TPTP Problem File: SLH0725^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Separation_Logic_Unbounded/0003_FixedPoint/prob_00852_025239__6905764_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1461 ( 517 unt; 181 typ; 0 def)
% Number of atoms : 3908 (1135 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10340 ( 515 ~; 51 |; 202 &;7744 @)
% ( 0 <=>;1828 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 1699 (1699 >; 0 *; 0 +; 0 <<)
% Number of symbols : 170 ( 167 usr; 22 con; 0-4 aty)
% Number of variables : 3413 ( 252 ^;3107 !; 54 ?;3413 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:37.146
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
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,
set_set_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
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 (167)
thf(sy_c_Filter_Ocofinite_001t__Nat__Onat,type,
cofinite_nat: filter_nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
finite3330819693523053784_set_a: set_c_d_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
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_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_001t__Nat__Onat_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
inj_on_nat_c_d_set_a: ( nat > ( c > d ) > set_a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $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__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
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_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
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__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
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_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_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
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__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
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_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
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__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
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_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
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_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__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
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_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
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__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
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_It__Nat__Onat_J,type,
bot_bot_filter_nat: filter_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
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__Nat__Onat,type,
ord_less_nat: nat > nat > $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__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $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__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $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_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $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_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $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_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__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
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_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
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__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $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__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
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__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
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_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
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_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
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__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
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_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
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_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__Nat__Onat_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_nat_c_d_set_a: ( nat > ( c > d ) > set_a ) > set_nat > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > 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_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_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_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__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
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_001t__Nat__Onat,type,
is_empty_nat: set_nat > $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__Nat__Onat,type,
is_singleton_nat: set_nat > $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__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $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__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
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__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
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__Nat__Onat,type,
member_nat: nat > set_nat > $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_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_A,type,
a2: set_c_d_set_a ).
thf(sy_v_z,type,
z: ( c > d ) > set_a ).
% Relevant facts (1279)
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_assms,axiom,
! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ a2 )
=> ( smaller_interp_c_d_a @ z @ X2 ) ) ).
% assms
thf(fact_8_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_9_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_10_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_11_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_12_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_13_inf__empty,axiom,
( ( inf_c_d_a @ bot_bo738396921950161403_set_a )
= full_interp_c_d_a ) ).
% inf_empty
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_empty__interp__def,axiom,
( empty_interp_c_d_a
= ( ^ [S2: c > d] : bot_bot_set_a ) ) ).
% empty_interp_def
thf(fact_19_sup__empty,axiom,
( ( sup_c_d_a @ bot_bo738396921950161403_set_a )
= empty_interp_c_d_a ) ).
% sup_empty
thf(fact_20_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_21_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_22_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_23_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_24_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_25_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_26_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_27_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_28_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_29_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_30_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_31_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_32_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_33_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_34_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_35_subset__empty,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_36_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_37_empty__subsetI,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A3 ) ).
% empty_subsetI
thf(fact_38_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_39_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_40_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_41_subsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_42_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_43_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_44_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_45_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_46_subset__antisym,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_47_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_48_empty__iff,axiom,
! [C: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ C @ bot_bo58555506362910043_set_a ) ).
% empty_iff
thf(fact_49_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_50_empty__iff,axiom,
! [C: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ C @ bot_bo738396921950161403_set_a ) ).
% empty_iff
thf(fact_51_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_52_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_53_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_54_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_55_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_56_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_57_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_58_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_59_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_60_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_61_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_62_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_63_bot__apply,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_64_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_65_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_66_dual__order_Orefl,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_67_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_70_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_71_order__refl,axiom,
! [X2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_72_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_73_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_74_order__refl,axiom,
! [X2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_75_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_76_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_77_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_78_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_79_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_80_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_81_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_82_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_83_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_84_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_87_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ 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_88_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_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_90_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ 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_92_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_93_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ 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_98_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_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_100_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_a > set_nat,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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ 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_102_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: nat > ( c > d ) > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_105_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_106_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_107_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_108_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_109_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_110_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_111_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_112_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_113_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_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_116_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_order__subst2,axiom,
! [A: nat,B: nat,F: nat > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_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_126_order__subst1,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_order__subst1,axiom,
! [A: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( ord_less_eq_set_nat @ 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_order__subst1,axiom,
! [A: nat,F: ( ( c > d ) > set_a ) > nat,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_nat @ 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_130_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [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_131_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_c_d_set_a,Z: set_c_d_set_a] : ( Y3 = Z ) )
= ( ^ [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_132_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_133_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_134_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y3 = Z ) )
= ( ^ [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_135_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_136_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_137_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_138_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_139_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_140_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_141_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_142_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_143_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_144_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_145_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_146_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_147_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_148_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_149_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_150_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_151_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_152_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_153_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_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [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_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_c_d_set_a,Z: set_c_d_set_a] : ( Y3 = Z ) )
= ( ^ [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_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y3 = Z ) )
= ( ^ [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_159_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_160_order__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_161_order__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ Z2 )
=> ( ord_le5982164083705284911_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_162_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_163_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_164_order__trans,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ Z2 )
=> ( ord_le8464990428230162895_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_165_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_166_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_167_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_168_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_169_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_170_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_171_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_172_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_173_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_174_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_175_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_176_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_177_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_178_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_179_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_180_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_181_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_182_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_183_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_184_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_185_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [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_186_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_c_d_set_a,Z: set_c_d_set_a] : ( Y3 = Z ) )
= ( ^ [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_187_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_188_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_189_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y3 = Z ) )
= ( ^ [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_190_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_191_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_192_bot__fun__def,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_193_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_194_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_195_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_196_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_197_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_198_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_199_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_200_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_201_equals0I,axiom,
! [A3: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_202_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_203_equals0I,axiom,
! [A3: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_204_equals0D,axiom,
! [A3: set_set_a,A: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_205_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_206_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_207_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_208_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_209_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_210_emptyE,axiom,
! [A: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) ).
