TPTP Problem File: SLH0970^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0021_Relational_Calculus/prob_01699_064670__17538332_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1411 ( 504 unt; 135 typ; 0 def)
% Number of atoms : 3751 (1166 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10975 ( 554 ~; 66 |; 236 &;8306 @)
% ( 0 <=>;1813 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 462 ( 462 >; 0 *; 0 +; 0 <<)
% Number of symbols : 128 ( 125 usr; 13 con; 0-4 aty)
% Number of variables : 3361 ( 160 ^;3070 !; 131 ?;3361 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:23.495
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
set_Re381260168593705685la_a_b: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relational_fmla_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
product_prod_b_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__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $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__a,type,
a: $tType ).
% Explicit typings (125)
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
finite5600759454172676150la_a_b: set_Re381260168593705685la_a_b > $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_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
fun_upd_nat_a: ( nat > a ) > nat > a > nat > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
minus_4077726661957047470la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > 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_It__List__Olist_Itf__a_J_J,type,
inf_inf_set_list_a: set_list_a > set_list_a > set_list_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_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
inf_in8483230781156617063la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
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_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
sup_su5130108678486352897la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
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_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
lattic5043722365632780795_a_nat: ( list_a > nat ) > set_list_a > list_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
lattic549851495318985137et_nat: ( list_a > set_nat ) > set_list_a > list_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
lattic1474851849222904177et_nat: ( nat > set_nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
lattic5380700691367270794_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Set__Oset_It__Nat__Onat_J,type,
lattic5577778279351311680et_nat: ( relational_fmla_a_b > set_nat ) > set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
lattic6146202810820454443et_nat: ( a > set_nat ) > set_a > 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__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3014633134055518761et_nat: set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3835124923745554447et_nat: set_set_nat > set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
size_s453432777765377587la_a_b: relational_fmla_a_b > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
bot_bo8852203127187332700_a_b_o: relational_fmla_a_b > $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__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
bot_bo4495933725496725865la_a_b: set_Re381260168593705685la_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
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_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $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_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
ord_le7152733262289451305la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $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_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_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
ord_le4112832032246704949la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $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_Relational__Calculus_Oadom_001tf__b_001tf__a,type,
relational_adom_b_a: ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Relational__Calculus_Oap_001tf__a_001tf__b,type,
relational_ap_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocov_001tf__a_001tf__b,type,
relational_cov_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocp_001tf__a_001tf__b,type,
relational_cp_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ocpropagated_001tf__a_001tf__b,type,
relati1591879772219623554ed_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocsts_001tf__a_001tf__b,type,
relational_csts_a_b: relational_fmla_a_b > set_a ).
thf(sy_c_Relational__Calculus_Oequiv_001tf__a_001tf__b,type,
relational_equiv_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oerase_001tf__a_001tf__b,type,
relational_erase_a_b: relational_fmla_a_b > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Oeval_001tf__a_001tf__b,type,
relational_eval_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oeval__on_001tf__a_001tf__b,type,
relati8814510239606734169on_a_b: set_nat > relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oexists_001tf__a_001tf__b,type,
relati3989891337220013914ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OBool_001tf__a_001tf__b,type,
relational_Bool_a_b: $o > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OConj_001tf__a_001tf__b,type,
relational_Conj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_ODisj_001tf__a_001tf__b,type,
relational_Disj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_ONeg_001tf__a_001tf__b,type,
relational_Neg_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofresh2_001tf__a_001tf__b,type,
relati2677767559083392098h2_a_b: nat > nat > relational_fmla_a_b > nat ).
thf(sy_c_Relational__Calculus_Ofv_001tf__a_001tf__b,type,
relational_fv_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Ogen_001tf__a_001tf__b,type,
relational_gen_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogen_H_001tf__a_001tf__b,type,
relational_gen_a_b2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogenempty_001tf__a_001tf__b,type,
relati5999705594545617851ty_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Onongens_001tf__a_001tf__b,type,
relati62690040636126068ns_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Oqp_001tf__a_001tf__b,type,
relational_qp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Orrb_001tf__a_001tf__b,type,
relational_rrb_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osat_001tf__a_001tf__b,type,
relational_sat_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o ).
thf(sy_c_Relational__Calculus_Osr_001tf__a_001tf__b,type,
relational_sr_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
collec3419995626248312948la_a_b: ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_Itf__a_J,type,
is_empty_list_a: set_list_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_emp6953259385542938189la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_Itf__a_J,type,
is_singleton_list_a: set_list_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_sin6594375743535830443la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_Itf__a_J,type,
remove_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
remove4261432235257513082la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_Itf__a_J,type,
the_elem_list_a: set_list_a > list_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v_Q_H,type,
q2: relational_fmla_a_b ).
% Relevant facts (1270)
thf(fact_0_sat__fv__cong,axiom,
! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
( ! [N: nat] :
( ( member_nat @ N @ ( relational_fv_a_b @ Phi ) )
=> ( ( Sigma @ N )
= ( Sigma2 @ N ) ) )
=> ( ( relational_sat_a_b @ Phi @ I @ Sigma )
= ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).
% sat_fv_cong
thf(fact_1_Relational__Calculus_Oeval__def,axiom,
( relational_eval_a_b
= ( ^ [Q: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q ) @ Q ) ) ) ).
% Relational_Calculus.eval_def
thf(fact_2_sat__fun__upd,axiom,
! [N2: nat,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
( ~ ( member_nat @ N2 @ ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_sat_a_b @ Q2 @ I @ ( fun_upd_nat_a @ Sigma @ N2 @ Z ) )
= ( relational_sat_a_b @ Q2 @ I @ Sigma ) ) ) ).
% sat_fun_upd
thf(fact_3_fresh2_I3_J,axiom,
! [X: nat,Y: nat,Q2: relational_fmla_a_b] :
~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q2 ) @ ( relational_fv_a_b @ Q2 ) ) ).
% fresh2(3)
thf(fact_4_sat__cp,axiom,
! [Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_cp_a_b @ Q2 ) @ I @ Sigma )
= ( relational_sat_a_b @ Q2 @ I @ Sigma ) ) ).
% sat_cp
thf(fact_5_finite__fv,axiom,
! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).
% finite_fv
thf(fact_6_sat_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
| ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(6)
thf(fact_7_sat_Osimps_I4_J,axiom,
! [Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Neg_a_b @ Phi ) @ I @ Sigma )
= ( ~ ( relational_sat_a_b @ Phi @ I @ Sigma ) ) ) ).
% sat.simps(4)
thf(fact_8_sat_Osimps_I5_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
& ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(5)
thf(fact_9_sat_Osimps_I2_J,axiom,
! [B: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Bool_a_b @ B ) @ I @ Sigma )
= B ) ).
% sat.simps(2)
thf(fact_10_fv_Osimps_I4_J,axiom,
! [Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
= ( relational_fv_a_b @ Phi ) ) ).
% fv.simps(4)
thf(fact_11_fmla_Oinject_I6_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
( ( ( relational_Disj_a_b @ X61 @ X62 )
= ( relational_Disj_a_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fmla.inject(6)
thf(fact_12_fmla_Oinject_I4_J,axiom,
! [X4: relational_fmla_a_b,Y4: relational_fmla_a_b] :
( ( ( relational_Neg_a_b @ X4 )
= ( relational_Neg_a_b @ Y4 ) )
= ( X4 = Y4 ) ) ).
% fmla.inject(4)
thf(fact_13_fmla_Oinject_I5_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
( ( ( relational_Conj_a_b @ X51 @ X52 )
= ( relational_Conj_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fmla.inject(5)
thf(fact_14_fmla_Oinject_I2_J,axiom,
! [X2: $o,Y2: $o] :
( ( ( relational_Bool_a_b @ X2 )
= ( relational_Bool_a_b @ Y2 ) )
= ( X2 = Y2 ) ) ).
% fmla.inject(2)
thf(fact_15_cp__idem,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relational_cp_a_b @ Q2 ) )
= ( relational_cp_a_b @ Q2 ) ) ).
% cp_idem
thf(fact_16_fresh2_I2_J,axiom,
! [Y: nat,X: nat,Q2: relational_fmla_a_b] :
( Y
!= ( relati2677767559083392098h2_a_b @ X @ Y @ Q2 ) ) ).
% fresh2(2)
thf(fact_17_fresh2_I1_J,axiom,
! [X: nat,Y: nat,Q2: relational_fmla_a_b] :
( X
!= ( relati2677767559083392098h2_a_b @ X @ Y @ Q2 ) ) ).
% fresh2(1)
thf(fact_18_cp_Osimps_I7_J,axiom,
! [V: $o] :
( ( relational_cp_a_b @ ( relational_Bool_a_b @ V ) )
= ( relational_Bool_a_b @ V ) ) ).
% cp.simps(7)
thf(fact_19_fmla_Odistinct_I37_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(37)
thf(fact_20_fmla_Odistinct_I33_J,axiom,
! [X4: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(33)
thf(fact_21_fmla_Odistinct_I31_J,axiom,
! [X4: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(31)
thf(fact_22_fmla_Odistinct_I19_J,axiom,
! [X2: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(19)
thf(fact_23_fmla_Odistinct_I17_J,axiom,
! [X2: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(17)
thf(fact_24_fmla_Odistinct_I15_J,axiom,
! [X2: $o,X4: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Neg_a_b @ X4 ) ) ).
% fmla.distinct(15)
thf(fact_25_fun__upd__upd,axiom,
! [F: nat > a,X: nat,Y: a,Z: a] :
( ( fun_upd_nat_a @ ( fun_upd_nat_a @ F @ X @ Y ) @ X @ Z )
= ( fun_upd_nat_a @ F @ X @ Z ) ) ).
% fun_upd_upd
thf(fact_26_fun__upd__triv,axiom,
! [F: nat > a,X: nat] :
( ( fun_upd_nat_a @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_27_fun__upd__apply,axiom,
( fun_upd_nat_a
= ( ^ [F2: nat > a,X3: nat,Y3: a,Z2: nat] : ( if_a @ ( Z2 = X3 ) @ Y3 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_28_fun__upd__def,axiom,
( fun_upd_nat_a
= ( ^ [F2: nat > a,A: nat,B2: a,X3: nat] : ( if_a @ ( X3 = A ) @ B2 @ ( F2 @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_29_fun__upd__eqD,axiom,
! [F: nat > a,X: nat,Y: a,G: nat > a,Z: a] :
( ( ( fun_upd_nat_a @ F @ X @ Y )
= ( fun_upd_nat_a @ G @ X @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_30_fun__upd__idem,axiom,
! [F: nat > a,X: nat,Y: a] :
( ( ( F @ X )
= Y )
=> ( ( fun_upd_nat_a @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_31_fun__upd__same,axiom,
! [F: nat > a,X: nat,Y: a] :
( ( fun_upd_nat_a @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_32_fun__upd__other,axiom,
! [Z: nat,X: nat,F: nat > a,Y: a] :
( ( Z != X )
=> ( ( fun_upd_nat_a @ F @ X @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_33_fun__upd__twist,axiom,
! [A2: nat,C: nat,M: nat > a,B: a,D: a] :
( ( A2 != C )
=> ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ M @ A2 @ B ) @ C @ D )
= ( fun_upd_nat_a @ ( fun_upd_nat_a @ M @ C @ D ) @ A2 @ B ) ) ) ).
% fun_upd_twist
thf(fact_34_fun__upd__idem__iff,axiom,
! [F: nat > a,X: nat,Y: a] :
( ( ( fun_upd_nat_a @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_35_finite__eval__on__Disj2D,axiom,
! [X5: set_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ I ) )
=> ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ Q22 @ I ) ) ) ) ).
% finite_eval_on_Disj2D
thf(fact_36_finite__eval__Disj2D,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ I ) )
=> ( finite_finite_list_a @ ( relational_eval_a_b @ Q22 @ I ) ) ) ).
% finite_eval_Disj2D
thf(fact_37_mem__Collect__eq,axiom,
! [A2: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_38_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_39_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_40_Collect__mem__eq,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_41_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_43_Collect__cong,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X6: nat] :
( ( P @ X6 )
= ( Q2 @ X6 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_44_genempty_Ointros_I3_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).
% genempty.intros(3)
thf(fact_45_genempty_Ointros_I4_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ) ).
% genempty.intros(4)
thf(fact_46_eval__on__False,axiom,
! [X5: set_nat,I: product_prod_b_nat > set_list_a] :
( ( relati8814510239606734169on_a_b @ X5 @ ( relational_Bool_a_b @ $false ) @ I )
= bot_bot_set_list_a ) ).
% eval_on_False
thf(fact_47_gen_H_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G2 )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G2 ) ) ).
% gen'.intros(4)
thf(fact_48_gen_H_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G2 )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G2 ) ) ).
% gen'.intros(5)
thf(fact_49_eval__Bool__False,axiom,
! [I: product_prod_b_nat > set_list_a] :
( ( relational_eval_a_b @ ( relational_Bool_a_b @ $false ) @ I )
= bot_bot_set_list_a ) ).
% eval_Bool_False
thf(fact_50_sr__Disj,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( ( relational_fv_a_b @ Q1 )
= ( relational_fv_a_b @ Q22 ) )
=> ( ( relational_sr_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( ( relational_sr_a_b @ Q1 )
& ( relational_sr_a_b @ Q22 ) ) ) ) ).
