TPTP Problem File: SLH0397^1.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 : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00184_006019__12098642_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1374 ( 550 unt; 108 typ; 0 def)
% Number of atoms : 3662 (1255 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11686 ( 462 ~; 59 |; 270 &;9161 @)
% ( 0 <=>;1734 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 603 ( 603 >; 0 *; 0 +; 0 <<)
% Number of symbols : 101 ( 100 usr; 15 con; 0-5 aty)
% Number of variables : 3454 ( 175 ^;3166 !; 113 ?;3454 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:19:54.676
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (100)
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_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
finite_finite_num: set_num > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001t__Nat__Onat,type,
group_group_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_001t__Nat__Onat,type,
group_monoid_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__a,type,
group_monoid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001t__Nat__Onat,type,
group_Units_nat: set_nat > ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Nat__Onat,type,
group_subgroup_nat: set_nat > set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $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__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
nO_MATCH_set_a_set_a: set_a > set_a > $o ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_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__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_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__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_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__Num__Onum,type,
lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).
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__Num__Onum,type,
lattic2897619205827179199_a_num: ( a > num ) > 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_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
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_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
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_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
bot_bot_set_num: set_num ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $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_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001t__Nat__Onat,type,
pluenn2073725187428264546up_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001tf__a,type,
pluenn1164192988769422572roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001t__Nat__Onat,type,
pluenn3669378163024332905et_nat: set_nat > ( nat > nat > nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001t__Nat__Onat,type,
pluenn5670965976768739049tp_nat: set_nat > ( nat > nat > nat ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $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_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_Aa____,type,
aa: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1264)
thf(fact_0_assms,axiom,
ord_less_eq_set_a @ a2 @ g ).
% assms
thf(fact_1_Suc_Ohyps_I2_J,axiom,
( ( suc @ n )
= ( finite_card_a @ aa ) ) ).
% Suc.hyps(2)
thf(fact_2_Suc_Oprems,axiom,
ord_less_eq_set_a @ aa @ g ).
% Suc.prems
thf(fact_3_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_4_sumsetdiff__sing,axiom,
! [A: set_a,B: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_5_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_6_sumset_Ocases,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A2
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_7_sumset_Osimps,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= ( ? [A4: a,B3: a] :
( ( A2
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_8_sumset_OsumsetI,axiom,
! [A2: a,A: set_a,B4: a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A2 @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_9_sumset__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_10_sumset__commute,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A ) ) ).
% sumset_commute
thf(fact_11_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_12_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_13_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_14_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_15_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_16_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_17_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_18_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_19_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_20_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_21_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_22_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_23_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_24_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_25_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_26_insert__absorb2,axiom,
! [X: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A ) )
= ( insert_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_27_insert__iff,axiom,
! [A2: a,B4: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B4 @ A ) )
= ( ( A2 = B4 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_28_insert__iff,axiom,
! [A2: nat,B4: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B4 @ A ) )
= ( ( A2 = B4 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_29_insertCI,axiom,
! [A2: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B4 ) )
=> ( member_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_30_insertCI,axiom,
! [A2: nat,B: set_nat,B4: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B4 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B4 @ B ) ) ) ).
% insertCI
thf(fact_31_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_32_Diff__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_33_Diff__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_34_DiffI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_35_DiffI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_36_sumset__subset__carrier,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_37_sumset__mono,axiom,
! [A5: set_a,A: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_mono
thf(fact_38_sumset__subset__insert_I1_J,axiom,
! [A: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_39_sumset__subset__insert_I2_J,axiom,
! [A: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_40_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_41_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_42_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_43_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_44_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_45_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_46_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_47_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B4: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A2 = B4 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_48_singleton__insert__inj__eq,axiom,
! [B4: a,A2: a,A: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B4 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_49_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_50_composition__closed,axiom,
! [A2: a,B4: a] :
( ( member_a @ A2 @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A2 @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_51_associative,axiom,
! [A2: a,B4: a,C2: a] :
( ( member_a @ A2 @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A2 @ B4 ) @ C2 )
= ( addition @ A2 @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_52_Suc_Ohyps_I1_J,axiom,
! [A: set_a] :
( ( n
= ( finite_card_a @ A ) )
=> ( ( ord_less_eq_set_a @ A @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ A ) ) @ ( binomial @ ( suc @ ( finite_card_a @ A ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Suc.hyps(1)
thf(fact_53_sumset__empty_I1_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_54_sumset__empty_I2_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_55_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_59_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_60_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_61_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_62_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_63_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_64_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_65_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_66_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_67_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_68_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A6 )
=> ( member_nat @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_69_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_70_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_71_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_72_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_73_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_74_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_75_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_76_subset__insertI2,axiom,
! [A: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_77_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_78_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_79_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_80_insert__mono,axiom,
! [C: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_81_double__diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_82_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_83_Diff__mono,axiom,
! [A: set_a,C: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_84_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_85_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_86_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_87_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_88_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_89_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_90_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_91_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_92_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B7: set_a] :
( ( A
= ( insert_a @ A2 @ B7 ) )
& ~ ( member_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_93_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat @ A2 @ B7 ) )
& ~ ( member_nat @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_94_insert__commute,axiom,
! [X: a,Y: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A ) )
= ( insert_a @ Y @ ( insert_a @ X @ A ) ) ) ).
% insert_commute
thf(fact_95_insert__eq__iff,axiom,
! [A2: a,A: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B4 @ B ) )
= ( ( ( A2 = B4 )
=> ( A = B ) )
& ( ( A2 != B4 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B4 @ C3 ) )
& ~ ( member_a @ B4 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_96_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B4: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B4 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B4 @ B ) )
= ( ( ( A2 = B4 )
=> ( A = B ) )
& ( ( A2 != B4 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B4 @ C3 ) )
& ~ ( member_nat @ B4 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_97_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_98_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_99_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_100_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_101_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B7: set_a] :
( ( A
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_102_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat @ X @ B7 ) )
=> ( member_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_103_insertI2,axiom,
! [A2: a,B: set_a,B4: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_104_insertI2,axiom,
! [A2: nat,B: set_nat,B4: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B4 @ B ) ) ) ).
% insertI2
thf(fact_105_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_106_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_107_insertE,axiom,
! [A2: a,B4: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B4 @ A ) )
=> ( ( A2 != B4 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_108_insertE,axiom,
! [A2: nat,B4: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B4 @ A ) )
=> ( ( A2 != B4 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_109_DiffD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_110_DiffD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_111_DiffD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_112_DiffD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_113_DiffE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_114_DiffE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_115_subset__singleton__iff,axiom,
! [X4: set_a,A2: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_116_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_117_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_118_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_119_Diff__single__insert,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_120_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_121_subset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_122_singleton__inject,axiom,
! [A2: a,B4: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A2 = B4 ) ) ).
% singleton_inject
thf(fact_123_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_124_doubleton__eq__iff,axiom,
! [A2: a,B4: a,C2: a,D2: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C2 )
& ( B4 = D2 ) )
| ( ( A2 = D2 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_125_singleton__iff,axiom,
! [B4: nat,A2: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B4 = A2 ) ) ).
% singleton_iff
thf(fact_126_singleton__iff,axiom,
! [B4: a,A2: a] :
( ( member_a @ B4 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B4 = A2 ) ) ).
% singleton_iff
thf(fact_127_singletonD,axiom,
! [B4: nat,A2: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B4 = A2 ) ) ).
% singletonD
thf(fact_128_singletonD,axiom,
! [B4: a,A2: a] :
( ( member_a @ B4 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B4 = A2 ) ) ).