% emptyE
thf(fact_211_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_212_emptyE,axiom,
! [A: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% emptyE
thf(fact_213_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_214_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_215_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_216_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_217_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_218_set__eq__subset,axiom,
( ( ^ [Y3: set_c_d_set_a,Z: set_c_d_set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A5 @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_219_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_220_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_221_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_222_subset__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_223_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_224_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_225_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_226_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_227_subset__refl,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_228_subset__refl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_229_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_230_subset__iff,axiom,
( ord_le7272806397018272911_set_a
= ( ^ [A5: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
! [T: set_c_d_set_a] :
( ( member_set_c_d_set_a @ T @ A5 )
=> ( member_set_c_d_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_231_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_232_subset__iff,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
! [T: ( c > d ) > set_a] :
( ( member_c_d_set_a @ T @ A5 )
=> ( member_c_d_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_233_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_234_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_235_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_236_equalityD2,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_237_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_238_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_239_equalityD1,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 = B3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_240_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_241_subset__eq,axiom,
( ord_le7272806397018272911_set_a
= ( ^ [A5: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A5 )
=> ( member_set_c_d_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_242_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_243_subset__eq,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A5 )
=> ( member_c_d_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_244_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_245_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_246_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_247_equalityE,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_248_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_249_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_250_subsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_251_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_252_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_253_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_254_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_255_in__mono,axiom,
! [A3: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_256_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_257_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_258_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_259_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_260_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_261_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_262_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_263_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_264_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_265_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_266_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_267_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_268_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_269_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_270_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_271_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_272_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_273_bot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% bot.extremum
thf(fact_274_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_275_bot_Oextremum,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A ) ).
% bot.extremum
thf(fact_276_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_277_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_278_bot_Oextremum,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ bot_bot_c_d_set_a @ A ) ).
% bot.extremum
thf(fact_279_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_280_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_281_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_282_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_283_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_284_full__interp__def,axiom,
( full_interp_c_d_a
= ( ^ [S2: c > d] : top_top_set_a ) ) ).
% full_interp_def
thf(fact_285_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_286_Set_Ois__empty__def,axiom,
( is_empty_c_d_set_a
= ( ^ [A5: set_c_d_set_a] : ( A5 = bot_bo738396921950161403_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_287_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A5: set_a] : ( A5 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_288_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_289_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_290_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_291_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_292_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_293_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_294_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_295_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_296_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_297_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_298_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_299_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_300_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_301_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_302_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_303_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_304_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_305_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_306_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_307_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_308_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X4: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_309_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_310_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_311_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_312_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_313_verit__comp__simplify1_I2_J,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_314_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_315_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_316_verit__comp__simplify1_I2_J,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_317_top__apply,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : top_top_o ) ) ).
% top_apply
thf(fact_318_top__apply,axiom,
( top_top_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : top_top_o ) ) ).
% top_apply
thf(fact_319_top__apply,axiom,
( top_top_a_o
= ( ^ [X3: a] : top_top_o ) ) ).
% top_apply
thf(fact_320_UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_321_UNIV__I,axiom,
! [X2: set_c_d_set_a] : ( member_set_c_d_set_a @ X2 @ top_to5717711934741766719_set_a ) ).
% UNIV_I
thf(fact_322_UNIV__I,axiom,
! [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ top_to4267977599310771935_set_a ) ).
% UNIV_I
thf(fact_323_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_324_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_325_UNIV__witness,axiom,
? [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_326_UNIV__witness,axiom,
? [X: set_c_d_set_a] : ( member_set_c_d_set_a @ X @ top_to5717711934741766719_set_a ) ).
% UNIV_witness
thf(fact_327_UNIV__witness,axiom,
? [X: ( c > d ) > set_a] : ( member_c_d_set_a @ X @ top_to4267977599310771935_set_a ) ).
% UNIV_witness
thf(fact_328_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_329_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_330_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_331_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_332_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_333_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X: a] : ( member_a @ X @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_334_UNIV__eq__I,axiom,
! [A3: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A3 )
=> ( top_top_set_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_335_top__greatest,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ top_top_nat_o ) ).
% top_greatest
thf(fact_336_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_337_top__greatest,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ top_top_a_o ) ).
% top_greatest
thf(fact_338_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_339_top__greatest,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ top_to4267977599310771935_set_a ) ).
% top_greatest
thf(fact_340_top__greatest,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ top_top_c_d_set_a ) ).
% top_greatest
thf(fact_341_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_342_top_Oextremum__unique,axiom,
! [A: nat > $o] :
( ( ord_less_eq_nat_o @ top_top_nat_o @ A )
= ( A = top_top_nat_o ) ) ).
% top.extremum_unique
thf(fact_343_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_344_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_345_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_346_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_347_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_348_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_349_top_Oextremum__uniqueI,axiom,
! [A: nat > $o] :
( ( ord_less_eq_nat_o @ top_top_nat_o @ A )
=> ( A = top_top_nat_o ) ) ).
% top.extremum_uniqueI
thf(fact_350_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_351_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_352_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_353_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_354_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_355_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_356_empty__not__UNIV,axiom,
bot_bo738396921950161403_set_a != top_to4267977599310771935_set_a ).
% empty_not_UNIV
thf(fact_357_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_358_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_359_subset__UNIV,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_360_subset__UNIV,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ top_to4267977599310771935_set_a ) ).
