TPTP Problem File: ITP073^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP073^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HF problem prob_473__5331678_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_473__5331678_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 406 ( 92 unt; 57 typ; 0 def)
% Number of atoms : 1190 ( 228 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3157 ( 104 ~; 22 |; 94 &;2290 @)
% ( 0 <=>; 647 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 294 ( 294 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 6 con; 0-2 aty)
% Number of variables : 1113 ( 76 ^; 968 !; 69 ?;1113 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:18.035
%------------------------------------------------------------------------------
% 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 (52)
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_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_OHCollect,type,
hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > 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_Ohmem,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__HF____Mirabelle____glliljednj__Ohf,type,
ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__HF____Mirabelle____glliljednj__Ohf_J,type,
ord_le2017230260lle_hf: ( $o > hF_Mirabelle_hf ) > ( $o > hF_Mirabelle_hf ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__HF____Mirabelle____glliljednj__Ohf,type,
ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_Oorder__class_OGreatest_001t__HF____Mirabelle____glliljednj__Ohf,type,
order_1640852850lle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
order_690706045lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
order_557194383hf_nat: ( hF_Mirabelle_hf > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
order_671905679lle_hf: ( nat > hF_Mirabelle_hf ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat,type,
order_1631207636at_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oordering_001t__HF____Mirabelle____glliljednj__Ohf,type,
orderi1737556723lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
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__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__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or1544565540an_nat: nat > nat > set_nat ).
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_A,type,
a: hF_Mirabelle_hf ).
thf(sy_v_B,type,
b: hF_Mirabelle_hf ).
% Relevant facts (347)
thf(fact_0_hf__equalityI,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A )
= ( hF_Mirabelle_hmem @ X @ B ) )
=> ( A = B ) ) ).
% hf_equalityI
thf(fact_1_hf__ext,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A2 )
= ( hF_Mirabelle_hmem @ X2 @ B2 ) ) ) ) ).
% hf_ext
thf(fact_2_replacement,axiom,
! [X3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ X3 )
=> ( ( R @ U @ V )
=> ( ( R @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ? [Z2: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X3 )
& ( R @ U2 @ V3 ) ) ) ) ) ).
% replacement
thf(fact_3_binary__union,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X3 )
| ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% binary_union
thf(fact_4_union__of__set,axiom,
! [X3: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ X3 )
& ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% union_of_set
thf(fact_5_comprehension,axiom,
! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X3 )
& ( P @ U3 ) ) ) ).
% comprehension
thf(fact_6_less__eq__hf__def,axiom,
( ord_le976219883lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A3 )
=> ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% less_eq_hf_def
thf(fact_7_replacement__fun,axiom,
! [X3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X3 )
& ( V3
= ( F @ U2 ) ) ) ) ) ).
% replacement_fun
thf(fact_8_order__refl,axiom,
! [X3: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X3 @ X3 ) ).
% order_refl
thf(fact_9_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_10_PrimReplace__iff,axiom,
! [A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ A4 )
=> ( ( R @ U @ V )
=> ( ( R @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A4 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( R @ U2 @ V4 ) ) ) ) ) ).
% PrimReplace_iff
thf(fact_11_HCollect__iff,axiom,
! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mir818139703ollect @ P @ A4 ) )
= ( ( P @ X3 )
& ( hF_Mirabelle_hmem @ X3 @ A4 ) ) ) ).
% HCollect_iff
thf(fact_12_HUnion__iff,axiom,
! [X3: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HUnion @ A4 ) )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A4 )
& ( hF_Mirabelle_hmem @ X3 @ Y3 ) ) ) ) ).
% HUnion_iff
thf(fact_13_Replace__iff,axiom,
! [V4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A4 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( R @ U2 @ V4 )
& ! [Y3: hF_Mirabelle_hf] :
( ( R @ U2 @ Y3 )
=> ( Y3 = V4 ) ) ) ) ) ).