% sr_Disj
thf(fact_51_sat__exists,axiom,
! [N2: nat,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relati3989891337220013914ts_a_b @ N2 @ Q2 ) @ I @ Sigma )
= ( ? [X3: a] : ( relational_sat_a_b @ Q2 @ I @ ( fun_upd_nat_a @ Sigma @ N2 @ X3 ) ) ) ) ).
% sat_exists
thf(fact_52_cp__exists,axiom,
! [X: nat,Q2: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q2 ) )
= ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q2 ) ) ) ).
% cp_exists
thf(fact_53_finite__has__minimal,axiom,
! [A3: set_set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A3 )
=> ( ( ord_less_eq_set_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_54_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_55_finite__has__maximal,axiom,
! [A3: set_set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A3 )
=> ( ( ord_less_eq_set_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_56_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_57_finite__psubset__induct,axiom,
! [A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A3 )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ! [B3: set_list_a] :
( ( ord_less_set_list_a @ B3 @ A4 )
=> ( P @ B3 ) )
=> ( P @ A4 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_58_finite__psubset__induct,axiom,
! [A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ! [B3: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
=> ( P @ B3 ) )
=> ( P @ A4 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_59_finite__psubset__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B3: set_nat] :
( ( ord_less_set_nat @ B3 @ A4 )
=> ( P @ B3 ) )
=> ( P @ A4 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_60_finite__has__maximal2,axiom,
! [A3: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( member_set_nat @ A2 @ A3 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A3 )
& ( ord_less_eq_set_nat @ A2 @ X6 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A3 )
=> ( ( ord_less_eq_set_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_61_finite__has__maximal2,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ( ord_less_eq_nat @ A2 @ X6 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_62_finite__has__minimal2,axiom,
! [A3: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( member_set_nat @ A2 @ A3 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A3 )
& ( ord_less_eq_set_nat @ X6 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A3 )
=> ( ( ord_less_eq_set_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_63_finite__has__minimal2,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ( ord_less_eq_nat @ X6 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_64_finite__subset,axiom,
! [A3: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B4 )
=> ( ( finite_finite_list_a @ B4 )
=> ( finite_finite_list_a @ A3 ) ) ) ).
% finite_subset
thf(fact_65_finite__subset,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( finite_finite_a @ B4 )
=> ( finite_finite_a @ A3 ) ) ) ).
% finite_subset
thf(fact_66_finite__subset,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_67_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_68_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_69_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_70_rev__finite__subset,axiom,
! [B4: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B4 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B4 )
=> ( finite_finite_list_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_71_rev__finite__subset,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( finite_finite_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_72_rev__finite__subset,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_73_gen_H_Ointros_I3_J,axiom,
! [X: nat,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q2 @ G2 )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) @ G2 ) ) ).
% gen'.intros(3)
thf(fact_74_gen_H_Ointros_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( ( relational_gen_a_b2 @ X @ Q1 @ G2 )
| ( relational_gen_a_b2 @ X @ Q22 @ G2 ) )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G2 ) ) ).
% gen'.intros(7)
thf(fact_75_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_76_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_77_finite_OemptyI,axiom,
finite5600759454172676150la_a_b @ bot_bo4495933725496725865la_a_b ).
% finite.emptyI
thf(fact_78_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_79_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_80_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_81_infinite__imp__nonempty,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% infinite_imp_nonempty
thf(fact_82_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_83_genempty_Ointros_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q1 )
=> ( ( relati5999705594545617851ty_a_b @ Q22 )
=> ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).
% genempty.intros(5)
thf(fact_84_genempty_Ointros_I2_J,axiom,
! [Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q2 )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) ) ) ).
% genempty.intros(2)
thf(fact_85_genempty_Ointros_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( ( relati5999705594545617851ty_a_b @ Q1 )
| ( relati5999705594545617851ty_a_b @ Q22 ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ).
% genempty.intros(6)
thf(fact_86_genempty_Ointros_I1_J,axiom,
relati5999705594545617851ty_a_b @ ( relational_Bool_a_b @ $false ) ).
% genempty.intros(1)
thf(fact_87_sr__False,axiom,
relational_sr_a_b @ ( relational_Bool_a_b @ $false ) ).
% sr_False
thf(fact_88_sr__cp,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_sr_a_b @ Q2 )
=> ( relational_sr_a_b @ ( relational_cp_a_b @ Q2 ) ) ) ).
% sr_cp
thf(fact_89_genempty__cp,axiom,
! [Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q2 )
=> ( ( relational_cp_a_b @ Q2 )
= ( relational_Bool_a_b @ $false ) ) ) ).
% genempty_cp
thf(fact_90_fv__cp,axiom,
! [Q2: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_cp_a_b @ Q2 ) ) @ ( relational_fv_a_b @ Q2 ) ) ).
% fv_cp
thf(fact_91_psubsetI,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_less_set_nat @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_92_subset__empty,axiom,
! [A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ bot_bot_set_list_a )
= ( A3 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_93_subset__empty,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ bot_bo4495933725496725865la_a_b )
= ( A3 = bot_bo4495933725496725865la_a_b ) ) ).
% subset_empty
thf(fact_94_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_95_empty__subsetI,axiom,
! [A3: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A3 ) ).
% empty_subsetI
thf(fact_96_empty__subsetI,axiom,
! [A3: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A3 ) ).
% empty_subsetI
thf(fact_97_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_98_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X6 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_99_ex__min__if__finite,axiom,
! [S: set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ S )
& ~ ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ S )
& ( ord_less_set_nat @ Xa @ X6 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_100_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X6 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_101_subsetI,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A3 )
=> ( member4680049679412964150la_a_b @ X6 @ B4 ) )
=> ( ord_le4112832032246704949la_a_b @ A3 @ B4 ) ) ).
% subsetI
thf(fact_102_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( member_a @ X6 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_103_subsetI,axiom,
! [A3: set_nat,B4: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( member_nat @ X6 @ B4 ) )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% subsetI
thf(fact_104_subset__antisym,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_105_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_106_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_107_empty__iff,axiom,
! [C: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_108_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_109_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_110_all__not__in__conv,axiom,
! [A3: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_111_all__not__in__conv,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A3 ) )
= ( A3 = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_112_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_113_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_114_Collect__empty__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: relational_fmla_a_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_115_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_116_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_117_empty__Collect__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X3: relational_fmla_a_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_118_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_119_fresh2__gt_I1_J,axiom,
! [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( ord_less_nat @ X @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q2 ) ) ).
% fresh2_gt(1)
thf(fact_120_fresh2__gt_I2_J,axiom,
! [Y: nat,X: nat,Q2: relational_fmla_a_b] : ( ord_less_nat @ Y @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q2 ) ) ).
% fresh2_gt(2)
thf(fact_121_fv_Osimps_I2_J,axiom,
! [B: $o] :
( ( relational_fv_a_b @ ( relational_Bool_a_b @ B ) )
= bot_bot_set_nat ) ).
% fv.simps(2)
thf(fact_122_fresh2__gt_I3_J,axiom,
! [Z: nat,Q2: relational_fmla_a_b,X: nat,Y: nat] :
( ( member_nat @ Z @ ( relational_fv_a_b @ Q2 ) )
=> ( ord_less_nat @ Z @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q2 ) ) ) ).
% fresh2_gt(3)
thf(fact_123_gen_H_Ointros_I1_J,axiom,
! [X: nat] : ( relational_gen_a_b2 @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen'.intros(1)
thf(fact_124_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_125_ex__in__conv,axiom,
! [A3: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_126_ex__in__conv,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( ? [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 ) )
= ( A3 != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_127_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_128_equals0I,axiom,
! [A3: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_129_equals0I,axiom,
! [A3: set_list_a] :
( ! [Y5: list_a] :
~ ( member_list_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_130_equals0I,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ! [Y5: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y5 @ A3 )
=> ( A3 = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_131_equals0I,axiom,
! [A3: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_132_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_133_equals0D,axiom,
! [A3: set_list_a,A2: list_a] :
( ( A3 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_134_equals0D,axiom,
! [A3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( A3 = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A2 @ A3 ) ) ).
% equals0D
thf(fact_135_equals0D,axiom,
! [A3: set_nat,A2: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A3 ) ) ).
% equals0D
thf(fact_136_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_137_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_138_emptyE,axiom,
! [A2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_139_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_140_Collect__mono__iff,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_141_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [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_142_subset__trans,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_143_Collect__mono,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X6: nat] :
( ( P @ X6 )
=> ( Q2 @ X6 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_144_subset__refl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_145_subset__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
! [T2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ T2 @ A5 )
=> ( member4680049679412964150la_a_b @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_146_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_147_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_148_Set_OequalityD2,axiom,
! [A3: set_nat,B4: set_nat] :
( ( A3 = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A3 ) ) ).
% Set.equalityD2
thf(fact_149_equalityD1,axiom,
! [A3: set_nat,B4: set_nat] :
( ( A3 = B4 )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_150_subset__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A5 )
=> ( member4680049679412964150la_a_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_151_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_152_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_153_equalityE,axiom,
! [A3: set_nat,B4: set_nat] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_154_subsetD,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ B4 )
=> ( ( member4680049679412964150la_a_b @ C @ A3 )
=> ( member4680049679412964150la_a_b @ C @ B4 ) ) ) ).
% subsetD
thf(fact_155_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_156_subsetD,axiom,
! [A3: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_157_in__mono,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ B4 )
=> ( ( member4680049679412964150la_a_b @ X @ A3 )
=> ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ).
% in_mono
thf(fact_158_in__mono,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_159_in__mono,axiom,
! [A3: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_160_psubset__trans,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ( ( ord_less_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_161_psubsetD,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le7152733262289451305la_a_b @ A3 @ B4 )
=> ( ( member4680049679412964150la_a_b @ C @ A3 )
=> ( member4680049679412964150la_a_b @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_162_psubsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_163_psubsetD,axiom,
! [A3: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_164_not__psubset__empty,axiom,
! [A3: set_list_a] :
~ ( ord_less_set_list_a @ A3 @ bot_bot_set_list_a ) ).
% not_psubset_empty
thf(fact_165_not__psubset__empty,axiom,
! [A3: set_Re381260168593705685la_a_b] :
~ ( ord_le7152733262289451305la_a_b @ A3 @ bot_bo4495933725496725865la_a_b ) ).
% not_psubset_empty
thf(fact_166_not__psubset__empty,axiom,
! [A3: set_nat] :
~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_167_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_168_subset__psubset__trans,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_169_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_170_psubset__subset__trans,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_171_psubset__imp__subset,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_172_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_173_psubsetE,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_174_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_175_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_176_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_177_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_178_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_179_arg__min__if__finite_I2_J,axiom,
! [S: set_a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ~ ? [X7: a] :
( ( member_a @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_180_arg__min__if__finite_I2_J,axiom,
! [S: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( S != bot_bot_set_list_a )
=> ~ ? [X7: list_a] :
( ( member_list_a @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic5043722365632780795_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_181_arg__min__if__finite_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( S != bot_bo4495933725496725865la_a_b )
=> ~ ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic5380700691367270794_b_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_182_arg__min__if__finite_I2_J,axiom,
! [S: set_a,F: a > set_nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ~ ? [X7: a] :
( ( member_a @ X7 @ S )
& ( ord_less_set_nat @ ( F @ X7 ) @ ( F @ ( lattic6146202810820454443et_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_183_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X7: nat] :
( ( member_nat @ X7 @ S )
& ( ord_less_set_nat @ ( F @ X7 ) @ ( F @ ( lattic1474851849222904177et_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_184_arg__min__if__finite_I2_J,axiom,
! [S: set_list_a,F: list_a > set_nat] :
( ( finite_finite_list_a @ S )
=> ( ( S != bot_bot_set_list_a )
=> ~ ? [X7: list_a] :
( ( member_list_a @ X7 @ S )
& ( ord_less_set_nat @ ( F @ X7 ) @ ( F @ ( lattic549851495318985137et_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_185_arg__min__if__finite_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( S != bot_bo4495933725496725865la_a_b )
=> ~ ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ S )
& ( ord_less_set_nat @ ( F @ X7 ) @ ( F @ ( lattic5577778279351311680et_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_186_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X7: nat] :
( ( member_nat @ X7 @ S )
& ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_187_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_188_arg__min__least,axiom,
! [S: set_list_a,Y: list_a,F: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( S != bot_bot_set_list_a )
=> ( ( member_list_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5043722365632780795_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_189_arg__min__least,axiom,
! [S: set_Re381260168593705685la_a_b,Y: relational_fmla_a_b,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( S != bot_bo4495933725496725865la_a_b )
=> ( ( member4680049679412964150la_a_b @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5380700691367270794_b_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_190_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_191_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X6: a] :
~ ( member_a @ X6 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_192_subset__emptyI,axiom,
! [A3: set_list_a] :
( ! [X6: list_a] :
~ ( member_list_a @ X6 @ A3 )
=> ( ord_le8861187494160871172list_a @ A3 @ bot_bot_set_list_a ) ) ).
% subset_emptyI
thf(fact_193_subset__emptyI,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ! [X6: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X6 @ A3 )
=> ( ord_le4112832032246704949la_a_b @ A3 @ bot_bo4495933725496725865la_a_b ) ) ).