% singletonD
thf(fact_129_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_130_insert__Diff__if,axiom,
! [X: a,B: set_a,A: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_131_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_132_Diff__insert__absorb,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_133_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_134_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_135_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_136_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_137_sumsetp_OsumsetI,axiom,
! [A: a > $o,A2: a,B: a > $o,B4: a] :
( ( A @ A2 )
=> ( ( member_a @ A2 @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ ( addition @ A2 @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_138_sumsetp_Osimps,axiom,
! [A: a > $o,B: a > $o,A2: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
= ( ? [A4: a,B3: a] :
( ( A2
= ( addition @ A4 @ B3 ) )
& ( A @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_139_sumsetp_Ocases,axiom,
! [A: a > $o,B: a > $o,A2: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
=> ~ ! [A3: a,B2: a] :
( ( A2
= ( addition @ A3 @ B2 ) )
=> ( ( A @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_140_card__sumset__0__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_141_sumset__empty_H_I1_J,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_142_sumset__empty_H_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_143_sumset__subset__Un_I1_J,axiom,
! [A: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_144_sumset__subset__Un_I2_J,axiom,
! [A: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_145_card__sumset__le,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ).
% card_sumset_le
thf(fact_146_finite__sumset,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% finite_sumset
thf(fact_147_card__le__sumset,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_148_IntI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_149_IntI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_150_Int__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_151_Int__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_152_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_153_UnCI,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_154_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_155_Un__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_156_sumset__subset__Un1,axiom,
! [A: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_157_sumset__subset__Un2,axiom,
! [A: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_158_finite__sumset_H,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% finite_sumset'
thf(fact_159_card__sumset__0__iff_H,axiom,
! [A: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_160_infinite__sumset__aux,axiom,
! [A: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_161_infinite__sumset__iff,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_162_Int__subset__iff,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_163_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_164_Int__insert__left__if0,axiom,
! [A2: nat,C: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_165_Int__insert__left__if0,axiom,
! [A2: a,C: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_166_Int__insert__left__if1,axiom,
! [A2: nat,C: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_167_Int__insert__left__if1,axiom,
! [A2: a,C: set_a,B: set_a] :
( ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_168_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_169_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_170_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_171_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_172_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_173_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_174_Un__insert__left,axiom,
! [A2: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_175_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_176_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_177_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_178_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_179_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_180_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_181_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_182_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_183_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_184_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_185_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_186_card__sumset__singleton__eq,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( ( member_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) )
& ( ~ ( member_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_187_disjoint__insert_I2_J,axiom,
! [A: set_nat,B4: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B4 @ B ) ) )
= ( ~ ( member_nat @ B4 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_188_disjoint__insert_I2_J,axiom,
! [A: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_189_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_190_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_191_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_192_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_193_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_194_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_195_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_196_sumset__Int__carrier__eq_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_197_sumset__Int__carrier__eq_I1_J,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_198_sumset__Int__carrier,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier
thf(fact_199_sumset__is__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_200_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_201_UnE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_202_IntE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_203_IntE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_204_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_205_UnI1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_206_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_207_UnI2,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_208_IntD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_209_IntD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% IntD1
thf(fact_210_IntD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_211_IntD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_212_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_213_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_214_Un__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_215_Int__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_216_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_217_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_218_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_219_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_220_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_221_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_222_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_223_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_224_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_225_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_226_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_227_Un__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_228_Int__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_229_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_230_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_231_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_232_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_233_Diff__Int,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C ) ) ) ).
% Diff_Int
thf(fact_234_Diff__Un,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C ) ) ) ).
% Diff_Un
thf(fact_235_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_236_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_237_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_238_subset__UnE,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C
!= ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_239_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_240_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_241_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_242_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_243_Un__least,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_244_Un__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_245_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_246_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_247_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_248_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_249_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_250_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_251_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_252_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_253_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_254_Int__greatest,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_255_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_256_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_257_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_258_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_259_Int__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_260_Int__insert__left,axiom,
! [A2: nat,C: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) )
& ( ~ ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_261_Int__insert__left,axiom,
! [A2: a,C: set_a,B: set_a] :
( ( ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_262_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_263_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_264_Un__Diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_265_Int__Diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_266_Diff__Int2,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_267_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_268_Diff__Int__distrib,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_269_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_270_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_271_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_272_singleton__Un__iff,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_273_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_274_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_275_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_276_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_277_sumset__insert1,axiom,
! [A: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_insert1
thf(fact_278_sumset__insert2,axiom,
! [B: set_a,A: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_insert2
thf(fact_279_card__insert__disjoint,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ~ ( member_a @ X @ A )
=> ( ( finite_card_a @ ( insert_a @ X @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_280_card__insert__disjoint,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_281_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_282_card__0__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_283_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_284_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_285_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_286_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_287_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_288_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_289_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_290_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_291_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_292_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_293_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_294_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_295_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_296_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_297_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_298_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_299_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_300_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_301_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( ord_less_eq_set_a @ A2 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_302_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_303_finite__has__maximal2,axiom,
! [A: set_num,A2: num] :
( ( finite_finite_num @ A )
=> ( ( member_num @ A2 @ A )
=> ? [X3: num] :
( ( member_num @ X3 @ A )
& ( ord_less_eq_num @ A2 @ X3 )
& ! [Xa: num] :
( ( member_num @ Xa @ A )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_304_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( ord_less_eq_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_305_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_306_finite__has__minimal2,axiom,
! [A: set_num,A2: num] :
( ( finite_finite_num @ A )
=> ( ( member_num @ A2 @ A )
=> ? [X3: num] :
( ( member_num @ X3 @ A )
& ( ord_less_eq_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_307_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_308_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_309_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_310_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_311_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_312_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_313_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_314_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_315_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_316_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_317_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_318_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_319_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_320_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_321_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_322_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_323_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_324_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_325_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_326_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_327_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_328_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_329_finite__has__minimal,axiom,
! [A: set_num] :
( ( finite_finite_num @ A )
=> ( ( A != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_330_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_331_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_332_finite__has__maximal,axiom,
! [A: set_num] :
( ( finite_finite_num @ A )
=> ( ( A != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_333_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_334_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_335_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_336_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_337_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_338_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_339_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A6: set_nat,B3: nat] :
( ( A4
= ( insert_nat @ B3 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_340_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A6: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_341_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A3: nat] :
( A2
= ( insert_nat @ A3 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_342_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A3: a] :
( A2
= ( insert_a @ A3 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_343_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_344_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_345_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_346_infinite__arbitrarily__large,axiom,
! [A: set_a,N: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N )
& ( ord_less_eq_set_a @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_347_card__insert__le,axiom,
! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X @ A ) ) ) ).
