TPTP Problem File: SLH0897^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00072_002799__16988396_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1504 ( 429 unt; 236 typ; 0 def)
% Number of atoms : 3730 (1133 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10710 ( 306 ~; 47 |; 208 &;8401 @)
% ( 0 <=>;1748 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 1597 (1597 >; 0 *; 0 +; 0 <<)
% Number of symbols : 211 ( 208 usr; 17 con; 0-4 aty)
% Number of variables : 3952 ( 326 ^;3504 !; 122 ?;3952 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:35:30.975
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_a_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
set_nat_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
set_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
set_int_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
set_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Int__Oint_J_J,type,
set_a_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
set_int_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_o_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (208)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_Fpow_nat_a: set_nat_a > set_set_nat_a ).
thf(sy_c_Finite__Set_OFpow_001_Eo,type,
finite_Fpow_o: set_o > set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__Int__Oint,type,
finite_Fpow_int: set_int > set_set_int ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_card_nat_a: set_nat_a > nat ).
thf(sy_c_Finite__Set_Ocard_001_Eo,type,
finite_card_o: set_o > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
inj_on_nat_a_nat_a: ( ( nat > a ) > nat > a ) > set_nat_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
inj_on_nat_a_nat: ( ( nat > a ) > nat ) > set_nat_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
inj_on_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
inj_on_o_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Int__Oint,type,
inj_on_o_int: ( $o > int ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Nat__Onat,type,
inj_on_o_nat: ( $o > nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001tf__a,type,
inj_on_o_a: ( $o > a ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001_Eo,type,
inj_on_int_o: ( int > $o ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
inj_on_int_nat: ( int > nat ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001tf__a,type,
inj_on_int_a: ( int > a ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
inj_on_nat_nat_a: ( nat > nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_Eo,type,
inj_on_nat_o: ( nat > $o ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > 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__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
inj_on6078662301222212017_nat_a: ( set_nat_a > set_nat_a ) > set_set_nat_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_int_set_a: ( set_int > set_a ) > set_set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
inj_on426556184350386907et_int: ( set_nat > set_int ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_Eo_J,type,
inj_on_set_a_set_o: ( set_a > set_o ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Int__Oint_J,type,
inj_on_set_a_set_int: ( set_a > set_int ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_set_a_set_nat: ( set_a > set_nat ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_a_set_a: ( set_a > set_a ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
inj_on_a_nat_a: ( a > nat > a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_Eo,type,
inj_on_a_o: ( a > $o ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Int__Oint,type,
inj_on_a_int: ( a > int ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
the_in381770415356814323_nat_a: set_nat_a > ( ( nat > a ) > nat > a ) > ( nat > a ) > nat > a ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001_Eo,type,
the_inv_into_o_o: set_o > ( $o > $o ) > $o > $o ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001t__Int__Oint,type,
the_inv_into_o_int: set_o > ( $o > int ) > int > $o ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001t__Nat__Onat,type,
the_inv_into_o_nat: set_o > ( $o > nat ) > nat > $o ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001tf__a,type,
the_inv_into_o_a: set_o > ( $o > a ) > a > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001_Eo,type,
the_inv_into_int_o: set_int > ( int > $o ) > $o > int ).
thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Int__Oint,type,
the_inv_into_int_int: set_int > ( int > int ) > int > int ).
thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Nat__Onat,type,
the_inv_into_int_nat: set_int > ( int > nat ) > nat > int ).
thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001tf__a,type,
the_inv_into_int_a: set_int > ( int > a ) > a > int ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Int__Oint,type,
the_inv_into_nat_int: set_nat > ( nat > int ) > int > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001_Eo,type,
the_inv_into_a_o: set_a > ( a > $o ) > $o > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Int__Oint,type,
the_inv_into_a_int: set_a > ( a > int ) > int > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Nat__Onat,type,
the_inv_into_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
piE_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_Eo,type,
piE_o_o: set_o > ( $o > set_o ) > set_o_o ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Nat__Onat,type,
piE_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001tf__a,type,
piE_o_a: set_o > ( $o > set_a ) > set_o_a ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001t__Int__Oint,type,
piE_int_int: set_int > ( int > set_int ) > set_int_int ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001t__Nat__Onat,type,
piE_int_nat: set_int > ( int > set_nat ) > set_int_nat ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001tf__a,type,
piE_int_a: set_int > ( int > set_a ) > set_int_a ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_Eo,type,
piE_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Int__Oint,type,
piE_nat_int: set_nat > ( nat > set_int ) > set_nat_int ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001tf__a,type,
piE_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPiE_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
piE_a_nat_a: set_a > ( a > set_nat_a ) > set_a_nat_a ).
thf(sy_c_FuncSet_OPiE_001tf__a_001_Eo,type,
piE_a_o: set_a > ( a > set_o ) > set_a_o ).
thf(sy_c_FuncSet_OPiE_001tf__a_001t__Int__Oint,type,
piE_a_int: set_a > ( a > set_int ) > set_a_int ).
thf(sy_c_FuncSet_OPiE_001tf__a_001t__Nat__Onat,type,
piE_a_nat: set_a > ( a > set_nat ) > set_a_nat ).
thf(sy_c_FuncSet_OPiE_001tf__a_001tf__a,type,
piE_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( nat > $o ) > nat > nat > $o ).
thf(sy_c_List_Ofolding__insort__key__axioms_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
foldin703030179759482504_nat_a: set_nat_a > ( ( nat > a ) > nat > a ) > $o ).