% subset_emptyI
thf(fact_194_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X6: nat] :
~ ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_195_bot_Onot__eq__extremum,axiom,
! [A2: set_list_a] :
( ( A2 != bot_bot_set_list_a )
= ( ord_less_set_list_a @ bot_bot_set_list_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_196_bot_Onot__eq__extremum,axiom,
! [A2: nat > $o] :
( ( A2 != bot_bot_nat_o )
= ( ord_less_nat_o @ bot_bot_nat_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_197_bot_Onot__eq__extremum,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
= ( ord_le7152733262289451305la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_198_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_199_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_200_bot_Oextremum__strict,axiom,
! [A2: set_list_a] :
~ ( ord_less_set_list_a @ A2 @ bot_bot_set_list_a ) ).
% bot.extremum_strict
thf(fact_201_bot_Oextremum__strict,axiom,
! [A2: nat > $o] :
~ ( ord_less_nat_o @ A2 @ bot_bot_nat_o ) ).
% bot.extremum_strict
thf(fact_202_bot_Oextremum__strict,axiom,
! [A2: set_Re381260168593705685la_a_b] :
~ ( ord_le7152733262289451305la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ).
% bot.extremum_strict
thf(fact_203_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_204_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_205_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
=> ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_206_bot_Oextremum__uniqueI,axiom,
! [A2: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ bot_bot_nat_o )
=> ( A2 = bot_bot_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_207_bot_Oextremum__uniqueI,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
=> ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_208_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_209_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_210_cpropagated__simps_I1_J,axiom,
! [B: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B ) ) ).
% cpropagated_simps(1)
thf(fact_211_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_212_bot__set__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ bot_bo8852203127187332700_a_b_o ) ) ).
% bot_set_def
thf(fact_213_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_214_cpropagated__cp,axiom,
! [Q2: relational_fmla_a_b] : ( relati1591879772219623554ed_a_b @ ( relational_cp_a_b @ Q2 ) ) ).
% cpropagated_cp
thf(fact_215_cpropagated__cp__triv,axiom,
! [Q2: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q2 )
=> ( ( relational_cp_a_b @ Q2 )
= Q2 ) ) ).
% cpropagated_cp_triv
thf(fact_216_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_217_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_218_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_219_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_220_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_221_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_222_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_223_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_224_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_225_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_226_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_227_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_228_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_229_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_230_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_231_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_232_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ A @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_233_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_234_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_235_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_236_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_237_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_238_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_239_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_240_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_241_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_243_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_244_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_245_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_246_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_247_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_248_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_249_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_250_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_251_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_252_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_257_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_258_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_259_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_260_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_261_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_262_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_263_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_264_less__imp__neq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_265_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_266_order_Oasym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ord_less_set_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_267_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_268_ord__eq__less__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_269_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_270_ord__less__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_271_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_272_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X6: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X6 )
=> ( P @ Y7 ) )
=> ( P @ X6 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_273_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_274_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_275_dual__order_Oasym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ~ ( ord_less_set_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_276_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_277_dual__order_Oirrefl,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_278_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_279_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_280_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_281_order_Ostrict__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_282_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_283_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_284_dual__order_Ostrict__trans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_285_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_286_order_Ostrict__implies__not__eq,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_287_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_288_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_289_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_290_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_291_order__less__asym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_292_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_293_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_294_order__less__asym_H,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ord_less_set_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_295_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_296_order__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_297_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_298_ord__eq__less__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_299_ord__eq__less__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_300_ord__eq__less__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_301_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_302_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_303_ord__less__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_304_ord__less__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_305_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_306_order__less__irrefl,axiom,
! [X: set_nat] :
~ ( ord_less_set_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_307_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_308_order__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_309_order__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_310_order__less__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_311_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_312_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_313_order__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_314_order__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_315_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_316_order__less__not__sym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_317_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_318_order__less__imp__triv,axiom,
! [X: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_319_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_320_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_321_order__less__imp__not__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_322_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_323_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_324_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_325_order__less__imp__not__less,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_326_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_327_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_328_arg__min__if__finite_I1_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( member_nat @ ( lattic7446932960582359483at_nat @ F @ S ) @ S ) ) ) ).
% arg_min_if_finite(1)
thf(fact_329_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_330_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_331_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_332_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_333_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_334_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_335_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_336_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_337_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_338_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_339_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_340_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_341_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
| ( A = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_342_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
| ( A = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_343_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
& ( A != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_344_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
& ( A != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_345_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_346_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_347_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_348_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_349_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_350_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_351_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
| ( A = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_352_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
| ( A = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_353_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
& ( A != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_354_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
& ( A != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_357_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_358_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_359_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
& ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_360_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
& ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_361_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_362_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_363_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_364_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_365_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_366_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_367_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_368_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_369_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_370_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_371_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_372_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_373_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_374_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_375_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_376_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_377_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_378_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_379_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_380_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_381_order__le__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_382_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_383_order__le__less__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_384_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_385_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_386_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_387_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_388_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_389_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_390_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_391_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_392_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_393_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_394_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_395_order__less__le__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X6: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X6 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_396_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_397_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_398_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_399_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_400_bot_Oextremum,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% bot.extremum
thf(fact_401_bot_Oextremum,axiom,
! [A2: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A2 ) ).
% bot.extremum
thf(fact_402_bot_Oextremum,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ).
% bot.extremum
thf(fact_403_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_404_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_405_bot_Oextremum__unique,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_406_bot_Oextremum__unique,axiom,
! [A2: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ bot_bot_nat_o )
= ( A2 = bot_bot_nat_o ) ) ).
% bot.extremum_unique
thf(fact_407_bot_Oextremum__unique,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_unique
thf(fact_408_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_409_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_410_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N2: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ N4 )
=> ( ord_less_nat @ X6 @ N2 ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_411_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_412_minf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_eq_nat @ T3 @ X7 ) ) ).
% minf(8)
thf(fact_413_minf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_eq_nat @ X7 @ T3 ) ) ).
% minf(6)
thf(fact_414_pinf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_eq_nat @ T3 @ X7 ) ) ).
% pinf(8)
thf(fact_415_pinf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T3 ) ) ).
% pinf(6)
thf(fact_416_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_417_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A2 @ X7 )
& ( ord_less_nat @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D2: nat] :
( ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ D2 ) )
=> ( P @ X6 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_418_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_419_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_420_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_421_verit__comp__simplify1_I1_J,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_422_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_423_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( Q2 @ X6 )
= ( Q3 @ X6 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P4 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_424_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( Q2 @ X6 )
= ( Q3 @ X6 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P4 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_425_pinf_I3_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T3 ) ) ).
% pinf(3)
thf(fact_426_pinf_I4_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T3 ) ) ).
% pinf(4)
thf(fact_427_pinf_I5_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T3 ) ) ).
% pinf(5)
thf(fact_428_pinf_I7_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_nat @ T3 @ X7 ) ) ).
% pinf(7)
thf(fact_429_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( Q2 @ X6 )
= ( Q3 @ X6 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P4 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_430_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( Q2 @ X6 )
= ( Q3 @ X6 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P4 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_431_minf_I3_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T3 ) ) ).
% minf(3)
thf(fact_432_minf_I4_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T3 ) ) ).
% minf(4)
thf(fact_433_minf_I5_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_nat @ X7 @ T3 ) ) ).
% minf(5)
thf(fact_434_minf_I7_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_nat @ T3 @ X7 ) ) ).
% minf(7)
thf(fact_435_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_436_nongens__cp,axiom,
! [Q2: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relati62690040636126068ns_a_b @ ( relational_cp_a_b @ Q2 ) ) @ ( relati62690040636126068ns_a_b @ Q2 ) ) ).
% nongens_cp
thf(fact_437_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_438_bot__empty__eq,axiom,
( bot_bot_list_a_o
= ( ^ [X3: list_a] : ( member_list_a @ X3 @ bot_bot_set_list_a ) ) ) ).
% bot_empty_eq
thf(fact_439_bot__empty__eq,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_440_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_441_Collect__empty__eq__bot,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( P = bot_bot_list_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_442_Collect__empty__eq__bot,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( P = bot_bo8852203127187332700_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_443_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_444_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K @ I2 )
=> ( P @ I2 ) )
=> ( P @ K ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_445_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_446_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_447_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_448_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_449_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_450_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_451_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_452_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_453_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_454_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_455_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_456_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_457_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_458_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_459_Set_Ois__empty__def,axiom,
( is_empty_list_a
= ( ^ [A5: set_list_a] : ( A5 = bot_bot_set_list_a ) ) ) ).
% Set.is_empty_def
thf(fact_460_Set_Ois__empty__def,axiom,
( is_emp6953259385542938189la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b] : ( A5 = bot_bo4495933725496725865la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_461_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_462_sr__def,axiom,
( relational_sr_a_b
= ( ^ [Q: relational_fmla_a_b] :
( ( ( relati62690040636126068ns_a_b @ Q )
= bot_bot_set_nat )
& ( relational_rrb_a_b @ Q ) ) ) ) ).
% sr_def
thf(fact_463_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_464_Inf__fin_Osubset__imp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_465_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_466_Sup__fin_Osubset__imp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_467_Max__mono,axiom,
! [M4: set_nat,N4: set_nat] :
( ( ord_less_eq_set_nat @ M4 @ N4 )
=> ( ( M4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N4 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M4 ) @ ( lattic8265883725875713057ax_nat @ N4 ) ) ) ) ) ).
% Max_mono
thf(fact_468_DiffI,axiom,
! [C: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A3 )
=> ( ~ ( member4680049679412964150la_a_b @ C @ B4 )
=> ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_469_DiffI,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_470_DiffI,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_471_Diff__iff,axiom,
! [C: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) )
= ( ( member4680049679412964150la_a_b @ C @ A3 )
& ~ ( member4680049679412964150la_a_b @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_472_Diff__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_473_Diff__iff,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_474_Diff__idemp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ B4 )
= ( minus_minus_set_nat @ A3 @ B4 ) ) ).
% Diff_idemp
thf(fact_475_Diff__cancel,axiom,
! [A3: set_list_a] :
( ( minus_646659088055828811list_a @ A3 @ A3 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_476_Diff__cancel,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A3 @ A3 )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_cancel
thf(fact_477_Diff__cancel,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ A3 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_478_empty__Diff,axiom,
! [A3: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A3 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_479_empty__Diff,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ bot_bo4495933725496725865la_a_b @ A3 )
= bot_bo4495933725496725865la_a_b ) ).
% empty_Diff
thf(fact_480_empty__Diff,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_481_Diff__empty,axiom,
! [A3: set_list_a] :
( ( minus_646659088055828811list_a @ A3 @ bot_bot_set_list_a )
= A3 ) ).
% Diff_empty
thf(fact_482_Diff__empty,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A3 @ bot_bo4495933725496725865la_a_b )
= A3 ) ).
% Diff_empty
thf(fact_483_Diff__empty,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Diff_empty
thf(fact_484_finite__Diff2,axiom,
! [B4: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B4 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B4 ) )
= ( finite_finite_list_a @ A3 ) ) ) ).
% finite_Diff2
thf(fact_485_finite__Diff2,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( finite_finite_a @ A3 ) ) ) ).
% finite_Diff2
thf(fact_486_finite__Diff2,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( finite_finite_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_487_finite__Diff,axiom,
! [A3: set_list_a,B4: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_488_finite__Diff,axiom,
! [A3: set_a,B4: set_a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_489_finite__Diff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_490_rrb__simps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( ( relational_rrb_a_b @ Q1 )
& ( relational_rrb_a_b @ Q22 ) ) ) ).
% rrb_simps(5)
thf(fact_491_rrb__simps_I4_J,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Neg_a_b @ Q2 ) )
= ( relational_rrb_a_b @ Q2 ) ) ).
% rrb_simps(4)
thf(fact_492_rrb__simps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
= ( ( relational_rrb_a_b @ Q1 )
& ( relational_rrb_a_b @ Q22 ) ) ) ).
% rrb_simps(6)
thf(fact_493_rrb__simps_I1_J,axiom,
! [B: $o] : ( relational_rrb_a_b @ ( relational_Bool_a_b @ B ) ) ).
% rrb_simps(1)
thf(fact_494_Diff__eq__empty__iff,axiom,
! [A3: set_list_a,B4: set_list_a] :
( ( ( minus_646659088055828811list_a @ A3 @ B4 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_495_Diff__eq__empty__iff,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ A3 @ B4 )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_496_Diff__eq__empty__iff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_497_Max_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_498_Max__less__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_499_DiffE,axiom,
! [C: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) )
=> ~ ( ( member4680049679412964150la_a_b @ C @ A3 )
=> ( member4680049679412964150la_a_b @ C @ B4 ) ) ) ).
% DiffE
thf(fact_500_DiffE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_501_DiffE,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_502_DiffD1,axiom,
! [C: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) )
=> ( member4680049679412964150la_a_b @ C @ A3 ) ) ).
% DiffD1
thf(fact_503_DiffD1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_504_DiffD1,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_505_DiffD2,axiom,
! [C: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) )
=> ~ ( member4680049679412964150la_a_b @ C @ B4 ) ) ).
% DiffD2
thf(fact_506_DiffD2,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_507_DiffD2,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_508_Sup__fin__Max,axiom,
lattic1093996805478795353in_nat = lattic8265883725875713057ax_nat ).