% card_insert_le
thf(fact_348_finite__subset__induct_H,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_349_finite__subset__induct_H,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_350_finite__subset__induct,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_351_finite__subset__induct,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_352_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_353_card__eq__0__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_354_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_355_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_356_infinite__coinduct,axiom,
! [X4: set_nat > $o,A: set_nat] :
( ( X4 @ A )
=> ( ! [A8: set_nat] :
( ( X4 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X4 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_357_infinite__coinduct,axiom,
! [X4: set_a > $o,A: set_a] :
( ( X4 @ A )
=> ( ! [A8: set_a] :
( ( X4 @ A8 )
=> ? [X5: a] :
( ( member_a @ X5 @ A8 )
& ( ( X4 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_358_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_359_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_360_card__insert__if,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( ( member_a @ X @ A )
=> ( ( finite_card_a @ ( insert_a @ X @ A ) )
= ( finite_card_a @ A ) ) )
& ( ~ ( member_a @ X @ A )
=> ( ( finite_card_a @ ( insert_a @ X @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_361_card__insert__if,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_362_card__Suc__eq__finite,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B3: a,B6: set_a] :
( ( A
= ( insert_a @ B3 @ B6 ) )
& ~ ( member_a @ B3 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_363_card__Suc__eq__finite,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B3: nat,B6: set_nat] :
( ( A
= ( insert_nat @ B3 @ B6 ) )
& ~ ( member_nat @ B3 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_364_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_365_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_366_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_367_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_368_exists__subset__between,axiom,
! [A: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_369_exists__subset__between,axiom,
! [A: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( finite_finite_a @ C )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A @ B7 )
& ( ord_less_eq_set_a @ B7 @ C )
& ( ( finite_card_a @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_370_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_371_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_372_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_373_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_374_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_375_card__le__sym__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_376_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_377_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_378_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_379_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_380_card__1__singleton__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: a] :
( A
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_381_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B2: nat,B7: set_nat] :
( ( A
= ( insert_nat @ B2 @ B7 ) )
& ~ ( member_nat @ B2 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_382_card__eq__SucD,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
=> ? [B2: a,B7: set_a] :
( ( A
= ( insert_a @ B2 @ B7 ) )
& ~ ( member_a @ B2 @ B7 )
& ( ( finite_card_a @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_383_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B3: nat,B6: set_nat] :
( ( A
= ( insert_nat @ B3 @ B6 ) )
& ~ ( member_nat @ B3 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_384_card__Suc__eq,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B3: a,B6: set_a] :
( ( A
= ( insert_a @ B3 @ B6 ) )
& ~ ( member_a @ B3 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_385_card__le__Suc0__iff__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ A )
=> ( X2 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_386_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( X2 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_387_card__le__Suc__iff,axiom,
! [N: nat,A: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A ) )
= ( ? [A4: a,B6: set_a] :
( ( A
= ( insert_a @ A4 @ B6 ) )
& ~ ( member_a @ A4 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B6 ) )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_388_card__le__Suc__iff,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A ) )
= ( ? [A4: nat,B6: set_nat] :
( ( A
= ( insert_nat @ A4 @ B6 ) )
& ~ ( member_nat @ A4 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_389_card__Diff1__le,axiom,
! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ).
% card_Diff1_le
thf(fact_390_card__Suc__Diff1,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_391_card__Suc__Diff1,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_392_card_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_393_card_Oinsert__remove,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( finite_card_a @ ( insert_a @ X @ A ) )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_394_card_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ A )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_395_card_Oremove,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( finite_card_a @ A )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_396_card__2__iff,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a,Y4: a] :
( ( S
= ( insert_a @ X2 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( X2 != Y4 ) ) ) ) ).
% card_2_iff
thf(fact_397_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_398_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_399_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_400_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_401_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_402_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_403_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_404_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_405_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_406_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_407_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_408_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_409_inf_Oright__idem,axiom,
! [A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ B4 )
= ( inf_inf_set_a @ A2 @ B4 ) ) ).
% inf.right_idem
thf(fact_410_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_411_inf_Oleft__idem,axiom,
! [A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B4 ) )
= ( inf_inf_set_a @ A2 @ B4 ) ) ).
% inf.left_idem
thf(fact_412_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_413_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_414_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_415_sup_Oright__idem,axiom,
! [A2: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B4 ) @ B4 )
= ( sup_sup_set_a @ A2 @ B4 ) ) ).
% sup.right_idem
thf(fact_416_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_417_sup_Oleft__idem,axiom,
! [A2: set_a,B4: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B4 ) )
= ( sup_sup_set_a @ A2 @ B4 ) ) ).
% sup.left_idem
thf(fact_418_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_419_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_420_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_421_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_422_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_423_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_424_inf_Obounded__iff,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_425_inf_Obounded__iff,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B4 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_426_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_427_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_428_sup_Obounded__iff,axiom,
! [B4: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_a @ B4 @ A2 )
& ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_429_sup_Obounded__iff,axiom,
! [B4: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B4 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_430_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_431_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_432_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_433_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_434_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_435_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_436_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_437_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_438_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A2 @ B4 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_439_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_440_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B4 ) )
= ( ( A2 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_441_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_442_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_443_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_444_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_445_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_446_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_447_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_448_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_449_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_450_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_451_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_452_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_453_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_454_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_455_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_456_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_457_diff__right__commute,axiom,
! [A2: nat,C2: nat,B4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B4 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_458_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_459_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_460_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_461_inf_Oleft__commute,axiom,
! [B4: set_a,A2: set_a,C2: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A2 @ C2 ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_462_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_463_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_464_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_465_inf_Oassoc,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_466_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_467_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_468_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_469_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_470_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_471_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_472_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_473_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_474_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_475_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_476_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_477_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_478_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_479_sup_Oleft__commute,axiom,
! [B4: set_a,A2: set_a,C2: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A2 @ C2 ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_480_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_481_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_482_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_483_sup_Oassoc,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B4 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_484_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_485_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_486_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_487_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_488_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_489_card__Diff__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_490_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_491_diff__card__le__card__Diff,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_492_card__Diff__subset__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_493_card__Diff__subset__Int,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_494_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_495_inf_OcoboundedI2,axiom,
! [B4: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_496_inf_OcoboundedI2,axiom,
! [B4: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_497_inf_OcoboundedI1,axiom,
! [A2: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_498_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_499_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_500_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_501_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_502_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_503_inf_Ocobounded2,axiom,
! [A2: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_504_inf_Ocobounded2,axiom,
! [A2: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_505_inf_Ocobounded1,axiom,
! [A2: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ A2 ) ).
% inf.cobounded1
thf(fact_506_inf_Ocobounded1,axiom,
! [A2: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ A2 ) ).
% inf.cobounded1
thf(fact_507_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_508_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_509_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_510_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_511_inf_OboundedI,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_512_inf_OboundedI,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_513_inf_OboundedE,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B4 )
=> ~ ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_514_inf_OboundedE,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B4 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_515_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_516_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_517_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_518_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_519_inf_Oabsorb2,axiom,
! [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_520_inf_Oabsorb2,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_521_inf_Oabsorb1,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= A2 ) ) ).
% inf.absorb1
thf(fact_522_inf_Oabsorb1,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( inf_inf_nat @ A2 @ B4 )
= A2 ) ) ).
% inf.absorb1
thf(fact_523_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_524_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_525_inf__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F3 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F3 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_526_inf__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_527_inf_OorderI,axiom,
! [A2: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B4 ) )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% inf.orderI
thf(fact_528_inf_OorderI,axiom,
! [A2: nat,B4: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B4 ) )
=> ( ord_less_eq_nat @ A2 @ B4 ) ) ).
% inf.orderI
thf(fact_529_inf_OorderE,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B4 ) ) ) ).
% inf.orderE
thf(fact_530_inf_OorderE,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( A2
= ( inf_inf_nat @ A2 @ B4 ) ) ) ).
% inf.orderE
thf(fact_531_le__infI2,axiom,
! [B4: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_532_le__infI2,axiom,
! [B4: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_533_le__infI1,axiom,
! [A2: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_534_le__infI1,axiom,
! [A2: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_535_inf__mono,axiom,
! [A2: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_536_inf__mono,axiom,
! [A2: nat,C2: nat,B4: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_537_le__infI,axiom,
! [X: set_a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% le_infI
thf(fact_538_le__infI,axiom,
! [X: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B4 ) ) ) ) ).
% le_infI
thf(fact_539_le__infE,axiom,
! [X: set_a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_540_le__infE,axiom,
! [X: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_541_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_542_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_543_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_544_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_545_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_546_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_547_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_548_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_549_sup_OcoboundedI2,axiom,
! [C2: set_a,B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_550_sup_OcoboundedI2,axiom,
! [C2: nat,B4: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_551_sup_OcoboundedI1,axiom,
! [C2: set_a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_552_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_553_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_554_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_555_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_556_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_557_sup_Ocobounded2,axiom,
! [B4: set_a,A2: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A2 @ B4 ) ) ).
