TPTP Problem File: ITP072^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP072^1 : 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 : 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 415 ( 154 unt; 63 typ; 0 def)
% Number of atoms : 1084 ( 383 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 2828 ( 164 ~; 16 |; 87 &;2035 @)
% ( 0 <=>; 526 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 221 ( 221 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 58 usr; 9 con; 0-2 aty)
% Number of variables : 898 ( 41 ^; 811 !; 46 ?; 898 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:17.372
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
set_se933006839lle_hf: $tType ).
thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
set_HF_Mirabelle_hf: $tType ).
thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
hF_Mirabelle_hf: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (58)
thf(sy_c_Finite__Set_Ocard_001t__HF____Mirabelle____glliljednj__Ohf,type,
finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
finite90088345lle_hf: set_se933006839lle_hf > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__HF____Mirabelle____glliljednj__Ohf,type,
finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
finite1450550360lle_hf: set_se933006839lle_hf > $o ).
thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $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__HF____Mirabelle____glliljednj__Ohf_J,type,
minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
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_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
minus_500612048lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
zero_z189798548lle_hf: hF_Mirabelle_hf ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
lattic710307446hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic1974000059at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Nat__Onat,type,
semiring_1_Nats_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_M_Eo_J,type,
bot_bo554042810e_hf_o: set_HF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
bot_bo2093393035lle_hf: set_se933006839lle_hf ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $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_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
ord_le2016357975lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > $o ).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit_001t__Nat__Onat,type,
semiri2115134414it_nat: nat > nat > nat ).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Nat__Onat,type,
semiri2013084963it_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
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__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
image_1514960916lle_hf: ( set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ) > set_se933006839lle_hf > set_se933006839lle_hf ).
thf(sy_c_Set_Oinsert_001t__HF____Mirabelle____glliljednj__Ohf,type,
insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
insert1636143089lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
thf(sy_c_Set_Ois__empty_001t__HF____Mirabelle____glliljednj__Ohf,type,
is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
thf(sy_c_Set_Ois__singleton_001t__HF____Mirabelle____glliljednj__Ohf,type,
is_sin1448700567lle_hf: set_HF_Mirabelle_hf > $o ).
thf(sy_c_Set_Othe__elem_001t__HF____Mirabelle____glliljednj__Ohf,type,
the_el1104322134lle_hf: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
thf(sy_v_z,type,
z: hF_Mirabelle_hf ).
% Relevant facts (351)
thf(fact_0_hf__ext,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A )
= ( hF_Mirabelle_hmem @ X @ B ) ) ) ) ).
% hf_ext
thf(fact_1_hemptyE,axiom,
! [A2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% hemptyE
thf(fact_2_hmem__hempty,axiom,
! [A2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% hmem_hempty
thf(fact_3_hf__cases,axiom,
! [Y2: hF_Mirabelle_hf] :
( ( Y2 != zero_z189798548lle_hf )
=> ~ ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( Y2
= ( hF_Mirabelle_hinsert @ A3 @ B2 ) )
=> ( hF_Mirabelle_hmem @ A3 @ B2 ) ) ) ).
% hf_cases
thf(fact_4_hmem__hinsert,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B3 @ C ) )
= ( ( A2 = B3 )
| ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% hmem_hinsert
thf(fact_5_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A @ ( hF_Mirabelle_hfset @ B ) ) ) ) ).
% hmem_def
thf(fact_6_zero__reorient,axiom,
! [X2: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X2 )
= ( X2 = zero_z189798548lle_hf ) ) ).
% zero_reorient
thf(fact_7_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_8_Abs__hf__0,axiom,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
= zero_z189798548lle_hf ) ).
% Abs_hf_0
thf(fact_9_Zero__hf__def,axiom,
( zero_z189798548lle_hf
= ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ).
% Zero_hf_def
thf(fact_10_push__bit__of__0,axiom,
! [N: nat] :
( ( semiri2013084963it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% push_bit_of_0
thf(fact_11_push__bit__eq__0__iff,axiom,
! [N: nat,A2: nat] :
( ( ( semiri2013084963it_nat @ N @ A2 )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% push_bit_eq_0_iff
thf(fact_12_drop__bit__of__0,axiom,
! [N: nat] :
( ( semiri2115134414it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% drop_bit_of_0
thf(fact_13_Nats__0,axiom,
member_nat @ zero_zero_nat @ semiring_1_Nats_nat ).
% Nats_0
thf(fact_14_HF__hfset,axiom,
! [A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
= A2 ) ).
% HF_hfset
thf(fact_15_empty__iff,axiom,
! [C: set_HF_Mirabelle_hf] :
~ ( member1490636632lle_hf @ C @ bot_bo2093393035lle_hf ) ).
% empty_iff
thf(fact_16_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_17_empty__iff,axiom,
! [C: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ C @ bot_bo53200981lle_hf ) ).
% empty_iff
thf(fact_18_all__not__in__conv,axiom,
! [A4: set_se933006839lle_hf] :
( ( ! [X: set_HF_Mirabelle_hf] :
~ ( member1490636632lle_hf @ X @ A4 ) )
= ( A4 = bot_bo2093393035lle_hf ) ) ).
% all_not_in_conv
thf(fact_19_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_20_all__not__in__conv,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( ! [X: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ X @ A4 ) )
= ( A4 = bot_bo53200981lle_hf ) ) ).
% all_not_in_conv
thf(fact_21_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_22_Collect__empty__eq,axiom,
! [P: set_HF_Mirabelle_hf > $o] :
( ( ( collec1758573718lle_hf @ P )
= bot_bo2093393035lle_hf )
= ( ! [X: set_HF_Mirabelle_hf] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_23_Collect__empty__eq,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( ( collec2046588256lle_hf @ P )
= bot_bo53200981lle_hf )
= ( ! [X: hF_Mirabelle_hf] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_24_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_25_empty__Collect__eq,axiom,
! [P: set_HF_Mirabelle_hf > $o] :
( ( bot_bo2093393035lle_hf
= ( collec1758573718lle_hf @ P ) )
= ( ! [X: set_HF_Mirabelle_hf] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_26_empty__Collect__eq,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( bot_bo53200981lle_hf
= ( collec2046588256lle_hf @ P ) )
= ( ! [X: hF_Mirabelle_hf] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_27_bot__apply,axiom,
( bot_bo1263054448e_hf_o
= ( ^ [X: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% bot_apply
thf(fact_28_bot__apply,axiom,
( bot_bo554042810e_hf_o
= ( ^ [X: set_HF_Mirabelle_hf] : bot_bot_o ) ) ).
% bot_apply
thf(fact_29_hinsert__def,axiom,
( hF_Mirabelle_hinsert
= ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert9649339lle_hf @ A @ ( hF_Mirabelle_hfset @ B ) ) ) ) ) ).
% hinsert_def
thf(fact_30_hfset__HF,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A4 ) )
= A4 ) ) ).
% hfset_HF
thf(fact_31_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_32_emptyE,axiom,
! [A2: set_HF_Mirabelle_hf] :
~ ( member1490636632lle_hf @ A2 @ bot_bo2093393035lle_hf ) ).
% emptyE
thf(fact_33_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_34_emptyE,axiom,
! [A2: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ A2 @ bot_bo53200981lle_hf ) ).
