
= ( image_set_a_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1064_fun__upd__image,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( image_5710119992958135237_set_a @ ( fun_up1723300674898315325_set_a @ F @ X2 @ Y ) @ A3 )
= ( insert_c_d_set_a @ Y @ ( image_5710119992958135237_set_a @ F @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ) ) )
& ( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( image_5710119992958135237_set_a @ ( fun_up1723300674898315325_set_a @ F @ X2 @ Y ) @ A3 )
= ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1065_fun__upd__image,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,Y: a] :
( ( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( image_c_d_set_a_a @ ( fun_upd_c_d_set_a_a @ F @ X2 @ Y ) @ A3 )
= ( insert_a @ Y @ ( image_c_d_set_a_a @ F @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ) ) )
& ( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( image_c_d_set_a_a @ ( fun_upd_c_d_set_a_a @ F @ X2 @ Y ) @ A3 )
= ( image_c_d_set_a_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1066_fun__upd__image,axiom,
! [X2: a,A3: set_a,F: a > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( member_a @ X2 @ A3 )
=> ( ( image_a_c_d_set_a @ ( fun_upd_a_c_d_set_a @ F @ X2 @ Y ) @ A3 )
= ( insert_c_d_set_a @ Y @ ( image_a_c_d_set_a @ F @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ( image_a_c_d_set_a @ ( fun_upd_a_c_d_set_a @ F @ X2 @ Y ) @ A3 )
= ( image_a_c_d_set_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1067_fun__upd__image,axiom,
! [X2: a,A3: set_a,F: a > a,Y: a] :
( ( ( member_a @ X2 @ A3 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F @ X2 @ Y ) @ A3 )
= ( insert_a @ Y @ ( image_a_a @ F @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F @ X2 @ Y ) @ A3 )
= ( image_a_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1068_in__image__insert__iff,axiom,
! [B3: set_se3584202636623819855_set_a,X2: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ! [C4: set_set_c_d_set_a] :
( ( member6574826897039512728_set_a @ C4 @ B3 )
=> ~ ( member_set_c_d_set_a @ X2 @ C4 ) )
=> ( ( member6574826897039512728_set_a @ A3 @ ( image_7012603752887491525_set_a @ ( insert_set_c_d_set_a @ X2 ) @ B3 ) )
= ( ( member_set_c_d_set_a @ X2 @ A3 )
& ( member6574826897039512728_set_a @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1069_in__image__insert__iff,axiom,
! [B3: set_set_set_a,X2: set_a,A3: set_set_a] :
( ! [C4: set_set_a] :
( ( member_set_set_a @ C4 @ B3 )
=> ~ ( member_set_a @ X2 @ C4 ) )
=> ( ( member_set_set_a @ A3 @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X2 ) @ B3 ) )
= ( ( member_set_a @ X2 @ A3 )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1070_in__image__insert__iff,axiom,
! [B3: set_set_a,X2: a,A3: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B3 )
=> ~ ( member_a @ X2 @ C4 ) )
=> ( ( member_set_a @ A3 @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B3 ) )
= ( ( member_a @ X2 @ A3 )
& ( member_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1071_in__image__insert__iff,axiom,
! [B3: set_set_c_d_set_a,X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ! [C4: set_c_d_set_a] :
( ( member_set_c_d_set_a @ C4 @ B3 )
=> ~ ( member_c_d_set_a @ X2 @ C4 ) )
=> ( ( member_set_c_d_set_a @ A3 @ ( image_5418612861375423429_set_a @ ( insert_c_d_set_a @ X2 ) @ B3 ) )
= ( ( member_c_d_set_a @ X2 @ A3 )
& ( member_set_c_d_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1072_surjD,axiom,
! [F: set_a > set_a,Y: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ? [X: set_a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_1073_surjD,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X: a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_1074_surjD,axiom,
! [F: a > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( image_a_c_d_set_a @ F @ top_top_set_a )
= top_to4267977599310771935_set_a )
=> ? [X: a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_1075_surjD,axiom,
! [F: ( ( c > d ) > set_a ) > a,Y: a] :
( ( ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a )
= top_top_set_a )
=> ? [X: ( c > d ) > set_a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_1076_surjD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
=> ? [X: ( c > d ) > set_a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_1077_surjE,axiom,
! [F: set_a > set_a,Y: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ~ ! [X: set_a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_1078_surjE,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X: a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_1079_surjE,axiom,
! [F: a > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( image_a_c_d_set_a @ F @ top_top_set_a )
= top_to4267977599310771935_set_a )
=> ~ ! [X: a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_1080_surjE,axiom,
! [F: ( ( c > d ) > set_a ) > a,Y: a] :
( ( ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a )
= top_top_set_a )
=> ~ ! [X: ( c > d ) > set_a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_1081_surjE,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
=> ~ ! [X: ( c > d ) > set_a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_1082_surjI,axiom,
! [G2: set_a > set_a,F: set_a > set_a] :
( ! [X: set_a] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_set_a_set_a @ G2 @ top_top_set_set_a )
= top_top_set_set_a ) ) ).