% Sup_fin_Max
thf(fact_509_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_510_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_511_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_512_double__diff,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_513_Diff__subset,axiom,
! [A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_514_Diff__mono,axiom,
! [A3: set_nat,C2: set_nat,D3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( ord_less_eq_set_nat @ D3 @ B4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_515_psubset__imp__ex__mem,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A3 @ B4 )
=> ? [B6: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ B6 @ ( minus_4077726661957047470la_a_b @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_516_psubset__imp__ex__mem,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_517_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_518_rrb__cp,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_rrb_a_b @ Q2 )
=> ( relational_rrb_a_b @ ( relational_cp_a_b @ Q2 ) ) ) ).
% rrb_cp
thf(fact_519_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A3 ) @ ( lattic3835124923745554447et_nat @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_520_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_521_Max_OcoboundedI,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ord_less_eq_nat @ A2 @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ).
% Max.coboundedI
thf(fact_522_Max__eq__if,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( ord_less_eq_nat @ X6 @ Xa ) ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A3 )
& ( ord_less_eq_nat @ X6 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A3 )
= ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_523_Max__eqI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ A3 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( lattic8265883725875713057ax_nat @ A3 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_524_Max__ge,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ).
% Max_ge
thf(fact_525_Max__in,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ A3 ) ) ) ).
% Max_in
thf(fact_526_Sup__fin_OcoboundedI,axiom,
! [A3: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( member_set_nat @ A2 @ A3 )
=> ( ord_less_eq_set_nat @ A2 @ ( lattic3835124923745554447et_nat @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_527_Sup__fin_OcoboundedI,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_528_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( member_set_nat @ A2 @ A3 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A3 ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_529_Inf__fin_OcoboundedI,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_530_Max_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ( ord_less_eq_nat @ A6 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_531_Max_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_532_eq__Max__iff,axiom,
! [A3: set_nat,M: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( M
= ( lattic8265883725875713057ax_nat @ A3 ) )
= ( ( member_nat @ M @ A3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_533_Max__ge__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_534_Max__eq__iff,axiom,
! [A3: set_nat,M: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A3 )
= M )
= ( ( member_nat @ M @ A3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_535_Max__gr__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_536_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ X )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( ord_less_eq_set_nat @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_537_Sup__fin_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_538_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( ord_less_eq_set_nat @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_539_Inf__fin_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_540_Sup__fin_OboundedI,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A3 )
=> ( ord_less_eq_set_nat @ A6 @ X ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_541_Sup__fin_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ( ord_less_eq_nat @ A6 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_542_Sup__fin_OboundedE,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ X )
=> ! [A8: set_nat] :
( ( member_set_nat @ A8 @ A3 )
=> ( ord_less_eq_set_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_543_Sup__fin_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_544_Inf__fin_OboundedI,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A3 )
=> ( ord_less_eq_set_nat @ X @ A6 ) )
=> ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_545_Inf__fin_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ( ord_less_eq_nat @ X @ A6 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_546_Inf__fin_OboundedE,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( A3 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) )
=> ! [A8: set_nat] :
( ( member_set_nat @ A8 @ A3 )
=> ( ord_less_eq_set_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_547_Inf__fin_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_548_Max_Osubset__imp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_549_diff__shunt__var,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( minus_646659088055828811list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_550_diff__shunt__var,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ( minus_minus_nat_o @ X @ Y )
= bot_bot_nat_o )
= ( ord_less_eq_nat_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_551_diff__shunt__var,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_552_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_553_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_554_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_555_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_556_less__diff__iff,axiom,
! [K2: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_557_fv__exists,axiom,
! [X: nat,Q2: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q2 ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Q2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% fv_exists
thf(fact_558_psubset__insert__iff,axiom,
! [A3: set_a,X: a,B4: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ( ( member_a @ X @ B4 )
=> ( ord_less_set_a @ A3 @ B4 ) )
& ( ~ ( member_a @ X @ B4 )
=> ( ( ( member_a @ X @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_559_psubset__insert__iff,axiom,
! [A3: set_list_a,X: list_a,B4: set_list_a] :
( ( ord_less_set_list_a @ A3 @ ( insert_list_a @ X @ B4 ) )
= ( ( ( member_list_a @ X @ B4 )
=> ( ord_less_set_list_a @ A3 @ B4 ) )
& ( ~ ( member_list_a @ X @ B4 )
=> ( ( ( member_list_a @ X @ A3 )
=> ( ord_less_set_list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B4 ) )
& ( ~ ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_560_psubset__insert__iff,axiom,
! [A3: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ B4 ) )
= ( ( ( member4680049679412964150la_a_b @ X @ B4 )
=> ( ord_le7152733262289451305la_a_b @ A3 @ B4 ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B4 )
=> ( ( ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ord_le7152733262289451305la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B4 ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ord_le4112832032246704949la_a_b @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_561_psubset__insert__iff,axiom,
! [A3: set_nat,X: nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ B4 )
=> ( ord_less_set_nat @ A3 @ B4 ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( ( member_nat @ X @ A3 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_562_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T4: set_a] :
( ( ord_less_set_a @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X7: a] :
( ( member_a @ X7 @ ( minus_minus_set_a @ S @ T4 ) )
& ( P @ ( insert_a @ X7 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_563_finite__induct__select,axiom,
! [S: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ S )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [T4: set_list_a] :
( ( ord_less_set_list_a @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X7: list_a] :
( ( member_list_a @ X7 @ ( minus_646659088055828811list_a @ S @ T4 ) )
& ( P @ ( insert_list_a @ X7 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_564_finite__induct__select,axiom,
! [S: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [T4: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ ( minus_4077726661957047470la_a_b @ S @ T4 ) )
& ( P @ ( insert7010464514620295119la_a_b @ X7 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_565_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ ( minus_minus_set_nat @ S @ T4 ) )
& ( P @ ( insert_nat @ X7 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_566_Min__antimono,axiom,
! [M4: set_nat,N4: set_nat] :
( ( ord_less_eq_set_nat @ M4 @ N4 )
=> ( ( M4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N4 ) @ ( lattic8721135487736765967in_nat @ M4 ) ) ) ) ) ).
% Min_antimono
thf(fact_567_Min_Osubset__imp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_568_remove__induct,axiom,
! [P: set_a > $o,B4: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B4 )
=> ( P @ B4 ) )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ! [X7: a] :
( ( member_a @ X7 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X7 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_569_remove__induct,axiom,
! [P: set_list_a > $o,B4: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B4 )
=> ( P @ B4 ) )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( A4 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
=> ( ! [X7: list_a] :
( ( member_list_a @ X7 @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X7 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_570_remove__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,B4: set_Re381260168593705685la_a_b] :
( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ( ~ ( finite5600759454172676150la_a_b @ B4 )
=> ( P @ B4 ) )
=> ( ! [A4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A4 )
=> ( ( A4 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A4 @ B4 )
=> ( ! [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ A4 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A4 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_571_remove__induct,axiom,
! [P: set_nat > $o,B4: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B4 )
=> ( P @ B4 ) )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_572_insertCI,axiom,
! [A2: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ B4 ) ) ) ).
% insertCI
thf(fact_573_insertCI,axiom,
! [A2: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_a @ A2 @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_574_insertCI,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_575_insert__iff,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ A3 ) )
= ( ( A2 = B )
| ( member4680049679412964150la_a_b @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_576_insert__iff,axiom,
! [A2: a,B: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A3 ) )
= ( ( A2 = B )
| ( member_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_577_insert__iff,axiom,
! [A2: nat,B: nat,A3: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A3 ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_578_insert__absorb2,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ X @ A3 ) )
= ( insert7010464514620295119la_a_b @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_579_insert__absorb2,axiom,
! [X: nat,A3: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A3 ) )
= ( insert_nat @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_580_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_581_singletonI,axiom,
! [A2: list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_582_singletonI,axiom,
! [A2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_583_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_584_finite__insert,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A3 ) )
= ( finite5600759454172676150la_a_b @ A3 ) ) ).
% finite_insert
thf(fact_585_finite__insert,axiom,
! [A2: list_a,A3: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A2 @ A3 ) )
= ( finite_finite_list_a @ A3 ) ) ).
% finite_insert
thf(fact_586_finite__insert,axiom,
! [A2: a,A3: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A3 ) )
= ( finite_finite_a @ A3 ) ) ).
% finite_insert
thf(fact_587_finite__insert,axiom,
! [A2: nat,A3: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) )
= ( finite_finite_nat @ A3 ) ) ).
% finite_insert
thf(fact_588_insert__subset,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ A3 ) @ B4 )
= ( ( member4680049679412964150la_a_b @ X @ B4 )
& ( ord_le4112832032246704949la_a_b @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_589_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_590_insert__subset,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( ( member_nat @ X @ B4 )
& ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_591_Diff__insert0,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ B4 ) )
= ( minus_4077726661957047470la_a_b @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_592_Diff__insert0,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_593_Diff__insert0,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ B4 ) )
= ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_594_insert__Diff1,axiom,
! [X: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ B4 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A3 ) @ B4 )
= ( minus_4077726661957047470la_a_b @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_595_insert__Diff1,axiom,
! [X: a,B4: set_a,A3: set_a] :
( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_596_insert__Diff1,axiom,
! [X: nat,B4: set_nat,A3: set_nat] :
( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_597_singleton__insert__inj__eq_H,axiom,
! [A2: list_a,A3: set_list_a,B: list_a] :
( ( ( insert_list_a @ A2 @ A3 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A2 = B )
& ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_598_singleton__insert__inj__eq_H,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ A3 )
= ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
= ( ( A2 = B )
& ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_599_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A3: set_nat,B: nat] :
( ( ( insert_nat @ A2 @ A3 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_600_singleton__insert__inj__eq,axiom,
! [B: list_a,A2: list_a,A3: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A2 @ A3 ) )
= ( ( A2 = B )
& ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_601_singleton__insert__inj__eq,axiom,
! [B: relational_fmla_a_b,A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ A2 @ A3 ) )
= ( ( A2 = B )
& ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_602_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A3: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A3 ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_603_insert__Diff__single,axiom,
! [A2: list_a,A3: set_list_a] :
( ( insert_list_a @ A2 @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_604_insert__Diff__single,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
= ( insert7010464514620295119la_a_b @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_605_insert__Diff__single,axiom,
! [A2: nat,A3: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_606_finite__Diff__insert,axiom,
! [A3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) ) )
= ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_607_finite__Diff__insert,axiom,
! [A3: set_list_a,A2: list_a,B4: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A2 @ B4 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_608_finite__Diff__insert,axiom,
! [A3: set_a,A2: a,B4: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ B4 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_609_finite__Diff__insert,axiom,
! [A3: set_nat,A2: nat,B4: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_610_Max__singleton,axiom,
! [X: nat] :
( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Max_singleton
thf(fact_611_Min__singleton,axiom,
! [X: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Min_singleton
thf(fact_612_Inf__fin_Osingleton,axiom,
! [X: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Inf_fin.singleton
thf(fact_613_Sup__fin_Osingleton,axiom,
! [X: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Sup_fin.singleton
thf(fact_614_Min_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_615_Min__gr__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_616_Min__insert2,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ A3 )
=> ( ord_less_eq_nat @ A2 @ B6 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A2 @ A3 ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_617_insertE,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ A3 ) )
=> ( ( A2 != B )
=> ( member4680049679412964150la_a_b @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_618_insertE,axiom,
! [A2: a,B: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A3 ) )
=> ( ( A2 != B )
=> ( member_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_619_insertE,axiom,
! [A2: nat,B: nat,A3: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A3 ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_620_insertI1,axiom,
! [A2: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) ) ).
% insertI1
thf(fact_621_insertI1,axiom,
! [A2: a,B4: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B4 ) ) ).
% insertI1
thf(fact_622_insertI1,axiom,
! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% insertI1
thf(fact_623_insertI2,axiom,
! [A2: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ B4 )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ B4 ) ) ) ).