% subset_UNIV
thf(fact_361_subset__UNIV,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_362_iso__tuple__UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_363_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_364_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_365_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_366_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_367_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_368_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_369_is__singletonI_H,axiom,
! [A3: set_nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_singleton_nat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_370_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_371_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_372_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_373_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_374_Diff__eq__empty__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( minus_minus_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_375_Diff__UNIV,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_376_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_377_Diff__UNIV,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_378_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_379_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_380_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A3: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_381_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_382_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_383_singleton__insert__inj__eq_H,axiom,
! [A: nat,A3: set_nat,B: nat] :
( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_384_insertCI,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_385_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_386_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_387_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_388_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_389_insert__iff,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
= ( ( A = B )
| ( member_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_390_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_391_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_392_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_393_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_394_insert__absorb2,axiom,
! [X2: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A3 ) )
= ( insert_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_395_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_396_insert__absorb2,axiom,
! [X2: nat,A3: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A3 ) )
= ( insert_nat @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_397_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_398_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_399_DiffI,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_400_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_401_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_402_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_403_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_404_Diff__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_405_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_406_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_407_Diff__idemp,axiom,
! [A3: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ B3 )
= ( minus_minus_set_nat @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_408_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_409_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_410_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_411_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_412_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_413_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_414_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_415_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_416_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_417_insert__subset,axiom,
! [X2: nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A3 ) @ B3 )
= ( ( member_nat @ X2 @ B3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_418_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_419_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_420_Diff__cancel,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ A3 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_421_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_422_Diff__cancel,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ A3 )
= bot_bo738396921950161403_set_a ) ).
% Diff_cancel
thf(fact_423_empty__Diff,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_424_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_425_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_426_Diff__empty,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Diff_empty
thf(fact_427_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_428_Diff__empty,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ bot_bo738396921950161403_set_a )
= A3 ) ).
% Diff_empty
thf(fact_429_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_430_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_431_Diff__insert0,axiom,
! [X2: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A3 )
=> ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ B3 ) )
= ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_432_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_433_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_434_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_435_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_436_insert__Diff1,axiom,
! [X2: nat,B3: set_nat,A3: set_nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B3 )
= ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_437_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_438_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_439_insert__Diff__single,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_440_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_441_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_442_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_443_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_444_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_445_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_446_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_447_Diff__insert__absorb,axiom,
! [X2: nat,A3: set_nat] :
( ~ ( member_nat @ X2 @ A3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_448_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_449_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_450_Diff__insert2,axiom,
! [A3: set_nat,A: nat,B3: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_451_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_452_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_453_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_454_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_455_insert__Diff,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_456_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_457_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_458_Diff__insert,axiom,
! [A3: set_nat,A: nat,B3: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_459_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_460_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_461_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_462_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_463_subset__Diff__insert,axiom,
! [A3: set_nat,B3: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C2 ) )
& ~ ( member_nat @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_464_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_465_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_466_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_467_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_468_DiffE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_469_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_470_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_471_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_472_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_473_DiffD1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_474_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_475_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_476_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_477_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_478_DiffD2,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_479_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_480_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_481_insertE,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_482_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_483_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_484_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_485_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_486_insertI1,axiom,
! [A: nat,B3: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B3 ) ) ).
% insertI1
thf(fact_487_insertI1,axiom,
! [A: set_a,B3: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_488_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_489_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_490_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_491_insertI2,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( member_nat @ A @ B3 )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_492_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_493_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_494_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_495_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_496_Set_Oset__insert,axiom,
! [X2: nat,A3: set_nat] :
( ( member_nat @ X2 @ A3 )
=> ~ ! [B6: set_nat] :
( ( A3
= ( insert_nat @ X2 @ B6 ) )
=> ( member_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_497_Set_Oset__insert,axiom,
! [X2: set_a,A3: set_set_a] :
( ( member_set_a @ X2 @ A3 )
=> ~ ! [B6: set_set_a] :
( ( A3
= ( insert_set_a @ X2 @ B6 ) )
=> ( member_set_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_498_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 )
=> ~ ! [B6: set_set_c_d_set_a] :
( ( A3
= ( insert_set_c_d_set_a @ X2 @ B6 ) )
=> ( member_set_c_d_set_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_499_Set_Oset__insert,axiom,
! [X2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ~ ! [B6: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ X2 @ B6 ) )
=> ( member_c_d_set_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_500_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X2 @ B6 ) )
=> ( member_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_501_insert__ident,axiom,
! [X2: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A3 )
=> ( ~ ( member_nat @ X2 @ B3 )
=> ( ( ( insert_nat @ X2 @ A3 )
= ( insert_nat @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_502_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_503_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_504_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_505_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_506_insert__absorb,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_507_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_508_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_509_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_510_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_511_insert__eq__iff,axiom,
! [A: nat,A3: set_nat,B: nat,B3: set_nat] :
( ~ ( member_nat @ A @ A3 )
=> ( ~ ( member_nat @ B @ B3 )
=> ( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A3
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B3
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_512_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_513_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_514_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_515_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_516_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_517_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_518_insert__Diff__if,axiom,
! [X2: nat,B3: set_nat,A3: set_nat] :
( ( ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B3 )
= ( minus_minus_set_nat @ A3 @ B3 ) ) )
& ( ~ ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B3 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_519_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_520_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_521_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_522_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_523_insert__commute,axiom,
! [X2: nat,Y: nat,A3: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y @ A3 ) )
= ( insert_nat @ Y @ ( insert_nat @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_524_mk__disjoint__insert,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ? [B6: set_nat] :
( ( A3
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_525_mk__disjoint__insert,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ? [B6: set_set_a] :
( ( A3
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_526_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 )
=> ? [B6: set_set_c_d_set_a] :
( ( A3
= ( insert_set_c_d_set_a @ A @ B6 ) )
& ~ ( member_set_c_d_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_527_mk__disjoint__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ? [B6: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ A @ B6 ) )
& ~ ( member_c_d_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_528_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_529_is__singletonE,axiom,
! [A3: set_nat] :
( ( is_singleton_nat @ A3 )
=> ~ ! [X: nat] :
( A3
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_530_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_531_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X: a] :
( A3
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_532_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
? [X3: nat] :
( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_533_is__singleton__def,axiom,
( is_sin6979784932356128547_set_a
= ( ^ [A5: set_c_d_set_a] :
? [X3: ( c > d ) > set_a] :
( A5
= ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_534_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
? [X3: a] :
( A5
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_535_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_536_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_537_Diff__single__insert,axiom,
! [A3: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_538_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_539_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_540_subset__insert__iff,axiom,
! [A3: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B3 ) )
= ( ( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_541_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_542_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_543_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_544_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_545_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_546_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_547_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_548_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_549_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_550_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_551_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_552_insert__UNIV,axiom,
! [X2: a] :
( ( insert_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_553_insert__UNIV,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_554_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_555_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_556_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_557_insert__not__empty,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ A3 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_558_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_559_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_560_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_561_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_562_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_563_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_564_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_565_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_566_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_567_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_568_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_569_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_570_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_571_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_572_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_573_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_574_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_575_subset__insertI2,axiom,
! [A3: set_nat,B3: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_576_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_577_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_578_subset__insertI,axiom,
! [B3: set_nat,A: nat] : ( ord_less_eq_set_nat @ B3 @ ( insert_nat @ A @ B3 ) ) ).