% Replace_iff
thf(fact_14_order__subst1,axiom,
! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_15_order__subst1,axiom,
! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_16_order__subst1,axiom,
! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_17_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_18_order__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_19_order__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_20_order__subst2,axiom,
! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_21_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_22_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_23_Replace__cong,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( A4 = B4 )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( P @ X @ Y4 )
= ( Q @ X @ Y4 ) ) )
=> ( ( hF_Mirabelle_Replace @ A4 @ P )
= ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% Replace_cong
thf(fact_24_dual__order_Oantisym,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( ord_le976219883lle_hf @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_25_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_26_dual__order_Oeq__iff,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B2 @ A2 )
& ( ord_le976219883lle_hf @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_27_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_28_dual__order_Otrans,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( ord_le976219883lle_hf @ C @ B )
=> ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_29_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_30_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_31_dual__order_Orefl,axiom,
! [A: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ A @ A ) ).
% dual_order.refl
thf(fact_32_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_33_order__trans,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ( ord_le976219883lle_hf @ Y2 @ Z3 )
=> ( ord_le976219883lle_hf @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_34_order__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_35_order__class_Oorder_Oantisym,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_le976219883lle_hf @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_36_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_37_ord__le__eq__trans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( B = C )
=> ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_38_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_39_ord__eq__le__trans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( A = B )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_40_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_41_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A2 @ B2 )
& ( ord_le976219883lle_hf @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_42_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_43_antisym__conv,axiom,
! [Y2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Y2 @ X3 )
=> ( ( ord_le976219883lle_hf @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv
thf(fact_44_antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv
thf(fact_45_le__cases3,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_46_order_Otrans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% order.trans
thf(fact_47_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_48_le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% le_cases
thf(fact_49_eq__refl,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( X3 = Y2 )
=> ( ord_le976219883lle_hf @ X3 @ Y2 ) ) ).
% eq_refl
thf(fact_50_eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% eq_refl
thf(fact_51_linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linear
thf(fact_52_antisym,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ( ord_le976219883lle_hf @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% antisym
thf(fact_53_antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% antisym
thf(fact_54_eq__iff,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X2 @ Y3 )
& ( ord_le976219883lle_hf @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_55_eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_56_ord__le__eq__subst,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__eq__le__subst,axiom,
! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( A
= ( F @ B ) )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_61_ord__eq__le__subst,axiom,
! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( A
= ( F @ B ) )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_62_ord__eq__le__subst,axiom,
! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_63_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_64_Greatest__equality,axiom,
! [P: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] :
( ( P @ X3 )
=> ( ! [Y4: hF_Mirabelle_hf] :
( ( P @ Y4 )
=> ( ord_le976219883lle_hf @ Y4 @ X3 ) )
=> ( ( order_1640852850lle_hf @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_65_Greatest__equality,axiom,
! [P: nat > $o,X3: nat] :
( ( P @ X3 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) )
=> ( ( order_Greatest_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_66_GreatestI2__order,axiom,
! [P: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf,Q: hF_Mirabelle_hf > $o] :
( ( P @ X3 )
=> ( ! [Y4: hF_Mirabelle_hf] :
( ( P @ Y4 )
=> ( ord_le976219883lle_hf @ Y4 @ X3 ) )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ( ! [Y5: hF_Mirabelle_hf] :
( ( P @ Y5 )
=> ( ord_le976219883lle_hf @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1640852850lle_hf @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_67_GreatestI2__order,axiom,
! [P: nat > $o,X3: nat,Q: nat > $o] :
( ( P @ X3 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_68_le__rel__bool__arg__iff,axiom,
( ord_le2017230260lle_hf
= ( ^ [X4: $o > hF_Mirabelle_hf,Y6: $o > hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le976219883lle_hf @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_69_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_70_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% hmem_def
thf(fact_71_mem__Collect__eq,axiom,
! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( member1490636632lle_hf @ A @ ( collec1758573718lle_hf @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( collec2046588256lle_hf
@ ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A4: set_se933006839lle_hf] :
( ( collec1758573718lle_hf
@ ^ [X2: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_75_Collect__cong,axiom,
! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
( ! [X: set_HF_Mirabelle_hf] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec1758573718lle_hf @ P )
= ( collec1758573718lle_hf @ Q ) ) ) ).