% surjI
thf(fact_1083_surjI,axiom,
! [G2: a > a,F: a > a] :
( ! [X: a] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_a_a @ G2 @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_1084_surjI,axiom,
! [G2: a > ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > a] :
( ! [X: ( c > d ) > set_a] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_a_c_d_set_a @ G2 @ top_top_set_a )
= top_to4267977599310771935_set_a ) ) ).
% surjI
thf(fact_1085_surjI,axiom,
! [G2: ( ( c > d ) > set_a ) > a,F: a > ( c > d ) > set_a] :
( ! [X: a] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_c_d_set_a_a @ G2 @ top_to4267977599310771935_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_1086_surjI,axiom,
! [G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X: ( c > d ) > set_a] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_5710119992958135237_set_a @ G2 @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a ) ) ).
% surjI
thf(fact_1087_surj__def,axiom,
! [F: set_a > set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
= ( ! [Y4: set_a] :
? [X3: set_a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1088_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y4: a] :
? [X3: a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1089_surj__def,axiom,
! [F: a > ( c > d ) > set_a] :
( ( ( image_a_c_d_set_a @ F @ top_top_set_a )
= top_to4267977599310771935_set_a )
= ( ! [Y4: ( c > d ) > set_a] :
? [X3: a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1090_surj__def,axiom,
! [F: ( ( c > d ) > set_a ) > a] :
( ( ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a )
= top_top_set_a )
= ( ! [Y4: a] :
? [X3: ( c > d ) > set_a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1091_surj__def,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
= ( ! [Y4: ( c > d ) > set_a] :
? [X3: ( c > d ) > set_a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1092_surj__Compl__image__subset,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ ( image_set_a_set_a @ F @ A3 ) ) @ ( image_set_a_set_a @ F @ ( uminus6103902357914783669_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1093_surj__Compl__image__subset,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_c_d_set_a_a @ F @ A3 ) ) @ ( image_c_d_set_a_a @ F @ ( uminus8771976365291672326_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1094_surj__Compl__image__subset,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A3 ) ) @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1095_surj__Compl__image__subset,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
=> ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) ) @ ( image_5710119992958135237_set_a @ F @ ( uminus8771976365291672326_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1096_surj__Compl__image__subset,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a] :
( ( ( image_a_c_d_set_a @ F @ top_top_set_a )
= top_to4267977599310771935_set_a )
=> ( ord_le5982164083705284911_set_a @ ( uminus8771976365291672326_set_a @ ( image_a_c_d_set_a @ F @ A3 ) ) @ ( image_a_c_d_set_a @ F @ ( uminus_uminus_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1097_these__not__empty__eq,axiom,
! [B3: set_option_c_d_set_a] :
( ( ( these_c_d_set_a @ B3 )
!= bot_bo738396921950161403_set_a )
= ( ( B3 != bot_bo6666349697208826049_set_a )
& ( B3
!= ( insert1935891768494221125_set_a @ none_c_d_set_a @ bot_bo6666349697208826049_set_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_1098_these__not__empty__eq,axiom,
! [B3: set_option_a] :
( ( ( these_a @ B3 )
!= bot_bot_set_a )
= ( ( B3 != bot_bot_set_option_a )
& ( B3
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_1099_these__empty__eq,axiom,
! [B3: set_option_c_d_set_a] :
( ( ( these_c_d_set_a @ B3 )
= bot_bo738396921950161403_set_a )
= ( ( B3 = bot_bo6666349697208826049_set_a )
| ( B3
= ( insert1935891768494221125_set_a @ none_c_d_set_a @ bot_bo6666349697208826049_set_a ) ) ) ) ).
% these_empty_eq
thf(fact_1100_these__empty__eq,axiom,
! [B3: set_option_a] :
( ( ( these_a @ B3 )
= bot_bot_set_a )
= ( ( B3 = bot_bot_set_option_a )
| ( B3
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_1101_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B3 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B3 ) @ ( lattic8209813465164889211_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1102_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A3 @ B3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ( ( finite457288119118821432_set_a @ B3 )
=> ( ord_le5982164083705284911_set_a @ ( lattic8453104748687127596_set_a @ B3 ) @ ( lattic8453104748687127596_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1103_Inf__fin_Osubset__imp,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( finite3330819693523053784_set_a @ B3 )
=> ( ord_le8464990428230162895_set_a @ ( lattic3893622604919961804_set_a @ B3 ) @ ( lattic3893622604919961804_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1104_Inf__fin_Osingleton,axiom,
! [X2: ( c > d ) > set_a] :
( ( lattic3893622604919961804_set_a @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_1105_these__empty,axiom,
( ( these_c_d_set_a @ bot_bo6666349697208826049_set_a )
= bot_bo738396921950161403_set_a ) ).