% subset_insertI
thf(fact_579_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_580_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_581_subset__insert,axiom,
! [X2: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B3 ) )
= ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_582_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_583_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_584_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_585_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_586_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_587_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_588_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_589_insert__subsetI,axiom,
! [X2: nat,A3: set_nat,X5: set_nat] :
( ( member_nat @ X2 @ A3 )
=> ( ( ord_less_eq_set_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_590_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_591_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_592_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_593_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_594_double__diff,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_595_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_596_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_597_Diff__subset,axiom,
! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_598_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_599_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_600_Diff__mono,axiom,
! [A3: set_nat,C2: set_nat,D2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_601_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_602_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_603_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_604_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_605_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_606_subset__singletonD,axiom,
! [A3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A3 = bot_bot_set_nat )
| ( A3
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_607_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_608_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_609_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_610_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_611_diff__shunt__var,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_612_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_613_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( A5
= ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_614_is__singleton__the__elem,axiom,
( is_sin6979784932356128547_set_a
= ( ^ [A5: set_c_d_set_a] :
( A5
= ( insert_c_d_set_a @ ( the_elem_c_d_set_a @ A5 ) @ bot_bo738396921950161403_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_615_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
( A5
= ( insert_a @ ( the_elem_a @ A5 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_616_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_617_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_618_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_619_top1I,axiom,
! [X2: nat] : ( top_top_nat_o @ X2 ) ).
% top1I
thf(fact_620_top1I,axiom,
! [X2: ( c > d ) > set_a] : ( top_top_c_d_set_a_o @ X2 ) ).
% top1I
thf(fact_621_top1I,axiom,
! [X2: a] : ( top_top_a_o @ X2 ) ).
% top1I
thf(fact_622_remove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A5: set_nat] : ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_623_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A5: set_a] : ( minus_minus_set_a @ A5 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_624_remove__def,axiom,
( remove_c_d_set_a
= ( ^ [X3: ( c > d ) > set_a,A5: set_c_d_set_a] : ( minus_1665977719694084726_set_a @ A5 @ ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% remove_def
thf(fact_625_top__conj_I2_J,axiom,
! [P: $o,X2: nat] :
( ( P
& ( top_top_nat_o @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_626_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_627_top__conj_I2_J,axiom,
! [P: $o,X2: a] :
( ( P
& ( top_top_a_o @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_628_top__conj_I1_J,axiom,
! [X2: nat,P: $o] :
( ( ( top_top_nat_o @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_629_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_630_top__conj_I1_J,axiom,
! [X2: a,P: $o] :
( ( ( top_top_a_o @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_631_member__remove,axiom,
! [X2: nat,Y: nat,A3: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y @ A3 ) )
= ( ( member_nat @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_632_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_633_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_634_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_635_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_636_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
=> ( R @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_637_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_638_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_639_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_640_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_641_psubset__insert__iff,axiom,
! [A3: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ ( insert_nat @ X2 @ B3 ) )
= ( ( ( member_nat @ X2 @ B3 )
=> ( ord_less_set_nat @ A3 @ B3 ) )
& ( ~ ( member_nat @ X2 @ B3 )
=> ( ( ( member_nat @ X2 @ A3 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_642_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_643_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_644_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_645_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_646_subset__Compl__singleton,axiom,
! [A3: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_647_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_648_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_649_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ( uminus_uminus_set_a @ X2 )
= ( uminus_uminus_set_a @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_650_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ( uminus8771976365291672326_set_a @ X2 )
= ( uminus8771976365291672326_set_a @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_651_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( uminus5710092332889474511et_nat @ X2 )
= ( uminus5710092332889474511et_nat @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_652_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_653_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_c_d_set_a] :
( ( uminus8771976365291672326_set_a @ ( uminus8771976365291672326_set_a @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_654_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_655_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_656_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_657_Compl__eq__Compl__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( uminus5710092332889474511et_nat @ A3 )
= ( uminus5710092332889474511et_nat @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_658_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_659_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_660_Compl__iff,axiom,
! [C: nat,A3: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
= ( ~ ( member_nat @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_661_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_662_Compl__iff,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
= ( ~ ( member_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_663_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_664_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_665_ComplI,axiom,
! [C: nat,A3: set_nat] :
( ~ ( member_nat @ C @ A3 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).
% ComplI
thf(fact_666_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_667_ComplI,axiom,
! [C: a,A3: set_a] :
( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_668_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_669_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_670_compl__le__compl__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_671_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_672_compl__less__compl__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_set_a @ Y @ X2 ) ) ).
% compl_less_compl_iff
thf(fact_673_compl__less__compl__iff,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ ( uminus8771976365291672326_set_a @ Y ) )
= ( ord_le3685282097655362107_set_a @ Y @ X2 ) ) ).
% compl_less_compl_iff
thf(fact_674_compl__less__compl__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_set_nat @ Y @ X2 ) ) ).
% compl_less_compl_iff
thf(fact_675_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_676_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_677_psubsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_nat @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_678_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_679_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_680_Compl__anti__mono,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B3 ) @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_681_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_682_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_683_Compl__subset__Compl__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B3 ) )
= ( ord_less_eq_set_nat @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_684_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_nat_o @ top_top_nat_o )
= bot_bot_nat_o ) ).