thf(sy_c_List_Ofolding__insort__key__axioms_001t__Nat__Onat_001t__Nat__Onat,type,
foldin1360219024038166634at_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $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_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_less_set_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
ord_less_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $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_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
ord_le3509452538356653652at_a_a: set_nat_a_a > set_nat_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_le4981610546006782297_o_nat: set_o_nat > set_o_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_set_o_a: set_o_a > set_o_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
ord_le2023132899490853297nt_nat: set_int_nat > set_int_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Int__Oint_Mtf__a_J_J,type,
ord_le943418215940126601_int_a: set_int_a > set_int_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le2508512696544544866_nat_a: set_a_nat_a > set_a_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le1612561287239139007_a_nat: set_a_nat > set_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
ord_less_eq_set_a_a: set_a_a > set_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le2390145808437456709_nat_a: set_set_nat_a > set_set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
top_top_nat_a_o: ( nat > a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
top_top_int_o: int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
top_top_set_int_int: set_int_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
top_top_set_int_nat: set_int_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
top_top_set_nat_int: set_nat_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
top_top_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
image_nat_a_nat_a: ( ( nat > a ) > nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
image_nat_a_nat: ( ( nat > a ) > nat ) > set_nat_a > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Int__Oint,type,
image_o_int: ( $o > int ) > set_o > set_int ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001_Eo,type,
image_int_o: ( int > $o ) > set_int > set_o ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001tf__a,type,
image_int_a: ( int > a ) > set_int > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
image_nat_nat_a: ( nat > nat > a ) > set_nat > set_nat_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
image_6965494298868581957_nat_a: ( set_nat_a > set_nat_a ) > set_set_nat_a > set_set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_Itf__a_J,type,
image_set_int_set_a: ( set_int > set_a ) > set_set_int > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
image_3739036796817536367et_int: ( set_nat > set_int ) > set_set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
image_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_Eo_J,type,
image_set_a_set_o: ( set_a > set_o ) > set_set_a > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Int__Oint_J,type,
image_set_a_set_int: ( set_a > set_int ) > set_set_a > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
image_a_nat_a: ( a > nat > a ) > set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
set_or7139685690850216873Than_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_or8677123885700112214_nat_a: set_nat_a > set_nat_a > set_set_nat_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
member_nat_int: ( nat > int ) > set_nat_int > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Int__Oint_J,type,
member_a_int: ( a > int ) > set_a_int > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
member_set_nat_a: set_nat_a > set_set_nat_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_f____,type,
f: ( nat > a ) > nat > a ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (1262)
thf(fact_0_assms,axiom,
member_a @ y @ b ).
% assms
thf(fact_1_c,axiom,
( inj_on_nat_a_nat_a @ f
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I: nat] : b ) ) ).
% c
thf(fact_2_image__ident,axiom,
! [Y: set_a] :
( ( image_a_a
@ ^ [X: a] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_3_image__ident,axiom,
! [Y: set_nat_a] :
( ( image_nat_a_nat_a
@ ^ [X: nat > a] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_4_image__ident,axiom,
! [Y: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_5_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_6_image__eqI,axiom,
! [B: nat,F: a > nat,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_7_image__eqI,axiom,
! [B: $o,F: a > $o,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_o @ B @ ( image_a_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_8_image__eqI,axiom,
! [B: int,F: nat > int,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_9_image__eqI,axiom,
! [B: a,F: nat > a,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_o @ B @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B: a,F: $o > a,X2: $o,A: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_a @ B @ ( image_o_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,A: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_image__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,A: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_15_f__def,axiom,
( f
= ( ^ [F2: nat > a,K: nat] : ( if_a @ ( ord_less_nat @ K @ n ) @ ( F2 @ K ) @ y ) ) ) ).
% f_def
thf(fact_16_Inf_OINF__identity__eq,axiom,
! [Inf: set_a > a,A: set_a] :
( ( Inf
@ ( image_a_a
@ ^ [X: a] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_17_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat_a > nat > a,A: set_nat_a] :
( ( Inf
@ ( image_nat_a_nat_a
@ ^ [X: nat > a] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_18_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_19_Sup_OSUP__identity__eq,axiom,
! [Sup: set_a > a,A: set_a] :
( ( Sup
@ ( image_a_a
@ ^ [X: a] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_20_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat_a > nat > a,A: set_nat_a] :
( ( Sup
@ ( image_nat_a_nat_a
@ ^ [X: nat > a] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_21_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_22_imageE,axiom,
! [B: int,F: nat > int,A: set_nat] :
( ( member_int @ B @ ( image_nat_int @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_23_imageE,axiom,
! [B: a,F: a > a,A: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_24_imageE,axiom,
! [B: a,F: nat > a,A: set_nat] :
( ( member_a @ B @ ( image_nat_a @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_25_imageE,axiom,
! [B: a,F: $o > a,A: set_o] :
( ( member_a @ B @ ( image_o_a @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_26_imageE,axiom,
! [B: nat,F: a > nat,A: set_a] :
( ( member_nat @ B @ ( image_a_nat @ F @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_27_imageE,axiom,
! [B: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_28_imageE,axiom,
! [B: nat,F: $o > nat,A: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_29_imageE,axiom,
! [B: $o,F: a > $o,A: set_a] :
( ( member_o @ B @ ( image_a_o @ F @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_30_imageE,axiom,
! [B: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_31_imageE,axiom,
! [B: $o,F: $o > $o,A: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_32_image__image,axiom,
! [F: a > a,G: nat > a,A: set_nat] :
( ( image_a_a @ F @ ( image_nat_a @ G @ A ) )
= ( image_nat_a
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_33_image__image,axiom,
! [F: a > int,G: nat > a,A: set_nat] :
( ( image_a_int @ F @ ( image_nat_a @ G @ A ) )
= ( image_nat_int
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_34_image__image,axiom,
! [F: a > nat,G: nat > a,A: set_nat] :
( ( image_a_nat @ F @ ( image_nat_a @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_35_image__image,axiom,
! [F: int > a,G: nat > int,A: set_nat] :
( ( image_int_a @ F @ ( image_nat_int @ G @ A ) )
= ( image_nat_a
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_36_image__image,axiom,
! [F: int > int,G: nat > int,A: set_nat] :
( ( image_int_int @ F @ ( image_nat_int @ G @ A ) )
= ( image_nat_int
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_37_image__image,axiom,
! [F: int > nat,G: nat > int,A: set_nat] :
( ( image_int_nat @ F @ ( image_nat_int @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_38_image__image,axiom,
! [F: nat > a,G: nat > nat,A: set_nat] :
( ( image_nat_a @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_a
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_39_image__image,axiom,
! [F: nat > int,G: nat > nat,A: set_nat] :
( ( image_nat_int @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_int
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_40_image__image,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_41_image__image,axiom,
! [F: ( nat > a ) > nat > a,G: ( nat > a ) > nat > a,A: set_nat_a] :
( ( image_nat_a_nat_a @ F @ ( image_nat_a_nat_a @ G @ A ) )
= ( image_nat_a_nat_a
@ ^ [X: nat > a] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_42_Compr__image__eq,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_43_Compr__image__eq,axiom,
! [F: $o > a,A: set_o,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_o_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_a @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_44_Compr__image__eq,axiom,