% Collect_cong
thf(fact_76_RepFun__iff,axiom,
! [V4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A4 @ F ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( V4
= ( F @ U2 ) ) ) ) ) ).
% RepFun_iff
thf(fact_77_less__hf__def,axiom,
( ord_le1310584031lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% less_hf_def
thf(fact_78_antimono__def,axiom,
( order_690706045lle_hf
= ( ^ [F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X2 @ Y3 )
=> ( ord_le976219883lle_hf @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimono_def
thf(fact_79_antimono__def,axiom,
( order_557194383hf_nat
= ( ^ [F2: hF_Mirabelle_hf > nat] :
! [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimono_def
thf(fact_80_antimono__def,axiom,
( order_671905679lle_hf
= ( ^ [F2: nat > hF_Mirabelle_hf] :
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le976219883lle_hf @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimono_def
thf(fact_81_antimono__def,axiom,
( order_1631207636at_nat
= ( ^ [F2: nat > nat] :
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimono_def
thf(fact_82_antimonoI,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ Y4 ) @ ( F @ X ) ) )
=> ( order_690706045lle_hf @ F ) ) ).
% antimonoI
thf(fact_83_antimonoI,axiom,
! [F: hF_Mirabelle_hf > nat] :
( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X ) ) )
=> ( order_557194383hf_nat @ F ) ) ).
% antimonoI
thf(fact_84_antimonoI,axiom,
! [F: nat > hF_Mirabelle_hf] :
( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ Y4 ) @ ( F @ X ) ) )
=> ( order_671905679lle_hf @ F ) ) ).
% antimonoI
thf(fact_85_antimonoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X ) ) )
=> ( order_1631207636at_nat @ F ) ) ).
% antimonoI
thf(fact_86_antimonoE,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( order_690706045lle_hf @ F )
=> ( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_87_antimonoE,axiom,
! [F: hF_Mirabelle_hf > nat,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( order_557194383hf_nat @ F )
=> ( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_88_antimonoE,axiom,
! [F: nat > hF_Mirabelle_hf,X3: nat,Y2: nat] :
( ( order_671905679lle_hf @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_89_antimonoE,axiom,
! [F: nat > nat,X3: nat,Y2: nat] :
( ( order_1631207636at_nat @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_90_antimonoD,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( order_690706045lle_hf @ F )
=> ( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_91_antimonoD,axiom,
! [F: hF_Mirabelle_hf > nat,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( order_557194383hf_nat @ F )
=> ( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_92_antimonoD,axiom,
! [F: nat > hF_Mirabelle_hf,X3: nat,Y2: nat] :
( ( order_671905679lle_hf @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_93_antimonoD,axiom,
! [F: nat > nat,X3: nat,Y2: nat] :
( ( order_1631207636at_nat @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_94_verit__comp__simplify1_I1_J,axiom,
! [A: hF_Mirabelle_hf] :
~ ( ord_le1310584031lle_hf @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_95_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_96_ord__eq__less__subst,axiom,
! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_ord__eq__less__subst,axiom,
! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_98_ord__eq__less__subst,axiom,
! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_99_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_100_ord__less__eq__subst,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_101_ord__less__eq__subst,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_102_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_103_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_104_order__less__subst1,axiom,
! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_105_order__less__subst1,axiom,
! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_106_order__less__subst1,axiom,
! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_107_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_108_order__less__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_109_order__less__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_110_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_111_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_112_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_113_neqE,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% neqE
thf(fact_114_neq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
= ( ( ord_less_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% neq_iff
thf(fact_115_order_Oasym,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ~ ( ord_le1310584031lle_hf @ B @ A ) ) ).
% order.asym
thf(fact_116_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_117_less__imp__neq,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_118_less__imp__neq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_119_less__asym,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% less_asym
thf(fact_120_less__asym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% less_asym
thf(fact_121_less__asym_H,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ~ ( ord_le1310584031lle_hf @ B @ A ) ) ).