% boolean_algebra.compl_one
thf(fact_685_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_686_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_a_o @ top_top_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.compl_one
thf(fact_687_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_688_boolean__algebra_Ocompl__one,axiom,
( ( uminus8771976365291672326_set_a @ top_to4267977599310771935_set_a )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.compl_one
thf(fact_689_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_690_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_nat_o @ bot_bot_nat_o )
= top_top_nat_o ) ).
% boolean_algebra.compl_zero
thf(fact_691_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_692_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_a_o @ bot_bot_a_o )
= top_top_a_o ) ).
% boolean_algebra.compl_zero
thf(fact_693_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_694_boolean__algebra_Ocompl__zero,axiom,
( ( uminus8771976365291672326_set_a @ bot_bo738396921950161403_set_a )
= top_to4267977599310771935_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_695_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_696_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_697_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_698_order__less__imp__not__less,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ~ ( ord_less_set_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_699_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_700_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_701_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_702_order__less__imp__not__eq2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_703_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_704_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_705_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_706_order__less__imp__not__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_707_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_708_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_709_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_710_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_711_order__less__imp__triv,axiom,
! [X2: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_712_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_713_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_714_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_715_order__less__not__sym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ~ ( ord_less_set_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_716_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_717_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_718_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_719_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_720_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_721_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_722_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_723_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_724_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_725_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_726_order__less__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > nat,C: nat] :
( ( ord_le3685282097655362107_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_727_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_728_order__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_729_order__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_730_order__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_731_order__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_732_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_733_order__less__subst1,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_734_order__less__subst1,axiom,
! [A: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_735_order__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_736_order__less__subst1,axiom,
! [A: set_c_d_set_a,F: nat > set_c_d_set_a,B: nat,C: nat] :
( ( ord_le3685282097655362107_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_le3685282097655362107_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3685282097655362107_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_737_order__less__irrefl,axiom,
! [X2: set_a] :
~ ( ord_less_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_738_order__less__irrefl,axiom,
! [X2: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_739_order__less__irrefl,axiom,
! [X2: set_nat] :
~ ( ord_less_set_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_740_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_741_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_742_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_743_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_744_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_745_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_746_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_747_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_748_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_749_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_750_ord__less__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > nat,C: nat] :
( ( 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_751_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_752_ord__eq__less__subst,axiom,
! [A: nat,F: set_a > nat,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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_753_ord__eq__less__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_754_ord__eq__less__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_755_ord__eq__less__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_756_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_757_ord__eq__less__subst,axiom,
! [A: set_nat,F: set_a > set_nat,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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_758_ord__eq__less__subst,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_759_ord__eq__less__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_760_ord__eq__less__subst,axiom,
! [A: nat,F: set_c_d_set_a > nat,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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_761_order__less__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_762_order__less__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ( ord_le3685282097655362107_set_a @ Y @ Z2 )
=> ( ord_le3685282097655362107_set_a @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_763_order__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_764_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_765_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_766_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_767_order__less__asym_H,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_768_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_769_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_770_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_771_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_772_order__less__asym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ~ ( ord_less_set_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_773_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_774_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_775_double__complement,axiom,
! [A3: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_776_double__complement,axiom,
! [A3: set_c_d_set_a] :
( ( uminus8771976365291672326_set_a @ ( uminus8771976365291672326_set_a @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_777_double__complement,axiom,
! [A3: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_778_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_779_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_780_psubset__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_781_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_782_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_783_pairwiseI,axiom,
! [S4: set_nat,R2: nat > nat > $o] :
( ! [X: nat,Y2: nat] :
( ( member_nat @ X @ S4 )
=> ( ( member_nat @ Y2 @ S4 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_nat @ R2 @ S4 ) ) ).
% pairwiseI
thf(fact_784_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_785_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_786_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_787_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_788_pairwiseD,axiom,
! [R2: nat > nat > $o,S4: set_nat,X2: nat,Y: nat] :
( ( pairwise_nat @ R2 @ S4 )
=> ( ( member_nat @ X2 @ S4 )
=> ( ( member_nat @ Y @ S4 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_789_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_790_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_791_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_792_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_793_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_794_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_795_psubsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_796_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_797_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_798_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_799_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_800_ComplD,axiom,
! [C: nat,A3: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
=> ~ ( member_nat @ C @ A3 ) ) ).
% ComplD
thf(fact_801_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_802_ComplD,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
=> ~ ( member_a @ C @ A3 ) ) ).
% ComplD
thf(fact_803_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_804_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_805_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_806_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_807_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_808_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_809_order_Ostrict__implies__not__eq,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_810_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_811_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_812_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_813_dual__order_Ostrict__trans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_814_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_815_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_816_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_817_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_818_order_Ostrict__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_819_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_820_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_821_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_822_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_823_dual__order_Oirrefl,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_824_dual__order_Oirrefl,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_825_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_826_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_827_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_828_dual__order_Oasym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_829_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_830_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_831_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_832_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_833_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_834_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_835_ord__less__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_836_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_837_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_838_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_839_ord__eq__less__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_840_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_841_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_842_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_843_order_Oasym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ B @ A ) ) ).
% order.asym
thf(fact_844_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_845_less__imp__neq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_846_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_847_less__imp__neq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_848_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_849_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_850_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_851_verit__comp__simplify1_I1_J,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_852_verit__comp__simplify1_I1_J,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_853_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_854_compl__less__swap2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ Y ) @ X2 )
=> ( ord_less_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) ) ).
% compl_less_swap2
thf(fact_855_compl__less__swap2,axiom,
! [Y: set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ ( uminus8771976365291672326_set_a @ Y ) @ X2 )
=> ( ord_le3685282097655362107_set_a @ ( uminus8771976365291672326_set_a @ X2 ) @ Y ) ) ).
% compl_less_swap2
thf(fact_856_compl__less__swap2,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X2 )
=> ( ord_less_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) ) ).
