TPTP Problem File: ITP072^2.p
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%------------------------------------------------------------------------------
% File : ITP072^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HF problem prob_113__5324706_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : HF/prob_113__5324706_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 340 ( 117 unt; 50 typ; 0 def)
% Number of atoms : 835 ( 258 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3500 ( 86 ~; 9 |; 46 &;2951 @)
% ( 0 <=>; 408 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 155 ( 155 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 48 usr; 2 con; 0-6 aty)
% Number of variables : 991 ( 48 ^; 875 !; 23 ?; 991 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:13.623
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
hF_Mirabelle_hf: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (47)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__bit__shifts,type,
semiring_bit_shifts:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Ounique__euclidean__semiring__with__bit__shifts,type,
unique788075200shifts:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic1704895705min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit,type,
semiri2097166173op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit,type,
semiri1924326578sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_z,type,
z: hF_Mirabelle_hf ).
% Relevant facts (254)
thf(fact_0_hf__ext,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A2 )
= ( hF_Mirabelle_hmem @ X @ B2 ) ) ) ) ).
% hf_ext
thf(fact_1_hemptyE,axiom,
! [A3: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% hemptyE
thf(fact_2_hmem__hempty,axiom,
! [A3: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% hmem_hempty
thf(fact_3_hf__cases,axiom,
! [Y2: hF_Mirabelle_hf] :
( ( Y2
!= ( zero_zero @ hF_Mirabelle_hf ) )
=> ~ ! [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( Y2
= ( hF_Mirabelle_hinsert @ A4 @ B3 ) )
=> ( hF_Mirabelle_hmem @ A4 @ B3 ) ) ) ).
% hf_cases
thf(fact_4_hmem__hinsert,axiom,
! [A3: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ A3 @ ( hF_Mirabelle_hinsert @ B4 @ C ) )
= ( ( A3 = B4 )
| ( hF_Mirabelle_hmem @ A3 @ C ) ) ) ).
% hmem_hinsert
thf(fact_5_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% hmem_def
thf(fact_6_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_7_Abs__hf__0,axiom,
( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
= ( zero_zero @ hF_Mirabelle_hf ) ) ).
% Abs_hf_0
thf(fact_8_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_9_Zero__hf__def,axiom,
( ( zero_zero @ hF_Mirabelle_hf )
= ( hF_Mirabelle_HF @ ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ) ).
% Zero_hf_def
thf(fact_10_push__bit__of__0,axiom,
! [A: $tType] :
( ( semiring_bit_shifts @ A )
=> ! [N: nat] :
( ( semiri1924326578sh_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_11_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( unique788075200shifts @ A )
=> ! [N: nat,A3: A] :
( ( ( semiri1924326578sh_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_12_drop__bit__of__0,axiom,
! [A: $tType] :
( ( semiring_bit_shifts @ A )
=> ! [N: nat] :
( ( semiri2097166173op_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_13_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_14_HF__hfset,axiom,
! [A3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A3 ) )
= A3 ) ).
% HF_hfset
thf(fact_15_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_16_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_17_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_18_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_19_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_20_hinsert__def,axiom,
( hF_Mirabelle_hinsert
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ) ).
% hinsert_def
thf(fact_21_hfset__HF,axiom,
! [A5: set @ hF_Mirabelle_hf] :
( ( finite_finite2 @ hF_Mirabelle_hf @ A5 )
=> ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A5 ) )
= A5 ) ) ).
% hfset_HF
thf(fact_22_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_23_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_24_equals0D,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A5 ) ) ).
% equals0D
thf(fact_25_insert__absorb2,axiom,
! [A: $tType,X2: A,A5: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A5 ) )
= ( insert @ A @ X2 @ A5 ) ) ).
% insert_absorb2
thf(fact_26_insert__iff,axiom,
! [A: $tType,A3: A,B4: A,A5: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A5 ) )
= ( ( A3 = B4 )
| ( member @ A @ A3 @ A5 ) ) ) ).