% these_empty
thf(fact_1106_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_1107_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1108_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( member_set_c_d_set_a @ A @ A3 )
=> ( ord_le5982164083705284911_set_a @ ( lattic8453104748687127596_set_a @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1109_Inf__fin_OcoboundedI,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ( ord_le8464990428230162895_set_a @ ( lattic3893622604919961804_set_a @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1110_Inf__fin_OboundedE,axiom,
! [A3: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A3 ) )
=> ! [A7: set_a] :
( ( member_set_a @ A7 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1111_Inf__fin_OboundedE,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ( ( ord_le5982164083705284911_set_a @ X2 @ ( lattic8453104748687127596_set_a @ A3 ) )
=> ! [A7: set_c_d_set_a] :
( ( member_set_c_d_set_a @ A7 @ A3 )
=> ( ord_le5982164083705284911_set_a @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1112_Inf__fin_OboundedE,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ ( lattic3893622604919961804_set_a @ A3 ) )
=> ! [A7: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A7 @ A3 )
=> ( ord_le8464990428230162895_set_a @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1113_Inf__fin_OboundedI,axiom,
! [A3: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ A6 ) )
=> ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1114_Inf__fin_OboundedI,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ( ! [A6: set_c_d_set_a] :
( ( member_set_c_d_set_a @ A6 @ A3 )
=> ( ord_le5982164083705284911_set_a @ X2 @ A6 ) )
=> ( ord_le5982164083705284911_set_a @ X2 @ ( lattic8453104748687127596_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1115_Inf__fin_OboundedI,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [A6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A6 @ A3 )
=> ( ord_le8464990428230162895_set_a @ X2 @ A6 ) )
=> ( ord_le8464990428230162895_set_a @ X2 @ ( lattic3893622604919961804_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1116_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A3 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1117_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ( ( ord_le5982164083705284911_set_a @ X2 @ ( lattic8453104748687127596_set_a @ A3 ) )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A3 )
=> ( ord_le5982164083705284911_set_a @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1118_Inf__fin_Obounded__iff,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ ( lattic3893622604919961804_set_a @ A3 ) )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( ord_le8464990428230162895_set_a @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1119_image__Fpow__mono,axiom,
! [F: set_a > set_a,A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ B3 )
=> ( ord_le5722252365846178494_set_a @ ( image_1042221919965026181_set_a @ ( image_set_a_set_a @ F ) @ ( finite_Fpow_set_a @ A3 ) ) @ ( finite_Fpow_set_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1120_image__Fpow__mono,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_4522397567451716500_set_a @ ( image_c_d_set_a_a @ F ) @ ( finite3010068450757450645_set_a @ A3 ) ) @ ( finite_Fpow_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1121_image__Fpow__mono,axiom,
! [F: a > a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A3 ) ) @ ( finite_Fpow_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1122_image__Fpow__mono,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ B3 )
=> ( ord_le7272806397018272911_set_a @ ( image_5418612861375423429_set_a @ ( image_5710119992958135237_set_a @ F ) @ ( finite3010068450757450645_set_a @ A3 ) ) @ ( finite3010068450757450645_set_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1123_image__Fpow__mono,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ B3 )
=> ( ord_le7272806397018272911_set_a @ ( image_3326132959979657270_set_a @ ( image_a_c_d_set_a @ F ) @ ( finite_Fpow_a @ A3 ) ) @ ( finite3010068450757450645_set_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1124_Inf__fin_Oinsert__remove,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( ( ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) )
= bot_bo58555506362910043_set_a )
=> ( ( lattic8453104748687127596_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) )
!= bot_bo58555506362910043_set_a )
=> ( ( lattic8453104748687127596_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) )
= ( inf_in754637537901350525_set_a @ X2 @ ( lattic8453104748687127596_set_a @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1125_Inf__fin_Oinsert__remove,axiom,
! [A3: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A3 ) )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1126_Inf__fin_Oinsert__remove,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic3893622604919961804_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic3893622604919961804_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) )
= ( inf_inf_c_d_set_a @ X2 @ ( lattic3893622604919961804_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1127_Inf__fin_Oremove,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( member_set_c_d_set_a @ X2 @ A3 )
=> ( ( ( ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) )
= bot_bo58555506362910043_set_a )
=> ( ( lattic8453104748687127596_set_a @ A3 )
= X2 ) )
& ( ( ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) )
!= bot_bo58555506362910043_set_a )
=> ( ( lattic8453104748687127596_set_a @ A3 )
= ( inf_in754637537901350525_set_a @ X2 @ ( lattic8453104748687127596_set_a @ ( minus_3753830358241515990_set_a @ A3 @ ( insert_set_c_d_set_a @ X2 @ bot_bo58555506362910043_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1128_Inf__fin_Oremove,axiom,
! [A3: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ X2 @ A3 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A3 )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A3 )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1129_Inf__fin_Oremove,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic3893622604919961804_set_a @ A3 )
= X2 ) )
& ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic3893622604919961804_set_a @ A3 )
= ( inf_inf_c_d_set_a @ X2 @ ( lattic3893622604919961804_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1130_Int__UNIV,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B3 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1131_Int__UNIV,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A3 @ B3 )
= top_to4267977599310771935_set_a )
= ( ( A3 = top_to4267977599310771935_set_a )
& ( B3 = top_to4267977599310771935_set_a ) ) ) ).