% compl_less_swap2
thf(fact_857_compl__less__swap1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_set_a @ Y @ ( uminus_uminus_set_a @ X2 ) )
=> ( ord_less_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_858_compl__less__swap1,axiom,
! [Y: set_c_d_set_a,X2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ Y @ ( uminus8771976365291672326_set_a @ X2 ) )
=> ( ord_le3685282097655362107_set_a @ X2 @ ( uminus8771976365291672326_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_859_compl__less__swap1,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X2 ) )
=> ( ord_less_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_less_swap1
thf(fact_860_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_861_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_862_compl__le__swap2,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_863_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_864_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_865_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_866_compl__le__swap1,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X2 ) )
=> ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_867_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_868_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_869_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_870_compl__mono,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X2 ) ) ) ).
% compl_mono
thf(fact_871_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_872_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_873_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_874_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_875_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_876_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_877_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_878_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_879_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_880_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_881_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_882_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_883_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_884_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_885_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_886_antisym__conv1,axiom,
! [X2: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_887_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_888_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_889_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_890_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_891_antisym__conv2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_892_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_893_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_894_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_895_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_896_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_897_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_898_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_899_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_900_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_901_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_902_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_903_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_904_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_905_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_906_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_907_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_908_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_909_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_910_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_911_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_912_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_913_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_914_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_915_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_916_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_917_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_918_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_919_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_920_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_921_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_922_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_923_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_924_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_925_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_926_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_927_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_928_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_929_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_930_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_931_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_932_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_933_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_934_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_935_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_936_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_937_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_938_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_939_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_940_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_941_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_942_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_943_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_944_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_945_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_946_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_947_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_948_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_949_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_950_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_951_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_952_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_953_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_954_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_955_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_956_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_957_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_958_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_959_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_960_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_961_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_962_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_963_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_964_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_965_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_966_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_967_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_968_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_969_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_970_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_971_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_972_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_973_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_974_order__less__imp__le,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_975_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_976_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_977_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_978_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_979_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_980_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_981_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_982_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_983_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_984_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_985_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_986_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_987_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_988_order__le__less__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y )
=> ( ( ord_le3685282097655362107_set_a @ Y @ Z2 )
=> ( ord_le3685282097655362107_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_989_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_990_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_991_order__le__less__trans,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y )
=> ( ( ord_less_c_d_set_a @ Y @ Z2 )
=> ( ord_less_c_d_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_992_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_993_order__less__le__trans,axiom,
! [X2: set_c_d_set_a,Y: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ X2 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ Z2 )
=> ( ord_le3685282097655362107_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_994_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_995_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_996_order__less__le__trans,axiom,
! [X2: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X2 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ Z2 )
=> ( ord_less_c_d_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_997_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_998_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_999_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1000_order__le__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1001_order__le__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1002_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_1003_order__le__less__subst1,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1004_order__le__less__subst1,axiom,
! [A: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1005_order__le__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1006_order__le__less__subst1,axiom,
! [A: ( c > d ) > set_a,F: nat > ( c > d ) > set_a,B: nat,C: nat] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ 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_1007_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1008_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1009_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1010_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1011_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1012_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_1013_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1014_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1015_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1016_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_c_d_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ 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_1017_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1018_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1019_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1020_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1021_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1022_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_1023_order__less__le__subst1,axiom,
! [A: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( ord_less_set_nat @ 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_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1024_order__less__le__subst1,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1025_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1026_order__less__le__subst1,axiom,
! [A: ( c > d ) > set_a,F: nat > ( c > d ) > set_a,B: nat,C: nat] :
( ( ord_less_c_d_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ 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_1027_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1028_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1029_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1030_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1031_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1032_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_1033_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1034_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1035_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1036_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ 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_1037_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1038_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_1039_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_1040_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1041_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1042_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_1043_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_1044_bot_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_1045_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1046_bot_Oextremum__strict,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% bot.extremum_strict
thf(fact_1047_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1048_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1049_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_1050_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_1051_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1052_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1053_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_1054_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_1055_psubset__imp__ex__mem,axiom,
! [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A3 @ B3 )
=> ? [B4: ( c > d ) > set_a] : ( member_c_d_set_a @ B4 @ ( minus_1665977719694084726_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1056_psubset__imp__ex__mem,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1057_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_1058_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_1059_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_1060_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_1061_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1062_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_1063_surj__Compl__image__subset,axiom,
! [F: nat > nat,A3: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_inf__top__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_1073_inf__top__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% inf_top_right
thf(fact_1074_inf__eq__top__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y )
= top_top_set_nat )
= ( ( X2 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1075_top__eq__inf__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X2 @ Y ) )
= ( ( X2 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1076_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1077_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1078_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B ) )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1079_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_1080_Int__UNIV,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= top_top_set_nat )
= ( ( A3 = top_top_set_nat )
& ( B3 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_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_1095_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_1096_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_1097_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X3: nat] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1098_surjI,axiom,
! [G2: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1099_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_1100_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X: nat] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_Int__UNIV__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
= A3 ) ).
% Int_UNIV_right
thf(fact_1126_Int__UNIV__left,axiom,
! [B3: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_1127_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_1128_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_1129_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_1130_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_1131_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_1132_range__eqI,axiom,
! [B: ( c > d ) > set_a,F: nat > ( c > d ) > set_a,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_c_d_set_a @ B @ ( image_nat_c_d_set_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1133_range__eqI,axiom,
! [B: a,F: nat > a,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1134_rangeI,axiom,
! [F: nat > ( c > d ) > set_a,X2: nat] : ( member_c_d_set_a @ ( F @ X2 ) @ ( image_nat_c_d_set_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1135_rangeI,axiom,
! [F: nat > a,X2: nat] : ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1136_range__subsetD,axiom,
! [F: nat > ( c > d ) > set_a,B3: set_c_d_set_a,I: nat] :
( ( ord_le5982164083705284911_set_a @ ( image_nat_c_d_set_a @ F @ top_top_set_nat ) @ B3 )
=> ( member_c_d_set_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1137_range__subsetD,axiom,
! [F: nat > a,B3: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B3 )
=> ( member_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1138_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_1139_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_1140_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1141_finite__induct__select,axiom,
! [S4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T2: set_nat] :
( ( ord_less_set_nat @ T2 @ S4 )
=> ( ( P @ T2 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ ( minus_minus_set_nat @ S4 @ T2 ) )
& ( P @ ( insert_nat @ X7 @ T2 ) ) ) ) )
=> ( P @ S4 ) ) ) ) ).