% sup.cobounded2
thf(fact_558_sup_Ocobounded2,axiom,
! [B4: nat,A2: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A2 @ B4 ) ) ).
% sup.cobounded2
thf(fact_559_sup_Ocobounded1,axiom,
! [A2: set_a,B4: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B4 ) ) ).
% sup.cobounded1
thf(fact_560_sup_Ocobounded1,axiom,
! [A2: nat,B4: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B4 ) ) ).
% sup.cobounded1
thf(fact_561_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_562_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_563_sup_OboundedI,axiom,
! [B4: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_564_sup_OboundedI,axiom,
! [B4: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_565_sup_OboundedE,axiom,
! [B4: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A2 )
=> ~ ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_566_sup_OboundedE,axiom,
! [B4: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B4 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_567_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_568_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_569_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_570_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_571_sup_Oabsorb2,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_572_sup_Oabsorb2,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( sup_sup_nat @ A2 @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_573_sup_Oabsorb1,axiom,
! [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= A2 ) ) ).
% sup.absorb1
thf(fact_574_sup_Oabsorb1,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B4 )
= A2 ) ) ).
% sup.absorb1
thf(fact_575_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z3 @ X3 )
=> ( ord_less_eq_set_a @ ( F3 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_576_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_577_sup_OorderI,axiom,
! [A2: set_a,B4: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A2 ) ) ).
% sup.orderI
thf(fact_578_sup_OorderI,axiom,
! [A2: nat,B4: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A2 ) ) ).
% sup.orderI
thf(fact_579_sup_OorderE,axiom,
! [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B4 ) ) ) ).
% sup.orderE
thf(fact_580_sup_OorderE,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% sup.orderE
thf(fact_581_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_582_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_583_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_584_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_585_sup__mono,axiom,
! [A2: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B4 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_586_sup__mono,axiom,
! [A2: nat,C2: nat,B4: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B4 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_587_sup_Omono,axiom,
! [C2: set_a,A2: set_a,D2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D2 ) @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ) ).
% sup.mono
thf(fact_588_sup_Omono,axiom,
! [C2: nat,A2: nat,D2: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A2 @ B4 ) ) ) ) ).
% sup.mono
thf(fact_589_le__supI2,axiom,
! [X: set_a,B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ).
% le_supI2
thf(fact_590_le__supI2,axiom,
! [X: nat,B4: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% le_supI2
thf(fact_591_le__supI1,axiom,
! [X: set_a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ).
% le_supI1
thf(fact_592_le__supI1,axiom,
! [X: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% le_supI1
thf(fact_593_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_594_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_595_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_596_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_597_le__supI,axiom,
! [A2: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_598_le__supI,axiom,
! [A2: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_599_le__supE,axiom,
! [A2: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_600_le__supE,axiom,
! [A2: nat,B4: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B4 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B4 @ X ) ) ) ).
% le_supE
thf(fact_601_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_602_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_603_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_604_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_605_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_606_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_607_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_608_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_609_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_610_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_611_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_612_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_613_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_614_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_615_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_616_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_617_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_618_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_619_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_620_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_621_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_622_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_623_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_624_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_625_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_626_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_627_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_628_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_629_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_630_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_631_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_632_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_633_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_634_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_635_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_636_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_637_lift__Suc__antimono__le,axiom,
! [F3: nat > set_a,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_638_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_639_lift__Suc__antimono__le,axiom,
! [F3: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_640_lift__Suc__mono__le,axiom,
! [F3: nat > set_a,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_641_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_642_lift__Suc__mono__le,axiom,
! [F3: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_643_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_644_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_645_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_646_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_647_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R2 @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B2: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_648_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R2 @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B2: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_649_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R2 @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B2: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_650_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R2 @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B2: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_651_card__2__iff_H,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ S )
& ? [Y4: a] :
( ( member_a @ Y4 @ S )
& ( X2 != Y4 )
& ! [Z4: a] :
( ( member_a @ Z4 @ S )
=> ( ( Z4 = X2 )
| ( Z4 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_652_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_653_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_654_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_655_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_656_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_657_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_658_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_659_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_660_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_661_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_662_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_663_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_664_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_665_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_666_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_667_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_668_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_669_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_670_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B4: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_671_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_672_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_673_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_674_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_675_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_676_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_677_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_678_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_679_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_680_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_681_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_682_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_683_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_684_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > num] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_num @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_685_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > num] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_num @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_686_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_687_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_688_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_689_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_690_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_691_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_692_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_693_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_694_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_695_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_696_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_697_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_698_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_699_ord__le__eq__subst,axiom,
! [A2: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_700_ord__le__eq__subst,axiom,
! [A2: set_a,B4: set_a,F3: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_701_ord__le__eq__subst,axiom,
! [A2: set_a,B4: set_a,F3: set_a > num,C2: num] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_702_ord__le__eq__subst,axiom,
! [A2: nat,B4: nat,F3: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_703_ord__le__eq__subst,axiom,
! [A2: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_704_ord__le__eq__subst,axiom,
! [A2: nat,B4: nat,F3: nat > num,C2: num] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_705_ord__le__eq__subst,axiom,
! [A2: num,B4: num,F3: num > set_a,C2: set_a] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_706_ord__le__eq__subst,axiom,
! [A2: num,B4: num,F3: num > nat,C2: nat] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_707_ord__le__eq__subst,axiom,
! [A2: num,B4: num,F3: num > num,C2: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_708_ord__eq__le__subst,axiom,
! [A2: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_709_ord__eq__le__subst,axiom,
! [A2: nat,F3: set_a > nat,B4: set_a,C2: set_a] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_710_ord__eq__le__subst,axiom,
! [A2: num,F3: set_a > num,B4: set_a,C2: set_a] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_711_ord__eq__le__subst,axiom,
! [A2: set_a,F3: nat > set_a,B4: nat,C2: nat] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_712_ord__eq__le__subst,axiom,
! [A2: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_713_ord__eq__le__subst,axiom,
! [A2: num,F3: nat > num,B4: nat,C2: nat] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_714_ord__eq__le__subst,axiom,
! [A2: set_a,F3: num > set_a,B4: num,C2: num] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_715_ord__eq__le__subst,axiom,
! [A2: nat,F3: num > nat,B4: num,C2: num] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_716_ord__eq__le__subst,axiom,
! [A2: num,F3: num > num,B4: num,C2: num] :
( ( A2
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_717_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_718_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_719_verit__la__disequality,axiom,
! [A2: nat,B4: nat] :
( ( A2 = B4 )
| ~ ( ord_less_eq_nat @ A2 @ B4 )
| ~ ( ord_less_eq_nat @ B4 @ A2 ) ) ).
% verit_la_disequality
thf(fact_720_verit__la__disequality,axiom,
! [A2: num,B4: num] :
( ( A2 = B4 )
| ~ ( ord_less_eq_num @ A2 @ B4 )
| ~ ( ord_less_eq_num @ B4 @ A2 ) ) ).