% Int_UNIV
thf(fact_1132_Int__subset__iff,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A3 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_1133_Int__subset__iff,axiom,
! [C2: set_c_d_set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C2 @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) )
= ( ( ord_le5982164083705284911_set_a @ C2 @ A3 )
& ( ord_le5982164083705284911_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_1134_Int__insert__left__if0,axiom,
! [A: set_c_d_set_a,C2: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ A @ C2 )
=> ( ( inf_in650668748222022109_set_a @ ( insert_set_c_d_set_a @ A @ B3 ) @ C2 )
= ( inf_in650668748222022109_set_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1135_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1136_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1137_Int__insert__left__if0,axiom,
! [A: ( c > d ) > set_a,C2: set_c_d_set_a,B3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ C2 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B3 ) @ C2 )
= ( inf_in754637537901350525_set_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1138_Int__insert__left__if1,axiom,
! [A: set_c_d_set_a,C2: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ C2 )
=> ( ( inf_in650668748222022109_set_a @ ( insert_set_c_d_set_a @ A @ B3 ) @ C2 )
= ( insert_set_c_d_set_a @ A @ ( inf_in650668748222022109_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1139_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B3: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B3 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1140_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1141_Int__insert__left__if1,axiom,
! [A: ( c > d ) > set_a,C2: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ C2 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B3 ) @ C2 )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1142_insert__inter__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A3 ) @ ( insert_c_d_set_a @ A @ B3 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1143_insert__inter__insert,axiom,
! [A: a,A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1144_Int__insert__right__if0,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ A @ A3 )
=> ( ( inf_in650668748222022109_set_a @ A3 @ ( insert_set_c_d_set_a @ A @ B3 ) )
= ( inf_in650668748222022109_set_a @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1145_Int__insert__right__if0,axiom,
! [A: set_a,A3: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B3 ) )
= ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1146_Int__insert__right__if0,axiom,
! [A: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1147_Int__insert__right__if0,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B3 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1148_Int__insert__right__if1,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ A3 )
=> ( ( inf_in650668748222022109_set_a @ A3 @ ( insert_set_c_d_set_a @ A @ B3 ) )
= ( insert_set_c_d_set_a @ A @ ( inf_in650668748222022109_set_a @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1149_Int__insert__right__if1,axiom,
! [A: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B3 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1150_Int__insert__right__if1,axiom,
! [A: a,A3: set_a,B3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1151_Int__insert__right__if1,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B3 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1152_boolean__algebra_Oconj__zero__right,axiom,
! [X2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X2 @ bot_bot_c_d_set_a )
= bot_bot_c_d_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1153_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X2 @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1154_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1155_boolean__algebra_Oconj__zero__right,axiom,
! [X2: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ X2 @ bot_bot_c_d_set_a_o )
= bot_bot_c_d_set_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1156_boolean__algebra_Oconj__zero__right,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ X2 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1157_boolean__algebra_Oconj__zero__left,axiom,
! [X2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ bot_bot_c_d_set_a @ X2 )
= bot_bot_c_d_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1158_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ X2 )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1159_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1160_boolean__algebra_Oconj__zero__left,axiom,
! [X2: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ bot_bot_c_d_set_a_o @ X2 )
= bot_bot_c_d_set_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1161_boolean__algebra_Oconj__zero__left,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X2 )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1162_insert__disjoint_I1_J,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( ( inf_in650668748222022109_set_a @ ( insert_set_c_d_set_a @ A @ A3 ) @ B3 )
= bot_bo58555506362910043_set_a )
= ( ~ ( member_set_c_d_set_a @ A @ B3 )
& ( ( inf_in650668748222022109_set_a @ A3 @ B3 )
= bot_bo58555506362910043_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1163_insert__disjoint_I1_J,axiom,
! [A: set_a,A3: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ B3 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B3 )
& ( ( inf_inf_set_set_a @ A3 @ B3 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1164_insert__disjoint_I1_J,axiom,
! [A: a,A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B3 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1165_insert__disjoint_I1_J,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A3 ) @ B3 )
= bot_bo738396921950161403_set_a )
= ( ~ ( member_c_d_set_a @ A @ B3 )
& ( ( inf_in754637537901350525_set_a @ A3 @ B3 )
= bot_bo738396921950161403_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1166_insert__disjoint_I2_J,axiom,
! [A: set_c_d_set_a,A3: set_set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( bot_bo58555506362910043_set_a