% finite_induct_select
thf(fact_1142_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1143_finite__insert,axiom,
! [A: nat,A3: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A3 ) )
= ( finite_finite_nat @ A3 ) ) ).
% finite_insert
thf(fact_1144_finite__Diff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% finite_Diff
thf(fact_1145_finite__Diff2,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B3 ) )
= ( finite_finite_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_1146_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1147_finite__Diff__insert,axiom,
! [A3: set_nat,A: nat,B3: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B3 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_1148_finite__compl,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A3 ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_1149_finite__surj,axiom,
! [A3: set_nat,B3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ( finite_finite_nat @ B3 ) ) ) ).
% finite_surj
thf(fact_1150_endo__inj__surj,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ A3 )
=> ( ( inj_on_nat_nat @ F @ A3 )
=> ( ( image_nat_nat @ F @ A3 )
= A3 ) ) ) ) ).
% endo_inj_surj
thf(fact_1151_inj__on__finite,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A3 ) ) ) ) ).
% inj_on_finite
thf(fact_1152_finite__surj__inj,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( image_nat_nat @ F @ A3 ) )
=> ( inj_on_nat_nat @ F @ A3 ) ) ) ).
% finite_surj_inj
thf(fact_1153_finite__subset__image,axiom,
! [B3: set_nat,F: nat > nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
& ( finite_finite_nat @ C4 )
& ( B3
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1154_ex__finite__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A3 )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1155_all__finite__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A3 ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1156_rev__finite__subset,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_1157_infinite__super,axiom,
! [S4: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ T3 )
=> ( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1158_finite__subset,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_1159_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_1160_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1161_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_1162_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1163_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1164_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1165_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_1166_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_1167_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_1168_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_1169_inj__img__insertE,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,X2: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( inj_on2268522623953733425_set_a @ F @ A3 )
=> ( ~ ( member_c_d_set_a @ X2 @ B3 )
=> ( ( ( insert_c_d_set_a @ X2 @ B3 )
= ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ~ ! [X8: ( c > d ) > set_a,A7: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X8 @ A7 )
=> ( ( A3
= ( insert_c_d_set_a @ X8 @ A7 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B3
!= ( image_5710119992958135237_set_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1170_inj__img__insertE,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,X2: ( c > d ) > set_a,B3: set_c_d_set_a] :
( ( inj_on_a_c_d_set_a @ F @ A3 )
=> ( ~ ( member_c_d_set_a @ X2 @ B3 )
=> ( ( ( insert_c_d_set_a @ X2 @ B3 )
= ( image_a_c_d_set_a @ F @ A3 ) )
=> ~ ! [X8: a,A7: set_a] :
( ~ ( member_a @ X8 @ A7 )
=> ( ( A3
= ( insert_a @ X8 @ A7 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B3
!= ( image_a_c_d_set_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1171_inj__img__insertE,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,X2: a,B3: set_a] :
( ( inj_on_c_d_set_a_a @ F @ A3 )
=> ( ~ ( member_a @ X2 @ B3 )
=> ( ( ( insert_a @ X2 @ B3 )
= ( image_c_d_set_a_a @ F @ A3 ) )
=> ~ ! [X8: ( c > d ) > set_a,A7: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X8 @ A7 )
=> ( ( A3
= ( insert_c_d_set_a @ X8 @ A7 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B3
!= ( image_c_d_set_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1172_inj__img__insertE,axiom,
! [F: a > a,A3: set_a,X2: a,B3: set_a] :
( ( inj_on_a_a @ F @ A3 )
=> ( ~ ( member_a @ X2 @ B3 )
=> ( ( ( insert_a @ X2 @ B3 )
= ( image_a_a @ F @ A3 ) )
=> ~ ! [X8: a,A7: set_a] :
( ~ ( member_a @ X8 @ A7 )
=> ( ( A3
= ( insert_a @ X8 @ A7 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B3
!= ( image_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1173_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_1174_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_1175_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_1176_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_1177_inj__image__mem__iff,axiom,
! [F: nat > ( c > d ) > set_a,A: nat,A3: set_nat] :
( ( inj_on_nat_c_d_set_a @ F @ top_top_set_nat )
=> ( ( member_c_d_set_a @ ( F @ A ) @ ( image_nat_c_d_set_a @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1178_inj__image__mem__iff,axiom,
! [F: nat > a,A: nat,A3: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ ( F @ A ) @ ( image_nat_a @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1179_range__ex1__eq,axiom,
! [F: nat > ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( inj_on_nat_c_d_set_a @ F @ top_top_set_nat )
=> ( ( member_c_d_set_a @ B @ ( image_nat_c_d_set_a @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y4: nat] :
( ( B
= ( F @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1180_range__ex1__eq,axiom,
! [F: nat > a,B: a] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y4: nat] :
( ( B
= ( F @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1181_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1182_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1183_ex__new__if__finite,axiom,
! [A3: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( ( finite3330819693523053784_set_a @ A3 )
=> ? [A4: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A4 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1184_ex__new__if__finite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A3 )
=> ? [A4: a] :
~ ( member_a @ A4 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1185_ex__new__if__finite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A3 )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1186_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1187_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1188_Diff__infinite__finite,axiom,
! [T3: set_nat,S4: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S4 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1189_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1190_infinite__imp__nonempty,axiom,
! [S4: set_nat] :
( ~ ( finite_finite_nat @ S4 )
=> ( S4 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1191_finite_OinsertI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_1192_infinite__finite__induct,axiom,
! [P: set_c_d_set_a > $o,A3: set_c_d_set_a] :
( ! [A8: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ A8 )
=> ( P @ A8 ) )
=> ( ( 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_1193_infinite__finite__induct,axiom,
! [P: set_a > $o,A3: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( 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_1194_infinite__finite__induct,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1195_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_1196_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_1197_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1198_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_1199_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_1200_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1201_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A2: set_nat] :
( ( A2 = bot_bot_set_nat )
| ? [A5: set_nat,B2: nat] :
( ( A2
= ( insert_nat @ B2 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1202_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A4: nat] :
( A
= ( insert_nat @ A4 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1203_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1204_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1205_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 )
=> ( ! [A4: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( member_c_d_set_a @ A4 @ A3 )
=> ( ~ ( member_c_d_set_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1206_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 )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A3 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1207_finite__subset__induct,axiom,
! [F4: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A3 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1208_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 )
=> ( ! [A4: ( c > d ) > set_a,F3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( member_c_d_set_a @ A4 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ F3 @ A3 )
=> ( ~ ( member_c_d_set_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_c_d_set_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1209_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 )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1210_finite__subset__induct_H,axiom,