% less_asym'
thf(fact_122_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_123_less__trans,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( ( ord_le1310584031lle_hf @ Y2 @ Z3 )
=> ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% less_trans
thf(fact_124_less__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% less_trans
thf(fact_125_less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% less_linear
thf(fact_126_less__irrefl,axiom,
! [X3: hF_Mirabelle_hf] :
~ ( ord_le1310584031lle_hf @ X3 @ X3 ) ).
% less_irrefl
thf(fact_127_less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% less_irrefl
thf(fact_128_ord__eq__less__trans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( A = B )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_129_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_130_ord__less__eq__trans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( B = C )
=> ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_131_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_132_dual__order_Oasym,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ A )
=> ~ ( ord_le1310584031lle_hf @ A @ B ) ) ).
% dual_order.asym
thf(fact_133_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_134_less__imp__not__eq,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_135_less__imp__not__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_136_less__not__sym,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% less_not_sym
thf(fact_137_less__not__sym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% less_not_sym
thf(fact_138_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_139_antisym__conv3,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_nat @ Y2 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_140_less__imp__not__eq2,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_141_less__imp__not__eq2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_142_less__imp__triv,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,P: $o] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( ( ord_le1310584031lle_hf @ Y2 @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_143_less__imp__triv,axiom,
! [X3: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_144_linorder__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_145_dual__order_Oirrefl,axiom,
! [A: hF_Mirabelle_hf] :
~ ( ord_le1310584031lle_hf @ A @ A ) ).
% dual_order.irrefl
thf(fact_146_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_147_order_Ostrict__trans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_148_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_149_less__imp__not__less,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% less_imp_not_less
thf(fact_150_less__imp__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% less_imp_not_less
thf(fact_151_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_152_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_153_dual__order_Ostrict__trans,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ A )
=> ( ( ord_le1310584031lle_hf @ C @ B )
=> ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_154_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_155_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_156_order_Ostrict__implies__not__eq,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_157_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_158_dual__order_Ostrict__implies__not__eq,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_159_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_160_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_161_leD,axiom,
! [Y2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Y2 @ X3 )
=> ~ ( ord_le1310584031lle_hf @ X3 @ Y2 ) ) ).
% leD
thf(fact_162_leD,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_163_leI,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% leI
thf(fact_164_le__less,axiom,
( ord_le976219883lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% le_less
thf(fact_165_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% le_less
thf(fact_166_less__le,axiom,
( ord_le1310584031lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% less_le
thf(fact_167_less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% less_le
thf(fact_168_order__le__less__subst1,axiom,
! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_169_order__le__less__subst1,axiom,
! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_170_order__le__less__subst1,axiom,
! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_171_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_172_order__le__less__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_173_order__le__less__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_174_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_175_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_176_order__less__le__subst1,axiom,
! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_177_order__less__le__subst1,axiom,
! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_178_order__less__le__subst1,axiom,
! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_179_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_180_order__less__le__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_181_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_182_order__less__le__subst2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_183_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_184_not__le,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X3 ) ) ).
% not_le
thf(fact_185_not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% not_less
thf(fact_186_le__neq__trans,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( A != B )
=> ( ord_le1310584031lle_hf @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_187_le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_188_antisym__conv1,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ~ ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( ( ord_le976219883lle_hf @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_189_antisym__conv1,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_190_antisym__conv2,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ( ~ ( ord_le1310584031lle_hf @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_191_antisym__conv2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_192_less__imp__le,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( ord_le976219883lle_hf @ X3 @ Y2 ) ) ).