% verit_la_disequality
thf(fact_721_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_722_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_723_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_724_order__subst2,axiom,
! [A2: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_725_order__subst2,axiom,
! [A2: set_a,B4: set_a,F3: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_726_order__subst2,axiom,
! [A2: set_a,B4: set_a,F3: set_a > num,C2: num] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_num @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_727_order__subst2,axiom,
! [A2: nat,B4: nat,F3: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_728_order__subst2,axiom,
! [A2: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_729_order__subst2,axiom,
! [A2: nat,B4: nat,F3: nat > num,C2: num] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_num @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_730_order__subst2,axiom,
! [A2: num,B4: num,F3: num > set_a,C2: set_a] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_731_order__subst2,axiom,
! [A2: num,B4: num,F3: num > nat,C2: nat] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_732_order__subst2,axiom,
! [A2: num,B4: num,F3: num > num,C2: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ord_less_eq_num @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F3 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_733_order__subst1,axiom,
! [A2: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_734_order__subst1,axiom,
! [A2: set_a,F3: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_735_order__subst1,axiom,
! [A2: set_a,F3: num > set_a,B4: num,C2: num] :
( ( ord_less_eq_set_a @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_736_order__subst1,axiom,
! [A2: nat,F3: set_a > nat,B4: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_737_order__subst1,axiom,
! [A2: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_738_order__subst1,axiom,
! [A2: nat,F3: num > nat,B4: num,C2: num] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_739_order__subst1,axiom,
! [A2: num,F3: set_a > num,B4: set_a,C2: set_a] :
( ( ord_less_eq_num @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_740_order__subst1,axiom,
! [A2: num,F3: nat > num,B4: nat,C2: nat] :
( ( ord_less_eq_num @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_741_order__subst1,axiom,
! [A2: num,F3: num > num,B4: num,C2: num] :
( ( ord_less_eq_num @ A2 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_742_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_743_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_744_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: num,Z: num] : ( Y2 = Z ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_745_antisym,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% antisym
thf(fact_746_antisym,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% antisym
thf(fact_747_antisym,axiom,
! [A2: num,B4: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ord_less_eq_num @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% antisym
thf(fact_748_dual__order_Otrans,axiom,
! [B4: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_749_dual__order_Otrans,axiom,
! [B4: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_750_dual__order_Otrans,axiom,
! [B4: num,A2: num,C2: num] :
( ( ord_less_eq_num @ B4 @ A2 )
=> ( ( ord_less_eq_num @ C2 @ B4 )
=> ( ord_less_eq_num @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_751_dual__order_Oantisym,axiom,
! [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_752_dual__order_Oantisym,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_753_dual__order_Oantisym,axiom,
! [B4: num,A2: num] :
( ( ord_less_eq_num @ B4 @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_754_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_755_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_756_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: num,Z: num] : ( Y2 = Z ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_757_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_758_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B4: num] :
( ! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_759_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_760_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_761_order__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_762_order_Otrans,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_763_order_Otrans,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_764_order_Otrans,axiom,
! [A2: num,B4: num,C2: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ord_less_eq_num @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_765_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_766_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_767_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_768_ord__le__eq__trans,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_769_ord__le__eq__trans,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_770_ord__le__eq__trans,axiom,
! [A2: num,B4: num,C2: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_num @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_771_ord__eq__le__trans,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( A2 = B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_772_ord__eq__le__trans,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( A2 = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_773_ord__eq__le__trans,axiom,
! [A2: num,B4: num,C2: num] :
( ( A2 = B4 )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ord_less_eq_num @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_774_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_775_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_776_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: num,Z: num] : ( Y2 = Z ) )
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_777_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_778_le__cases3,axiom,
! [X: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_779_nle__le,axiom,
! [A2: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A2 )
& ( B4 != A2 ) ) ) ).
% nle_le
thf(fact_780_nle__le,axiom,
! [A2: num,B4: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B4 ) )
= ( ( ord_less_eq_num @ B4 @ A2 )
& ( B4 != A2 ) ) ) ).
% nle_le
thf(fact_781_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_782_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_783_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_784_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_785_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_786_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_787_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_788_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_789_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_790_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_791_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_792_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_793_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_794_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_795_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_796_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_797_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_798_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X2: a] :
( A6
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_799_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_800_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ A2 @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ ( inf_inf_set_nat @ A @ G ) ) ) )
& ( ~ ( member_nat @ A2 @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_801_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( ( member_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ G ) ) ) )
& ( ~ ( member_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_802_arg__min__least,axiom,
! [S: set_nat,Y: nat,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F3 @ ( lattic7446932960582359483at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_803_arg__min__least,axiom,
! [S: set_a,Y: a,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F3 @ ( lattic6340287419671400565_a_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_804_arg__min__least,axiom,
! [S: set_nat,Y: nat,F3: nat > num] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_num @ ( F3 @ ( lattic4004264746738138117at_num @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_805_arg__min__least,axiom,
! [S: set_a,Y: a,F3: a > num] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_num @ ( F3 @ ( lattic2897619205827179199_a_num @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_806_insert__subsetI,axiom,
! [X: nat,A: set_nat,X4: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_807_insert__subsetI,axiom,
! [X: a,A: set_a,X4: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_808_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,B: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ A2 )
=> ~ ! [A3: nat,B2: nat] :
( ( A2
= ( Addition @ A3 @ B2 ) )
=> ( ( A @ A3 )
=> ( ( member_nat @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_809_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,B: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ A2 )
=> ~ ! [A3: a,B2: a] :
( ( A2
= ( Addition @ A3 @ B2 ) )
=> ( ( A @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_810_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,B: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ A2 )
= ( ? [A4: nat,B3: nat] :
( ( A2
= ( Addition @ A4 @ B3 ) )
& ( A @ A4 )
& ( member_nat @ A4 @ G )
& ( B @ B3 )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_811_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,B: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ A2 )
= ( ? [A4: a,B3: a] :
( ( A2
= ( Addition @ A4 @ B3 ) )
& ( A @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_812_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat,B: nat > $o,B4: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( A @ A2 )
=> ( ( member_nat @ A2 @ G )
=> ( ( B @ B4 )
=> ( ( member_nat @ B4 @ G )
=> ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_813_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A @ A2 )
=> ( ( member_a @ A2 @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_814_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
=> ~ ! [A3: nat,B2: nat] :
( ( A2
= ( Addition @ A3 @ B2 ) )
=> ( ( member_nat @ A3 @ A )
=> ( ( member_nat @ A3 @ G )
=> ( ( member_nat @ B2 @ B )
=> ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_815_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A2
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_816_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
= ( ? [A4: nat,B3: nat] :
( ( A2
= ( Addition @ A4 @ B3 ) )
& ( member_nat @ A4 @ A )
& ( member_nat @ A4 @ G )
& ( member_nat @ B3 @ B )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_817_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= ( ? [A4: a,B3: a] :
( ( A2
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_818_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_819_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B4: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_nat @ A2 @ G )
=> ( ( member_nat @ B4 @ B )
=> ( ( member_nat @ B4 @ G )
=> ( member_nat @ ( Addition @ A2 @ B4 ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_820_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A2 @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_821_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_822_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_823_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_824_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_825_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_826_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_827_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_828_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_829_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_830_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_831_additive__abelian__group_Osumset__subset__Un2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,B5: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B5 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un2
thf(fact_832_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_833_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_834_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_835_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_836_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_837_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_838_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_839_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_840_additive__abelian__group_Osumset__subset__Un_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( sup_sup_set_a @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un(1)
thf(fact_841_additive__abelian__group_Osumset__subset__Un_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A @ C ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_Un(2)
thf(fact_842_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
& ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) )
| ( ( ( inf_inf_set_nat @ A @ G )
!= bot_bot_set_nat )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_843_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
& ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A @ G )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_844_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) )
= ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_845_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) )
= ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_846_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A )
= ( inf_inf_set_a @ A @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_847_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_848_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A @ G )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_849_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_850_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ G ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_851_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_nat @ A2 @ G )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_852_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ G )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_853_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_854_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_855_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_856_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_857_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_858_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_859_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_860_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_861_remove__def,axiom,
( remove_a
= ( ^ [X2: a,A6: set_a] : ( minus_minus_set_a @ A6 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_862_sumset__D_I2_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A )
= ( inf_inf_set_a @ A @ g ) ) ).