= ( inf_in650668748222022109_set_a @ ( insert_set_c_d_set_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member_set_c_d_set_a @ A @ B3 )
& ( bot_bo58555506362910043_set_a
= ( inf_in650668748222022109_set_a @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1167_insert__disjoint_I2_J,axiom,
! [A: set_a,A3: set_set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member_set_a @ A @ B3 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1168_insert__disjoint_I2_J,axiom,
! [A: a,A3: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member_a @ A @ B3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1169_insert__disjoint_I2_J,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member_c_d_set_a @ A @ B3 )
& ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1170_disjoint__insert_I1_J,axiom,
! [B3: set_set_c_d_set_a,A: set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( ( inf_in650668748222022109_set_a @ B3 @ ( insert_set_c_d_set_a @ A @ A3 ) )
= bot_bo58555506362910043_set_a )
= ( ~ ( member_set_c_d_set_a @ A @ B3 )
& ( ( inf_in650668748222022109_set_a @ B3 @ A3 )
= bot_bo58555506362910043_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1171_disjoint__insert_I1_J,axiom,
! [B3: set_set_a,A: set_a,A3: set_set_a] :
( ( ( inf_inf_set_set_a @ B3 @ ( insert_set_a @ A @ A3 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B3 )
& ( ( inf_inf_set_set_a @ B3 @ A3 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1172_disjoint__insert_I1_J,axiom,
! [B3: set_a,A: a,A3: set_a] :
( ( ( inf_inf_set_a @ B3 @ ( insert_a @ A @ A3 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ B3 @ A3 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1173_disjoint__insert_I1_J,axiom,
! [B3: set_c_d_set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ B3 @ ( insert_c_d_set_a @ A @ A3 ) )
= bot_bo738396921950161403_set_a )
= ( ~ ( member_c_d_set_a @ A @ B3 )
& ( ( inf_in754637537901350525_set_a @ B3 @ A3 )
= bot_bo738396921950161403_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1174_disjoint__insert_I2_J,axiom,
! [A3: set_set_c_d_set_a,B: set_c_d_set_a,B3: set_set_c_d_set_a] :
( ( bot_bo58555506362910043_set_a
= ( inf_in650668748222022109_set_a @ A3 @ ( insert_set_c_d_set_a @ B @ B3 ) ) )
= ( ~ ( member_set_c_d_set_a @ B @ A3 )
& ( bot_bo58555506362910043_set_a
= ( inf_in650668748222022109_set_a @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1175_disjoint__insert_I2_J,axiom,
! [A3: set_set_a,B: set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ B @ B3 ) ) )
= ( ~ ( member_set_a @ B @ A3 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1176_disjoint__insert_I2_J,axiom,
! [A3: set_a,B: a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B @ B3 ) ) )
= ( ~ ( member_a @ B @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1177_disjoint__insert_I2_J,axiom,
! [A3: set_c_d_set_a,B: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ B @ B3 ) ) )
= ( ~ ( member_c_d_set_a @ B @ A3 )
& ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1178_Diff__disjoint,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B3 @ A3 ) )
= bot_bo738396921950161403_set_a ) ).
% Diff_disjoint
thf(fact_1179_Diff__disjoint,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1180_Compl__disjoint2,axiom,
! [A3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( uminus8771976365291672326_set_a @ A3 ) @ A3 )
= bot_bo738396921950161403_set_a ) ).
% Compl_disjoint2
thf(fact_1181_Compl__disjoint2,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A3 ) @ A3 )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_1182_Compl__disjoint,axiom,
! [A3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ ( uminus8771976365291672326_set_a @ A3 ) )
= bot_bo738396921950161403_set_a ) ).
% Compl_disjoint
thf(fact_1183_Compl__disjoint,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_1184_Diff__Compl,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ ( uminus8771976365291672326_set_a @ B3 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ).
% Diff_Compl
thf(fact_1185_Diff__Compl,axiom,
! [A3: set_a,B3: set_a] :
( ( minus_minus_set_a @ A3 @ ( uminus_uminus_set_a @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% Diff_Compl
thf(fact_1186_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X2 @ ( uminus3002763893361803174_set_a @ X2 ) )
= bot_bot_c_d_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1187_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ X2 @ ( uminus6307618635820417879et_a_o @ X2 ) )
= bot_bot_c_d_set_a_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1188_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ X2 @ ( uminus_uminus_a_o @ X2 ) )
= bot_bot_a_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1189_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X2 @ ( uminus8771976365291672326_set_a @ X2 ) )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1190_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ ( uminus_uminus_set_a @ X2 ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1191_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ X2 )
= bot_bot_c_d_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1192_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ ( uminus6307618635820417879et_a_o @ X2 ) @ X2 )
= bot_bot_c_d_set_a_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1193_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ ( uminus_uminus_a_o @ X2 ) @ X2 )
= bot_bot_a_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1194_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ X2 )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1195_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1196_inf__compl__bot__right,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X2 @ ( inf_inf_c_d_set_a @ Y @ ( uminus3002763893361803174_set_a @ X2 ) ) )
= bot_bot_c_d_set_a ) ).
% inf_compl_bot_right
thf(fact_1197_inf__compl__bot__right,axiom,
! [X2: ( ( c > d ) > set_a ) > $o,Y: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ X2 @ ( inf_inf_c_d_set_a_o @ Y @ ( uminus6307618635820417879et_a_o @ X2 ) ) )
= bot_bot_c_d_set_a_o ) ).