% emptyE
thf(fact_35_equals0D,axiom,
! [A4: set_se933006839lle_hf,A2: set_HF_Mirabelle_hf] :
( ( A4 = bot_bo2093393035lle_hf )
=> ~ ( member1490636632lle_hf @ A2 @ A4 ) ) ).
% equals0D
thf(fact_36_equals0D,axiom,
! [A4: set_nat,A2: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A4 ) ) ).
% equals0D
thf(fact_37_equals0D,axiom,
! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( A4 = bot_bo53200981lle_hf )
=> ~ ( member1367349282lle_hf @ A2 @ A4 ) ) ).
% equals0D
thf(fact_38_insert__absorb2,axiom,
! [X2: nat,A4: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A4 ) )
= ( insert_nat @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_39_insert__absorb2,axiom,
! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( insert1636143089lle_hf @ X2 @ ( insert1636143089lle_hf @ X2 @ A4 ) )
= ( insert1636143089lle_hf @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_40_insert__absorb2,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( insert9649339lle_hf @ X2 @ ( insert9649339lle_hf @ X2 @ A4 ) )
= ( insert9649339lle_hf @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_41_insert__iff,axiom,
! [A2: nat,B3: nat,A4: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( member_nat @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_42_insert__iff,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( member1490636632lle_hf @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_43_insert__iff,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( member1367349282lle_hf @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_44_insertCI,axiom,
! [A2: nat,B4: set_nat,B3: nat] :
( ( ~ ( member_nat @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_45_insertCI,axiom,
! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf] :
( ( ~ ( member1490636632lle_hf @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_46_insertCI,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( ~ ( member1367349282lle_hf @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_47_singletonI,axiom,
! [A2: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) ) ).
% singletonI
thf(fact_48_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_49_singletonI,axiom,
! [A2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ).
% singletonI
thf(fact_50_mk__disjoint__insert,axiom,
! [A2: nat,A4: set_nat] :
( ( member_nat @ A2 @ A4 )
=> ? [B5: set_nat] :
( ( A4
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_51_mk__disjoint__insert,axiom,
! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( member1490636632lle_hf @ A2 @ A4 )
=> ? [B5: set_se933006839lle_hf] :
( ( A4
= ( insert1636143089lle_hf @ A2 @ B5 ) )
& ~ ( member1490636632lle_hf @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_52_mk__disjoint__insert,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ A4 )
=> ? [B5: set_HF_Mirabelle_hf] :
( ( A4
= ( insert9649339lle_hf @ A2 @ B5 ) )
& ~ ( member1367349282lle_hf @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_53_insert__commute,axiom,
! [X2: nat,Y2: nat,A4: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y2 @ A4 ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_54_insert__commute,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( insert1636143089lle_hf @ X2 @ ( insert1636143089lle_hf @ Y2 @ A4 ) )
= ( insert1636143089lle_hf @ Y2 @ ( insert1636143089lle_hf @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_55_insert__commute,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( insert9649339lle_hf @ X2 @ ( insert9649339lle_hf @ Y2 @ A4 ) )
= ( insert9649339lle_hf @ Y2 @ ( insert9649339lle_hf @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_56_insert__eq__iff,axiom,
! [A2: nat,A4: set_nat,B3: nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A4 )
=> ( ~ ( member_nat @ B3 @ B4 )
=> ( ( ( insert_nat @ A2 @ A4 )
= ( insert_nat @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A4 = B4 ) )
& ( ( A2 != B3 )
=> ? [C2: set_nat] :
( ( A4
= ( insert_nat @ B3 @ C2 ) )
& ~ ( member_nat @ B3 @ C2 )
& ( B4
= ( insert_nat @ A2 @ C2 ) )
& ~ ( member_nat @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_57_insert__eq__iff,axiom,
! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf] :
( ~ ( member1490636632lle_hf @ A2 @ A4 )
=> ( ~ ( member1490636632lle_hf @ B3 @ B4 )
=> ( ( ( insert1636143089lle_hf @ A2 @ A4 )
= ( insert1636143089lle_hf @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A4 = B4 ) )
& ( ( A2 != B3 )
=> ? [C2: set_se933006839lle_hf] :
( ( A4
= ( insert1636143089lle_hf @ B3 @ C2 ) )
& ~ ( member1490636632lle_hf @ B3 @ C2 )
& ( B4
= ( insert1636143089lle_hf @ A2 @ C2 ) )
& ~ ( member1490636632lle_hf @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_58_insert__eq__iff,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ A2 @ A4 )
=> ( ~ ( member1367349282lle_hf @ B3 @ B4 )
=> ( ( ( insert9649339lle_hf @ A2 @ A4 )
= ( insert9649339lle_hf @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A4 = B4 ) )
& ( ( A2 != B3 )
=> ? [C2: set_HF_Mirabelle_hf] :
( ( A4
= ( insert9649339lle_hf @ B3 @ C2 ) )
& ~ ( member1367349282lle_hf @ B3 @ C2 )
& ( B4
= ( insert9649339lle_hf @ A2 @ C2 ) )
& ~ ( member1367349282lle_hf @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_59_insert__absorb,axiom,
! [A2: nat,A4: set_nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( insert_nat @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_60_insert__absorb,axiom,
! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( member1490636632lle_hf @ A2 @ A4 )
=> ( ( insert1636143089lle_hf @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_61_insert__absorb,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ A4 )
=> ( ( insert9649339lle_hf @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_62_insert__ident,axiom,
! [X2: nat,A4: set_nat,B4: set_nat] :
( ~ ( member_nat @ X2 @ A4 )
=> ( ~ ( member_nat @ X2 @ B4 )
=> ( ( ( insert_nat @ X2 @ A4 )
= ( insert_nat @ X2 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_63_insert__ident,axiom,
! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
( ~ ( member1490636632lle_hf @ X2 @ A4 )
=> ( ~ ( member1490636632lle_hf @ X2 @ B4 )
=> ( ( ( insert1636143089lle_hf @ X2 @ A4 )
= ( insert1636143089lle_hf @ X2 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_64_insert__ident,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ X2 @ A4 )
=> ( ~ ( member1367349282lle_hf @ X2 @ B4 )
=> ( ( ( insert9649339lle_hf @ X2 @ A4 )
= ( insert9649339lle_hf @ X2 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_65_Set_Oset__insert,axiom,
! [X2: nat,A4: set_nat] :
( ( member_nat @ X2 @ A4 )
=> ~ ! [B5: set_nat] :
( ( A4
= ( insert_nat @ X2 @ B5 ) )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_66_Set_Oset__insert,axiom,
! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( member1490636632lle_hf @ X2 @ A4 )
=> ~ ! [B5: set_se933006839lle_hf] :
( ( A4
= ( insert1636143089lle_hf @ X2 @ B5 ) )
=> ( member1490636632lle_hf @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_67_Set_Oset__insert,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X2 @ A4 )
=> ~ ! [B5: set_HF_Mirabelle_hf] :
( ( A4
= ( insert9649339lle_hf @ X2 @ B5 ) )
=> ( member1367349282lle_hf @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_68_insertI2,axiom,
! [A2: nat,B4: set_nat,B3: nat] :
( ( member_nat @ A2 @ B4 )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_69_insertI2,axiom,
! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ A2 @ B4 )
=> ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_70_insertI2,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ B4 )
=> ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_71_insertI1,axiom,
! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% insertI1
thf(fact_72_insertI1,axiom,
! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf] : ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ A2 @ B4 ) ) ).