! [F: a > $o,A: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_a_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_o @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_45_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_46_Compr__image__eq,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_nat_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_47_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_48_Compr__image__eq,axiom,
! [F: int > a,A: set_int,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_int_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_int_a @ F
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_49_Compr__image__eq,axiom,
! [F: int > $o,A: set_int,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_int_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_int_o @ F
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_50_Compr__image__eq,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_51_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_o_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_52_PiE__ext,axiom,
! [X2: nat > a,K2: set_nat,S: nat > set_a,Y2: nat > a] :
( ( member_nat_a @ X2 @ ( piE_nat_a @ K2 @ S ) )
=> ( ( member_nat_a @ Y2 @ ( piE_nat_a @ K2 @ S ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ( X2 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_53_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_54_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_55_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_56_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_57_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_58_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_59_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_60_rev__image__eqI,axiom,
! [X2: a,A: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_61_rev__image__eqI,axiom,
! [X2: a,A: set_a,B: nat,F: a > nat] :
( ( member_a @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_62_rev__image__eqI,axiom,
! [X2: a,A: set_a,B: $o,F: a > $o] :
( ( member_a @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_a_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_63_rev__image__eqI,axiom,
! [X2: a,A: set_a,B: int,F: a > int] :
( ( member_a @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_a_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_64_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: a,F: nat > a] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_65_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_66_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_67_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: int,F: nat > int] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_68_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B: a,F: $o > a] :
( ( member_o @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_o_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_69_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B: nat,F: $o > nat] :
( ( member_o @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_70_ball__imageD,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,P: ( nat > a ) > $o] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ ( image_nat_a_nat_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_71_ball__imageD,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_nat_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_72_ball__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_73_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_74_ball__imageD,axiom,
! [F: int > a,A: set_int,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_int_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_75_ball__imageD,axiom,
! [F: a > $o,A: set_a,P: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_a_o @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_76_ball__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_77_ball__imageD,axiom,
! [F: a > int,A: set_a,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_a_int @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_78_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_79_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > $o,G: a > $o] :
( ( M2 = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_o @ F @ M2 )
= ( image_a_o @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_80_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > nat,G: a > nat] :
( ( M2 = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_nat @ F @ M2 )
= ( image_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_81_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > int,G: a > int] :
( ( M2 = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_int @ F @ M2 )
= ( image_a_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_82_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > a,G: a > a] :
( ( M2 = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_a @ F @ M2 )
= ( image_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_83_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > a,G: nat > a] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_a @ F @ M2 )
= ( image_nat_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_84_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > int,G: nat > int] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_int @ F @ M2 )
= ( image_nat_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_85_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_86_image__cong,axiom,
! [M2: set_int,N2: set_int,F: int > a,G: int > a] :
( ( M2 = N2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_int_a @ F @ M2 )
= ( image_int_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_87_image__cong,axiom,
! [M2: set_nat_a,N2: set_nat_a,F: ( nat > a ) > nat > a,G: ( nat > a ) > nat > a] :
( ( M2 = N2 )
=> ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_a_nat_a @ F @ M2 )
= ( image_nat_a_nat_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_88_bex__imageD,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,P: ( nat > a ) > $o] :
( ? [X4: nat > a] :
( ( member_nat_a @ X4 @ ( image_nat_a_nat_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_89_bex__imageD,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_nat_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_90_bex__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_91_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_92_bex__imageD,axiom,
! [F: int > a,A: set_int,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_int_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_93_bex__imageD,axiom,
! [F: a > $o,A: set_a,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_a_o @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_94_bex__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_a_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_95_bex__imageD,axiom,
! [F: a > int,A: set_a,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_a_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_96_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_97_image__iff,axiom,
! [Z: a,F: nat > a,A: set_nat] :
( ( member_a @ Z @ ( image_nat_a @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_98_image__iff,axiom,
! [Z: a,F: int > a,A: set_int] :
( ( member_a @ Z @ ( image_int_a @ F @ A ) )
= ( ? [X: int] :
( ( member_int @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_99_image__iff,axiom,
! [Z: a,F: a > a,A: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_100_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_101_image__iff,axiom,
! [Z: nat,F: a > nat,A: set_a] :
( ( member_nat @ Z @ ( image_a_nat @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_102_image__iff,axiom,
! [Z: $o,F: a > $o,A: set_a] :
( ( member_o @ Z @ ( image_a_o @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_103_image__iff,axiom,
! [Z: int,F: nat > int,A: set_nat] :
( ( member_int @ Z @ ( image_nat_int @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_104_image__iff,axiom,
! [Z: int,F: a > int,A: set_a] :
( ( member_int @ Z @ ( image_a_int @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_105_image__iff,axiom,
! [Z: nat > a,F: ( nat > a ) > nat > a,A: set_nat_a] :
( ( member_nat_a @ Z @ ( image_nat_a_nat_a @ F @ A ) )
= ( ? [X: nat > a] :
( ( member_nat_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_106_imageI,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_107_imageI,axiom,
! [X2: a,A: set_a,F: a > nat] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_108_imageI,axiom,
! [X2: a,A: set_a,F: a > $o] :
( ( member_a @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_a_o @ F @ A ) ) ) ).
% imageI
thf(fact_109_imageI,axiom,
! [X2: a,A: set_a,F: a > int] :
( ( member_a @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ ( image_a_int @ F @ A ) ) ) ).
% imageI
thf(fact_110_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_111_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_112_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_113_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ ( image_nat_int @ F @ A ) ) ) ).
% imageI
thf(fact_114_imageI,axiom,
! [X2: $o,A: set_o,F: $o > a] :
( ( member_o @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_o_a @ F @ A ) ) ) ).