% sumset_D(2)
thf(fact_863_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_864_member__remove,axiom,
! [X: a,Y: a,A: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A ) )
= ( ( member_a @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_865_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_866_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_867_right__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ g )
=> ( ( addition @ A2 @ zero )
= A2 ) ) ).
% right_unit
thf(fact_868_left__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ g )
=> ( ( addition @ zero @ A2 )
= A2 ) ) ).
% left_unit
thf(fact_869_sumset__D_I1_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A @ g ) ) ).
% sumset_D(1)
thf(fact_870_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_871_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_872_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_873_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_874_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_875_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ g )
& ( ( addition @ U @ X2 )
= zero )
& ( ( addition @ X2 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_876_invertible__right__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_877_invertible__left__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_878_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_879_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_880_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_881_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_882_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_883_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_884_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_885_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_886_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_887_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_888_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_889_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_890_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_891_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_892_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_893_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_894_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_895_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_896_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% invertible_left_inverse
thf(fact_897_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% invertible_right_inverse
thf(fact_898_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,X: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ X @ G )
=> ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_899_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ X @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_900_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G3 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_901_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_902_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_903_inverse__subgroupD,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H2 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_904_monoid_OsubgroupI,axiom,
! [M2: set_nat,Composition: nat > nat > nat,Unit: nat,G: set_nat] :
( ( group_monoid_nat @ M2 @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ G @ M2 )
=> ( ( member_nat @ Unit @ G )
=> ( ! [G3: nat,H: nat] :
( ( member_nat @ G3 @ G )
=> ( ( member_nat @ H @ G )
=> ( member_nat @ ( Composition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G )
=> ( group_invertible_nat @ M2 @ Composition @ Unit @ G3 ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G )
=> ( member_nat @ ( group_inverse_nat @ M2 @ Composition @ Unit @ G3 ) @ G ) )
=> ( group_subgroup_nat @ G @ M2 @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_905_monoid_OsubgroupI,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_monoid_a @ M2 @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G @ M2 )
=> ( ( member_a @ Unit @ G )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( Composition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( group_invertible_a @ M2 @ Composition @ Unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( member_a @ ( group_inverse_a @ M2 @ Composition @ Unit @ G3 ) @ G ) )
=> ( group_subgroup_a @ G @ M2 @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_906_inverse__subgroupI,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ H2 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_907_image__eqI,axiom,
! [B4: a,F3: a > a,X: a,A: set_a] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_908_image__eqI,axiom,
! [B4: nat,F3: a > nat,X: a,A: set_a] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_nat @ B4 @ ( image_a_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_909_image__eqI,axiom,
! [B4: a,F3: nat > a,X: nat,A: set_nat] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ B4 @ ( image_nat_a @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_910_image__eqI,axiom,
! [B4: nat,F3: nat > nat,X: nat,A: set_nat] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B4 @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_911_image__empty,axiom,
! [F3: nat > nat] :
( ( image_nat_nat @ F3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_912_image__empty,axiom,
! [F3: a > a] :
( ( image_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_913_empty__is__image,axiom,
! [F3: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_914_empty__is__image,axiom,
! [F3: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_915_image__is__empty,axiom,
! [F3: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_916_image__is__empty,axiom,
! [F3: a > a,A: set_a] :
( ( ( image_a_a @ F3 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_917_finite__imageI,axiom,
! [F: set_a,H3: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_918_finite__imageI,axiom,
! [F: set_a,H3: a > nat] :
( ( finite_finite_a @ F )
=> ( finite_finite_nat @ ( image_a_nat @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_919_finite__imageI,axiom,
! [F: set_nat,H3: nat > a] :
( ( finite_finite_nat @ F )
=> ( finite_finite_a @ ( image_nat_a @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_920_finite__imageI,axiom,
! [F: set_nat,H3: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_921_insert__image,axiom,
! [X: a,A: set_a,F3: a > a] :
( ( member_a @ X @ A )
=> ( ( insert_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ A ) )
= ( image_a_a @ F3 @ A ) ) ) ).
% insert_image
thf(fact_922_insert__image,axiom,
! [X: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ A ) )
= ( image_nat_nat @ F3 @ A ) ) ) ).
% insert_image
thf(fact_923_insert__image,axiom,
! [X: nat,A: set_nat,F3: nat > a] :
( ( member_nat @ X @ A )
=> ( ( insert_a @ ( F3 @ X ) @ ( image_nat_a @ F3 @ A ) )
= ( image_nat_a @ F3 @ A ) ) ) ).
% insert_image
thf(fact_924_image__insert,axiom,
! [F3: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F3 @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ ( F3 @ A2 ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_insert
thf(fact_925_image__insert,axiom,
! [F3: a > a,A2: a,B: set_a] :
( ( image_a_a @ F3 @ ( insert_a @ A2 @ B ) )
= ( insert_a @ ( F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_insert
thf(fact_926_subset__image__iff,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_927_subset__image__iff,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_928_image__subset__iff,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F3 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_929_image__subset__iff,axiom,
! [F3: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F3 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_930_subset__imageE,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_nat @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_931_subset__imageE,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_932_image__subsetI,axiom,
! [A: set_a,F3: a > nat,B: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_933_image__subsetI,axiom,
! [A: set_nat,F3: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_934_image__subsetI,axiom,
! [A: set_a,F3: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_935_image__subsetI,axiom,
! [A: set_nat,F3: nat > a,B: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_936_image__mono,axiom,
! [A: set_nat,B: set_nat,F3: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_mono
thf(fact_937_image__mono,axiom,
! [A: set_a,B: set_a,F3: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_mono
thf(fact_938_all__subset__image,axiom,
! [F3: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A )
=> ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_939_all__subset__image,axiom,
! [F3: a > a,A: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F3 @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A )
=> ( P @ ( image_a_a @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_940_image__Un,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F3 @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_Un
thf(fact_941_image__Un,axiom,
! [F3: a > a,A: set_a,B: set_a] :
( ( image_a_a @ F3 @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F3 @ A ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_Un
thf(fact_942_imageI,axiom,
! [X: a,A: set_a,F3: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ A ) ) ) ).
% imageI
thf(fact_943_imageI,axiom,
! [X: a,A: set_a,F3: a > nat] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ ( image_a_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_944_imageI,axiom,
! [X: nat,A: set_nat,F3: nat > a] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F3 @ X ) @ ( image_nat_a @ F3 @ A ) ) ) ).
% imageI
thf(fact_945_imageI,axiom,
! [X: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_946_image__iff,axiom,
! [Z2: a,F3: a > a,A: set_a] :
( ( member_a @ Z2 @ ( image_a_a @ F3 @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z2
= ( F3 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_947_image__iff,axiom,
! [Z2: nat,F3: nat > nat,A: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F3 @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z2
= ( F3 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_948_bex__imageD,axiom,
! [F3: a > a,A: set_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( image_a_a @ F3 @ A ) )
& ( P @ X5 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_949_bex__imageD,axiom,
! [F3: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F3 @ A ) )
& ( P @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_950_image__cong,axiom,
! [M2: set_a,N5: set_a,F3: a > a,G4: a > a] :
( ( M2 = N5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N5 )
=> ( ( F3 @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_a_a @ F3 @ M2 )
= ( image_a_a @ G4 @ N5 ) ) ) ) ).
% image_cong
thf(fact_951_image__cong,axiom,
! [M2: set_nat,N5: set_nat,F3: nat > nat,G4: nat > nat] :
( ( M2 = N5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ( F3 @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_nat_nat @ F3 @ M2 )
= ( image_nat_nat @ G4 @ N5 ) ) ) ) ).