% insertI1
thf(fact_73_insertI1,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% insertI1
thf(fact_74_insertE,axiom,
! [A2: nat,B3: nat,A4: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( member_nat @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_75_insertE,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( member1490636632lle_hf @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_76_insertE,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( member1367349282lle_hf @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_77_finite__cases,axiom,
! [F: set_se933006839lle_hf] :
( ( finite1450550360lle_hf @ F )
=> ( ( F != bot_bo2093393035lle_hf )
=> ~ ! [A5: set_se933006839lle_hf,X3: set_HF_Mirabelle_hf] :
( ( F
= ( insert1636143089lle_hf @ X3 @ A5 ) )
=> ( ~ ( member1490636632lle_hf @ X3 @ A5 )
=> ~ ( finite1450550360lle_hf @ A5 ) ) ) ) ) ).
% finite_cases
thf(fact_78_finite__cases,axiom,
! [F: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ~ ! [A5: set_nat,X3: nat] :
( ( F
= ( insert_nat @ X3 @ A5 ) )
=> ( ~ ( member_nat @ X3 @ A5 )
=> ~ ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite_cases
thf(fact_79_finite__cases,axiom,
! [F: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F )
=> ( ( F != bot_bo53200981lle_hf )
=> ~ ! [A5: set_HF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( F
= ( insert9649339lle_hf @ X3 @ A5 ) )
=> ( ~ ( member1367349282lle_hf @ X3 @ A5 )
=> ~ ( finite586181922lle_hf @ A5 ) ) ) ) ) ).
% finite_cases
thf(fact_80_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_81_bot__set__def,axiom,
( bot_bo2093393035lle_hf
= ( collec1758573718lle_hf @ bot_bo554042810e_hf_o ) ) ).
% bot_set_def
thf(fact_82_bot__set__def,axiom,
( bot_bo53200981lle_hf
= ( collec2046588256lle_hf @ bot_bo1263054448e_hf_o ) ) ).
% bot_set_def
thf(fact_83_singleton__inject,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf )
= ( insert1636143089lle_hf @ B3 @ bot_bo2093393035lle_hf ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_84_singleton__inject,axiom,
! [A2: nat,B3: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_85_singleton__inject,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf )
= ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_86_insert__not__empty,axiom,
! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( insert1636143089lle_hf @ A2 @ A4 )
!= bot_bo2093393035lle_hf ) ).
% insert_not_empty
thf(fact_87_insert__not__empty,axiom,
! [A2: nat,A4: set_nat] :
( ( insert_nat @ A2 @ A4 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_88_insert__not__empty,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( insert9649339lle_hf @ A2 @ A4 )
!= bot_bo53200981lle_hf ) ).
% insert_not_empty
thf(fact_89_doubleton__eq__iff,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf,D: set_HF_Mirabelle_hf] :
( ( ( insert1636143089lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ bot_bo2093393035lle_hf ) )
= ( insert1636143089lle_hf @ C @ ( insert1636143089lle_hf @ D @ bot_bo2093393035lle_hf ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_90_doubleton__eq__iff,axiom,
! [A2: nat,B3: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_91_doubleton__eq__iff,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
( ( ( insert9649339lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
= ( insert9649339lle_hf @ C @ ( insert9649339lle_hf @ D @ bot_bo53200981lle_hf ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_92_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( member1367349282lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A2: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( member1490636632lle_hf @ A2 @ ( collec1758573718lle_hf @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( collec2046588256lle_hf
@ ^ [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A4: set_se933006839lle_hf] :
( ( collec1758573718lle_hf
@ ^ [X: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_98_Collect__cong,axiom,
! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
( ! [X3: hF_Mirabelle_hf] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec2046588256lle_hf @ P )
= ( collec2046588256lle_hf @ Q ) ) ) ).
% Collect_cong
thf(fact_99_Collect__cong,axiom,
! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
( ! [X3: set_HF_Mirabelle_hf] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1758573718lle_hf @ P )
= ( collec1758573718lle_hf @ Q ) ) ) ).
% Collect_cong
thf(fact_100_singleton__iff,axiom,
! [B3: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ B3 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_101_singleton__iff,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_102_singleton__iff,axiom,
! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ B3 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_103_singletonD,axiom,
! [B3: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ B3 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_104_singletonD,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_105_singletonD,axiom,
! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ B3 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_106_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_107_finite__hfset,axiom,
! [A2: hF_Mirabelle_hf] : ( finite586181922lle_hf @ ( hF_Mirabelle_hfset @ A2 ) ) ).
% finite_hfset
thf(fact_108_bot__fun__def,axiom,
( bot_bo1263054448e_hf_o
= ( ^ [X: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_109_bot__fun__def,axiom,
( bot_bo554042810e_hf_o
= ( ^ [X: set_HF_Mirabelle_hf] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_110_ex__in__conv,axiom,
! [A4: set_se933006839lle_hf] :
( ( ? [X: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X @ A4 ) )
= ( A4 != bot_bo2093393035lle_hf ) ) ).
% ex_in_conv
thf(fact_111_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_112_ex__in__conv,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( ? [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ A4 ) )
= ( A4 != bot_bo53200981lle_hf ) ) ).
% ex_in_conv
thf(fact_113_equals0I,axiom,
! [A4: set_se933006839lle_hf] :
( ! [Y3: set_HF_Mirabelle_hf] :
~ ( member1490636632lle_hf @ Y3 @ A4 )
=> ( A4 = bot_bo2093393035lle_hf ) ) ).
% equals0I
thf(fact_114_equals0I,axiom,
! [A4: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_115_equals0I,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ! [Y3: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ Y3 @ A4 )
=> ( A4 = bot_bo53200981lle_hf ) ) ).
% equals0I
thf(fact_116_finite__insert,axiom,
! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( finite1450550360lle_hf @ ( insert1636143089lle_hf @ A2 @ A4 ) )
= ( finite1450550360lle_hf @ A4 ) ) ).
% finite_insert
thf(fact_117_finite__insert,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ ( insert9649339lle_hf @ A2 @ A4 ) )
= ( finite586181922lle_hf @ A4 ) ) ).
% finite_insert
thf(fact_118_finite__insert,axiom,
! [A2: nat,A4: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ).
% finite_insert
thf(fact_119_finite_Ocases,axiom,
! [A2: set_se933006839lle_hf] :
( ( finite1450550360lle_hf @ A2 )
=> ( ( A2 != bot_bo2093393035lle_hf )
=> ~ ! [A5: set_se933006839lle_hf] :
( ? [A3: set_HF_Mirabelle_hf] :
( A2
= ( insert1636143089lle_hf @ A3 @ A5 ) )
=> ~ ( finite1450550360lle_hf @ A5 ) ) ) ) ).
% finite.cases
thf(fact_120_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A5: set_nat] :
( ? [A3: nat] :
( A2
= ( insert_nat @ A3 @ A5 ) )
=> ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% finite.cases
thf(fact_121_finite_Ocases,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A2 )
=> ( ( A2 != bot_bo53200981lle_hf )
=> ~ ! [A5: set_HF_Mirabelle_hf] :
( ? [A3: hF_Mirabelle_hf] :
( A2
= ( insert9649339lle_hf @ A3 @ A5 ) )
=> ~ ( finite586181922lle_hf @ A5 ) ) ) ) ).