% imageI
thf(fact_115_imageI,axiom,
! [X2: $o,A: set_o,F: $o > nat] :
( ( member_o @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_116_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > $o,D: a > $o,Sup: set_o > $o] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_o @ C @ A ) )
= ( Sup @ ( image_a_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_117_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > nat,D: a > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_nat @ C @ A ) )
= ( Sup @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_118_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > int,D: a > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_int @ C @ A ) )
= ( Sup @ ( image_a_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_119_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > a,D: a > a,Sup: set_a > a] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_a @ C @ A ) )
= ( Sup @ ( image_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_120_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > a,D: nat > a,Sup: set_a > a] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_a @ C @ A ) )
= ( Sup @ ( image_nat_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_121_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > int,D: nat > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_int @ C @ A ) )
= ( Sup @ ( image_nat_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_122_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_123_Sup_OSUP__cong,axiom,
! [A: set_int,B2: set_int,C: int > a,D: int > a,Sup: set_a > a] :
( ( A = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_int_a @ C @ A ) )
= ( Sup @ ( image_int_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_124_Sup_OSUP__cong,axiom,
! [A: set_nat_a,B2: set_nat_a,C: ( nat > a ) > nat > a,D: ( nat > a ) > nat > a,Sup: set_nat_a > nat > a] :
( ( A = B2 )
=> ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_a_nat_a @ C @ A ) )
= ( Sup @ ( image_nat_a_nat_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_125_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C: a > $o,D: a > $o,Inf: set_o > $o] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_o @ C @ A ) )
= ( Inf @ ( image_a_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_126_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C: a > nat,D: a > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_nat @ C @ A ) )
= ( Inf @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_127_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C: a > int,D: a > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_int @ C @ A ) )
= ( Inf @ ( image_a_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_128_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C: a > a,D: a > a,Inf: set_a > a] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_a @ C @ A ) )
= ( Inf @ ( image_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_129_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > a,D: nat > a,Inf: set_a > a] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_a @ C @ A ) )
= ( Inf @ ( image_nat_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_130_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > int,D: nat > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_int @ C @ A ) )
= ( Inf @ ( image_nat_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_131_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_132_Inf_OINF__cong,axiom,
! [A: set_int,B2: set_int,C: int > a,D: int > a,Inf: set_a > a] :
( ( A = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_int_a @ C @ A ) )
= ( Inf @ ( image_int_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_133_Inf_OINF__cong,axiom,
! [A: set_nat_a,B2: set_nat_a,C: ( nat > a ) > nat > a,D: ( nat > a ) > nat > a,Inf: set_nat_a > nat > a] :
( ( A = B2 )
=> ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_a_nat_a @ C @ A ) )
= ( Inf @ ( image_nat_a_nat_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_134_PiE__cong,axiom,
! [I3: set_nat,A: nat > set_a,B2: nat > set_a] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I3 )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_a @ I3 @ A )
= ( piE_nat_a @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_135_PiE__mem,axiom,
! [F: a > a,S2: set_a,T: a > set_a,X2: a] :
( ( member_a_a @ F @ ( piE_a_a @ S2 @ T ) )
=> ( ( member_a @ X2 @ S2 )
=> ( member_a @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_136_PiE__mem,axiom,
! [F: a > nat,S2: set_a,T: a > set_nat,X2: a] :
( ( member_a_nat @ F @ ( piE_a_nat @ S2 @ T ) )
=> ( ( member_a @ X2 @ S2 )
=> ( member_nat @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_137_PiE__mem,axiom,
! [F: a > $o,S2: set_a,T: a > set_o,X2: a] :
( ( member_a_o @ F @ ( piE_a_o @ S2 @ T ) )
=> ( ( member_a @ X2 @ S2 )
=> ( member_o @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_138_PiE__mem,axiom,
! [F: a > int,S2: set_a,T: a > set_int,X2: a] :
( ( member_a_int @ F @ ( piE_a_int @ S2 @ T ) )
=> ( ( member_a @ X2 @ S2 )
=> ( member_int @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_139_PiE__mem,axiom,
! [F: nat > nat,S2: set_nat,T: nat > set_nat,X2: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S2 @ T ) )
=> ( ( member_nat @ X2 @ S2 )
=> ( member_nat @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_140_PiE__mem,axiom,
! [F: nat > $o,S2: set_nat,T: nat > set_o,X2: nat] :
( ( member_nat_o @ F @ ( piE_nat_o @ S2 @ T ) )
=> ( ( member_nat @ X2 @ S2 )
=> ( member_o @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_141_PiE__mem,axiom,
! [F: nat > int,S2: set_nat,T: nat > set_int,X2: nat] :
( ( member_nat_int @ F @ ( piE_nat_int @ S2 @ T ) )
=> ( ( member_nat @ X2 @ S2 )
=> ( member_int @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_142_PiE__mem,axiom,
! [F: $o > a,S2: set_o,T: $o > set_a,X2: $o] :
( ( member_o_a @ F @ ( piE_o_a @ S2 @ T ) )
=> ( ( member_o @ X2 @ S2 )
=> ( member_a @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_143_PiE__mem,axiom,
! [F: $o > nat,S2: set_o,T: $o > set_nat,X2: $o] :
( ( member_o_nat @ F @ ( piE_o_nat @ S2 @ T ) )
=> ( ( member_o @ X2 @ S2 )
=> ( member_nat @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_144_PiE__mem,axiom,
! [F: $o > $o,S2: set_o,T: $o > set_o,X2: $o] :
( ( member_o_o @ F @ ( piE_o_o @ S2 @ T ) )
=> ( ( member_o @ X2 @ S2 )
=> ( member_o @ ( F @ X2 ) @ ( T @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_145_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_146_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_147_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_148_pigeonhole,axiom,
! [F: int > a,A: set_int] :
( ( ord_less_nat @ ( finite_card_a @ ( image_int_a @ F @ A ) ) @ ( finite_card_int @ A ) )
=> ~ ( inj_on_int_a @ F @ A ) ) ).
% pigeonhole
thf(fact_149_pigeonhole,axiom,
! [F: a > $o,A: set_a] :
( ( ord_less_nat @ ( finite_card_o @ ( image_a_o @ F @ A ) ) @ ( finite_card_a @ A ) )
=> ~ ( inj_on_a_o @ F @ A ) ) ).
% pigeonhole
thf(fact_150_pigeonhole,axiom,
! [F: a > int,A: set_a] :
( ( ord_less_nat @ ( finite_card_int @ ( image_a_int @ F @ A ) ) @ ( finite_card_a @ A ) )
=> ~ ( inj_on_a_int @ F @ A ) ) ).
% pigeonhole
thf(fact_151_pigeonhole,axiom,
! [F: a > a,A: set_a] :
( ( ord_less_nat @ ( finite_card_a @ ( image_a_a @ F @ A ) ) @ ( finite_card_a @ A ) )
=> ~ ( inj_on_a_a @ F @ A ) ) ).
% pigeonhole
thf(fact_152_pigeonhole,axiom,
! [F: nat > a,A: set_nat] :
( ( ord_less_nat @ ( finite_card_a @ ( image_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on_nat_a @ F @ A ) ) ).
% pigeonhole
thf(fact_153_pigeonhole,axiom,
! [F: nat > int,A: set_nat] :
( ( ord_less_nat @ ( finite_card_int @ ( image_nat_int @ F @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on_nat_int @ F @ A ) ) ).
% pigeonhole
thf(fact_154_pigeonhole,axiom,
! [F: a > nat,A: set_a] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) )
=> ~ ( inj_on_a_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_155_pigeonhole,axiom,
! [F: nat > nat,A: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on_nat_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_156_pigeonhole,axiom,
! [F: nat > nat > a,A: set_nat] :
( ( ord_less_nat @ ( finite_card_nat_a @ ( image_nat_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on_nat_nat_a @ F @ A ) ) ).