% insertI2
thf(fact_624_insertI2,axiom,
! [A2: a,B4: set_a,B: a] :
( ( member_a @ A2 @ B4 )
=> ( member_a @ A2 @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_625_insertI2,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( member_nat @ A2 @ B4 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_626_Set_Oset__insert,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A3 )
=> ~ ! [B8: set_Re381260168593705685la_a_b] :
( ( A3
= ( insert7010464514620295119la_a_b @ X @ B8 ) )
=> ( member4680049679412964150la_a_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_627_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B8: set_a] :
( ( A3
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_628_Set_Oset__insert,axiom,
! [X: nat,A3: set_nat] :
( ( member_nat @ X @ A3 )
=> ~ ! [B8: set_nat] :
( ( A3
= ( insert_nat @ X @ B8 ) )
=> ( member_nat @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_629_insert__ident,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B4 )
=> ( ( ( insert7010464514620295119la_a_b @ X @ A3 )
= ( insert7010464514620295119la_a_b @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_630_insert__ident,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B4 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_631_insert__ident,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ A3 )
= ( insert_nat @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_632_insert__absorb,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A3 )
=> ( ( insert7010464514620295119la_a_b @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_633_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_634_insert__absorb,axiom,
! [A2: nat,A3: set_nat] :
( ( member_nat @ A2 @ A3 )
=> ( ( insert_nat @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_635_insert__eq__iff,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A3 )
=> ( ~ ( member4680049679412964150la_a_b @ B @ B4 )
=> ( ( ( insert7010464514620295119la_a_b @ A2 @ A3 )
= ( insert7010464514620295119la_a_b @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A3 = B4 ) )
& ( ( A2 != B )
=> ? [C4: set_Re381260168593705685la_a_b] :
( ( A3
= ( insert7010464514620295119la_a_b @ B @ C4 ) )
& ~ ( member4680049679412964150la_a_b @ B @ C4 )
& ( B4
= ( insert7010464514620295119la_a_b @ A2 @ C4 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_636_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A3 = B4 ) )
& ( ( A2 != B )
=> ? [C4: set_a] :
( ( A3
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B4
= ( insert_a @ A2 @ C4 ) )
& ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_637_insert__eq__iff,axiom,
! [A2: nat,A3: set_nat,B: nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A3 )
=> ( ~ ( member_nat @ B @ B4 )
=> ( ( ( insert_nat @ A2 @ A3 )
= ( insert_nat @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A3 = B4 ) )
& ( ( A2 != B )
=> ? [C4: set_nat] :
( ( A3
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B4
= ( insert_nat @ A2 @ C4 ) )
& ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_638_insert__commute,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ Y @ A3 ) )
= ( insert7010464514620295119la_a_b @ Y @ ( insert7010464514620295119la_a_b @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_639_insert__commute,axiom,
! [X: nat,Y: nat,A3: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A3 ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_640_mk__disjoint__insert,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A3 )
=> ? [B8: set_Re381260168593705685la_a_b] :
( ( A3
= ( insert7010464514620295119la_a_b @ A2 @ B8 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_641_mk__disjoint__insert,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ? [B8: set_a] :
( ( A3
= ( insert_a @ A2 @ B8 ) )
& ~ ( member_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_642_mk__disjoint__insert,axiom,
! [A2: nat,A3: set_nat] :
( ( member_nat @ A2 @ A3 )
=> ? [B8: set_nat] :
( ( A3
= ( insert_nat @ A2 @ B8 ) )
& ~ ( member_nat @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_643_singleton__inject,axiom,
! [A2: list_a,B: list_a] :
( ( ( insert_list_a @ A2 @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_644_singleton__inject,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_645_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_646_insert__not__empty,axiom,
! [A2: list_a,A3: set_list_a] :
( ( insert_list_a @ A2 @ A3 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_647_insert__not__empty,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A2 @ A3 )
!= bot_bo4495933725496725865la_a_b ) ).
% insert_not_empty
thf(fact_648_insert__not__empty,axiom,
! [A2: nat,A3: set_nat] :
( ( insert_nat @ A2 @ A3 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_649_doubleton__eq__iff,axiom,
! [A2: list_a,B: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_650_doubleton__eq__iff,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b,C: relational_fmla_a_b,D: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_651_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_652_singleton__iff,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_653_singleton__iff,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_654_singleton__iff,axiom,
! [B: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_655_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_656_singletonD,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_657_singletonD,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_658_singletonD,axiom,
! [B: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_659_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_660_finite_OinsertI,axiom,
! [A3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A3 )
=> ( finite5600759454172676150la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_661_finite_OinsertI,axiom,
! [A3: set_list_a,A2: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( finite_finite_list_a @ ( insert_list_a @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_662_finite_OinsertI,axiom,
! [A3: set_a,A2: a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_a @ ( insert_a @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_663_finite_OinsertI,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_664_subset__insertI2,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ B4 )
=> ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_665_subset__insertI2,axiom,
! [A3: set_nat,B4: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_666_subset__insertI,axiom,
! [B4: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ B4 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_667_subset__insertI,axiom,
! [B4: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_668_subset__insert,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ B4 ) )
= ( ord_le4112832032246704949la_a_b @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_669_subset__insert,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_670_subset__insert,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B4 ) )
= ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_671_insert__mono,axiom,
! [C2: set_Re381260168593705685la_a_b,D3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ D3 )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ C2 ) @ ( insert7010464514620295119la_a_b @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_672_insert__mono,axiom,
! [C2: set_nat,D3: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_673_insert__subsetI,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ( ord_le4112832032246704949la_a_b @ X5 @ A3 )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_674_insert__subsetI,axiom,
! [X: a,A3: set_a,X5: set_a] :
( ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X5 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_675_insert__subsetI,axiom,
! [X: nat,A3: set_nat,X5: set_nat] :
( ( member_nat @ X @ A3 )
=> ( ( ord_less_eq_set_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_676_insert__Diff__if,axiom,
! [X: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ B4 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A3 ) @ B4 )
= ( minus_4077726661957047470la_a_b @ A3 @ B4 ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B4 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A3 ) @ B4 )
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_677_insert__Diff__if,axiom,
! [X: a,B4: set_a,A3: set_a] :
( ( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) )
& ( ~ ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( insert_a @ X @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_678_insert__Diff__if,axiom,
! [X: nat,B4: set_nat,A3: set_nat] :
( ( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( minus_minus_set_nat @ A3 @ B4 ) ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_679_Inf__fin__Min,axiom,
lattic5238388535129920115in_nat = lattic8721135487736765967in_nat ).
% Inf_fin_Min
thf(fact_680_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A4: set_a] :
( ? [A6: a] :
( A2
= ( insert_a @ A6 @ A4 ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_681_finite_Ocases,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( A2 != bot_bot_set_list_a )
=> ~ ! [A4: set_list_a] :
( ? [A6: list_a] :
( A2
= ( insert_list_a @ A6 @ A4 ) )
=> ~ ( finite_finite_list_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_682_finite_Ocases,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ~ ! [A4: set_Re381260168593705685la_a_b] :
( ? [A6: relational_fmla_a_b] :
( A2
= ( insert7010464514620295119la_a_b @ A6 @ A4 ) )
=> ~ ( finite5600759454172676150la_a_b @ A4 ) ) ) ) ).
% finite.cases
thf(fact_683_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A4: set_nat] :
( ? [A6: nat] :
( A2
= ( insert_nat @ A6 @ A4 ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_684_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A: set_a] :
( ( A = bot_bot_set_a )
| ? [A5: set_a,B2: a] :
( ( A
= ( insert_a @ B2 @ A5 ) )
& ( finite_finite_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_685_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A: set_list_a] :
( ( A = bot_bot_set_list_a )
| ? [A5: set_list_a,B2: list_a] :
( ( A
= ( insert_list_a @ B2 @ A5 ) )
& ( finite_finite_list_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_686_finite_Osimps,axiom,
( finite5600759454172676150la_a_b
= ( ^ [A: set_Re381260168593705685la_a_b] :
( ( A = bot_bo4495933725496725865la_a_b )
| ? [A5: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ B2 @ A5 ) )
& ( finite5600759454172676150la_a_b @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_687_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A: set_nat] :
( ( A = bot_bot_set_nat )
| ? [A5: set_nat,B2: nat] :
( ( A
= ( insert_nat @ B2 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_688_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X6 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_689_finite__induct,axiom,
! [F3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ~ ( member_list_a @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ X6 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_690_finite__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X6: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ~ ( member4680049679412964150la_a_b @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X6 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_691_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X6 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_692_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X6: a] : ( P @ ( insert_a @ X6 @ bot_bot_set_a ) )
=> ( ! [X6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_693_finite__ne__induct,axiom,
! [F3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ! [X6: list_a] : ( P @ ( insert_list_a @ X6 @ bot_bot_set_list_a ) )
=> ( ! [X6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( F4 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ X6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_694_finite__ne__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( F3 != bot_bo4495933725496725865la_a_b )
=> ( ! [X6: relational_fmla_a_b] : ( P @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) )
=> ( ! [X6: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( F4 != bot_bo4495933725496725865la_a_b )
=> ( ~ ( member4680049679412964150la_a_b @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_695_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X6: nat] : ( P @ ( insert_nat @ X6 @ bot_bot_set_nat ) )
=> ( ! [X6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_696_infinite__finite__induct,axiom,
! [P: set_a > $o,A3: set_a] :
( ! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X6 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_697_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A3: set_list_a] :
( ! [A4: set_list_a] :
( ~ ( finite_finite_list_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ~ ( member_list_a @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ X6 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_698_infinite__finite__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A3: set_Re381260168593705685la_a_b] :
( ! [A4: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X6: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ~ ( member4680049679412964150la_a_b @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X6 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_699_infinite__finite__induct,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X6 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_700_subset__singleton__iff,axiom,
! [X5: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_701_subset__singleton__iff,axiom,
! [X5: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( ( X5 = bot_bo4495933725496725865la_a_b )
| ( X5
= ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_702_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_703_subset__singletonD,axiom,
! [A3: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A3 = bot_bot_set_list_a )
| ( A3
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_704_subset__singletonD,axiom,
! [A3: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
=> ( ( A3 = bot_bo4495933725496725865la_a_b )
| ( A3
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_705_subset__singletonD,axiom,
! [A3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A3 = bot_bot_set_nat )
| ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_706_Min_OcoboundedI,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ A2 ) ) ) ).
% Min.coboundedI
thf(fact_707_Min__eqI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ A3 )
=> ( ord_less_eq_nat @ X @ Y5 ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( lattic8721135487736765967in_nat @ A3 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_708_Min__le,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X ) ) ) ).
% Min_le
thf(fact_709_Diff__insert__absorb,axiom,
! [X: a,A3: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_710_Diff__insert__absorb,axiom,
! [X: list_a,A3: set_list_a] :
( ~ ( member_list_a @ X @ A3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A3 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_711_Diff__insert__absorb,axiom,
! [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A3 ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_712_Diff__insert__absorb,axiom,
! [X: nat,A3: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_713_Diff__insert2,axiom,
! [A3: set_list_a,A2: list_a,B4: set_list_a] :
( ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A2 @ B4 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_714_Diff__insert2,axiom,
! [A3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_715_Diff__insert2,axiom,
! [A3: set_nat,A2: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_716_insert__Diff,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_717_insert__Diff,axiom,
! [A2: list_a,A3: set_list_a] :
( ( member_list_a @ A2 @ A3 )
=> ( ( insert_list_a @ A2 @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_718_insert__Diff,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A3 )
=> ( ( insert7010464514620295119la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_719_insert__Diff,axiom,
! [A2: nat,A3: set_nat] :
( ( member_nat @ A2 @ A3 )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_720_Diff__insert,axiom,
! [A3: set_list_a,A2: list_a,B4: set_list_a] :
( ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A2 @ B4 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A3 @ B4 ) @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_721_Diff__insert,axiom,
! [A3: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ B4 ) @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% Diff_insert
thf(fact_722_Diff__insert,axiom,
! [A3: set_nat,A2: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_723_Min__in,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ A3 ) ) ) ).
% Min_in
thf(fact_724_subset__Diff__insert,axiom,
! [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ ( minus_4077726661957047470la_a_b @ B4 @ ( insert7010464514620295119la_a_b @ X @ C2 ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A3 @ ( minus_4077726661957047470la_a_b @ B4 @ C2 ) )
& ~ ( member4680049679412964150la_a_b @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_725_subset__Diff__insert,axiom,
! [A3: set_a,B4: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B4 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B4 @ C2 ) )
& ~ ( member_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_726_subset__Diff__insert,axiom,
! [A3: set_nat,B4: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ C2 ) )
& ~ ( member_nat @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_727_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X6: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y7: a] :
( ( member_a @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X6 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_728_finite__ranking__induct,axiom,
! [S: set_list_a,P: set_list_a > $o,F: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X6: list_a,S3: set_list_a] :
( ( finite_finite_list_a @ S3 )
=> ( ! [Y7: list_a] :
( ( member_list_a @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_list_a @ X6 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_729_finite__ranking__induct,axiom,
! [S: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X6: relational_fmla_a_b,S3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ S3 )
=> ( ! [Y7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert7010464514620295119la_a_b @ X6 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_730_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X6: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X6 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X6 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_731_finite__linorder__min__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A4 )
=> ( ord_less_nat @ B6 @ X7 ) )
=> ( ( P @ A4 )
=> ( P @ ( insert_nat @ B6 @ A4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_732_finite__linorder__max__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A4 )
=> ( ord_less_nat @ X7 @ B6 ) )
=> ( ( P @ A4 )
=> ( P @ ( insert_nat @ B6 @ A4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_733_finite__subset__induct_H,axiom,
! [F3: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A6 @ A3 )
=> ( ( ord_less_eq_set_a @ F4 @ A3 )
=> ( ~ ( member_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_734_finite__subset__induct_H,axiom,
! [F3: set_list_a,A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( member_list_a @ A6 @ A3 )
=> ( ( ord_le8861187494160871172list_a @ F4 @ A3 )
=> ( ~ ( member_list_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_735_finite__subset__induct_H,axiom,
! [F3: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A3 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A6: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( member4680049679412964150la_a_b @ A6 @ A3 )
=> ( ( ord_le4112832032246704949la_a_b @ F4 @ A3 )
=> ( ~ ( member4680049679412964150la_a_b @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_736_finite__subset__induct_H,axiom,
! [F3: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A6 @ A3 )
=> ( ( ord_less_eq_set_nat @ F4 @ A3 )
=> ( ~ ( member_nat @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_737_finite__subset__induct,axiom,
! [F3: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A6 @ A3 )
=> ( ~ ( member_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_738_finite__subset__induct,axiom,
! [F3: set_list_a,A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( member_list_a @ A6 @ A3 )
=> ( ~ ( member_list_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_739_finite__subset__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A3 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A6: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( member4680049679412964150la_a_b @ A6 @ A3 )
=> ( ~ ( member4680049679412964150la_a_b @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_740_finite__subset__induct,axiom,
! [F3: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A6 @ A3 )
=> ( ~ ( member_nat @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_741_Min_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ( ord_less_eq_nat @ X @ A6 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).