% finite.cases
thf(fact_122_finite_Osimps,axiom,
( finite1450550360lle_hf
= ( ^ [A: set_se933006839lle_hf] :
( ( A = bot_bo2093393035lle_hf )
| ? [A6: set_se933006839lle_hf,B: set_HF_Mirabelle_hf] :
( ( A
= ( insert1636143089lle_hf @ B @ A6 ) )
& ( finite1450550360lle_hf @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_123_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A: set_nat] :
( ( A = bot_bot_set_nat )
| ? [A6: set_nat,B: nat] :
( ( A
= ( insert_nat @ B @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_124_finite_Osimps,axiom,
( finite586181922lle_hf
= ( ^ [A: set_HF_Mirabelle_hf] :
( ( A = bot_bo53200981lle_hf )
| ? [A6: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A
= ( insert9649339lle_hf @ B @ A6 ) )
& ( finite586181922lle_hf @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_125_finite__induct,axiom,
! [F: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
( ( finite1450550360lle_hf @ F )
=> ( ( P @ bot_bo2093393035lle_hf )
=> ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
( ( finite1450550360lle_hf @ F2 )
=> ( ~ ( member1490636632lle_hf @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_126_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_127_finite__induct,axiom,
! [F: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ F )
=> ( ( P @ bot_bo53200981lle_hf )
=> ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F2 )
=> ( ~ ( member1367349282lle_hf @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_128_finite_Oinducts,axiom,
! [X2: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
( ( finite1450550360lle_hf @ X2 )
=> ( ( P @ bot_bo2093393035lle_hf )
=> ( ! [A5: set_se933006839lle_hf,A3: set_HF_Mirabelle_hf] :
( ( finite1450550360lle_hf @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert1636143089lle_hf @ A3 @ A5 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_129_finite_Oinducts,axiom,
! [X2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ X2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: set_nat,A3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ A3 @ A5 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_130_finite_Oinducts,axiom,
! [X2: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ X2 )
=> ( ( P @ bot_bo53200981lle_hf )
=> ( ! [A5: set_HF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert9649339lle_hf @ A3 @ A5 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_131_finite__ne__induct,axiom,
! [F: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
( ( finite1450550360lle_hf @ F )
=> ( ( F != bot_bo2093393035lle_hf )
=> ( ! [X3: set_HF_Mirabelle_hf] : ( P @ ( insert1636143089lle_hf @ X3 @ bot_bo2093393035lle_hf ) )
=> ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
( ( finite1450550360lle_hf @ F2 )
=> ( ( F2 != bot_bo2093393035lle_hf )
=> ( ~ ( member1490636632lle_hf @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_132_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_133_finite__ne__induct,axiom,
! [F: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ F )
=> ( ( F != bot_bo53200981lle_hf )
=> ( ! [X3: hF_Mirabelle_hf] : ( P @ ( insert9649339lle_hf @ X3 @ bot_bo53200981lle_hf ) )
=> ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F2 )
=> ( ( F2 != bot_bo53200981lle_hf )
=> ( ~ ( member1367349282lle_hf @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_134_infinite__finite__induct,axiom,
! [P: set_se933006839lle_hf > $o,A4: set_se933006839lle_hf] :
( ! [A5: set_se933006839lle_hf] :
( ~ ( finite1450550360lle_hf @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo2093393035lle_hf )
=> ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
( ( finite1450550360lle_hf @ F2 )
=> ( ~ ( member1490636632lle_hf @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_135_infinite__finite__induct,axiom,
! [P: set_nat > $o,A4: set_nat] :
( ! [A5: set_nat] :
( ~ ( finite_finite_nat @ A5 )
=> ( P @ A5 ) )
=> ( ( 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 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_136_infinite__finite__induct,axiom,
! [P: set_HF_Mirabelle_hf > $o,A4: set_HF_Mirabelle_hf] :
( ! [A5: set_HF_Mirabelle_hf] :
( ~ ( finite586181922lle_hf @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo53200981lle_hf )
=> ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F2 )
=> ( ~ ( member1367349282lle_hf @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_137_the__elem__eq,axiom,
! [X2: hF_Mirabelle_hf] :
( ( the_el1104322134lle_hf @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
= X2 ) ).
% the_elem_eq
thf(fact_138_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_139_finite_OemptyI,axiom,
finite586181922lle_hf @ bot_bo53200981lle_hf ).
% finite.emptyI
thf(fact_140_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_141_infinite__imp__nonempty,axiom,
! [S: set_HF_Mirabelle_hf] :
( ~ ( finite586181922lle_hf @ S )
=> ( S != bot_bo53200981lle_hf ) ) ).
% infinite_imp_nonempty
thf(fact_142_finite_OinsertI,axiom,
! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( finite586181922lle_hf @ ( insert9649339lle_hf @ A2 @ A4 ) ) ) ).
% finite.insertI
thf(fact_143_finite_OinsertI,axiom,
! [A4: set_nat,A2: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A4 ) ) ) ).
% finite.insertI
thf(fact_144_bot__empty__eq,axiom,
( bot_bo1263054448e_hf_o
= ( ^ [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ bot_bo53200981lle_hf ) ) ) ).
% bot_empty_eq
thf(fact_145_Collect__empty__eq__bot,axiom,
! [P: set_HF_Mirabelle_hf > $o] :
( ( ( collec1758573718lle_hf @ P )
= bot_bo2093393035lle_hf )
= ( P = bot_bo554042810e_hf_o ) ) ).
% Collect_empty_eq_bot
thf(fact_146_Collect__empty__eq__bot,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( ( collec2046588256lle_hf @ P )
= bot_bo53200981lle_hf )
= ( P = bot_bo1263054448e_hf_o ) ) ).
% Collect_empty_eq_bot
thf(fact_147_is__singleton__the__elem,axiom,
( is_sin1448700567lle_hf
= ( ^ [A6: set_HF_Mirabelle_hf] :
( A6
= ( insert9649339lle_hf @ ( the_el1104322134lle_hf @ A6 ) @ bot_bo53200981lle_hf ) ) ) ) ).
% is_singleton_the_elem
thf(fact_148_is__singletonI,axiom,
! [X2: hF_Mirabelle_hf] : ( is_sin1448700567lle_hf @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ).
% is_singletonI
thf(fact_149_Set_Ois__empty__def,axiom,
( is_emp566801209lle_hf
= ( ^ [A6: set_HF_Mirabelle_hf] : A6 = bot_bo53200981lle_hf ) ) ).
% Set.is_empty_def
thf(fact_150_is__singleton__def,axiom,
( is_sin1448700567lle_hf
= ( ^ [A6: set_HF_Mirabelle_hf] :
? [X: hF_Mirabelle_hf] :
( A6
= ( insert9649339lle_hf @ X @ bot_bo53200981lle_hf ) ) ) ) ).
% is_singleton_def
thf(fact_151_is__singletonE,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( is_sin1448700567lle_hf @ A4 )
=> ~ ! [X3: hF_Mirabelle_hf] :
( A4
!= ( insert9649339lle_hf @ X3 @ bot_bo53200981lle_hf ) ) ) ).