% less_imp_le
thf(fact_193_less__imp__le,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% less_imp_le
thf(fact_194_le__less__trans,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ( ord_le1310584031lle_hf @ Y2 @ Z3 )
=> ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% le_less_trans
thf(fact_195_le__less__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% le_less_trans
thf(fact_196_less__le__trans,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ X3 @ Y2 )
=> ( ( ord_le976219883lle_hf @ Y2 @ Z3 )
=> ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% less_le_trans
thf(fact_197_less__le__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% less_le_trans
thf(fact_198_le__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% le_less_linear
thf(fact_199_le__imp__less__or__eq,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X3 @ Y2 )
=> ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_200_le__imp__less__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_201_less__le__not__le,axiom,
( ord_le1310584031lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X2 @ Y3 )
& ~ ( ord_le976219883lle_hf @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_202_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_203_not__le__imp__less,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ord_less_nat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_204_order_Ostrict__trans1,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_le1310584031lle_hf @ B @ C )
=> ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_205_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_206_order_Ostrict__trans2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ( ord_le976219883lle_hf @ B @ C )
=> ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_207_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_208_order_Oorder__iff__strict,axiom,
( ord_le976219883lle_hf
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_209_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_210_order_Ostrict__iff__order,axiom,
( ord_le1310584031lle_hf
= ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_211_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_212_dual__order_Ostrict__trans1,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( ord_le1310584031lle_hf @ C @ B )
=> ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_213_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_214_dual__order_Ostrict__trans2,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ A )
=> ( ( ord_le976219883lle_hf @ C @ B )
=> ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_215_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_216_order_Ostrict__implies__order,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ B )
=> ( ord_le976219883lle_hf @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_217_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_218_dual__order_Oorder__iff__strict,axiom,
( ord_le976219883lle_hf
= ( ^ [B2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_219_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_220_dual__order_Ostrict__iff__order,axiom,
( ord_le1310584031lle_hf
= ( ^ [B2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_221_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_222_dual__order_Ostrict__implies__order,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ A )
=> ( ord_le976219883lle_hf @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_223_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_224_order_Onot__eq__order__implies__strict,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A != B )
=> ( ( ord_le976219883lle_hf @ A @ B )
=> ( ord_le1310584031lle_hf @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_225_order_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_226_RepFun__cong,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( A4 = B4 )
=> ( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( hF_Mirabelle_RepFun @ A4 @ F )
= ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% RepFun_cong
thf(fact_227_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_228_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_229_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_230_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_231_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_232_HF__hfset,axiom,
! [A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
= A ) ).
% HF_hfset
thf(fact_233_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_234_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_235_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_236_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_237_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_238_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_239_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_240_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_241_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_242_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_243_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_244_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_245_hfset__HF,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A4 ) )
= A4 ) ) ).
% hfset_HF
thf(fact_246_hmem__HF__iff,axiom,
! [X3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HF @ A4 ) )
= ( ( member1367349282lle_hf @ X3 @ A4 )
& ( finite586181922lle_hf @ A4 ) ) ) ).
% hmem_HF_iff
thf(fact_247_finite__hfset,axiom,
! [A: hF_Mirabelle_hf] : ( finite586181922lle_hf @ ( hF_Mirabelle_hfset @ A ) ) ).
% finite_hfset
thf(fact_248_finite__has__minimal2,axiom,
! [A4: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( member1367349282lle_hf @ A @ A4 )
=> ? [X: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X @ A4 )
& ( ord_le976219883lle_hf @ X @ A )
& ! [Xa: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ Xa @ A4 )
=> ( ( ord_le976219883lle_hf @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_249_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_250_finite__has__maximal2,axiom,
! [A4: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( member1367349282lle_hf @ A @ A4 )
=> ? [X: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X @ A4 )
& ( ord_le976219883lle_hf @ A @ X )
& ! [Xa: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ Xa @ A4 )
=> ( ( ord_le976219883lle_hf @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_251_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_252_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_253_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_254_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_255_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_256_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_257_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_258_finite__psubset__induct,axiom,
! [A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ A4 )
=> ( ! [A7: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A7 )
=> ( ! [B7: set_HF_Mirabelle_hf] :
( ( ord_le1344122901lle_hf @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_259_finite__psubset__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_260_card__psubset,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ A4 @ B4 )
=> ( ( ord_less_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
=> ( ord_le1344122901lle_hf @ A4 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_261_card__psubset,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_set_nat @ A4 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_262_inj__on__HF,axiom,
inj_on811196232lle_hf @ hF_Mirabelle_HF @ ( collec1758573718lle_hf @ finite586181922lle_hf ) ).