% Min.boundedI
thf(fact_742_Min_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_743_eq__Min__iff,axiom,
! [A3: set_nat,M: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A3 ) )
= ( ( member_nat @ M @ A3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_744_Min__le__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_745_Min__eq__iff,axiom,
! [A3: set_nat,M: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A3 )
= M )
= ( ( member_nat @ M @ A3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_746_finite__empty__induct,axiom,
! [A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: a,A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_747_finite__empty__induct,axiom,
! [A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: list_a,A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( member_list_a @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ A6 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_748_finite__empty__induct,axiom,
! [A3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: relational_fmla_a_b,A4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A4 )
=> ( ( member4680049679412964150la_a_b @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A4 @ ( insert7010464514620295119la_a_b @ A6 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ( P @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% finite_empty_induct
thf(fact_749_finite__empty__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_750_infinite__coinduct,axiom,
! [X5: set_a > $o,A3: set_a] :
( ( X5 @ A3 )
=> ( ! [A4: set_a] :
( ( X5 @ A4 )
=> ? [X7: a] :
( ( member_a @ X7 @ A4 )
& ( ( X5 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X7 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X7 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_751_infinite__coinduct,axiom,
! [X5: set_list_a > $o,A3: set_list_a] :
( ( X5 @ A3 )
=> ( ! [A4: set_list_a] :
( ( X5 @ A4 )
=> ? [X7: list_a] :
( ( member_list_a @ X7 @ A4 )
& ( ( X5 @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X7 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X7 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_752_infinite__coinduct,axiom,
! [X5: set_Re381260168593705685la_a_b > $o,A3: set_Re381260168593705685la_a_b] :
( ( X5 @ A3 )
=> ( ! [A4: set_Re381260168593705685la_a_b] :
( ( X5 @ A4 )
=> ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ A4 )
& ( ( X5 @ ( minus_4077726661957047470la_a_b @ A4 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) )
| ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A4 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ~ ( finite5600759454172676150la_a_b @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_753_infinite__coinduct,axiom,
! [X5: set_nat > $o,A3: set_nat] :
( ( X5 @ A3 )
=> ( ! [A4: set_nat] :
( ( X5 @ A4 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A4 )
& ( ( X5 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_754_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_755_infinite__remove,axiom,
! [S: set_list_a,A2: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_756_infinite__remove,axiom,
! [S: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ S @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% infinite_remove
thf(fact_757_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_758_Min__less__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_759_Diff__single__insert,axiom,
! [A3: set_list_a,X: list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B4 )
=> ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_760_Diff__single__insert,axiom,
! [A3: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B4 )
=> ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_761_Diff__single__insert,axiom,
! [A3: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_762_subset__insert__iff,axiom,
! [A3: set_a,X: a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_763_subset__insert__iff,axiom,
! [A3: set_list_a,X: list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ B4 ) )
= ( ( ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B4 ) )
& ( ~ ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_764_subset__insert__iff,axiom,
! [A3: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ B4 ) )
= ( ( ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B4 ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A3 )
=> ( ord_le4112832032246704949la_a_b @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_765_subset__insert__iff,axiom,
! [A3: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_766_Max__insert2,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ A3 )
=> ( ord_less_eq_nat @ B6 @ A2 ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A2 @ A3 ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_767_finite__remove__induct,axiom,
! [B4: set_a,P: set_a > $o] :
( ( finite_finite_a @ B4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ! [X7: a] :
( ( member_a @ X7 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X7 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_768_finite__remove__induct,axiom,
! [B4: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B4 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( A4 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
=> ( ! [X7: list_a] :
( ( member_list_a @ X7 @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X7 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_769_finite__remove__induct,axiom,
! [B4: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ B4 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A4 )
=> ( ( A4 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A4 @ B4 )
=> ( ! [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ A4 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A4 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_770_finite__remove__induct,axiom,
! [B4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_771_insert__remove__id,axiom,
! [X: a,X5: set_a] :
( ( member_a @ X @ X5 )
=> ( X5
= ( insert_a @ X @ ( minus_minus_set_a @ X5 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% insert_remove_id
thf(fact_772_insert__remove__id,axiom,
! [X: list_a,X5: set_list_a] :
( ( member_list_a @ X @ X5 )
=> ( X5
= ( insert_list_a @ X @ ( minus_646659088055828811list_a @ X5 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ).
% insert_remove_id
thf(fact_773_insert__remove__id,axiom,
! [X: relational_fmla_a_b,X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ X5 )
=> ( X5
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_774_insert__remove__id,axiom,
! [X: nat,X5: set_nat] :
( ( member_nat @ X @ X5 )
=> ( X5
= ( insert_nat @ X @ ( minus_minus_set_nat @ X5 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_775_the__elem__eq,axiom,
! [X: list_a] :
( ( the_elem_list_a @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= X ) ).
% the_elem_eq
thf(fact_776_the__elem__eq,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_777_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_778_is__singletonI,axiom,
! [X: list_a] : ( is_singleton_list_a @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ).
% is_singletonI
thf(fact_779_is__singletonI,axiom,
! [X: relational_fmla_a_b] : ( is_sin6594375743535830443la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% is_singletonI
thf(fact_780_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_781_diff__diff__less,axiom,
! [I3: nat,M: nat,N2: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N2 ) ) )
= ( ( ord_less_nat @ I3 @ M )
& ( ord_less_nat @ I3 @ N2 ) ) ) ).
% diff_diff_less
thf(fact_782_remove__def,axiom,
( remove_list_a
= ( ^ [X3: list_a,A5: set_list_a] : ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% remove_def
thf(fact_783_remove__def,axiom,
( remove4261432235257513082la_a_b
= ( ^ [X3: relational_fmla_a_b,A5: set_Re381260168593705685la_a_b] : ( minus_4077726661957047470la_a_b @ A5 @ ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% remove_def
thf(fact_784_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_785_member__remove,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( remove4261432235257513082la_a_b @ Y @ A3 ) )
= ( ( member4680049679412964150la_a_b @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_786_member__remove,axiom,
! [X: a,Y: a,A3: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A3 ) )
= ( ( member_a @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_787_member__remove,axiom,
! [X: nat,Y: nat,A3: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A3 ) )
= ( ( member_nat @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_788_is__singleton__the__elem,axiom,
( is_singleton_list_a
= ( ^ [A5: set_list_a] :
( A5
= ( insert_list_a @ ( the_elem_list_a @ A5 ) @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_789_is__singleton__the__elem,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b] :
( A5
= ( insert7010464514620295119la_a_b @ ( the_el6350558617753882986la_a_b @ A5 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_790_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_791_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X6: a,Y5: a] :
( ( member_a @ X6 @ A3 )
=> ( ( member_a @ Y5 @ A3 )
=> ( X6 = Y5 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_792_is__singletonI_H,axiom,
! [A3: set_list_a] :
( ( A3 != bot_bot_set_list_a )
=> ( ! [X6: list_a,Y5: list_a] :
( ( member_list_a @ X6 @ A3 )
=> ( ( member_list_a @ Y5 @ A3 )
=> ( X6 = Y5 ) ) )
=> ( is_singleton_list_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_793_is__singletonI_H,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( A3 != bot_bo4495933725496725865la_a_b )
=> ( ! [X6: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A3 )
=> ( ( member4680049679412964150la_a_b @ Y5 @ A3 )
=> ( X6 = Y5 ) ) )
=> ( is_sin6594375743535830443la_a_b @ A3 ) ) ) ).
% is_singletonI'
thf(fact_794_is__singletonI_H,axiom,
! [A3: set_nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [X6: nat,Y5: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ( member_nat @ Y5 @ A3 )
=> ( X6 = Y5 ) ) )
=> ( is_singleton_nat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_795_is__singletonE,axiom,
! [A3: set_list_a] :
( ( is_singleton_list_a @ A3 )
=> ~ ! [X6: list_a] :
( A3
!= ( insert_list_a @ X6 @ bot_bot_set_list_a ) ) ) ).
% is_singletonE
thf(fact_796_is__singletonE,axiom,
! [A3: set_Re381260168593705685la_a_b] :
( ( is_sin6594375743535830443la_a_b @ A3 )
=> ~ ! [X6: relational_fmla_a_b] :
( A3
!= ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% is_singletonE
thf(fact_797_is__singletonE,axiom,
! [A3: set_nat] :
( ( is_singleton_nat @ A3 )
=> ~ ! [X6: nat] :
( A3
!= ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_798_is__singleton__def,axiom,
( is_singleton_list_a
= ( ^ [A5: set_list_a] :
? [X3: list_a] :
( A5
= ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_def
thf(fact_799_is__singleton__def,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b] :
? [X3: relational_fmla_a_b] :
( A5
= ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_800_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_801_fv__erase,axiom,
! [Q2: relational_fmla_a_b,X: nat] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_erase_a_b @ Q2 @ X ) ) @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% fv_erase
thf(fact_802_cov__fv__aux,axiom,
! [X: nat,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q2 @ G2 )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( relational_fv_a_b @ Q2 ) ) ) ) ) ).
% cov_fv_aux
thf(fact_803_gen_H_Ointros_I2_J,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( relational_gen_a_b2 @ X @ Q2 @ ( insert7010464514620295119la_a_b @ Q2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen'.intros(2)
thf(fact_804_Inf__fin_Oremove,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( member_set_nat @ X @ A3 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A3 )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A3 )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_805_Inf__fin_Oremove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_806_Inf__fin_Oinsert__remove,axiom,
! [A3: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A3 ) )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A3 ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_807_Inf__fin_Oinsert__remove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_808_finite__Int,axiom,
! [F3: set_list_a,G2: set_list_a] :
( ( ( finite_finite_list_a @ F3 )
| ( finite_finite_list_a @ G2 ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_809_finite__Int,axiom,
! [F3: set_a,G2: set_a] :
( ( ( finite_finite_a @ F3 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_810_finite__Int,axiom,
! [F3: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_811_Int__insert__left__if0,axiom,
! [A2: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ C2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) @ C2 )
= ( inf_in8483230781156617063la_a_b @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_812_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B4: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_813_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_814_Int__insert__left__if1,axiom,
! [A2: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ C2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) @ C2 )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_815_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B4: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B4 ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_816_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_817_insert__inter__insert,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A3 ) @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_818_insert__inter__insert,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_819_Int__insert__right__if0,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A3 )
=> ( ( inf_in8483230781156617063la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) )
= ( inf_in8483230781156617063la_a_b @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_820_Int__insert__right__if0,axiom,
! [A2: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B4 ) )
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_821_Int__insert__right__if0,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_822_Int__insert__right__if1,axiom,
! [A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A3 )
=> ( ( inf_in8483230781156617063la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ A2 @ B4 ) )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_823_Int__insert__right__if1,axiom,
! [A2: a,A3: set_a,B4: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B4 ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_824_Int__insert__right__if1,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_825_Int__subset__iff,axiom,
! [C2: set_nat,A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B4 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A3 )
& ( ord_less_eq_set_nat @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_826_disjoint__insert_I2_J,axiom,
! [A3: set_a,B: a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B @ B4 ) ) )
= ( ~ ( member_a @ B @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_827_disjoint__insert_I2_J,axiom,
! [A3: set_list_a,B: list_a,B4: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A3 @ ( insert_list_a @ B @ B4 ) ) )
= ( ~ ( member_list_a @ B @ A3 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_828_disjoint__insert_I2_J,axiom,
! [A3: set_Re381260168593705685la_a_b,B: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ B @ B4 ) ) )
= ( ~ ( member4680049679412964150la_a_b @ B @ A3 )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_829_disjoint__insert_I2_J,axiom,
! [A3: set_nat,B: nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ ( insert_nat @ B @ B4 ) ) )
= ( ~ ( member_nat @ B @ A3 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_830_disjoint__insert_I1_J,axiom,
! [B4: set_list_a,A2: list_a,A3: set_list_a] :
( ( ( inf_inf_set_list_a @ B4 @ ( insert_list_a @ A2 @ A3 ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A2 @ B4 )
& ( ( inf_inf_set_list_a @ B4 @ A3 )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_831_disjoint__insert_I1_J,axiom,
! [B4: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ B4 @ ( insert7010464514620295119la_a_b @ A2 @ A3 ) )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A2 @ B4 )
& ( ( inf_in8483230781156617063la_a_b @ B4 @ A3 )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_832_disjoint__insert_I1_J,axiom,
! [B4: set_nat,A2: nat,A3: set_nat] :
( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A2 @ A3 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B4 )
& ( ( inf_inf_set_nat @ B4 @ A3 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_833_insert__disjoint_I2_J,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B4 ) )
= ( ~ ( member_nat @ A2 @ B4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_834_insert__disjoint_I1_J,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B4 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B4 )
& ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_835_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_836_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_837_Diff__disjoint,axiom,
! [A3: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ A3 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_838_inf__Sup__absorb,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A3 ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_839_Inf__fin_Oinsert,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_840_cov_ONeg,axiom,
! [X: nat,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q2 @ G2 )
=> ( relational_cov_a_b @ X @ ( relational_Neg_a_b @ Q2 ) @ G2 ) ) ).