% is_singletonE
thf(fact_152_is__singletonI_H,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( A4 != bot_bo53200981lle_hf )
=> ( ! [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A4 )
=> ( ( member1367349282lle_hf @ Y3 @ A4 )
=> ( X3 = Y3 ) ) )
=> ( is_sin1448700567lle_hf @ A4 ) ) ) ).
% is_singletonI'
thf(fact_153_card__0__eq,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( finite_card_nat @ A4 )
= zero_zero_nat )
= ( A4 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_154_card__0__eq,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( ( finite1213132899lle_hf @ A4 )
= zero_zero_nat )
= ( A4 = bot_bo53200981lle_hf ) ) ) ).
% card_0_eq
thf(fact_155_finite__subset__induct,axiom,
! [F: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_156_finite__subset__induct,axiom,
! [F: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ F )
=> ( ( ord_le432112161lle_hf @ F @ A4 )
=> ( ( P @ bot_bo53200981lle_hf )
=> ( ! [A3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F2 )
=> ( ( member1367349282lle_hf @ A3 @ A4 )
=> ( ~ ( member1367349282lle_hf @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert9649339lle_hf @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_157_finite__subset__induct_H,axiom,
! [F: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( ord_less_eq_set_nat @ F2 @ A4 )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_158_finite__subset__induct_H,axiom,
! [F: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ F )
=> ( ( ord_le432112161lle_hf @ F @ A4 )
=> ( ( P @ bot_bo53200981lle_hf )
=> ( ! [A3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F2 )
=> ( ( member1367349282lle_hf @ A3 @ A4 )
=> ( ( ord_le432112161lle_hf @ F2 @ A4 )
=> ( ~ ( member1367349282lle_hf @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert9649339lle_hf @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_159_inj__on__HF,axiom,
inj_on811196232lle_hf @ hF_Mirabelle_HF @ ( collec1758573718lle_hf @ finite586181922lle_hf ) ).
% inj_on_HF
thf(fact_160_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_161_subsetI,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A4 )
=> ( member1367349282lle_hf @ X3 @ B4 ) )
=> ( ord_le432112161lle_hf @ A4 @ B4 ) ) ).
% subsetI
thf(fact_162_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_163_subset__empty,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ bot_bo53200981lle_hf )
= ( A4 = bot_bo53200981lle_hf ) ) ).
% subset_empty
thf(fact_164_empty__subsetI,axiom,
! [A4: set_HF_Mirabelle_hf] : ( ord_le432112161lle_hf @ bot_bo53200981lle_hf @ A4 ) ).
% empty_subsetI
thf(fact_165_insert__subset,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ B4 )
= ( ( member1367349282lle_hf @ X2 @ B4 )
& ( ord_le432112161lle_hf @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_166_singleton__insert__inj__eq_H,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( ( insert9649339lle_hf @ A2 @ A4 )
= ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
= ( ( A2 = B3 )
& ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_167_singleton__insert__inj__eq,axiom,
! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf )
= ( insert9649339lle_hf @ A2 @ A4 ) )
= ( ( A2 = B3 )
& ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_168_card_Oempty,axiom,
( ( finite1213132899lle_hf @ bot_bo53200981lle_hf )
= zero_zero_nat ) ).
% card.empty
thf(fact_169_card_Oinfinite,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ~ ( finite586181922lle_hf @ A4 )
=> ( ( finite1213132899lle_hf @ A4 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_170_card_Oinfinite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_card_nat @ A4 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_171_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_172_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_173_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_174_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: 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 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_175_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_176_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_177_order__class_Oorder_Oantisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_178_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_179_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_180_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_181_antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_182_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_183_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_184_le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_185_eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_186_linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linear
thf(fact_187_antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_188_eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% eq_iff
thf(fact_189_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_190_ord__eq__le__subst,axiom,
! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_191_order__subst2,axiom,
! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_192_order__subst1,axiom,
! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_193_Collect__mono__iff,axiom,
! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
( ( ord_le2016357975lle_hf @ ( collec1758573718lle_hf @ P ) @ ( collec1758573718lle_hf @ Q ) )
= ( ! [X: set_HF_Mirabelle_hf] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_194_Collect__mono,axiom,
! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
( ! [X3: set_HF_Mirabelle_hf] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2016357975lle_hf @ ( collec1758573718lle_hf @ P ) @ ( collec1758573718lle_hf @ Q ) ) ) ).
% Collect_mono
thf(fact_195_subset__iff,axiom,
( ord_le432112161lle_hf
= ( ^ [A6: set_HF_Mirabelle_hf,B6: set_HF_Mirabelle_hf] :
! [T: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ T @ A6 )
=> ( member1367349282lle_hf @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_196_subset__eq,axiom,
( ord_le432112161lle_hf
= ( ^ [A6: set_HF_Mirabelle_hf,B6: set_HF_Mirabelle_hf] :
! [X: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X @ A6 )
=> ( member1367349282lle_hf @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_197_subsetD,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( member1367349282lle_hf @ C @ A4 )
=> ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% subsetD
thf(fact_198_in__mono,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( member1367349282lle_hf @ X2 @ A4 )
=> ( member1367349282lle_hf @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_199_card__subset__eq,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( ( finite1213132899lle_hf @ A4 )
= ( finite1213132899lle_hf @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_200_card__subset__eq,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ( finite_card_nat @ A4 )
= ( finite_card_nat @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_201_infinite__arbitrarily__large,axiom,
! [A4: set_HF_Mirabelle_hf,N: nat] :
( ~ ( finite586181922lle_hf @ A4 )
=> ? [B5: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B5 )
& ( ( finite1213132899lle_hf @ B5 )
= N )
& ( ord_le432112161lle_hf @ B5 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_202_infinite__arbitrarily__large,axiom,
! [A4: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A4 )
=> ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ( finite_card_nat @ B5 )
= N )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_203_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_204_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_205_bot_Oextremum,axiom,
! [A2: set_HF_Mirabelle_hf] : ( ord_le432112161lle_hf @ bot_bo53200981lle_hf @ A2 ) ).
% bot.extremum
thf(fact_206_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_207_bot_Oextremum__unique,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A2 @ bot_bo53200981lle_hf )
= ( A2 = bot_bo53200981lle_hf ) ) ).
% bot.extremum_unique
thf(fact_208_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_209_bot_Oextremum__uniqueI,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A2 @ bot_bo53200981lle_hf )
=> ( A2 = bot_bo53200981lle_hf ) ) ).
% bot.extremum_uniqueI
thf(fact_210_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_211_finite__has__maximal2,axiom,
! [A4: set_nat,A2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A2 @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_212_finite__has__minimal2,axiom,
! [A4: set_nat,A2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A2 @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_213_finite__subset,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( finite586181922lle_hf @ B4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_subset
thf(fact_214_finite__subset,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_215_infinite__super,axiom,
! [S: set_HF_Mirabelle_hf,T2: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ S @ T2 )
=> ( ~ ( finite586181922lle_hf @ S )
=> ~ ( finite586181922lle_hf @ T2 ) ) ) ).
% infinite_super
thf(fact_216_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_217_rev__finite__subset,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_218_rev__finite__subset,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_219_insert__mono,axiom,
! [C3: set_HF_Mirabelle_hf,D2: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ C3 @ D2 )
=> ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ A2 @ C3 ) @ ( insert9649339lle_hf @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_220_subset__insert,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ X2 @ A4 )
=> ( ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ B4 ) )
= ( ord_le432112161lle_hf @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_221_subset__insertI,axiom,
! [B4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] : ( ord_le432112161lle_hf @ B4 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_222_subset__insertI2,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_223_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_224_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_225_subset__singleton__iff,axiom,
! [X4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ X4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
= ( ( X4 = bot_bo53200981lle_hf )
| ( X4
= ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ) ).