% insert_iff
thf(fact_27_insertCI,axiom,
! [A: $tType,A3: A,B5: set @ A,B4: A] :
( ( ~ ( member @ A @ A3 @ B5 )
=> ( A3 = B4 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% insertCI
thf(fact_28_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_29_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ? [B6: set @ A] :
( ( A5
= ( insert @ A @ A3 @ B6 ) )
& ~ ( member @ A @ A3 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_30_insert__commute,axiom,
! [A: $tType,X2: A,Y2: A,A5: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y2 @ A5 ) )
= ( insert @ A @ Y2 @ ( insert @ A @ X2 @ A5 ) ) ) ).
% insert_commute
thf(fact_31_insert__eq__iff,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: A,B5: set @ A] :
( ~ ( member @ A @ A3 @ A5 )
=> ( ~ ( member @ A @ B4 @ B5 )
=> ( ( ( insert @ A @ A3 @ A5 )
= ( insert @ A @ B4 @ B5 ) )
= ( ( ( A3 = B4 )
=> ( A5 = B5 ) )
& ( ( A3 != B4 )
=> ? [C3: set @ A] :
( ( A5
= ( insert @ A @ B4 @ C3 ) )
& ~ ( member @ A @ B4 @ C3 )
& ( B5
= ( insert @ A @ A3 @ C3 ) )
& ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_32_insert__absorb,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( insert @ A @ A3 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_33_insert__ident,axiom,
! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X2 @ A5 )
=> ( ~ ( member @ A @ X2 @ B5 )
=> ( ( ( insert @ A @ X2 @ A5 )
= ( insert @ A @ X2 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_34_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A5: set @ A] :
( ( member @ A @ X2 @ A5 )
=> ~ ! [B6: set @ A] :
( ( A5
= ( insert @ A @ X2 @ B6 ) )
=> ( member @ A @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_35_insertI2,axiom,
! [A: $tType,A3: A,B5: set @ A,B4: A] :
( ( member @ A @ A3 @ B5 )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% insertI2
thf(fact_36_insertI1,axiom,
! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% insertI1
thf(fact_37_insertE,axiom,
! [A: $tType,A3: A,B4: A,A5: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A5 ) )
=> ( ( A3 != B4 )
=> ( member @ A @ A3 @ A5 ) ) ) ).
% insertE
thf(fact_38_finite__cases,axiom,
! [A: $tType,F: set @ A] :
( ( finite_finite2 @ A @ F )
=> ( ( F
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A6: set @ A,X3: A] :
( ( F
= ( insert @ A @ X3 @ A6 ) )
=> ( ~ ( member @ A @ X3 @ A6 )
=> ~ ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% finite_cases
thf(fact_39_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_40_singleton__inject,axiom,
! [A: $tType,A3: A,B4: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B4 ) ) ).
% singleton_inject
thf(fact_41_insert__not__empty,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert @ A @ A3 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_42_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B4: A,C: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C )
& ( B4 = D2 ) )
| ( ( A3 = D2 )
& ( B4 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_47_singleton__iff,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_48_singletonD,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_49_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_50_finite__hfset,axiom,
! [A3: hF_Mirabelle_hf] : ( finite_finite2 @ hF_Mirabelle_hf @ ( hF_Mirabelle_hfset @ A3 ) ) ).
% finite_hfset
thf(fact_51_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_52_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_53_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_54_finite__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A3 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ).
% finite_insert
thf(fact_55_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A6: set @ A] :
( ? [A4: A] :
( A3
= ( insert @ A @ A4 @ A6 ) )
=> ~ ( finite_finite2 @ A @ A6 ) ) ) ) ).
% finite.cases
thf(fact_56_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A2: set @ A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B2: A] :
( ( A2
= ( insert @ A @ B2 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_57_finite__induct,axiom,
! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_58_finite_Oinducts,axiom,
! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: set @ A,A4: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( insert @ A @ A4 @ A6 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_59_finite__ne__induct,axiom,
! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( F
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( F3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_60_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
( ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_61_the__elem__eq,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ).