% inf_compl_bot_right
thf(fact_1198_inf__compl__bot__right,axiom,
! [X2: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X2 @ ( inf_inf_a_o @ Y @ ( uminus_uminus_a_o @ X2 ) ) )
= bot_bot_a_o ) ).
% inf_compl_bot_right
thf(fact_1199_inf__compl__bot__right,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X2 @ ( inf_in754637537901350525_set_a @ Y @ ( uminus8771976365291672326_set_a @ X2 ) ) )
= bot_bo738396921950161403_set_a ) ).
% inf_compl_bot_right
thf(fact_1200_inf__compl__bot__right,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X2 ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_1201_inf__compl__bot__left2,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X2 @ ( inf_inf_c_d_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ Y ) )
= bot_bot_c_d_set_a ) ).
% inf_compl_bot_left2
thf(fact_1202_inf__compl__bot__left2,axiom,
! [X2: ( ( c > d ) > set_a ) > $o,Y: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ X2 @ ( inf_inf_c_d_set_a_o @ ( uminus6307618635820417879et_a_o @ X2 ) @ Y ) )
= bot_bot_c_d_set_a_o ) ).
% inf_compl_bot_left2
thf(fact_1203_inf__compl__bot__left2,axiom,
! [X2: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X2 @ ( inf_inf_a_o @ ( uminus_uminus_a_o @ X2 ) @ Y ) )
= bot_bot_a_o ) ).
% inf_compl_bot_left2
thf(fact_1204_inf__compl__bot__left2,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X2 @ ( inf_in754637537901350525_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ Y ) )
= bot_bo738396921950161403_set_a ) ).
% inf_compl_bot_left2
thf(fact_1205_inf__compl__bot__left2,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_1206_inf__compl__bot__left1,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ ( inf_inf_c_d_set_a @ X2 @ Y ) )
= bot_bot_c_d_set_a ) ).
% inf_compl_bot_left1
thf(fact_1207_inf__compl__bot__left1,axiom,
! [X2: ( ( c > d ) > set_a ) > $o,Y: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ ( uminus6307618635820417879et_a_o @ X2 ) @ ( inf_inf_c_d_set_a_o @ X2 @ Y ) )
= bot_bot_c_d_set_a_o ) ).
% inf_compl_bot_left1
thf(fact_1208_inf__compl__bot__left1,axiom,
! [X2: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ ( uminus_uminus_a_o @ X2 ) @ ( inf_inf_a_o @ X2 @ Y ) )
= bot_bot_a_o ) ).
% inf_compl_bot_left1
thf(fact_1209_inf__compl__bot__left1,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ ( inf_in754637537901350525_set_a @ X2 @ Y ) )
= bot_bo738396921950161403_set_a ) ).
% inf_compl_bot_left1
thf(fact_1210_inf__compl__bot__left1,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( inf_inf_set_a @ X2 @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_1211_Inf__fin_Oinsert,axiom,
! [A3: set_set_c_d_set_a,X2: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ( ( lattic8453104748687127596_set_a @ ( insert_set_c_d_set_a @ X2 @ A3 ) )
= ( inf_in754637537901350525_set_a @ X2 @ ( lattic8453104748687127596_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1212_Inf__fin_Oinsert,axiom,
! [A3: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A3 ) )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1213_Inf__fin_Oinsert,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( lattic3893622604919961804_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) )
= ( inf_inf_c_d_set_a @ X2 @ ( lattic3893622604919961804_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1214_inf__cancel__left2,axiom,
! [X2: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ ( inf_inf_c_d_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ A ) @ ( inf_inf_c_d_set_a @ X2 @ B ) )
= bot_bot_c_d_set_a ) ).
% inf_cancel_left2
thf(fact_1215_inf__cancel__left2,axiom,
! [X2: ( ( c > d ) > set_a ) > $o,A: ( ( c > d ) > set_a ) > $o,B: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ ( inf_inf_c_d_set_a_o @ ( uminus6307618635820417879et_a_o @ X2 ) @ A ) @ ( inf_inf_c_d_set_a_o @ X2 @ B ) )
= bot_bot_c_d_set_a_o ) ).
% inf_cancel_left2
thf(fact_1216_inf__cancel__left2,axiom,
! [X2: a > $o,A: a > $o,B: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ ( uminus_uminus_a_o @ X2 ) @ A ) @ ( inf_inf_a_o @ X2 @ B ) )
= bot_bot_a_o ) ).
% inf_cancel_left2
thf(fact_1217_inf__cancel__left2,axiom,
! [X2: set_c_d_set_a,A: set_c_d_set_a,B: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ A ) @ ( inf_in754637537901350525_set_a @ X2 @ B ) )
= bot_bo738396921950161403_set_a ) ).
% inf_cancel_left2
thf(fact_1218_inf__cancel__left2,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ A ) @ ( inf_inf_set_a @ X2 @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_1219_inf__cancel__left1,axiom,
! [X2: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ ( inf_inf_c_d_set_a @ X2 @ A ) @ ( inf_inf_c_d_set_a @ ( uminus3002763893361803174_set_a @ X2 ) @ B ) )
= bot_bot_c_d_set_a ) ).