% pigeonhole
thf(fact_157_pigeonhole,axiom,
! [F: ( nat > a ) > nat,A: set_nat_a] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_a_nat @ F @ A ) ) @ ( finite_card_nat_a @ A ) )
=> ~ ( inj_on_nat_a_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_158_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_159_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_160_card__image,axiom,
! [F: int > a,A: set_int] :
( ( inj_on_int_a @ F @ A )
=> ( ( finite_card_a @ ( image_int_a @ F @ A ) )
= ( finite_card_int @ A ) ) ) ).
% card_image
thf(fact_161_card__image,axiom,
! [F: a > $o,A: set_a] :
( ( inj_on_a_o @ F @ A )
=> ( ( finite_card_o @ ( image_a_o @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_162_card__image,axiom,
! [F: a > int,A: set_a] :
( ( inj_on_a_int @ F @ A )
=> ( ( finite_card_int @ ( image_a_int @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_163_card__image,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( finite_card_a @ ( image_a_a @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_164_card__image,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( finite_card_a @ ( image_nat_a @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_165_card__image,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( finite_card_int @ ( image_nat_int @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_166_card__image,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_167_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_168_card__image,axiom,
! [F: nat > nat > a,A: set_nat] :
( ( inj_on_nat_nat_a @ F @ A )
=> ( ( finite_card_nat_a @ ( image_nat_nat_a @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_169_card__image,axiom,
! [F: ( nat > a ) > nat,A: set_nat_a] :
( ( inj_on_nat_a_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_a_nat @ F @ A ) )
= ( finite_card_nat_a @ A ) ) ) ).
% card_image
thf(fact_170_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_171_atLeastLessThan__eq__iff,axiom,
! [A2: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ( set_or4662586982721622107an_int @ A2 @ B )
= ( set_or4662586982721622107an_int @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_172_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_173_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_174_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( A2 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_175_atLeastLessThan__inj_I1_J,axiom,
! [A2: int,B: int,C2: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A2 @ B )
= ( set_or4662586982721622107an_int @ C2 @ D2 ) )
=> ( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( A2 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_176_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_177_atLeastLessThan__inj_I2_J,axiom,
! [A2: int,B: int,C2: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A2 @ B )
= ( set_or4662586982721622107an_int @ C2 @ D2 ) )
=> ( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_178_inj__on__image__iff,axiom,
! [A: set_nat_a,G: ( nat > a ) > nat > a,F: ( nat > a ) > nat > a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ! [Xa: nat > a] :
( ( member_nat_a @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( inj_on_nat_a_nat_a @ G @ ( image_nat_a_nat_a @ F @ A ) )
= ( inj_on_nat_a_nat_a @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_179_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_180_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_181_image__strict__mono,axiom,
! [F: nat > a,B2: set_nat,A: set_nat] :
( ( inj_on_nat_a @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_182_image__strict__mono,axiom,
! [F: nat > int,B2: set_nat,A: set_nat] :
( ( inj_on_nat_int @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_183_image__strict__mono,axiom,
! [F: int > a,B2: set_int,A: set_int] :
( ( inj_on_int_a @ F @ B2 )
=> ( ( ord_less_set_int @ A @ B2 )
=> ( ord_less_set_a @ ( image_int_a @ F @ A ) @ ( image_int_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_184_image__strict__mono,axiom,
! [F: a > $o,B2: set_a,A: set_a] :
( ( inj_on_a_o @ F @ B2 )
=> ( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_set_o @ ( image_a_o @ F @ A ) @ ( image_a_o @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_185_image__strict__mono,axiom,
! [F: a > nat,B2: set_a,A: set_a] :
( ( inj_on_a_nat @ F @ B2 )
=> ( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_186_image__strict__mono,axiom,
! [F: a > int,B2: set_a,A: set_a] :
( ( inj_on_a_int @ F @ B2 )
=> ( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_set_int @ ( image_a_int @ F @ A ) @ ( image_a_int @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_187_image__strict__mono,axiom,
! [F: a > a,B2: set_a,A: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_188_image__strict__mono,axiom,
! [F: ( nat > a ) > nat > a,B2: set_nat_a,A: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ B2 )
=> ( ( ord_less_set_nat_a @ A @ B2 )
=> ( ord_less_set_nat_a @ ( image_nat_a_nat_a @ F @ A ) @ ( image_nat_a_nat_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_189_image__strict__mono,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_190_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_191_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_192_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_193_mem__Collect__eq,axiom,
! [A2: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A2 @ ( collect_nat_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_195_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_196_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_197_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_198_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X: nat > a] : ( member_nat_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_200_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_201_Collect__cong,axiom,
! [P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ! [X3: nat > a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat_a @ P )
= ( collect_nat_a @ Q ) ) ) ).
% Collect_cong
thf(fact_202_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_203_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_204_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_205_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_206_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_207_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_208_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_209_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_210_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_211_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_212_inj__on__inverseI,axiom,
! [A: set_nat_a,G: ( nat > a ) > nat > a,F: ( nat > a ) > nat > a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_a_nat_a @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_213_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X2 != Y2 )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_214_inj__on__contraD,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,X2: nat > a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( X2 != Y2 )
=> ( ( member_nat_a @ X2 @ A )
=> ( ( member_nat_a @ Y2 @ A )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_215_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_216_inj__on__eq__iff,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,X2: nat > a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( member_nat_a @ X2 @ A )
=> ( ( member_nat_a @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_217_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_218_inj__on__cong,axiom,
! [A: set_nat_a,F: ( nat > a ) > nat > a,G: ( nat > a ) > nat > a] :
( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_a_nat_a @ F @ A )
= ( inj_on_nat_a_nat_a @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_219_inj__on__def,axiom,
( inj_on_nat_a_nat_a
= ( ^ [F2: ( nat > a ) > nat > a,A4: set_nat_a] :
! [X: nat > a] :
( ( member_nat_a @ X @ A4 )
=> ! [Y3: nat > a] :
( ( member_nat_a @ Y3 @ A4 )
=> ( ( ( F2 @ X )
= ( F2 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_220_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A4 )
=> ( ( ( F2 @ X )
= ( F2 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_221_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_222_inj__onI,axiom,
! [A: set_nat_a,F: ( nat > a ) > nat > a] :
( ! [X3: nat > a,Y4: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ( ( member_nat_a @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_nat_a_nat_a @ F @ A ) ) ).
% inj_onI
thf(fact_223_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_224_inj__onD,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,X2: nat > a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_nat_a @ X2 @ A )
=> ( ( member_nat_a @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_225_inj__on__id2,axiom,
! [A: set_nat_a] :
( inj_on_nat_a_nat_a
@ ^ [X: nat > a] : X
@ A ) ).