% subset_singleton_iff
thf(fact_226_subset__singletonD,axiom,
! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
=> ( ( A4 = bot_bo53200981lle_hf )
| ( A4
= ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ) ) ).
% subset_singletonD
thf(fact_227_card__eq__0__iff,axiom,
! [A4: set_nat] :
( ( ( finite_card_nat @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_228_card__eq__0__iff,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( ( finite1213132899lle_hf @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bo53200981lle_hf )
| ~ ( finite586181922lle_hf @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_229_inj__on__empty,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] : ( inj_on811196232lle_hf @ F3 @ bot_bo2093393035lle_hf ) ).
% inj_on_empty
thf(fact_230_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_231_finite__ranking__induct,axiom,
! [S: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o,F3: hF_Mirabelle_hf > nat] :
( ( finite586181922lle_hf @ S )
=> ( ( P @ bot_bo53200981lle_hf )
=> ( ! [X3: hF_Mirabelle_hf,S2: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ S2 )
=> ( ! [Y5: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert9649339lle_hf @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_232_inj__on__subset,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( ord_le2016357975lle_hf @ B4 @ A4 )
=> ( inj_on811196232lle_hf @ F3 @ B4 ) ) ) ).
% inj_on_subset
thf(fact_233_subset__inj__on,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_se933006839lle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ B4 )
=> ( ( ord_le2016357975lle_hf @ A4 @ B4 )
=> ( inj_on811196232lle_hf @ F3 @ A4 ) ) ) ).
% subset_inj_on
thf(fact_234_insert__subsetI,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,X4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X2 @ A4 )
=> ( ( ord_le432112161lle_hf @ X4 @ A4 )
=> ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ X2 @ X4 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_235_subset__emptyI,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ X3 @ A4 )
=> ( ord_le432112161lle_hf @ A4 @ bot_bo53200981lle_hf ) ) ).
% subset_emptyI
thf(fact_236_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_237_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_238_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_239_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_240_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_241_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_242_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_HF_Mirabelle_hf,C3: nat] :
( ! [G: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ G @ F )
=> ( ( finite586181922lle_hf @ G )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ G ) @ C3 ) ) )
=> ( ( finite586181922lle_hf @ F )
& ( ord_less_eq_nat @ ( finite1213132899lle_hf @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_243_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C3: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_244_card__seteq,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ B4 ) @ ( finite1213132899lle_hf @ A4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_seteq
thf(fact_245_card__seteq,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_seteq
thf(fact_246_card__mono,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% card_mono
thf(fact_247_card__mono,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_248_card__insert__le,axiom,
! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) ) ) ) ).
% card_insert_le
thf(fact_249_card__insert__le,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A4 ) ) ) ) ).
% card_insert_le
thf(fact_250_inj__on__inverseI,axiom,
! [A4: set_se933006839lle_hf,G2: hF_Mirabelle_hf > set_HF_Mirabelle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
( ! [X3: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ X3 @ A4 )
=> ( ( G2 @ ( F3 @ X3 ) )
= X3 ) )
=> ( inj_on811196232lle_hf @ F3 @ A4 ) ) ).
% inj_on_inverseI
thf(fact_251_inj__on__contraD,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( X2 != Y2 )
=> ( ( member1490636632lle_hf @ X2 @ A4 )
=> ( ( member1490636632lle_hf @ Y2 @ A4 )
=> ( ( F3 @ X2 )
!= ( F3 @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_252_inj__on__eq__iff,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( member1490636632lle_hf @ X2 @ A4 )
=> ( ( member1490636632lle_hf @ Y2 @ A4 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_253_inj__on__cong,axiom,
! [A4: set_se933006839lle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,G2: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
( ! [A3: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ A3 @ A4 )
=> ( ( F3 @ A3 )
= ( G2 @ A3 ) ) )
=> ( ( inj_on811196232lle_hf @ F3 @ A4 )
= ( inj_on811196232lle_hf @ G2 @ A4 ) ) ) ).
% inj_on_cong
thf(fact_254_inj__on__def,axiom,
( inj_on811196232lle_hf
= ( ^ [F4: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A6: set_se933006839lle_hf] :
! [X: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ X @ A6 )
=> ! [Y4: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ Y4 @ A6 )
=> ( ( ( F4 @ X )
= ( F4 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_255_inj__onI,axiom,
! [A4: set_se933006839lle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
( ! [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ X3 @ A4 )
=> ( ( member1490636632lle_hf @ Y3 @ A4 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on811196232lle_hf @ F3 @ A4 ) ) ).
% inj_onI
thf(fact_256_inj__onD,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y2 ) )
=> ( ( member1490636632lle_hf @ X2 @ A4 )
=> ( ( member1490636632lle_hf @ Y2 @ A4 )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_257_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_HF_Mirabelle_hf] :
( ( ord_less_eq_nat @ N @ ( finite1213132899lle_hf @ S ) )
=> ~ ! [T3: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ T3 @ S )
=> ( ( ( finite1213132899lle_hf @ T3 )
= N )
=> ~ ( finite586181922lle_hf @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_258_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_259_card__le__if__inj__on__rel,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ B4 )
=> ( ! [A3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A3 @ A4 )
=> ? [B7: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ B7 @ B4 )
& ( R @ A3 @ B7 ) ) )
=> ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A1 @ A4 )
=> ( ( member1367349282lle_hf @ A22 @ A4 )
=> ( ( member1367349282lle_hf @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_260_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A4: set_HF_Mirabelle_hf,R: hF_Mirabelle_hf > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A3 @ A4 )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B4 )
& ( R @ A3 @ B7 ) ) )
=> ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B2: nat] :
( ( member1367349282lle_hf @ A1 @ A4 )
=> ( ( member1367349282lle_hf @ A22 @ A4 )
=> ( ( member_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_261_arg__min__least,axiom,
! [S: set_nat,Y2: nat,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F3 @ ( lattic1974000059at_nat @ F3 @ S ) ) @ ( F3 @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_262_arg__min__least,axiom,
! [S: set_HF_Mirabelle_hf,Y2: hF_Mirabelle_hf,F3: hF_Mirabelle_hf > nat] :
( ( finite586181922lle_hf @ S )
=> ( ( S != bot_bo53200981lle_hf )
=> ( ( member1367349282lle_hf @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F3 @ ( lattic710307446hf_nat @ F3 @ S ) ) @ ( F3 @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_263_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 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_264_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_265_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_266_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_267_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_268_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_269_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_270_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_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_271_card__Diff1__le,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ).
% card_Diff1_le
thf(fact_272_card__Diff1__le,axiom,
! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ) @ ( finite1213132899lle_hf @ A4 ) ) ) ).