% inf_cancel_left1
thf(fact_1220_inf__cancel__left1,axiom,
! [X2: ( ( c > d ) > set_a ) > $o,A: ( ( c > d ) > set_a ) > $o,B: ( ( c > d ) > set_a ) > $o] :
( ( inf_inf_c_d_set_a_o @ ( inf_inf_c_d_set_a_o @ X2 @ A ) @ ( inf_inf_c_d_set_a_o @ ( uminus6307618635820417879et_a_o @ X2 ) @ B ) )
= bot_bot_c_d_set_a_o ) ).
% inf_cancel_left1
thf(fact_1221_inf__cancel__left1,axiom,
! [X2: a > $o,A: a > $o,B: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X2 @ A ) @ ( inf_inf_a_o @ ( uminus_uminus_a_o @ X2 ) @ B ) )
= bot_bot_a_o ) ).
% inf_cancel_left1
thf(fact_1222_inf__cancel__left1,axiom,
! [X2: set_c_d_set_a,A: set_c_d_set_a,B: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ X2 @ A ) @ ( inf_in754637537901350525_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ B ) )
= bot_bo738396921950161403_set_a ) ).
% inf_cancel_left1
thf(fact_1223_inf__cancel__left1,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ A ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_1224_image__Int__subset,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,B3: set_c_d_set_a] : ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ ( image_c_d_set_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1225_image__Int__subset,axiom,
! [F: a > a,A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1226_image__Int__subset,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) @ ( inf_in754637537901350525_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ ( image_5710119992958135237_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1227_image__Int__subset,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,B3: set_a] : ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_in754637537901350525_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ ( image_a_c_d_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1228_Int__emptyI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ~ ( member_a @ X @ B3 ) )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1229_Int__emptyI,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ~ ( member_c_d_set_a @ X @ B3 ) )
=> ( ( inf_in754637537901350525_set_a @ A3 @ B3 )
= bot_bo738396921950161403_set_a ) ) ).
% Int_emptyI
thf(fact_1230_disjoint__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ~ ( member_a @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1231_disjoint__iff,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A3 @ B3 )
= bot_bo738396921950161403_set_a )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ~ ( member_c_d_set_a @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1232_Int__Collect__mono,axiom,
! [A3: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1233_Int__Collect__mono,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A3 @ ( collect_c_d_set_a @ P ) ) @ ( inf_in754637537901350525_set_a @ B3 @ ( collect_c_d_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1234_Int__insert__left,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1235_Int__insert__left,axiom,
! [A: ( c > d ) > set_a,C2: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( member_c_d_set_a @ A @ C2 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B3 ) @ C2 )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ B3 @ C2 ) ) ) )
& ( ~ ( member_c_d_set_a @ A @ C2 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B3 ) @ C2 )
= ( inf_in754637537901350525_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1236_Int__insert__right,axiom,
! [A: a,A3: set_a,B3: set_a] :
( ( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B3 ) ) ) )
& ( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1237_Int__insert__right,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B3 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) )
& ( ~ ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B3 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1238_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( lattic3893622604919961804_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) )
= ( inf_inf_c_d_set_a @ X2 @ ( lattic3893622604919961804_set_a @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1239_Inf__fin_Oclosed,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( member_c_d_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ ( insert_c_d_set_a @ X @ ( insert_c_d_set_a @ Y2 @ bot_bo738396921950161403_set_a ) ) )
=> ( member_c_d_set_a @ ( lattic3893622604919961804_set_a @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1240_IntI,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_1241_IntI,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ A3 )
=> ( ( member_c_d_set_a @ C @ B3 )
=> ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_1242_Int__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1243_Int__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) )
= ( ( member_c_d_set_a @ C @ A3 )
& ( member_c_d_set_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1244_IntE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_1245_IntE,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) )
=> ~ ( ( member_c_d_set_a @ C @ A3 )
=> ~ ( member_c_d_set_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_1246_IntD1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ A3 ) ) ).
% IntD1
thf(fact_1247_IntD1,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) )
=> ( member_c_d_set_a @ C @ A3 ) ) ).
% IntD1
thf(fact_1248_IntD2,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_1249_IntD2,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B3 ) )
=> ( member_c_d_set_a @ C @ B3 ) ) ).