% inj_on_id2
thf(fact_226_inj__on__id2,axiom,
! [A: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A ) ).
% inj_on_id2
thf(fact_227_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_228_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_229_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_230_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_231_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_232_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_233_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_234_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I2: nat,J: nat] :
( ( member_nat @ I2 @ A )
=> ( ( member_nat @ J @ A )
=> ( ( ord_less_nat @ I2 @ J )
=> ( ( F @ I2 )
!= ( F @ J ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_235_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_236_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_237_calculation,axiom,
( ( finite_card_nat_a
@ ( collect_nat_a
@ ^ [F2: nat > a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ b )
& ! [X: nat] :
( ( ord_less_eq_nat @ n @ X )
=> ( ( F2 @ X )
= y ) ) ) ) )
= ( finite_card_nat_a
@ ( image_nat_a_nat_a @ f
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I: nat] : b ) ) ) ) ).
% calculation
thf(fact_238_inj__on__fixpoints,axiom,
! [F: nat > nat] :
( inj_on_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( F @ X )
= X ) ) ) ).
% inj_on_fixpoints
thf(fact_239_inj__on__fixpoints,axiom,
! [F: ( nat > a ) > nat > a] :
( inj_on_nat_a_nat_a @ F
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( F @ X )
= X ) ) ) ).
% inj_on_fixpoints
thf(fact_240_inj__on__fixpoints,axiom,
! [F: int > int] :
( inj_on_int_int @ F
@ ( collect_int
@ ^ [X: int] :
( ( F @ X )
= X ) ) ) ).
% inj_on_fixpoints
thf(fact_241_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_242_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_243_the__inv__into__onto,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( image_int_nat @ ( the_inv_into_nat_int @ A @ F ) @ ( image_nat_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_244_the__inv__into__onto,axiom,
! [F: a > $o,A: set_a] :
( ( inj_on_a_o @ F @ A )
=> ( ( image_o_a @ ( the_inv_into_a_o @ A @ F ) @ ( image_a_o @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_245_the__inv__into__onto,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( image_nat_a @ ( the_inv_into_a_nat @ A @ F ) @ ( image_a_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_246_the__inv__into__onto,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( image_nat_int @ ( the_inv_into_int_nat @ A @ F ) @ ( image_int_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_247_the__inv__into__onto,axiom,
! [F: a > int,A: set_a] :
( ( inj_on_a_int @ F @ A )
=> ( ( image_int_a @ ( the_inv_into_a_int @ A @ F ) @ ( image_a_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_248_the__inv__into__onto,axiom,
! [F: $o > a,A: set_o] :
( ( inj_on_o_a @ F @ A )
=> ( ( image_a_o @ ( the_inv_into_o_a @ A @ F ) @ ( image_o_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_249_the__inv__into__onto,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( image_a_nat @ ( the_inv_into_nat_a @ A @ F ) @ ( image_nat_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_250_the__inv__into__onto,axiom,
! [F: int > a,A: set_int] :
( ( inj_on_int_a @ F @ A )
=> ( ( image_a_int @ ( the_inv_into_int_a @ A @ F ) @ ( image_int_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_251_the__inv__into__onto,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( image_a_a @ ( the_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_252_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_253_all__interval__nat__def,axiom,
( all_interval_nat
= ( ^ [P2: nat > $o,I: nat,J2: nat] :
! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
=> ( P2 @ X ) ) ) ) ).
% all_interval_nat_def
thf(fact_254_inj__on__image__Fpow,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_255_inj__on__image__Fpow,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_256_inj__on__image__Fpow,axiom,
! [F: int > a,A: set_int] :
( ( inj_on_int_a @ F @ A )
=> ( inj_on_set_int_set_a @ ( image_int_a @ F ) @ ( finite_Fpow_int @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_257_inj__on__image__Fpow,axiom,
! [F: a > $o,A: set_a] :
( ( inj_on_a_o @ F @ A )
=> ( inj_on_set_a_set_o @ ( image_a_o @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_258_inj__on__image__Fpow,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_259_inj__on__image__Fpow,axiom,
! [F: a > int,A: set_a] :
( ( inj_on_a_int @ F @ A )
=> ( inj_on_set_a_set_int @ ( image_a_int @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_260_inj__on__image__Fpow,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_261_inj__on__image__Fpow,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( inj_on6078662301222212017_nat_a @ ( image_nat_a_nat_a @ F ) @ ( finite_Fpow_nat_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_262_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_263_folding__insort__key__axioms__def,axiom,
( foldin703030179759482504_nat_a
= ( ^ [S3: set_nat_a,F2: ( nat > a ) > nat > a] : ( inj_on_nat_a_nat_a @ F2 @ S3 ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_264_folding__insort__key__axioms__def,axiom,
( foldin1360219024038166634at_nat
= ( ^ [S3: set_nat,F2: nat > nat] : ( inj_on_nat_nat @ F2 @ S3 ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_265_folding__insort__key__axioms_Ointro,axiom,
! [F: ( nat > a ) > nat > a,S2: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ S2 )
=> ( foldin703030179759482504_nat_a @ S2 @ F ) ) ).
% folding_insort_key_axioms.intro
thf(fact_266_folding__insort__key__axioms_Ointro,axiom,
! [F: nat > nat,S2: set_nat] :
( ( inj_on_nat_nat @ F @ S2 )
=> ( foldin1360219024038166634at_nat @ S2 @ F ) ) ).
% folding_insort_key_axioms.intro
thf(fact_267_a,axiom,
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [F2: nat > a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ b )
& ! [X: nat] :
( ( ord_less_eq_nat @ n @ X )
=> ( ( F2 @ X )
= y ) ) ) )
@ ( image_nat_a_nat_a @ f
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I: nat] : b ) ) ) ).
% a
thf(fact_268_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_269_subset__antisym,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_270_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_271_subsetI,axiom,
! [A: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B2 ) )
=> ( ord_less_eq_set_o @ A @ B2 ) ) ).
% subsetI
thf(fact_272_subsetI,axiom,
! [A: set_int,B2: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_int @ X3 @ B2 ) )
=> ( ord_less_eq_set_int @ A @ B2 ) ) ).
% subsetI
thf(fact_273_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_274_subsetI,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ( member_nat_a @ X3 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A @ B2 ) ) ).
% subsetI
thf(fact_275_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_276_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_277_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_278_UNIV__I,axiom,
! [X2: nat > a] : ( member_nat_a @ X2 @ top_top_set_nat_a ) ).