% the_elem_eq
thf(fact_62_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_63_finite__set__choice,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [X_1: B] : ( P @ X3 @ X_1 ) )
=> ? [F4: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( P @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_64_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% finite
thf(fact_65_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_66_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_67_finite_OinsertI,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( insert @ A @ A3 @ A5 ) ) ) ).
% finite.insertI
thf(fact_68_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_69_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_70_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_71_is__singletonI,axiom,
! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_72_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_73_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X: A] :
( A7
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_74_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_75_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_76_card__0__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_77_finite__subset__induct,axiom,
! [A: $tType,F: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ F @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( member @ A @ A4 @ A5 )
=> ( ~ ( member @ A @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_78_finite__subset__induct_H,axiom,
! [A: $tType,F: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ F @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( member @ A @ A4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F3 @ A5 )
=> ( ~ ( member @ A @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_79_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R: A > A > $o,F5: B > A,X: B,Y4: B] : ( R @ ( F5 @ X ) @ ( F5 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_80_inj__on__HF,axiom,
inj_on @ ( set @ hF_Mirabelle_hf ) @ hF_Mirabelle_hf @ hF_Mirabelle_HF @ ( collect @ ( set @ hF_Mirabelle_hf ) @ ( finite_finite2 @ hF_Mirabelle_hf ) ) ).
% inj_on_HF
thf(fact_81_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_82_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_83_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_84_subsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% subsetI
thf(fact_85_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_86_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_87_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_88_insert__subset,axiom,
! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ B5 )
= ( ( member @ A @ X2 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_89_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: A] :
( ( ( insert @ A @ A3 @ A5 )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_90_singleton__insert__inj__eq,axiom,
! [A: $tType,B4: A,A3: A,A5: set @ A] :
( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A5 ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_91_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_92_card_Oinfinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_93_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
=> ( A3 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_94_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_95_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_96_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B4: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_97_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_98_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_99_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_100_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_101_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C: A] :
( ( A3 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_102_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_103_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_104_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_105_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_106_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_107_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_108_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_109_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_110_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_111_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B4: A,F2: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ( F2 @ B4 )
= C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B4: B,C: B] :
( ( A3
= ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B4: A,F2: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ C2 @ ( F2 @ B4 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C2 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_115_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_116_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F5: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F5 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_117_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_118_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_119_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_120_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_121_subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_122_Collect__mono,axiom,
! [A: $tType,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_123_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_124_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T: A] :
( ( member @ A @ T @ A7 )
=> ( member @ A @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_125_equalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_126_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_127_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_128_equalityE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_129_subsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B5 ) ) ) ).
% subsetD
thf(fact_130_in__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_131_card__subset__eq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ B5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_132_infinite__arbitrarily__large,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
& ( ( finite_card @ A @ B6 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_133_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_134_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_135_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_136_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_137_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_138_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ A3 @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_139_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_140_finite__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_141_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_142_rev__finite__subset,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_143_insert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_144_subset__insert,axiom,
! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_145_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A3 @ B5 ) ) ).
% subset_insertI
thf(fact_146_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_147_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_148_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_149_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_150_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_151_card__eq__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% card_eq_0_iff
thf(fact_152_inj__on__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_153_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S2: set @ B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S2 )
=> ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert @ B @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_154_inj__on__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( inj_on @ A @ B @ F2 @ B5 ) ) ) ).
% inj_on_subset
thf(fact_155_subset__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( inj_on @ A @ B @ F2 @ A5 ) ) ) ).
% subset_inj_on
thf(fact_156_insert__subsetI,axiom,
! [A: $tType,X2: A,A5: set @ A,X5: set @ A] :
( ( member @ A @ X2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_157_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_158_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_159_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_160_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_161_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_162_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_163_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_164_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F: set @ A,C4: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F )
=> ( ( finite_finite2 @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C4 ) ) )
=> ( ( finite_finite2 @ A @ F )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_165_card__seteq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_seteq
thf(fact_166_card__mono,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% card_mono
thf(fact_167_card__insert__le,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X2 @ A5 ) ) ) ) ).