% image_cong
thf(fact_952_ball__imageD,axiom,
! [F3: a > a,A: set_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F3 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_953_ball__imageD,axiom,
! [F3: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F3 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_954_rev__image__eqI,axiom,
! [X: a,A: set_a,B4: a,F3: a > a] :
( ( member_a @ X @ A )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_955_rev__image__eqI,axiom,
! [X: a,A: set_a,B4: nat,F3: a > nat] :
( ( member_a @ X @ A )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_nat @ B4 @ ( image_a_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_956_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B4: a,F3: nat > a] :
( ( member_nat @ X @ A )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_a @ B4 @ ( image_nat_a @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_957_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B4: nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_nat @ B4 @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_958_finite__surj,axiom,
! [A: set_a,B: set_nat,F3: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_959_finite__surj,axiom,
! [A: set_nat,B: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_960_finite__surj,axiom,
! [A: set_a,B: set_a,F3: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_961_finite__surj,axiom,
! [A: set_nat,B: set_a,F3: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_962_finite__subset__image,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_963_finite__subset__image,axiom,
! [B: set_nat,F3: a > nat,A: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_964_finite__subset__image,axiom,
! [B: set_a,F3: nat > a,A: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_965_finite__subset__image,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_966_ex__finite__subset__image,axiom,
! [F3: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A )
& ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_967_ex__finite__subset__image,axiom,
! [F3: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F3 @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A )
& ( P @ ( image_a_nat @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_968_ex__finite__subset__image,axiom,
! [F3: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F3 @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A )
& ( P @ ( image_nat_a @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_969_ex__finite__subset__image,axiom,
! [F3: a > a,A: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F3 @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A )
& ( P @ ( image_a_a @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_970_all__finite__subset__image,axiom,
! [F3: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A ) )
=> ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_971_all__finite__subset__image,axiom,
! [F3: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F3 @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A ) )
=> ( P @ ( image_a_nat @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_972_all__finite__subset__image,axiom,
! [F3: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F3 @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A ) )
=> ( P @ ( image_nat_a @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_973_all__finite__subset__image,axiom,
! [F3: a > a,A: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F3 @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A ) )
=> ( P @ ( image_a_a @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_974_image__Int__subset,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_975_image__Int__subset,axiom,
! [F3: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F3 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F3 @ A ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_976_image__diff__subset,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_977_image__diff__subset,axiom,
! [F3: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F3 @ A ) @ ( image_a_a @ F3 @ B ) ) @ ( image_a_a @ F3 @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_978_the__elem__image__unique,axiom,
! [A: set_nat,F3: nat > nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A )
=> ( ( F3 @ Y3 )
= ( F3 @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F3 @ A ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_979_the__elem__image__unique,axiom,
! [A: set_a,F3: a > a,X: a] :
( ( A != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A )
=> ( ( F3 @ Y3 )
= ( F3 @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F3 @ A ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_980_card__image__le,axiom,
! [A: set_a,F3: a > a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F3 @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_981_card__image__le,axiom,
! [A: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F3 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_982_card__image__le,axiom,
! [A: set_nat,F3: nat > a] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F3 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_983_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_984_surj__card__le,axiom,
! [A: set_a,B: set_a,F3: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_985_surj__card__le,axiom,
! [A: set_nat,B: set_a,F3: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_986_group_Oinverse__subgroupD,axiom,
! [G: set_nat,Composition: nat > nat > nat,Unit: nat,H2: set_nat] :
( ( group_group_nat @ G @ Composition @ Unit )
=> ( ( group_subgroup_nat @ ( image_nat_nat @ ( group_inverse_nat @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ H2 @ ( group_Units_nat @ G @ Composition @ Unit ) )
=> ( group_subgroup_nat @ H2 @ G @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_987_group_Oinverse__subgroupD,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ G @ Composition @ Unit ) )
=> ( group_subgroup_a @ H2 @ G @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_988_Sup__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_989_Sup__fin_Oremove,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_990_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_991_Sup__fin_Oinsert__remove,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_992_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_993_inf__Sup__absorb,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_994_Sup__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_995_Sup__fin_Oinsert,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_996_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_997_Sup__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_998_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_999_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1000_Sup__fin_Oin__idem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X @ A )
=> ( ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1001_in__image__insert__iff,axiom,
! [B: set_set_nat,X: nat,A: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X @ C4 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1002_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X @ C4 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1003_Sup__fin_OboundedE,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A )
=> ( ord_less_eq_set_a @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1004_Sup__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A )
=> ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1005_Sup__fin_OboundedI,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_a @ A3 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1006_Sup__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ A3 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1007_Sup__fin_Obounded__iff,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1008_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1009_Sup__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1010_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1011_Sup__fin_Ohom__commute,axiom,
! [H3: nat > nat,N5: set_nat] :
( ! [X3: nat,Y3: nat] :
( ( H3 @ ( sup_sup_nat @ X3 @ Y3 ) )
= ( sup_sup_nat @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_nat @ N5 )
=> ( ( N5 != bot_bot_set_nat )
=> ( ( H3 @ ( lattic1093996805478795353in_nat @ N5 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H3 @ N5 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1012_Sup__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N5: set_set_a] :
( ! [X3: set_a,Y3: set_a] :
( ( H3 @ ( sup_sup_set_a @ X3 @ Y3 ) )
= ( sup_sup_set_a @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N5 )
=> ( ( N5 != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic2918178356826803221_set_a @ N5 ) )
= ( lattic2918178356826803221_set_a @ ( image_set_a_set_a @ H3 @ N5 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1013_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1014_Sup__fin_Osubset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B ) @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1015_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1016_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ~ ( member_set_a @ X @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1017_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1018_Sup__fin_Oclosed,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] : ( member_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic2918178356826803221_set_a @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1019_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1020_Sup__fin_Ounion,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1021_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1022_Inf__fin_Oinsert__remove,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1023_Inf__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1024_Inf__fin_Oremove,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1025_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_1026_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1027_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1028_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1029_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1030_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1031_sup__Inf__absorb,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1032_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1033_card__Diff__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ A2 @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1034_card__Diff__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ~ ( member_a @ A2 @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1035_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1036_Inf__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1037_Inf__fin_Oinsert,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1038_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1039_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_1040_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1041_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1042_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1043_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1044_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1045_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1046_Inf__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1047_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1048_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1049_Inf__fin_Oin__idem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X @ A )
=> ( ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) )
= ( lattic8209813465164889211_set_a @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1050_is__singleton__altdef,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( ( finite_card_a @ A6 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1051_card__1__singletonE,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= one_one_nat )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1052_Inf__fin_Obounded__iff,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1053_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1054_Inf__fin_OboundedI,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_a @ X @ A3 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1055_Inf__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ X @ A3 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1056_Inf__fin_OboundedE,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A )
=> ( ord_less_eq_set_a @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1057_Inf__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A )
=> ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1058_Inf__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1059_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1060_Inf__fin_Ohom__commute,axiom,
! [H3: nat > nat,N5: set_nat] :
( ! [X3: nat,Y3: nat] :
( ( H3 @ ( inf_inf_nat @ X3 @ Y3 ) )
= ( inf_inf_nat @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_nat @ N5 )
=> ( ( N5 != bot_bot_set_nat )
=> ( ( H3 @ ( lattic5238388535129920115in_nat @ N5 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H3 @ N5 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1061_Inf__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N5: set_set_a] :
( ! [X3: set_a,Y3: set_a] :
( ( H3 @ ( inf_inf_set_a @ X3 @ Y3 ) )
= ( inf_inf_set_a @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N5 )
=> ( ( N5 != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic8209813465164889211_set_a @ N5 ) )
= ( lattic8209813465164889211_set_a @ ( image_set_a_set_a @ H3 @ N5 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1062_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1063_Inf__fin_Osubset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A ) )
= ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1064_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1065_Inf__fin_Oclosed,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] : ( member_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1066_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1067_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ~ ( member_set_a @ X @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1068_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1069_Inf__fin_Ounion,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ A ) @ ( lattic8209813465164889211_set_a @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1070_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1071_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1072_card__Diff__singleton,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1073_card__Diff__singleton,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1074_card__Diff__singleton__if,axiom,
! [X: nat,A: set_nat] :
( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1075_card__Diff__singleton__if,axiom,
! [X: a,A: set_a] :
( ( ( member_a @ X @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1076_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_1077_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_1078_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1079_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1080_card__insert__le__m1,axiom,
! [N: nat,Y: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1081_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1082_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1083_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1084_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1085_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1086_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1087_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1088_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_1089_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1090_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1091_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1092_Compl__disjoint2,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ A )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_1093_Compl__disjoint,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_1094_Diff__Compl,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( uminus_uminus_set_a @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Compl
thf(fact_1095_Compl__Diff__eq,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1096_inf__compl__bot__left1,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( inf_inf_set_a @ X @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_1097_inf__compl__bot__left2,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_1098_inf__compl__bot__right,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_1099_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1100_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1101_subset__Compl__singleton,axiom,
! [A: set_nat,B4: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B4 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1102_subset__Compl__singleton,axiom,
! [A: set_a,B4: a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ ( insert_a @ B4 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B4 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1103_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1104_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1105_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ X @ Y ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1106_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ X @ Y ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1107_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1108_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1109_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1110_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1111_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1112_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1113_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1114_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1115_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1116_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1117_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1118_diff__eq,axiom,
( minus_minus_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ X2 @ ( uminus_uminus_set_a @ Y4 ) ) ) ) ).