% inj_on_HF
thf(fact_263_order_Oordering__axioms,axiom,
orderi1737556723lle_hf @ ord_le976219883lle_hf @ ord_le1310584031lle_hf ).
% order.ordering_axioms
thf(fact_264_order_Oordering__axioms,axiom,
ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.ordering_axioms
thf(fact_265_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_266_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_267_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_268_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_269_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_270_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_271_infinite__arbitrarily__large,axiom,
! [A4: set_HF_Mirabelle_hf,N2: nat] :
( ~ ( finite586181922lle_hf @ A4 )
=> ? [B8: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B8 )
& ( ( finite1213132899lle_hf @ B8 )
= N2 )
& ( ord_le432112161lle_hf @ B8 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_272_infinite__arbitrarily__large,axiom,
! [A4: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A4 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N2 )
& ( ord_less_eq_set_nat @ B8 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_273_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_HF_Mirabelle_hf,C3: nat] :
( ! [G2: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ G2 @ F3 )
=> ( ( finite586181922lle_hf @ G2 )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ G2 ) @ C3 ) ) )
=> ( ( finite586181922lle_hf @ F3 )
& ( ord_less_eq_nat @ ( finite1213132899lle_hf @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_274_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C3: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F3 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_275_psubset__card__mono,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le1344122901lle_hf @ A4 @ B4 )
=> ( ord_less_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_276_psubset__card__mono,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_set_nat @ A4 @ B4 )
=> ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_277_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_HF_Mirabelle_hf] :
( ( ord_less_eq_nat @ N2 @ ( finite1213132899lle_hf @ S ) )
=> ~ ! [T3: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ T3 @ S )
=> ( ( ( finite1213132899lle_hf @ T3 )
= N2 )
=> ~ ( finite586181922lle_hf @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_278_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_279_linorder__inj__onI,axiom,
! [A4: set_se933006839lle_hf,F: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
( ! [X: set_HF_Mirabelle_hf,Y4: set_HF_Mirabelle_hf] :
( ( ord_le1344122901lle_hf @ X @ Y4 )
=> ( ( member1490636632lle_hf @ X @ A4 )
=> ( ( member1490636632lle_hf @ Y4 @ A4 )
=> ( ( F @ X )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X: set_HF_Mirabelle_hf,Y4: set_HF_Mirabelle_hf] :
( ( member1490636632lle_hf @ X @ A4 )
=> ( ( member1490636632lle_hf @ Y4 @ A4 )
=> ( ( ord_le432112161lle_hf @ X @ Y4 )
| ( ord_le432112161lle_hf @ Y4 @ X ) ) ) )
=> ( inj_on811196232lle_hf @ F @ A4 ) ) ) ).
% linorder_inj_onI
thf(fact_280_card__le__if__inj__on__rel,axiom,
! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( finite586181922lle_hf @ B4 )
=> ( ! [A5: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A5 @ A4 )
=> ? [B9: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ B9 @ B4 )
& ( R2 @ A5 @ B9 ) ) )
=> ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A1 @ A4 )
=> ( ( member1367349282lle_hf @ A22 @ A4 )
=> ( ( member1367349282lle_hf @ B5 @ B4 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_281_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A4: set_HF_Mirabelle_hf,R2: hF_Mirabelle_hf > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A5: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A5 @ A4 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B4 )
& ( R2 @ A5 @ B9 ) ) )
=> ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B5: nat] :
( ( member1367349282lle_hf @ A1 @ A4 )
=> ( ( member1367349282lle_hf @ A22 @ A4 )
=> ( ( member_nat @ B5 @ B4 )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_282_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_283_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ N2 ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_284_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_285_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 ) )
=> ? [F4: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F4 @ A4 ) @ B4 )
& ( inj_on811196232lle_hf @ F4 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_286_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 ) )
=> ? [F4: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F4 @ A4 ) @ B4 )
& ( inj_on755450110lle_hf @ F4 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_287_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 ) )
=> ? [F4: hF_Mirabelle_hf > nat] :
( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F4 @ A4 ) @ B4 )
& ( inj_on1874279374hf_nat @ F4 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_288_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 ) )