% cov.Neg
thf(fact_841_erase_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X )
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q22 @ X ) ) ) ).
% erase.simps(6)
thf(fact_842_erase_Osimps_I4_J,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Neg_a_b @ Q2 ) @ X )
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).
% erase.simps(4)
thf(fact_843_Int__emptyI,axiom,
! [A3: set_nat,B4: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ~ ( member_nat @ X6 @ B4 ) )
=> ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_844_disjoint__iff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ~ ( member_nat @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_845_Int__empty__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_846_Int__empty__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_847_disjoint__iff__not__equal,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_848_Int__Collect__mono,axiom,
! [A3: set_nat,B4: set_nat,P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ( P @ X6 )
=> ( Q2 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_849_Int__insert__right,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) )
& ( ~ ( member_nat @ A2 @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_850_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_851_Diff__Int__distrib2,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_852_Diff__Int__distrib,axiom,
! [C2: set_nat,A3: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_853_Diff__Diff__Int,axiom,
! [A3: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( inf_inf_set_nat @ A3 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_854_Diff__Int2,axiom,
! [A3: set_nat,C2: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_855_Int__Diff,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ C2 ) ) ) ).
% Int_Diff
thf(fact_856_cov_ODisjR,axiom,
! [X: nat,Q22: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q22 @ G2 )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q22 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ G2 ) ) ) ).
% cov.DisjR
thf(fact_857_cov_ODisjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G2 )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ G2 ) ) ) ).
% cov.DisjL
thf(fact_858_cov_OConjR,axiom,
! [X: nat,Q22: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q22 @ G2 )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q22 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G2 ) ) ) ).
% cov.ConjR
thf(fact_859_cov_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G2 )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G2 ) ) ) ).
% cov.ConjL
thf(fact_860_ap__cp__triv,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q2 )
=> ( ( relational_cp_a_b @ Q2 )
= Q2 ) ) ).
% ap_cp_triv
thf(fact_861_ap__cp,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q2 )
=> ( relational_ap_a_b @ ( relational_cp_a_b @ Q2 ) ) ) ).
% ap_cp
thf(fact_862_ap__cp__erase,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% ap_cp_erase
thf(fact_863_cov_Oap,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( relational_cov_a_b @ X @ Q2 @ ( insert7010464514620295119la_a_b @ Q2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov.ap
thf(fact_864_cov_Ononfree,axiom,
! [X: nat,Q2: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( relational_cov_a_b @ X @ Q2 @ bot_bo4495933725496725865la_a_b ) ) ).
% cov.nonfree
thf(fact_865_Int__Diff__disjoint,axiom,
! [A3: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ A3 @ B4 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_866_Diff__triv,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A3 @ B4 )
= A3 ) ) ).
% Diff_triv
thf(fact_867_Inf__fin_Oin__idem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_868_ex__cov,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_rrb_a_b @ Q2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_cov_a_b @ X @ Q2 @ X_1 ) ) ) ).
% ex_cov
thf(fact_869_exists__cp__erase,axiom,
! [X: nat,Q2: relational_fmla_a_b] :
( ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) ) )
= ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).
% exists_cp_erase
thf(fact_870_cov__fv,axiom,
! [X: nat,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q2 @ G2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q2 ) ) ) ) ) ) ).
% cov_fv
thf(fact_871_sat__erase,axiom,
! [Q2: relational_fmla_a_b,X: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
( ( relational_sat_a_b @ ( relational_erase_a_b @ Q2 @ X ) @ I @ ( fun_upd_nat_a @ Sigma @ X @ Z ) )
= ( relational_sat_a_b @ ( relational_erase_a_b @ Q2 @ X ) @ I @ Sigma ) ) ).
% sat_erase
thf(fact_872_Inf__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X6: nat,Y5: nat] : ( member_nat @ ( inf_inf_nat @ X6 @ Y5 ) @ ( insert_nat @ X6 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_873_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_874_Inf__fin_Osubset,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_875_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_876_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_877_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_878_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_879_IntI,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_880_Int__iff,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B4 ) )
= ( ( member_nat @ C @ A3 )
& ( member_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_881_IntE,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B4 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ~ ( member_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_882_IntD1,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B4 ) )
=> ( member_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_883_IntD2,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B4 ) )
=> ( member_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_884_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_885_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_886_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A: nat] :
( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_887_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B2: nat] :
( ( inf_inf_nat @ A @ B2 )
= A ) ) ) ).
% inf.absorb_iff1
thf(fact_888_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_889_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_890_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B2: nat] :
( A
= ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_891_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_892_inf_OboundedI,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_893_inf_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_894_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_895_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_896_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_897_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_898_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_899_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X6: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X6 @ Y5 ) @ X6 )
=> ( ! [X6: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X6 @ Y5 ) @ Y5 )
=> ( ! [X6: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
=> ( ( ord_less_eq_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_900_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_901_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_902_le__infI2,axiom,
! [B: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_903_le__infI1,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_904_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_905_le__infI,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_906_le__infE,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_907_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_908_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_909_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_910_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_911_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_912_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_913_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
& ( A != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_914_inf_Ostrict__boundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_915_inf_Oabsorb4,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_916_inf_Oabsorb3,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_917_less__infI2,axiom,
! [B: nat,X: nat,A2: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% less_infI2
thf(fact_918_less__infI1,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_nat @ A2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% less_infI1
thf(fact_919_Sup__fin_Oremove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_920_Sup__fin_Oinsert__remove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_921_Un__empty,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ( A3 = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_922_finite__Un,axiom,
! [F3: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G2 ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_923_Un__insert__right,axiom,
! [A3: set_nat,A2: nat,B4: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_924_Un__insert__left,axiom,
! [A2: nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_925_Un__Diff__cancel2,axiom,
! [B4: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B4 @ A3 ) @ A3 )
= ( sup_sup_set_nat @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_926_Un__Diff__cancel,axiom,
! [A3: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ A3 ) )
= ( sup_sup_set_nat @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_927_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_928_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_929_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_930_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_931_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_932_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_933_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_934_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_935_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_936_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_937_sup__Inf__absorb,axiom,
! [A3: set_nat,A2: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_938_Sup__fin_Oinsert,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_939_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_940_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_941_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B2: nat] :
( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_942_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A: nat] :
( ( sup_sup_nat @ A @ B2 )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_943_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_944_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_945_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A: nat] :
( A
= ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_946_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_947_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_948_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_949_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_950_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_951_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_952_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X6: nat,Y5: nat] : ( ord_less_eq_nat @ X6 @ ( F @ X6 @ Y5 ) )
=> ( ! [X6: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X6 @ Y5 ) )
=> ( ! [X6: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y5 @ X6 )
=> ( ( ord_less_eq_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z4 ) @ X6 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_953_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_954_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_955_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_956_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_957_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_958_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_959_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_960_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_961_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_962_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_963_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_964_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_965_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_966_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_967_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_968_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_969_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
& ( A != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_970_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_971_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_972_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_973_less__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_974_less__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_975_equiv__eval__on__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q2: relational_fmla_a_b,Q3: relational_fmla_a_b,X5: set_nat] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( relational_equiv_a_b @ Q2 @ Q3 )
=> ( ( relati8814510239606734169on_a_b @ X5 @ Q2 @ I )
= ( relati8814510239606734169on_a_b @ X5 @ Q3 @ I ) ) ) ) ).
% equiv_eval_on_eqI
thf(fact_976_Sup__fin_Ounion,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A3 @ B4 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_977_Un__empty__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_978_Un__empty__right,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Un_empty_right
thf(fact_979_finite__UnI,axiom,
! [F3: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_980_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_981_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_982_Un__Diff,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C2 ) @ ( minus_minus_set_nat @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_983_Relational__Calculus_Oequiv__def,axiom,
( relational_equiv_a_b
= ( ^ [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
! [I5: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( ( finite_finite_a @ ( relational_adom_b_a @ I5 ) )
=> ( ( relational_sat_a_b @ Q12 @ I5 @ Sigma3 )
= ( relational_sat_a_b @ Q23 @ I5 @ Sigma3 ) ) ) ) ) ).
% Relational_Calculus.equiv_def
thf(fact_984_equiv__Disj__cong,axiom,
! [Q1: relational_fmla_a_b,Q13: relational_fmla_a_b,Q22: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q13 )
=> ( ( relational_equiv_a_b @ Q22 @ Q24 )
=> ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) ) ).
% equiv_Disj_cong
thf(fact_985_equiv__Disj__Assoc,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,Q32: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_Disj_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ Q32 ) @ ( relational_Disj_a_b @ Q1 @ ( relational_Disj_a_b @ Q22 @ Q32 ) ) ) ).
% equiv_Disj_Assoc
thf(fact_986_equiv__Neg__cong,axiom,
! [Q2: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q2 @ Q3 )
=> ( relational_equiv_a_b @ ( relational_Neg_a_b @ Q2 ) @ ( relational_Neg_a_b @ Q3 ) ) ) ).
% equiv_Neg_cong
thf(fact_987_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_988_equiv__cp,axiom,
! [Q2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_cp_a_b @ Q2 ) @ Q2 ) ).
% equiv_cp
thf(fact_989_equiv__cp__cong,axiom,
! [Q2: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q2 @ Q3 )
=> ( relational_equiv_a_b @ ( relational_cp_a_b @ Q2 ) @ ( relational_cp_a_b @ Q3 ) ) ) ).
% equiv_cp_cong
thf(fact_990_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_991_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_992_equiv__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q2: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ( relational_fv_a_b @ Q2 )
= ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_equiv_a_b @ Q2 @ Q3 )
=> ( ( relational_eval_a_b @ Q2 @ I )
= ( relational_eval_a_b @ Q3 @ I ) ) ) ) ) ).
% equiv_eval_eqI
thf(fact_993_singleton__Un__iff,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A3 @ B4 ) )
= ( ( ( A3 = bot_bot_set_nat )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4 = bot_bot_set_nat ) )
| ( ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_994_Un__singleton__iff,axiom,
! [A3: set_nat,B4: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A3 @ B4 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A3 = bot_bot_set_nat )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4 = bot_bot_set_nat ) )
| ( ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_995_insert__is__Un,axiom,
( insert_nat
= ( ^ [A: nat] : ( sup_sup_set_nat @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_996_Diff__partition,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_997_Diff__subset__conv,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ C2 )
= ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_998_Diff__Un,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B4 @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_999_Diff__Int,axiom,
! [A3: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1000_Int__Diff__Un,axiom,
! [A3: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ A3 @ B4 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1001_Un__Diff__Int,axiom,
! [A3: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( inf_inf_set_nat @ A3 @ B4 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1002_Sup__fin_Oin__idem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) )
= ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1003_fv_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(6)
thf(fact_1004_fv_Osimps_I5_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(5)
thf(fact_1005_cov_ODisj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q22 @ G22 )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov.Disj
thf(fact_1006_gen_H_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b2 @ X @ Q22 @ G22 )
=> ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen'.intros(6)
thf(fact_1007_equiv__eval__on__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q2: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q2 ) @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_equiv_a_b @ Q2 @ Q3 )
=> ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q3 ) @ Q2 @ I )
= ( relational_eval_a_b @ Q3 @ I ) ) ) ) ) ).
% equiv_eval_on_eval_eqI
thf(fact_1008_Sup__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X6: nat,Y5: nat] : ( member_nat @ ( sup_sup_nat @ X6 @ Y5 ) @ ( insert_nat @ X6 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1009_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1010_Sup__fin_Osubset,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B4 ) @ ( lattic1093996805478795353in_nat @ A3 ) )
= ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1011_Inf__fin_Ounion,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A3 @ B4 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic5238388535129920115in_nat @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1012_ap__fresh__val,axiom,
! [Q2: relational_fmla_a_b,Sigma: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_ap_a_b @ Q2 )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_csts_a_b @ Q2 ) )
=> ( ( relational_sat_a_b @ Q2 @ I @ Sigma )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) ) ) ) ) ) ).
% ap_fresh_val
thf(fact_1013_card__Diff1__less__iff,axiom,
! [A3: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) )
= ( ( finite_finite_nat @ A3 )
& ( member_nat @ X @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1014_UnCI,axiom,
! [C: nat,B4: set_nat,A3: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A3 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_1015_Un__iff,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B4 ) )
= ( ( member_nat @ C @ A3 )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_1016_csts_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q22 ) ) ) ).
% csts.simps(6)
thf(fact_1017_UnE,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B4 ) )
=> ( ~ ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_1018_UnI1,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1019_UnI2,axiom,
! [C: nat,B4: set_nat,A3: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1020_csts_Osimps_I4_J,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Neg_a_b @ Q2 ) )
= ( relational_csts_a_b @ Q2 ) ) ).