% IntD2
thf(fact_1250_Sup__fin_OcoboundedI,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ( ord_le8464990428230162895_set_a @ A @ ( lattic8365952737566729574_set_a @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1251_Sup__fin_OboundedI,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [A6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A6 @ A3 )
=> ( ord_le8464990428230162895_set_a @ A6 @ X2 ) )
=> ( ord_le8464990428230162895_set_a @ ( lattic8365952737566729574_set_a @ A3 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1252_Sup__fin_OboundedE,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( ord_le8464990428230162895_set_a @ ( lattic8365952737566729574_set_a @ A3 ) @ X2 )
=> ! [A7: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A7 @ A3 )
=> ( ord_le8464990428230162895_set_a @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1253_Sup__fin_Oremove,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ X2 @ A3 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic8365952737566729574_set_a @ A3 )
= X2 ) )
& ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic8365952737566729574_set_a @ A3 )
= ( sup_sup_c_d_set_a @ X2 @ ( lattic8365952737566729574_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1254_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ~ ( member_c_d_set_a @ X2 @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( lattic8365952737566729574_set_a @ ( insert_c_d_set_a @ X2 @ A3 ) )
= ( sup_sup_c_d_set_a @ X2 @ ( lattic8365952737566729574_set_a @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1255_Sup__fin_Oclosed,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( member_c_d_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ ( insert_c_d_set_a @ X @ ( insert_c_d_set_a @ Y2 @ bot_bo738396921950161403_set_a ) ) )
=> ( member_c_d_set_a @ ( lattic8365952737566729574_set_a @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1256_Un__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1257_Un__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B3 ) )
= ( ( member_c_d_set_a @ C @ A3 )
| ( member_c_d_set_a @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1258_UnCI,axiom,
! [C: a,B3: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_1259_UnCI,axiom,
! [C: ( c > d ) > set_a,B3: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ~ ( member_c_d_set_a @ C @ B3 )
=> ( member_c_d_set_a @ C @ A3 ) )
=> ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_1260_UnI2,axiom,
! [C: a,B3: set_a,A3: set_a] :
( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1261_UnI2,axiom,
! [C: ( c > d ) > set_a,B3: set_c_d_set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ B3 )
=> ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1262_UnI1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1263_UnI1,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1264_UnE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% UnE
thf(fact_1265_UnE,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B3 ) )
=> ( ~ ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B3 ) ) ) ).
% UnE
thf(fact_1266_inj__image__mem__iff,axiom,
! [F: a > a,A: a,A3: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A3 ) )
= ( member_a @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1267_inj__image__mem__iff,axiom,
! [F: ( ( c > d ) > set_a ) > a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( inj_on_c_d_set_a_a @ F @ top_to4267977599310771935_set_a )
=> ( ( member_a @ ( F @ A ) @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( member_c_d_set_a @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1268_inj__image__mem__iff,axiom,
! [F: a > ( c > d ) > set_a,A: a,A3: set_a] :
( ( inj_on_a_c_d_set_a @ F @ top_top_set_a )
=> ( ( member_c_d_set_a @ ( F @ A ) @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( member_a @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1269_inj__image__mem__iff,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( inj_on2268522623953733425_set_a @ F @ top_to4267977599310771935_set_a )
=> ( ( member_c_d_set_a @ ( F @ A ) @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( member_c_d_set_a @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1270_inj__on__image__mem__iff,axiom,
! [F: a > a,B3: set_a,A: a,A3: set_a] :
( ( inj_on_a_a @ F @ B3 )
=> ( ( member_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A3 ) )
= ( member_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1271_inj__on__image__mem__iff,axiom,
! [F: a > ( c > d ) > set_a,B3: set_a,A: a,A3: set_a] :
( ( inj_on_a_c_d_set_a @ F @ B3 )
=> ( ( member_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_c_d_set_a @ ( F @ A ) @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( member_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1272_inj__on__image__mem__iff,axiom,
! [F: ( ( c > d ) > set_a ) > a,B3: set_c_d_set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( inj_on_c_d_set_a_a @ F @ B3 )
=> ( ( member_c_d_set_a @ A @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( member_a @ ( F @ A ) @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( member_c_d_set_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1273_inj__on__image__mem__iff,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B3: set_c_d_set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( inj_on2268522623953733425_set_a @ F @ B3 )
=> ( ( member_c_d_set_a @ A @ B3 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
=> ( ( member_c_d_set_a @ ( F @ A ) @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( member_c_d_set_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1274_semilattice__order__set_OboundedE,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A3: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ( ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A3 ) )
=> ! [A7: a] :
( ( member_a @ A7 @ A3 )
=> ( Less_eq @ X2 @ A7 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1275_semilattice__order__set_OboundedE,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Less: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( lattic1995125144389820681_set_a @ F @ Less_eq @ Less )
=> ( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( Less_eq @ X2 @ ( lattic5899416730317975049_set_a @ F @ A3 ) )
=> ! [A7: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A7 @ A3 )
=> ( Less_eq @ X2 @ A7 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1276_semilattice__order__set_OboundedI,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A3: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A3 )
=> ( Less_eq @ X2 @ A6 ) )
=> ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1277_semilattice__order__set_OboundedI,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Less: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( lattic1995125144389820681_set_a @ F @ Less_eq @ Less )
=> ( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [A6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A6 @ A3 )
=> ( Less_eq @ X2 @ A6 ) )
=> ( Less_eq @ X2 @ ( lattic5899416730317975049_set_a @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
% Conjectures (1)
thf(conj_0,conjecture,
smaller_interp_c_d_a @ x @ x ).
%------------------------------------------------------------------------------