=> ? [F4: nat > hF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ ( image_246164834lle_hf @ F4 @ A4 ) @ B4 )
& ( inj_on1988990670lle_hf @ F4 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_289_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 ) )
=> ? [F4: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A4 ) @ B4 )
& ( inj_on_nat_nat @ F4 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_290_card__inj__on__le,axiom,
! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
( ( inj_on811196232lle_hf @ F @ A4 )
=> ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F @ A4 ) @ B4 )
=> ( ( finite586181922lle_hf @ B4 )
=> ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_291_finite__imageI,axiom,
! [F3: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ F3 )
=> ( finite586181922lle_hf @ ( image_1743964010lle_hf @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_292_finite__imageI,axiom,
! [F3: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > nat] :
( ( finite586181922lle_hf @ F3 )
=> ( finite_finite_nat @ ( image_131453538hf_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_293_finite__imageI,axiom,
! [F3: set_nat,H: nat > hF_Mirabelle_hf] :
( ( finite_finite_nat @ F3 )
=> ( finite586181922lle_hf @ ( image_246164834lle_hf @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_294_finite__imageI,axiom,
! [F3: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_295_all__finite__subset__image,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( ! [B3: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ ( image_1743964010lle_hf @ F @ A4 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ A4 ) )
=> ( P @ ( image_1743964010lle_hf @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_296_all__finite__subset__image,axiom,
! [F: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
( ( ! [B3: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ ( image_246164834lle_hf @ F @ A4 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) )
=> ( P @ ( image_246164834lle_hf @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_297_all__finite__subset__image,axiom,
! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_131453538hf_nat @ F @ A4 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_HF_Mirabelle_hf] :
( ( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ A4 ) )
=> ( P @ ( image_131453538hf_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_298_all__finite__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_299_ex__finite__subset__image,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
( ( ? [B3: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ ( image_1743964010lle_hf @ F @ A4 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ A4 )
& ( P @ ( image_1743964010lle_hf @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_300_ex__finite__subset__image,axiom,
! [F: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
( ( ? [B3: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ ( image_246164834lle_hf @ F @ A4 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 )
& ( P @ ( image_246164834lle_hf @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_301_ex__finite__subset__image,axiom,
! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_131453538hf_nat @ F @ A4 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B3 )
& ( ord_le432112161lle_hf @ B3 @ A4 )
& ( P @ ( image_131453538hf_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_302_ex__finite__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_303_finite__subset__image,axiom,
! [B4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F @ A4 ) )
=> ? [C4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ C4 @ A4 )
& ( finite586181922lle_hf @ C4 )
& ( B4
= ( image_1743964010lle_hf @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_304_finite__subset__image,axiom,
! [B4: set_HF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,A4: set_nat] :
( ( finite586181922lle_hf @ B4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F @ A4 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A4 )
& ( finite_finite_nat @ C4 )
& ( B4
= ( image_246164834lle_hf @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_305_finite__subset__image,axiom,
! [B4: set_nat,F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F @ A4 ) )
=> ? [C4: set_HF_Mirabelle_hf] :
( ( ord_le432112161lle_hf @ C4 @ A4 )
& ( finite586181922lle_hf @ C4 )
& ( B4
= ( image_131453538hf_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_306_finite__subset__image,axiom,
! [B4: set_nat,F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A4 )
& ( finite_finite_nat @ C4 )
& ( B4
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_307_finite__surj,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F @ A4 ) )
=> ( finite586181922lle_hf @ B4 ) ) ) ).
% finite_surj
thf(fact_308_finite__surj,axiom,
! [A4: set_HF_Mirabelle_hf,B4: set_nat,F: hF_Mirabelle_hf > nat] :
( ( finite586181922lle_hf @ A4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F @ A4 ) )
=> ( finite_finite_nat @ B4 ) ) ) ).