% diff_eq
thf(fact_1119_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1120_inf__cancel__left2,axiom,
! [X: set_a,A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ A2 ) @ ( inf_inf_set_a @ X @ B4 ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_1121_inf__cancel__left1,axiom,
! [X: set_a,A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ A2 ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ B4 ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_1122_subset__Compl__self__eq,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= ( A = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1123_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1124_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1125_Compl__Int,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) ) ) ).
% Compl_Int
thf(fact_1126_Compl__Un,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) ) ) ).
% Compl_Un
thf(fact_1127_Diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ A6 @ ( uminus_uminus_set_a @ B6 ) ) ) ) ).
% Diff_eq
thf(fact_1128_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1129_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1130_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1131_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1132_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1133_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1134_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1135_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1136_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1137_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1138_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1139_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1140_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1141_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N6: nat] : ( ord_less_eq_nat @ ( suc @ N6 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1142_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1143_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1144_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1145_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1146_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1147_diff__less__mono,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1148_verit__comp__simplify1_I3_J,axiom,
! [B10: nat,A10: nat] :
( ( ~ ( ord_less_eq_nat @ B10 @ A10 ) )
= ( ord_less_nat @ A10 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1149_verit__comp__simplify1_I3_J,axiom,
! [B10: num,A10: num] :
( ( ~ ( ord_less_eq_num @ B10 @ A10 ) )
= ( ord_less_num @ A10 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1150_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_1151_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1152_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_1153_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1154_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_1155_nless__le,axiom,
! [A2: set_a,B4: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B4 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B4 )
| ( A2 = B4 ) ) ) ).
% nless_le
thf(fact_1156_nless__le,axiom,
! [A2: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A2 @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B4 )
| ( A2 = B4 ) ) ) ).
% nless_le
thf(fact_1157_nless__le,axiom,
! [A2: num,B4: num] :
( ( ~ ( ord_less_num @ A2 @ B4 ) )
= ( ~ ( ord_less_eq_num @ A2 @ B4 )
| ( A2 = B4 ) ) ) ).
% nless_le
thf(fact_1158_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1159_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1160_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1161_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1162_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1163_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1164_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1165_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1166_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1167_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_1168_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1169_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1170_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1171_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1172_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1173_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1174_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1175_order_Ostrict__trans1,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1176_order_Ostrict__trans1,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1177_order_Ostrict__trans1,axiom,
! [A2: num,B4: num,C2: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( ord_less_num @ B4 @ C2 )
=> ( ord_less_num @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1178_order_Ostrict__trans2,axiom,
! [A2: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1179_order_Ostrict__trans2,axiom,
! [A2: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1180_order_Ostrict__trans2,axiom,
! [A2: num,B4: num,C2: num] :
( ( ord_less_num @ A2 @ B4 )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ord_less_num @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1181_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1182_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1183_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1184_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1185_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1186_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_num @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1187_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1188_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1189_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1190_dual__order_Ostrict__trans1,axiom,
! [B4: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( ord_less_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1191_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1192_dual__order_Ostrict__trans1,axiom,
! [B4: num,A2: num,C2: num] :
( ( ord_less_eq_num @ B4 @ A2 )
=> ( ( ord_less_num @ C2 @ B4 )
=> ( ord_less_num @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1193_dual__order_Ostrict__trans2,axiom,
! [B4: set_a,A2: set_a,C2: set_a] :
( ( ord_less_set_a @ B4 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1194_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1195_dual__order_Ostrict__trans2,axiom,
! [B4: num,A2: num,C2: num] :
( ( ord_less_num @ B4 @ A2 )
=> ( ( ord_less_eq_num @ C2 @ B4 )
=> ( ord_less_num @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1196_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1197_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1198_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1199_order_Ostrict__implies__order,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_set_a @ A2 @ B4 )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1200_order_Ostrict__implies__order,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_nat @ A2 @ B4 )
=> ( ord_less_eq_nat @ A2 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1201_order_Ostrict__implies__order,axiom,
! [A2: num,B4: num] :
( ( ord_less_num @ A2 @ B4 )
=> ( ord_less_eq_num @ A2 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1202_dual__order_Ostrict__implies__order,axiom,
! [B4: set_a,A2: set_a] :
( ( ord_less_set_a @ B4 @ A2 )
=> ( ord_less_eq_set_a @ B4 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1203_dual__order_Ostrict__implies__order,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_nat @ B4 @ A2 )
=> ( ord_less_eq_nat @ B4 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1204_dual__order_Ostrict__implies__order,axiom,
! [B4: num,A2: num] :
( ( ord_less_num @ B4 @ A2 )
=> ( ord_less_eq_num @ B4 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1205_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1206_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1207_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_num @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1208_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1209_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1210_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1211_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1212_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1213_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1214_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1215_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1216_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_1217_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1218_order__le__neq__trans,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less_set_a @ A2 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1219_order__le__neq__trans,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less_nat @ A2 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1220_order__le__neq__trans,axiom,
! [A2: num,B4: num] :
( ( ord_less_eq_num @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less_num @ A2 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1221_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F3 @ I4 ) @ ( F3 @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1222_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1223_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1224_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
| ( M4 = N6 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1225_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1226_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
& ( M4 != N6 ) ) ) ) ).
% nat_less_le
thf(fact_1227_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1228_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1229_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1230_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1231_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1232_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1233_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1234_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1235_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1236_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1237_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1238_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1239_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1240_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1241_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1242_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1243_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1244_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1245_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1246_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1247_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1248_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1249_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1250_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1251_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1252_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1253_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1254_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1255_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1256_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1257_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1258_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less_nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_1259_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1260_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1261_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1262_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_1263_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
% Conjectures (2)
thf(conj_0,hypothesis,
! [A9: a,A11: set_a] :
( ( member_a @ A9 @ aa )
=> ( ( A11
= ( minus_minus_set_a @ aa @ ( insert_a @ A9 @ bot_bot_set_a ) ) )
=> ( ( member_a @ A9 @ g )
=> thesis ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------