% card_insert_le
thf(fact_168_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A5: set @ A,G: B > A,F2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( G @ ( F2 @ X3 ) )
= X3 ) )
=> ( inj_on @ A @ B @ F2 @ A5 ) ) ).
% inj_on_inverseI
thf(fact_169_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( X2 != Y2 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_170_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_171_inj__on__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,F2: A > B,G: A > B] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on @ A @ B @ F2 @ A5 )
= ( inj_on @ A @ B @ G @ A5 ) ) ) ).
% inj_on_cong
thf(fact_172_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F5: A > B,A7: set @ A] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A7 )
=> ( ( ( F5 @ X )
= ( F5 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_173_inj__onI,axiom,
! [B: $tType,A: $tType,A5: set @ A,F2: A > B] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A5 ) ) ).
% inj_onI
thf(fact_174_inj__onD,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_175_obtain__subset__with__card__n,axiom,
! [A: $tType,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_finite2 @ A @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_176_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set @ A,A5: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ? [B8: A] :
( ( member @ A @ B8 @ B5 )
& ( R2 @ A4 @ B8 ) ) )
=> ( ! [A1: B,A22: B,B3: A] :
( ( member @ B @ A1 @ A5 )
=> ( ( member @ B @ A22 @ A5 )
=> ( ( member @ A @ B3 @ B5 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_177_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y2: A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y2 @ S )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic1704895705min_on @ A @ B @ F2 @ S ) ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% arg_min_least
thf(fact_178_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_179_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M3: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M3 @ X3 ) @ ( M3 @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_180_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_181_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
| ( ord_less_eq @ nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_182_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ( ord_less_eq @ nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_183_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq @ nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_184_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_185_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_186_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N2: set @ nat] :
? [M4: nat] :
! [X: nat] :
( ( member @ nat @ X @ N2 )
=> ( ord_less_eq @ nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_187_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic1704895705min_on @ A @ B @ F2 @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_188_card__Diff1__le,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% card_Diff1_le
thf(fact_189_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) )
=> ? [F4: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A5 ) @ B5 )
& ( inj_on @ A @ B @ F4 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_190_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_191_image__eqI,axiom,
! [A: $tType,B: $tType,B4: A,F2: B > A,X2: B,A5: set @ B] :
( ( B4
= ( F2 @ X2 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( member @ A @ B4 @ ( image @ B @ A @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_192_DiffI,axiom,
! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C @ A5 )
=> ( ~ ( member @ A @ C @ B5 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_193_Diff__iff,axiom,
! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( ( member @ A @ C @ A5 )
& ~ ( member @ A @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_194_Diff__idemp,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ).
% Diff_idemp
thf(fact_195_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_196_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_197_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_198_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_199_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_200_image__empty,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_201_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F2 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_202_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
( ( ( image @ B @ A @ F2 @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_203_finite__imageI,axiom,
! [B: $tType,A: $tType,F: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H @ F ) ) ) ).
% finite_imageI
thf(fact_204_image__insert,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: B,B5: set @ B] :
( ( image @ B @ A @ F2 @ ( insert @ B @ A3 @ B5 ) )
= ( insert @ A @ ( F2 @ A3 ) @ ( image @ B @ A @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_205_insert__image,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,F2: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( ( insert @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A5 ) )
= ( image @ A @ B @ F2 @ A5 ) ) ) ).
% insert_image
thf(fact_206_Diff__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Diff_empty
thf(fact_207_empty__Diff,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_208_Diff__cancel,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_209_finite__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% finite_Diff
thf(fact_210_finite__Diff2,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_Diff2
thf(fact_211_Diff__insert0,axiom,
! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X2 @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_212_insert__Diff1,axiom,
! [A: $tType,X2: A,B5: set @ A,A5: set @ A] :
( ( member @ A @ X2 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_213_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B4 ) )
= ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_214_Diff__eq__empty__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_215_insert__Diff__single,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A5 ) ) ).