% UNIV_I
thf(fact_279_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_280_UNIV__I,axiom,
! [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_I
thf(fact_281_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_282_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_283_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_284_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_285_psubsetI,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_286_psubsetI,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat_a @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_287_psubsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_288_PiE__UNIV,axiom,
( ( piE_nat_a @ top_top_set_nat
@ ^ [I: nat] : top_top_set_a )
= top_top_set_nat_a ) ).
% PiE_UNIV
thf(fact_289_PiE__UNIV,axiom,
( ( piE_nat_nat @ top_top_set_nat
@ ^ [I: nat] : top_top_set_nat )
= top_top_set_nat_nat ) ).
% PiE_UNIV
thf(fact_290_PiE__UNIV,axiom,
( ( piE_nat_int @ top_top_set_nat
@ ^ [I: nat] : top_top_set_int )
= top_top_set_nat_int ) ).
% PiE_UNIV
thf(fact_291_PiE__UNIV,axiom,
( ( piE_int_nat @ top_top_set_int
@ ^ [I: int] : top_top_set_nat )
= top_top_set_int_nat ) ).
% PiE_UNIV
thf(fact_292_PiE__UNIV,axiom,
( ( piE_int_int @ top_top_set_int
@ ^ [I: int] : top_top_set_int )
= top_top_set_int_int ) ).
% PiE_UNIV
thf(fact_293_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_294_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_295_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_296_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_297_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_298_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_299_atLeastLessThan__iff,axiom,
! [I4: $o,L: $o,U: $o] :
( ( member_o @ I4 @ ( set_or7139685690850216873Than_o @ L @ U ) )
= ( ( ord_less_eq_o @ L @ I4 )
& ( ord_less_o @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_300_atLeastLessThan__iff,axiom,
! [I4: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I4 @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I4 )
& ( ord_less_set_a @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_301_atLeastLessThan__iff,axiom,
! [I4: set_nat_a,L: set_nat_a,U: set_nat_a] :
( ( member_set_nat_a @ I4 @ ( set_or8677123885700112214_nat_a @ L @ U ) )
= ( ( ord_le871467723717165285_nat_a @ L @ I4 )
& ( ord_less_set_nat_a @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_302_atLeastLessThan__iff,axiom,
! [I4: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I4 @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I4 )
& ( ord_less_set_nat @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_303_atLeastLessThan__iff,axiom,
! [I4: nat,L: nat,U: nat] :
( ( member_nat @ I4 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I4 )
& ( ord_less_nat @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_304_atLeastLessThan__iff,axiom,
! [I4: int,L: int,U: int] :
( ( member_int @ I4 @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I4 )
& ( ord_less_int @ I4 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_305_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_306_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_307_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_308_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_309_ivl__subset,axiom,
! [I4: nat,J3: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I4 @ J3 ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J3 @ I4 )
| ( ( ord_less_eq_nat @ M @ I4 )
& ( ord_less_eq_nat @ J3 @ N ) ) ) ) ).
% ivl_subset
thf(fact_310_ivl__subset,axiom,
! [I4: int,J3: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I4 @ J3 ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J3 @ I4 )
| ( ( ord_less_eq_int @ M @ I4 )
& ( ord_less_eq_int @ J3 @ N ) ) ) ) ).
% ivl_subset
thf(fact_311_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_312_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_313_b,axiom,
( ord_le871467723717165285_nat_a
@ ( image_nat_a_nat_a @ f
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I: nat] : b ) )
@ ( collect_nat_a
@ ^ [F2: nat > a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ b )
& ! [X: nat] :
( ( ord_less_eq_nat @ n @ X )
=> ( ( F2 @ X )
= y ) ) ) ) ) ).
% b
thf(fact_314_inj__of__nat,axiom,
inj_on_nat_nat @ semiri1316708129612266289at_nat @ top_top_set_nat ).
% inj_of_nat
thf(fact_315_inj__of__nat,axiom,
inj_on_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ).
% inj_of_nat
thf(fact_316_inj__on__of__nat,axiom,
! [N2: set_nat] : ( inj_on_nat_nat @ semiri1316708129612266289at_nat @ N2 ) ).
% inj_on_of_nat
thf(fact_317_inj__on__of__nat,axiom,
! [N2: set_nat] : ( inj_on_nat_int @ semiri1314217659103216013at_int @ N2 ) ).
% inj_on_of_nat
thf(fact_318_psubsetD,axiom,
! [A: set_a,B2: set_a,C2: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_319_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_320_psubsetD,axiom,
! [A: set_o,B2: set_o,C2: $o] :
( ( ord_less_set_o @ A @ B2 )
=> ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_321_psubsetD,axiom,
! [A: set_int,B2: set_int,C2: int] :
( ( ord_less_set_int @ A @ B2 )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_322_psubsetD,axiom,
! [A: set_nat_a,B2: set_nat_a,C2: nat > a] :
( ( ord_less_set_nat_a @ A @ B2 )
=> ( ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_323_psubsetE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_324_psubsetE,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B2 )
=> ~ ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ord_le871467723717165285_nat_a @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_325_psubsetE,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_326_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_327_psubset__eq,axiom,
( ord_less_set_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_328_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_329_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ord_less_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_set_def
thf(fact_330_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ).
% less_set_def
thf(fact_331_less__set__def,axiom,
( ord_less_set_o
= ( ^ [A4: set_o,B3: set_o] :
( ord_less_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ).
% less_set_def
thf(fact_332_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A4: set_int,B3: set_int] :
( ord_less_int_o
@ ^ [X: int] : ( member_int @ X @ A4 )
@ ^ [X: int] : ( member_int @ X @ B3 ) ) ) ) ).
% less_set_def
thf(fact_333_less__set__def,axiom,
( ord_less_set_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ord_less_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A4 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B3 ) ) ) ) ).