% card_Diff1_le
thf(fact_273_card__le__inj,axiom,
! [A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
( ( finite1450550360lle_hf @ A4 )
=> ( ( finite586181922lle_hf @ B4 )
=> ( ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
=> ? [F5: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F5 @ A4 ) @ B4 )
& ( inj_on811196232lle_hf @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_274_card__le__inj,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( finite586181922lle_hf @ B4 )
=> ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
=> ? [F5: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F5 @ A4 ) @ B4 )
& ( inj_on755450110lle_hf @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_275_card__le__inj,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_nat] :
( ( finite586181922lle_hf @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) )
=> ? [F5: hF_Mirabelle_hf > nat] :
( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F5 @ A4 ) @ B4 )
& ( inj_on1874279374hf_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_276_card__le__inj,axiom,
! [A4: set_nat,B4: set_HF_Mirabelle_hf] :
( ( finite_finite_nat @ A4 )
=> ( ( finite586181922lle_hf @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
=> ? [F5: nat > hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( image_246164834lle_hf @ F5 @ A4 ) @ B4 )
& ( inj_on1988990670lle_hf @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_277_card__le__inj,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
=> ? [F5: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A4 ) @ B4 )
& ( inj_on_nat_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_278_card__inj__on__le,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) @ B4 )
=> ( ( finite586181922lle_hf @ B4 )
=> ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_279_image__eqI,axiom,
! [B3: hF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( B3
= ( F3 @ X2 ) )
=> ( ( member1367349282lle_hf @ X2 @ A4 )
=> ( member1367349282lle_hf @ B3 @ ( image_1743964010lle_hf @ F3 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_280_DiffI,axiom,
! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ A4 )
=> ( ~ ( member1367349282lle_hf @ C @ B4 )
=> ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_281_Diff__iff,axiom,
! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
= ( ( member1367349282lle_hf @ C @ A4 )
& ~ ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_282_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_283_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_284_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_285_image__empty,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( image_1743964010lle_hf @ F3 @ bot_bo53200981lle_hf )
= bot_bo53200981lle_hf ) ).
% image_empty
thf(fact_286_empty__is__image,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( bot_bo53200981lle_hf
= ( image_1743964010lle_hf @ F3 @ A4 ) )
= ( A4 = bot_bo53200981lle_hf ) ) ).
% empty_is_image
thf(fact_287_image__is__empty,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( ( image_1743964010lle_hf @ F3 @ A4 )
= bot_bo53200981lle_hf )
= ( A4 = bot_bo53200981lle_hf ) ) ).
% image_is_empty
thf(fact_288_finite__imageI,axiom,
! [F: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F )
=> ( finite586181922lle_hf @ ( image_1743964010lle_hf @ H @ F ) ) ) ).
% finite_imageI
thf(fact_289_finite__imageI,axiom,
! [F: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > nat] :
( ( finite586181922lle_hf @ F )
=> ( finite_finite_nat @ ( image_131453538hf_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_290_finite__imageI,axiom,
! [F: set_nat,H: nat > hF_Mirabelle_hf] :
( ( finite_finite_nat @ F )
=> ( finite586181922lle_hf @ ( image_246164834lle_hf @ H @ F ) ) ) ).
% finite_imageI
thf(fact_291_finite__imageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_292_image__insert,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( image_1743964010lle_hf @ F3 @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( insert9649339lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ B4 ) ) ) ).
% image_insert
thf(fact_293_insert__image,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X2 @ A4 )
=> ( ( insert9649339lle_hf @ ( F3 @ X2 ) @ ( image_1743964010lle_hf @ F3 @ A4 ) )
= ( image_1743964010lle_hf @ F3 @ A4 ) ) ) ).
% insert_image
thf(fact_294_Diff__empty,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A4 @ bot_bo53200981lle_hf )
= A4 ) ).
% Diff_empty
thf(fact_295_empty__Diff,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ bot_bo53200981lle_hf @ A4 )
= bot_bo53200981lle_hf ) ).
% empty_Diff
thf(fact_296_Diff__cancel,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A4 @ A4 )
= bot_bo53200981lle_hf ) ).
% Diff_cancel
thf(fact_297_finite__Diff,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% finite_Diff
thf(fact_298_finite__Diff,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% finite_Diff
thf(fact_299_finite__Diff2,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
= ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_Diff2
thf(fact_300_finite__Diff2,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_301_Diff__insert0,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ X2 @ A4 )
=> ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ B4 ) )
= ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_302_insert__Diff1,axiom,
! [X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X2 @ B4 )
=> ( ( minus_1450406810lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ B4 )
= ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_303_Diff__eq__empty__iff,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( ( minus_1450406810lle_hf @ A4 @ B4 )
= bot_bo53200981lle_hf )
= ( ord_le432112161lle_hf @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_304_insert__Diff__single,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( insert9649339lle_hf @ A2 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) )
= ( insert9649339lle_hf @ A2 @ A4 ) ) ).
% insert_Diff_single
thf(fact_305_finite__Diff__insert,axiom,
! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) ) )
= ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_306_finite__Diff__insert,axiom,
! [A4: set_nat,A2: nat,B4: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A2 @ B4 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_307_inj__on__insert,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ ( insert1636143089lle_hf @ A2 @ A4 ) )
= ( ( inj_on811196232lle_hf @ F3 @ A4 )
& ~ ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_899003828lle_hf @ F3 @ ( minus_500612048lle_hf @ A4 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_308_inj__on__insert,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( inj_on755450110lle_hf @ F3 @ ( insert9649339lle_hf @ A2 @ A4 ) )
= ( ( inj_on755450110lle_hf @ F3 @ A4 )
& ~ ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_309_inj__on__image__set__diff,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,C3: set_se933006839lle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ C3 )
=> ( ( ord_le2016357975lle_hf @ ( minus_500612048lle_hf @ A4 @ B4 ) @ C3 )
=> ( ( ord_le2016357975lle_hf @ B4 @ C3 )
=> ( ( image_899003828lle_hf @ F3 @ ( minus_500612048lle_hf @ A4 @ B4 ) )
= ( minus_1450406810lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) @ ( image_899003828lle_hf @ F3 @ B4 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_310_image__subsetI,axiom,
! [A4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A4 )
=> ( member1367349282lle_hf @ ( F3 @ X3 ) @ B4 ) )
=> ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_311_finite__surj,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
=> ( finite586181922lle_hf @ B4 ) ) ) ).
% finite_surj
thf(fact_312_finite__surj,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_nat,F3: hF_Mirabelle_hf > nat] :
( ( finite586181922lle_hf @ A4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F3 @ A4 ) )
=> ( finite_finite_nat @ B4 ) ) ) ).
% finite_surj
thf(fact_313_finite__surj,axiom,
! [A4: set_nat,B4: set_HF_Mirabelle_hf,F3: nat > hF_Mirabelle_hf] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F3 @ A4 ) )
=> ( finite586181922lle_hf @ B4 ) ) ) ).
% finite_surj
thf(fact_314_finite__surj,axiom,
! [A4: set_nat,B4: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A4 ) )
=> ( finite_finite_nat @ B4 ) ) ) ).