% csts.simps(4)
thf(fact_1021_infinite__arbitrarily__large,axiom,
! [A3: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A3 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N2 )
& ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1022_card__subset__eq,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ( finite_card_nat @ A3 )
= ( finite_card_nat @ B4 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_1023_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A3: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B4 )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A3 )
=> ( ( member_nat @ A22 @ A3 )
=> ( ( member_nat @ B6 @ B4 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1024_card__insert__le,axiom,
! [A3: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( insert_nat @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_1025_card__mono,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_1026_card__seteq,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A3 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_seteq
thf(fact_1027_exists__subset__between,axiom,
! [A3: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1028_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1029_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F3 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1030_card__less__sym__Diff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1031_card__le__sym__Diff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1032_psubset__card__mono,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_set_nat @ A3 @ B4 )
=> ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_1033_card__Diff1__le,axiom,
! [A3: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ).
% card_Diff1_le
thf(fact_1034_card__Diff__subset,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1035_diff__card__le__card__Diff,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1036_card__psubset,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_set_nat @ A3 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_1037_card__Diff__subset__Int,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A3 @ B4 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1038_card__Diff1__less,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_1039_card__Diff2__less,axiom,
! [A3: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y @ A3 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1040_card__Diff__singleton__if,axiom,
! [X: nat,A3: set_nat] :
( ( ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1041_card__Diff__singleton,axiom,
! [X: nat,A3: set_nat] :
( ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1042_card__Diff__insert,axiom,
! [A2: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ A3 )
=> ( ~ ( member_nat @ A2 @ B4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B4 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1043_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( ( finite_card_nat @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1044_card__1__singletonE,axiom,
! [A3: set_nat] :
( ( ( finite_card_nat @ A3 )
= one_one_nat )
=> ~ ! [X6: nat] :
( A3
!= ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1045_card__insert__le__m1,axiom,
! [N2: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1046_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1047_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1048_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1049_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1050_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1051_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1052_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1053_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1054_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1055_card_Oinfinite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_card_nat @ A3 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1056_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1057_card__0__eq,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( finite_card_nat @ A3 )
= zero_zero_nat )
= ( A3 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1058_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1059_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1060_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1061_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1062_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1063_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1064_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1065_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1066_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1067_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1068_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P @ N )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1069_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1070_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1071_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1072_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_1073_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1074_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1075_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_1076_card__eq__0__iff,axiom,
! [A3: set_nat] :
( ( ( finite_card_nat @ A3 )
= zero_zero_nat )
= ( ( A3 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_1077_card__ge__0__finite,axiom,
! [A3: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
=> ( finite_finite_nat @ A3 ) ) ).
% card_ge_0_finite
thf(fact_1078_card__gt__0__iff,axiom,
! [A3: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
= ( ( A3 != bot_bot_set_nat )
& ( finite_finite_nat @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_1079_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1080_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1081_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1082_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1083_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1084_forall__finite_I1_J,axiom,
! [P: nat > $o,I2: nat] :
( ( ord_less_nat @ I2 @ zero_zero_nat )
=> ( P @ I2 ) ) ).
% forall_finite(1)
thf(fact_1085_card__Un__disjoint,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B4 ) )
= ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1086_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1087_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1088_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_1089_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_1090_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1091_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1092_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1093_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1094_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1095_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1096_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1097_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1098_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1099_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1100_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1101_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1102_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1103_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1104_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I3 @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1105_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1106_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1107_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1108_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1109_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1110_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1111_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1112_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1113_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1114_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1115_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_1116_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1117_add__le__add__imp__diff__le,axiom,
! [I3: nat,K2: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1118_add__le__imp__le__diff,axiom,
! [I3: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1119_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1120_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1121_less__diff__conv,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1122_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1123_add__lessD1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_1124_add__less__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1125_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_1126_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1127_add__less__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1128_trans__less__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1129_trans__less__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1130_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1131_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1132_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1133_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1134_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1135_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1136_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1137_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1138_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1139_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1140_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1141_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1142_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1143_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1144_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_1145_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1146_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B2: nat] :
? [C5: nat] :
( B2
= ( plus_plus_nat @ A @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_1147_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_1148_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_1149_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_1150_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1151_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1152_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1153_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1154_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1155_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1156_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1157_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1158_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1159_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1160_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1161_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1162_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1163_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1164_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1165_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1166_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1167_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1168_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1169_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1170_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1171_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1172_less__diff__conv2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1173_card__Un__le,axiom,
! [A3: set_nat,B4: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B4 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ).
% card_Un_le
thf(fact_1174_card__Un__Int,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
= ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B4 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1175_card_Oremove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ A3 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1176_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_1177_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1178_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1179_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1180_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1181_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1182_card__insert__disjoint,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1183_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1184_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ X ) )
=> ( P @ ( suc @ I6 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_1185_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ zero_zero_nat ) )
=> ( P @ I6 ) ) )
= ( P @ zero_zero_nat ) ) ).
% forall_finite(2)
thf(fact_1186_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ X ) )
=> ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ X )
=> ( P @ ( suc @ I6 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_1187_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q4: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1188_less__add__Suc1,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1189_less__add__Suc2,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1190_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1191_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1192_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1193_strict__inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_1194_less__Suc__induct,axiom,
! [I3: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I4 @ K ) ) ) ) )
=> ( P @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1195_less__trans__Suc,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1196_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1197_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1198_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1199_Nat_OAll__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N2 ) )
=> ( P @ I6 ) ) )
= ( ( P @ N2 )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( P @ I6 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_1200_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1201_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1202_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N2 ) )
& ( P @ I6 ) ) )
= ( ( P @ N2 )
| ? [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
& ( P @ I6 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1203_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1204_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1205_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1206_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1207_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1208_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1209_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1210_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1211_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1212_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1213_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1214_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1215_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1216_inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ J )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ ( suc @ N ) )
=> ( P @ N ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_1217_dec__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ I3 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1218_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_1219_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1220_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1221_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1222_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1223_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1224_card__le__Suc__Max,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_1225_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1226_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1227_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N2 ) )
=> ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( P @ ( suc @ I6 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1228_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1229_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N2 ) )
& ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
& ( P @ ( suc @ I6 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1230_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1231_diff__Suc__less,axiom,
! [N2: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1232_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1233_card__insert__if,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( finite_card_nat @ A3 ) ) )
& ( ~ ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1234_card__Suc__eq__finite,axiom,
! [A3: set_nat,K2: nat] :
( ( ( finite_card_nat @ A3 )
= ( suc @ K2 ) )
= ( ? [B2: nat,B5: set_nat] :
( ( A3
= ( insert_nat @ B2 @ B5 ) )
& ~ ( member_nat @ B2 @ B5 )
& ( ( finite_card_nat @ B5 )
= K2 )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1235_fresh2__def,axiom,
( relati2677767559083392098h2_a_b
= ( ^ [X3: nat,Y3: nat,Q: relational_fmla_a_b] : ( suc @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ) ).
% fresh2_def
thf(fact_1236_card__1__singleton__iff,axiom,
! [A3: set_nat] :
( ( ( finite_card_nat @ A3 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( A3
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1237_card__eq__SucD,axiom,
! [A3: set_nat,K2: nat] :
( ( ( finite_card_nat @ A3 )
= ( suc @ K2 ) )
=> ? [B6: nat,B8: set_nat] :
( ( A3
= ( insert_nat @ B6 @ B8 ) )
& ~ ( member_nat @ B6 @ B8 )
& ( ( finite_card_nat @ B8 )
= K2 )
& ( ( K2 = zero_zero_nat )
=> ( B8 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1238_card__Suc__eq,axiom,
! [A3: set_nat,K2: nat] :
( ( ( finite_card_nat @ A3 )
= ( suc @ K2 ) )
= ( ? [B2: nat,B5: set_nat] :
( ( A3
= ( insert_nat @ B2 @ B5 ) )
& ~ ( member_nat @ B2 @ B5 )
& ( ( finite_card_nat @ B5 )
= K2 )
& ( ( K2 = zero_zero_nat )
=> ( B5 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1239_card__le__Suc0__iff__eq,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1240_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1241_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1242_card__le__Suc__iff,axiom,
! [N2: nat,A3: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_nat @ A3 ) )
= ( ? [A: nat,B5: set_nat] :
( ( A3
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 )
& ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ B5 ) )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1243_card__Suc__Diff1,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1244_card_Oinsert__remove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1245_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K2 ) @ ( insert_nat @ ( suc @ K2 ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_1246_greaterThan__iff,axiom,
! [I3: nat,K2: nat] :
( ( member_nat @ I3 @ ( set_or1210151606488870762an_nat @ K2 ) )
= ( ord_less_nat @ K2 @ I3 ) ) ).
% greaterThan_iff
thf(fact_1247_greaterThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X ) @ ( set_or1210151606488870762an_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_1248_greaterThan__non__empty,axiom,
! [X: nat] :
( ( set_or1210151606488870762an_nat @ X )
!= bot_bot_set_nat ) ).
% greaterThan_non_empty
thf(fact_1249_infinite__Ioi,axiom,
! [A2: nat] :
~ ( finite_finite_nat @ ( set_or1210151606488870762an_nat @ A2 ) ) ).
% infinite_Ioi
thf(fact_1250_fmla_Osize_I13_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( size_s453432777765377587la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s453432777765377587la_a_b @ X61 ) @ ( size_s453432777765377587la_a_b @ X62 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fmla.size(13)
thf(fact_1251_fmla_Osize_I11_J,axiom,
! [X4: relational_fmla_a_b] :
( ( size_s453432777765377587la_a_b @ ( relational_Neg_a_b @ X4 ) )
= ( plus_plus_nat @ ( size_s453432777765377587la_a_b @ X4 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fmla.size(11)
thf(fact_1252_qp__fresh__val,axiom,
! [Q2: relational_fmla_a_b,Sigma: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_qp_a_b @ Q2 )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_csts_a_b @ Q2 ) )
=> ( ( relational_sat_a_b @ Q2 @ I @ Sigma )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) ) ) ) ) ) ).
% qp_fresh_val
thf(fact_1253_qp__Disj,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ).
% qp_Disj
thf(fact_1254_qp__Neg,axiom,
! [Q2: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Neg_a_b @ Q2 ) ) ).
% qp_Neg
thf(fact_1255_qp__cp,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q2 )
=> ( relational_qp_a_b @ ( relational_cp_a_b @ Q2 ) ) ) ).
% qp_cp
thf(fact_1256_qp__cp__triv,axiom,
! [Q2: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q2 )
=> ( ( relational_cp_a_b @ Q2 )
= Q2 ) ) ).
% qp_cp_triv
thf(fact_1257_qp__cp__erase,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% qp_cp_erase
thf(fact_1258_binomial__addition__formula,axiom,
! [N2: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( binomial @ N2 @ ( suc @ K2 ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K2 ) ) ) ) ).
% binomial_addition_formula
thf(fact_1259_binomial__eq__0__iff,axiom,
! [N2: nat,K2: nat] :
( ( ( binomial @ N2 @ K2 )
= zero_zero_nat )
= ( ord_less_nat @ N2 @ K2 ) ) ).
% binomial_eq_0_iff
thf(fact_1260_zero__less__binomial__iff,axiom,
! [N2: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K2 ) )
= ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% zero_less_binomial_iff
thf(fact_1261_rrb__simps_I8_J,axiom,
! [Y: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y @ Qy ) )
= ( ( ( member_nat @ Y @ ( relational_fv_a_b @ Qy ) )
=> ? [X9: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y @ Qy @ X9 ) )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(8)
thf(fact_1262_qp__Gen,axiom,
! [Q2: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q2 )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q2 @ X_1 ) ) ) ).
% qp_Gen
thf(fact_1263_binomial__eq__0,axiom,
! [N2: nat,K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( ( binomial @ N2 @ K2 )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1264_zero__less__binomial,axiom,
! [K2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K2 ) ) ) ).
% zero_less_binomial
thf(fact_1265_gen_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b @ X @ Q22 @ G22 )
=> ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen.intros(6)
thf(fact_1266_gen__sat__erase,axiom,
! [Y: nat,Q2: relational_fmla_a_b,Gy: set_Re381260168593705685la_a_b,X: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_gen_a_b @ Y @ Q2 @ Gy )
=> ( ( relational_sat_a_b @ ( relational_erase_a_b @ Q2 @ X ) @ I @ Sigma )
=> ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ Gy )
& ( relational_sat_a_b @ X6 @ I @ Sigma ) ) ) ) ).
% gen_sat_erase
thf(fact_1267_gen__sat,axiom,
! [X: nat,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_gen_a_b @ X @ Q2 @ G2 )
=> ( ( relational_sat_a_b @ Q2 @ I @ Sigma )
=> ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ G2 )
& ( relational_sat_a_b @ X6 @ I @ Sigma ) ) ) ) ).
% gen_sat
thf(fact_1268_gen__Gen__cp,axiom,
! [X: nat,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q2 @ G2 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q2 ) @ X_1 ) ) ).
% gen_Gen_cp
thf(fact_1269_Gen__cp,axiom,
! [X: nat,Q2: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q2 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q2 ) @ X_1 ) ) ).
% Gen_cp
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
( ( relational_fv_a_b @ q )
= ( relational_fv_a_b @ q2 ) ) ).
thf(conj_1,hypothesis,
! [Sigma4: nat > a] :
( ( relational_sat_a_b @ q @ i @ Sigma4 )
= ( relational_sat_a_b @ q2 @ i @ Sigma4 ) ) ).
thf(conj_2,conjecture,
( ( relational_eval_a_b @ q @ i )
= ( relational_eval_a_b @ q2 @ i ) ) ).
%------------------------------------------------------------------------------