% insert_Diff_single
thf(fact_216_finite__Diff__insert,axiom,
! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_217_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A3 @ A5 ) )
= ( ( inj_on @ A @ B @ F2 @ A5 )
& ~ ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_218_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A5 ) @ ( image @ B @ A @ F2 @ B5 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_219_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_220_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B4: A,D2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ D2 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_221_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B4: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B4 ) ) ) ) ).
% diff_left_mono
thf(fact_222_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B4 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_223_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B4: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B4 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_224_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
=> ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_225_image__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_226_image__subsetI,axiom,
! [A: $tType,B: $tType,A5: set @ A,F2: A > B,B5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_227_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
=> ~ ! [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
=> ( B5
!= ( image @ B @ A @ F2 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_228_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A5 ) @ B5 )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( member @ A @ ( F2 @ X ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_229_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
& ( B5
= ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_230_Diff__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,D3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_231_Diff__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ A5 ) ).
% Diff_subset
thf(fact_232_double__diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_233_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_234_finite__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F2 @ A5 ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_surj
thf(fact_235_finite__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
=> ? [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
& ( finite_finite2 @ B @ C5 )
& ( B5
= ( image @ B @ A @ F2 @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_236_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set @ B] :
( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
& ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_237_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A5 ) )
=> ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_238_finite__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A5 ) )
=> ( ( inj_on @ B @ A @ F2 @ A5 )
=> ( finite_finite2 @ B @ A5 ) ) ) ).
% finite_imageD
thf(fact_239_finite__image__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_image_iff
thf(fact_240_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( ( image @ A @ B @ F2 @ A5 )
= ( image @ A @ B @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_241_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F2 @ B5 )
=> ( ( member @ A @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ A5 ) )
= ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_242_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: B,B5: set @ B] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ~ ( member @ B @ X2 @ B5 )
=> ( ( ( insert @ B @ X2 @ B5 )
= ( image @ A @ B @ F2 @ A5 ) )
=> ~ ! [X6: A,A8: set @ A] :
( ~ ( member @ A @ X6 @ A8 )
=> ( ( A5
= ( insert @ A @ X6 @ A8 ) )
=> ( ( X2
= ( F2 @ X6 ) )
=> ( B5
!= ( image @ A @ B @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_243_card__image,axiom,
! [B: $tType,A: $tType,F2: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F2 @ A5 )
=> ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% card_image
thf(fact_244_Diff__insert__absorb,axiom,
! [A: $tType,X2: A,A5: set @ A] :
( ~ ( member @ A @ X2 @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_245_Diff__insert2,axiom,
! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_246_insert__Diff,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_247_Diff__insert,axiom,
! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_248_subset__Diff__insert,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X2: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X2 @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
& ~ ( member @ A @ X2 @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_249_in__image__insert__iff,axiom,
! [A: $tType,B5: set @ ( set @ A ),X2: A,A5: set @ A] :
( ! [C5: set @ A] :
( ( member @ ( set @ A ) @ C5 @ B5 )
=> ~ ( member @ A @ X2 @ C5 ) )
=> ( ( member @ ( set @ A ) @ A5 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B5 ) )
= ( ( member @ A @ X2 @ A5 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_250_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B4: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B4 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( A3 = B4 )
= ( C = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_251_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,C: A,B4: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B4 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B4 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_252_DiffE,axiom,
! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
=> ~ ( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B5 ) ) ) ).
% DiffE
thf(fact_253_DiffD1,axiom,
! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
=> ( member @ A @ C @ A5 ) ) ).
% DiffD1
% Type constructors (35)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A9: $tType,A10: $tType] :
( ( order_bot @ A10 )
=> ( order_bot @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 )
=> ( bot @ ( A9 > A10 ) ) ) ).
thf(tcon_Nat_Onat___Parity_Ounique__euclidean__semiring__with__bit__shifts,axiom,
unique788075200shifts @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__bit__shifts,axiom,
semiring_bit_shifts @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
bot @ $o ).
thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_19,axiom,
zero @ hF_Mirabelle_hf ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( z
= ( zero_zero @ hF_Mirabelle_hf ) )
= ( ! [X: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X @ z ) ) ) ).
%------------------------------------------------------------------------------