% finite_surj
thf(fact_315_finite__subset__image,axiom,
! [B4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
=> ? [C4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ C4 @ A4 )
& ( finite586181922lle_hf @ C4 )
& ( B4
= ( image_1743964010lle_hf @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_316_finite__subset__image,axiom,
! [B4: set_HF_Mirabelle_hf,F3: nat > hF_Mirabelle_hf,A4: set_nat] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F3 @ A4 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A4 )
& ( finite_finite_nat @ C4 )
& ( B4
= ( image_246164834lle_hf @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_317_finite__subset__image,axiom,
! [B4: set_nat,F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F3 @ A4 ) )
=> ? [C4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ C4 @ A4 )
& ( finite586181922lle_hf @ C4 )
& ( B4
= ( image_131453538hf_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_318_finite__subset__image,axiom,
! [B4: set_nat,F3: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A4 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A4 )
& ( finite_finite_nat @ C4 )
& ( B4
= ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_319_ex__finite__subset__image,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( ? [B6: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ A4 )
& ( P @ ( image_1743964010lle_hf @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_320_ex__finite__subset__image,axiom,
! [F3: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
( ( ? [B6: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ ( image_246164834lle_hf @ F3 @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P @ ( image_246164834lle_hf @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_321_ex__finite__subset__image,axiom,
! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_131453538hf_nat @ F3 @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ A4 )
& ( P @ ( image_131453538hf_nat @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_322_ex__finite__subset__image,axiom,
! [F3: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_323_all__finite__subset__image,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( ! [B6: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ ( image_1743964010lle_hf @ F3 @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ A4 ) )
=> ( P @ ( image_1743964010lle_hf @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_324_all__finite__subset__image,axiom,
! [F3: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
( ( ! [B6: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ ( image_246164834lle_hf @ F3 @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P @ ( image_246164834lle_hf @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_325_all__finite__subset__image,axiom,
! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_131453538hf_nat @ F3 @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B6 )
& ( ord_le432112161lle_hf @ B6 @ A4 ) )
=> ( P @ ( image_131453538hf_nat @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_326_all__finite__subset__image,axiom,
! [F3: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_327_finite__imageD,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
=> ( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( finite1450550360lle_hf @ A4 ) ) ) ).
% finite_imageD
thf(fact_328_finite__imageD,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) )
=> ( ( inj_on755450110lle_hf @ F3 @ A4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_imageD
thf(fact_329_finite__imageD,axiom,
! [F3: nat > hF_Mirabelle_hf,A4: set_nat] :
( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F3 @ A4 ) )
=> ( ( inj_on1988990670lle_hf @ F3 @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_330_finite__imageD,axiom,
! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
( ( finite_finite_nat @ ( image_131453538hf_nat @ F3 @ A4 ) )
=> ( ( inj_on1874279374hf_nat @ F3 @ A4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_imageD
thf(fact_331_finite__imageD,axiom,
! [F3: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A4 ) )
=> ( ( inj_on_nat_nat @ F3 @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_332_finite__image__iff,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
= ( finite1450550360lle_hf @ A4 ) ) ) ).
% finite_image_iff
thf(fact_333_finite__image__iff,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( inj_on755450110lle_hf @ F3 @ A4 )
=> ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) )
= ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_image_iff
thf(fact_334_finite__image__iff,axiom,
! [F3: nat > hF_Mirabelle_hf,A4: set_nat] :
( ( inj_on1988990670lle_hf @ F3 @ A4 )
=> ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F3 @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_335_finite__image__iff,axiom,
! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
( ( inj_on1874279374hf_nat @ F3 @ A4 )
=> ( ( finite_finite_nat @ ( image_131453538hf_nat @ F3 @ A4 ) )
= ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_image_iff
thf(fact_336_finite__image__iff,axiom,
! [F3: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F3 @ A4 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_337_inj__on__image__eq__iff,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,C3: set_se933006839lle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ C3 )
=> ( ( ord_le2016357975lle_hf @ A4 @ C3 )
=> ( ( ord_le2016357975lle_hf @ B4 @ C3 )
=> ( ( ( image_899003828lle_hf @ F3 @ A4 )
= ( image_899003828lle_hf @ F3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_338_inj__on__image__mem__iff,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_se933006839lle_hf,A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ B4 )
=> ( ( member1490636632lle_hf @ A2 @ B4 )
=> ( ( ord_le2016357975lle_hf @ A4 @ B4 )
=> ( ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_899003828lle_hf @ F3 @ A4 ) )
= ( member1490636632lle_hf @ A2 @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_339_inj__on__image__mem__iff,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( inj_on755450110lle_hf @ F3 @ B4 )
=> ( ( member1367349282lle_hf @ A2 @ B4 )
=> ( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ A4 ) )
= ( member1367349282lle_hf @ A2 @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_340_inj__img__insertE,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ~ ( member1367349282lle_hf @ X2 @ B4 )
=> ( ( ( insert9649339lle_hf @ X2 @ B4 )
= ( image_899003828lle_hf @ F3 @ A4 ) )
=> ~ ! [X6: set_HF_Mirabelle_hf,A7: set_se933006839lle_hf] :
( ~ ( member1490636632lle_hf @ X6 @ A7 )
=> ( ( A4
= ( insert1636143089lle_hf @ X6 @ A7 ) )
=> ( ( X2
= ( F3 @ X6 ) )
=> ( B4
!= ( image_899003828lle_hf @ F3 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_341_inj__img__insertE,axiom,
! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( inj_on755450110lle_hf @ F3 @ A4 )
=> ( ~ ( member1367349282lle_hf @ X2 @ B4 )
=> ( ( ( insert9649339lle_hf @ X2 @ B4 )
= ( image_1743964010lle_hf @ F3 @ A4 ) )
=> ~ ! [X6: hF_Mirabelle_hf,A7: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ X6 @ A7 )
=> ( ( A4
= ( insert9649339lle_hf @ X6 @ A7 ) )
=> ( ( X2
= ( F3 @ X6 ) )
=> ( B4
!= ( image_1743964010lle_hf @ F3 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_342_card__image,axiom,
! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F3 @ A4 )
=> ( ( finite1213132899lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
= ( finite90088345lle_hf @ A4 ) ) ) ).
% card_image
thf(fact_343_Diff__insert__absorb,axiom,
! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ X2 @ A4 )
=> ( ( minus_1450406810lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_344_Diff__insert2,axiom,
! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_345_insert__Diff,axiom,
! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ A4 )
=> ( ( insert9649339lle_hf @ A2 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_346_Diff__insert,axiom,
! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ).
% Diff_insert
thf(fact_347_subset__Diff__insert,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf,C3: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ A4 @ ( minus_1450406810lle_hf @ B4 @ ( insert9649339lle_hf @ X2 @ C3 ) ) )
= ( ( ord_le432112161lle_hf @ A4 @ ( minus_1450406810lle_hf @ B4 @ C3 ) )
& ~ ( member1367349282lle_hf @ X2 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_348_in__image__insert__iff,axiom,
! [B4: set_se933006839lle_hf,X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ! [C4: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ C4 @ B4 )
=> ~ ( member1367349282lle_hf @ X2 @ C4 ) )
=> ( ( member1490636632lle_hf @ A4 @ ( image_1514960916lle_hf @ ( insert9649339lle_hf @ X2 ) @ B4 ) )
= ( ( member1367349282lle_hf @ X2 @ A4 )
& ( member1490636632lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_349_DiffE,axiom,
! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
=> ~ ( ( member1367349282lle_hf @ C @ A4 )
=> ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% DiffE
thf(fact_350_DiffD1,axiom,
! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
=> ( member1367349282lle_hf @ C @ A4 ) ) ).
% DiffD1
% Conjectures (1)
thf(conj_0,conjecture,
( ( z = zero_z189798548lle_hf )
= ( ! [X: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X @ z ) ) ) ).
%------------------------------------------------------------------------------