% finite_surj
thf(fact_309_finite__surj,axiom,
! [A4: set_nat,B4: set_HF_Mirabelle_hf,F: nat > hF_Mirabelle_hf] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F @ A4 ) )
=> ( finite586181922lle_hf @ B4 ) ) ) ).
% finite_surj
thf(fact_310_finite__surj,axiom,
! [A4: set_nat,B4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
=> ( finite_finite_nat @ B4 ) ) ) ).
% finite_surj
thf(fact_311_finite__image__iff,axiom,
! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F @ A4 )
=> ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
= ( finite1450550360lle_hf @ A4 ) ) ) ).
% finite_image_iff
thf(fact_312_finite__image__iff,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( inj_on755450110lle_hf @ F @ A4 )
=> ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F @ A4 ) )
= ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_image_iff
thf(fact_313_finite__image__iff,axiom,
! [F: nat > hF_Mirabelle_hf,A4: set_nat] :
( ( inj_on1988990670lle_hf @ F @ A4 )
=> ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_314_finite__image__iff,axiom,
! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
( ( inj_on1874279374hf_nat @ F @ A4 )
=> ( ( finite_finite_nat @ ( image_131453538hf_nat @ F @ A4 ) )
= ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_image_iff
thf(fact_315_finite__image__iff,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_316_finite__imageD,axiom,
! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
=> ( ( inj_on811196232lle_hf @ F @ A4 )
=> ( finite1450550360lle_hf @ A4 ) ) ) ).
% finite_imageD
thf(fact_317_finite__imageD,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F @ A4 ) )
=> ( ( inj_on755450110lle_hf @ F @ A4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_imageD
thf(fact_318_finite__imageD,axiom,
! [F: nat > hF_Mirabelle_hf,A4: set_nat] :
( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F @ A4 ) )
=> ( ( inj_on1988990670lle_hf @ F @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_319_finite__imageD,axiom,
! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
( ( finite_finite_nat @ ( image_131453538hf_nat @ F @ A4 ) )
=> ( ( inj_on1874279374hf_nat @ F @ A4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ).
% finite_imageD
thf(fact_320_finite__imageD,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( inj_on_nat_nat @ F @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_321_card__image,axiom,
! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
( ( inj_on811196232lle_hf @ F @ A4 )
=> ( ( finite1213132899lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
= ( finite90088345lle_hf @ A4 ) ) ) ).
% card_image
thf(fact_322_finite__surj__inj,axiom,
! [A4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( finite586181922lle_hf @ A4 )
=> ( ( ord_le432112161lle_hf @ A4 @ ( image_1743964010lle_hf @ F @ A4 ) )
=> ( inj_on755450110lle_hf @ F @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_323_finite__surj__inj,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ A4 ) )
=> ( inj_on_nat_nat @ F @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_324_inj__on__finite,axiom,
! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,B4: set_nat] :
( ( inj_on1874279374hf_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F @ A4 ) @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite586181922lle_hf @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_325_inj__on__finite,axiom,
! [F: nat > nat,A4: set_nat,B4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_326_finite__greaterThanLessThan,axiom,
! [L: nat,U4: nat] : ( finite_finite_nat @ ( set_or1544565540an_nat @ L @ U4 ) ) ).
% finite_greaterThanLessThan
thf(fact_327_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_328_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_329_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_330_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_331_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_332_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_333_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_334_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_335_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_336_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ~ ( P @ I ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_337_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_338_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N5 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_339_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_340_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_341_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_342_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_343_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_344_RepFun__0,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
= zero_z189798548lle_hf ) ).
% RepFun_0
thf(fact_345_HUnion__hempty,axiom,
( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HUnion_hempty
thf(fact_346_Replace__0,axiom,
! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
= zero_z189798548lle_hf ) ).
% Replace_0
% Conjectures (2)
thf(conj_0,hypothesis,
! [X6: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X6 @ a )
=> ( hF_Mirabelle_hmem @ X6 @ b ) ) ).
thf(conj_1,conjecture,
ord_le976219883lle_hf @ a @ b ).
%------------------------------------------------------------------------------