! [F4: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1211_finite__empty__induct,axiom,
! [A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: ( c > d ) > set_a,A8: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A8 )
=> ( ( member_c_d_set_a @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ A4 @ bot_bo738396921950161403_set_a ) ) ) ) ) )
=> ( P @ bot_bo738396921950161403_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1212_finite__empty__induct,axiom,
! [A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1213_finite__empty__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1214_infinite__coinduct,axiom,
! [X5: set_nat > $o,A3: set_nat] :
( ( X5 @ A3 )
=> ( ! [A8: set_nat] :
( ( X5 @ A8 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A8 )
& ( ( X5 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_1215_infinite__remove,axiom,
! [S4: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1216_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 )
=> ( ! [A8: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A8 )
=> ( ( A8 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ A8 @ B3 )
=> ( ! [X7: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X7 @ A8 )
=> ( P @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ X7 @ bot_bo738396921950161403_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_1217_finite__remove__induct,axiom,
! [B3: set_a,P: set_a > $o] :
( ( finite_finite_a @ B3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B3 )
=> ( ! [X7: a] :
( ( member_a @ X7 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X7 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_1218_finite__remove__induct,axiom,
! [B3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B3 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_1219_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 ) )
=> ( ! [A8: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A8 )
=> ( ( A8 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ A8 @ B3 )
=> ( ! [X7: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X7 @ A8 )
=> ( P @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ X7 @ bot_bo738396921950161403_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_1220_remove__induct,axiom,
! [P: set_a > $o,B3: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B3 )
=> ( ! [X7: a] :
( ( member_a @ X7 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X7 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_1221_remove__induct,axiom,
! [P: set_nat > $o,B3: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B3 )
=> ( P @ B3 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B3 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_1222_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1223_infinite__iff__countable__subset,axiom,
! [S4: set_nat] :
( ( ~ ( finite_finite_nat @ S4 ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S4 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1224_infinite__countable__subset,axiom,
! [S4: set_nat] :
( ~ ( finite_finite_nat @ S4 )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S4 ) ) ) ).
% infinite_countable_subset
thf(fact_1225_finite__linorder__max__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A8 )
=> ( ord_less_nat @ X7 @ B4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1226_finite__linorder__min__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A8 )
=> ( ord_less_nat @ B4 @ X7 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1227_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1228_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_1229_ex__min__if__finite,axiom,
! [S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ( S4 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ S4 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S4 )
& ( ord_less_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1230_cofinite__bot,axiom,
( ( cofinite_nat = bot_bot_filter_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% cofinite_bot
thf(fact_1231_Inf__fin_Oinsert__remove,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A3 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1232_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_1233_Inf__fin_Oremove,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X2 @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1234_Inf__fin_Oinsert,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A3 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1235_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_1236_Inf__fin_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1237_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 ) )
=> ! [A9: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A9 @ A3 )
=> ( ord_le8464990428230162895_set_a @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1238_Inf__fin_OboundedE,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A3 ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A3 )
=> ( ord_less_eq_nat @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1239_Inf__fin_OboundedI,axiom,
! [A3: set_c_d_set_a,X2: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [A4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A4 @ A3 )
=> ( ord_le8464990428230162895_set_a @ X2 @ A4 ) )
=> ( ord_le8464990428230162895_set_a @ X2 @ ( lattic3893622604919961804_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1240_Inf__fin_OboundedI,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ord_less_eq_nat @ X2 @ A4 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1241_Inf__fin_Obounded__iff,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1242_Inf__fin_Osubset__imp,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B3 ) @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1243_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( inf_inf_nat @ X @ Y2 ) )
= ( inf_inf_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N2 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N2 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1244_Inf__fin_Osubset,axiom,
! [A3: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( B3 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B3 ) @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1245_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_1246_Inf__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y2 ) @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1247_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_1248_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X2 @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A3 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1249_Inf__fin__le__Sup__fin,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1250_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_1251_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_1252_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_1253_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_1254_sup__top__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_1255_sup__top__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_1256_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_1257_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_1258_sup__compl__top__left1,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( sup_sup_set_nat @ X2 @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_1259_sup__compl__top__left2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_1260_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1261_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1262_Sup__fin_Oinsert,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A3 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1263_sup__cancel__left2,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A ) @ ( sup_sup_set_nat @ X2 @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_1264_sup__cancel__left1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ A ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_1265_Compl__partition,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_1266_Compl__partition2,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ A3 )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_1267_Un__UNIV__right,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_1268_Un__UNIV__left,axiom,
! [B3: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B3 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_1269_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_1270_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_1271_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_1272_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_1273_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_1274_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_1275_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( sup_sup_nat @ X @ Y2 ) )
= ( sup_sup_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N2 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N2 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1276_Sup__fin_Oinsert__remove,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A3 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1277_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_1278_Sup__fin_Oremove,axiom,
! [A3: set_nat,X2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X2 @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
% Conjectures (1)
thf(conj_0,conjecture,
smaller_interp_c_d_a @ z @ ( inf_c_d_a @ a2 ) ).
%------------------------------------------------------------------------------