% less_set_def
thf(fact_334_psubset__imp__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_335_psubset__imp__subset,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B2 )
=> ( ord_le871467723717165285_nat_a @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_336_psubset__imp__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_337_psubset__subset__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_338_psubset__subset__trans,axiom,
! [A: set_nat_a,B2: set_nat_a,C: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ C )
=> ( ord_less_set_nat_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_339_psubset__subset__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_340_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_341_subset__not__subset__eq,axiom,
( ord_less_set_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A4 @ B3 )
& ~ ( ord_le871467723717165285_nat_a @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_342_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_343_subset__psubset__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_344_subset__psubset__trans,axiom,
! [A: set_nat_a,B2: set_nat_a,C: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( ord_less_set_nat_a @ B2 @ C )
=> ( ord_less_set_nat_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_345_subset__psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_346_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_347_subset__iff__psubset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ( ord_less_set_nat_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_348_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_349_image__Fpow__mono,axiom,
! [F: nat > int,A: set_nat,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B2 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_int @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_350_image__Fpow__mono,axiom,
! [F: a > $o,A: set_a,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_set_a_set_o @ ( image_a_o @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_o @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_351_image__Fpow__mono,axiom,
! [F: a > int,A: set_a,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ B2 )
=> ( ord_le4403425263959731960et_int @ ( image_set_a_set_int @ ( image_a_int @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_int @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_352_image__Fpow__mono,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_353_image__Fpow__mono,axiom,
! [F: int > a,A: set_int,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_int_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_int_set_a @ ( image_int_a @ F ) @ ( finite_Fpow_int @ A ) ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_354_image__Fpow__mono,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_355_image__Fpow__mono,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( image_nat_a_nat_a @ F @ A ) @ B2 )
=> ( ord_le2390145808437456709_nat_a @ ( image_6965494298868581957_nat_a @ ( image_nat_a_nat_a @ F ) @ ( finite_Fpow_nat_a @ A ) ) @ ( finite_Fpow_nat_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_356_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_357_image__Fpow__mono,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_358_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_359_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_360_Fpow__mono,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% Fpow_mono
thf(fact_361_Fpow__mono,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ord_le2390145808437456709_nat_a @ ( finite_Fpow_nat_a @ A ) @ ( finite_Fpow_nat_a @ B2 ) ) ) ).
% Fpow_mono
thf(fact_362_Fpow__mono,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% Fpow_mono
thf(fact_363_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_364_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_365_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_366_le__trans,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I4 @ K2 ) ) ) ).
% le_trans
thf(fact_367_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_368_of__nat__mono,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ ( semiri1316708129612266289at_nat @ J3 ) ) ) ).
% of_nat_mono
thf(fact_369_of__nat__mono,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I4 ) @ ( semiri1314217659103216013at_int @ J3 ) ) ) ).
% of_nat_mono
thf(fact_370_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X: int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_371_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_372_Collect__mono__iff,axiom,
! [P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ( ord_le871467723717165285_nat_a @ ( collect_nat_a @ P ) @ ( collect_nat_a @ Q ) )
= ( ! [X: nat > a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_373_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_374_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_375_set__eq__subset,axiom,
( ( ^ [Y6: set_nat_a,Z2: set_nat_a] : ( Y6 = Z2 ) )
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A4 @ B3 )
& ( ord_le871467723717165285_nat_a @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_376_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_377_subset__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_378_subset__trans,axiom,
! [A: set_nat_a,B2: set_nat_a,C: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ C )
=> ( ord_le871467723717165285_nat_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_379_subset__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_380_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_381_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_382_UNIV__witness,axiom,
? [X3: nat > a] : ( member_nat_a @ X3 @ top_top_set_nat_a ) ).
% UNIV_witness
thf(fact_383_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_384_UNIV__witness,axiom,
? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_385_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_386_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_387_Collect__mono,axiom,
! [P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ! [X3: nat > a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le871467723717165285_nat_a @ ( collect_nat_a @ P ) @ ( collect_nat_a @ Q ) ) ) ).
% Collect_mono
thf(fact_388_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_389_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_390_subset__refl,axiom,
! [A: set_nat_a] : ( ord_le871467723717165285_nat_a @ A @ A ) ).
% subset_refl
thf(fact_391_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_392_subset__UNIV,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% subset_UNIV
thf(fact_393_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_394_subset__UNIV,axiom,
! [A: set_nat_a] : ( ord_le871467723717165285_nat_a @ A @ top_top_set_nat_a ) ).
% subset_UNIV
thf(fact_395_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_396_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B3: set_o] :
! [T3: $o] :
( ( member_o @ T3 @ A4 )
=> ( member_o @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_397_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B3: set_int] :
! [T3: int] :
( ( member_int @ T3 @ A4 )
=> ( member_int @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_398_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_399_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
! [T3: nat > a] :
( ( member_nat_a @ T3 @ A4 )
=> ( member_nat_a @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_400_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_401_equalityD2,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_402_equalityD2,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( A = B2 )
=> ( ord_le871467723717165285_nat_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_403_equalityD2,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_404_equalityD1,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_405_equalityD1,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( A = B2 )
=> ( ord_le871467723717165285_nat_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_406_equalityD1,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_407_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B3: set_o] :
! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_o @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_408_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B3: set_int] :
! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_int @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_409_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_410_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
! [X: nat > a] :
( ( member_nat_a @ X @ A4 )
=> ( member_nat_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_411_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_412_equalityE,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_413_equalityE,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( A = B2 )
=> ~ ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ~ ( ord_le871467723717165285_nat_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_414_equalityE,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_415_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_416_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_417_UNIV__eq__I,axiom,
! [A: set_nat_a] :
( ! [X3: nat > a] : ( member_nat_a @ X3 @ A )
=> ( top_top_set_nat_a = A ) ) ).
% UNIV_eq_I
thf(fact_418_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_419_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X3: int] : ( member_int @ X3 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_420_subsetD,axiom,
! [A: set_o,B2: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_421_subsetD,axiom,
! [A: set_int,B2: set_int,C2: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_422_subsetD,axiom,
! [A: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_423_subsetD,axiom,
! [A: set_nat_a,B2: set_nat_a,C2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_424_subsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_425_in__mono,axiom,
! [A: set_o,B2: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_426_in__mono,axiom,
! [A: set_int,B2: set_int,X2: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_427_in__mono,axiom,
! [A: set_a,B2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_428_in__mono,axiom,
! [A: set_nat_a,B2: set_nat_a,X2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( member_nat_a @ X2 @ A )
=> ( member_nat_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_429_in__mono,axiom,
! [A: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_430_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_431_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_432_range__subsetD,axiom,
! [F: a > $o,B2: set_o,I4: a] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ top_top_set_a ) @ B2 )
=> ( member_o @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_433_range__subsetD,axiom,
! [F: a > int,B2: set_int,I4: a] :
( ( ord_less_eq_set_int @ ( image_a_int @ F @ top_top_set_a ) @ B2 )
=> ( member_int @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_434_range__subsetD,axiom,
! [F: nat > $o,B2: set_o,I4: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B2 )
=> ( member_o @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_435_range__subsetD,axiom,
! [F: nat > int,B2: set_int,I4: nat] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ top_top_set_nat ) @ B2 )
=> ( member_int @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_436_range__subsetD,axiom,
! [F: int > $o,B2: set_o,I4: int] :
( ( ord_less_eq_set_o @ ( image_int_o @ F @ top_top_set_int ) @ B2 )
=> ( member_o @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_437_range__subsetD,axiom,
! [F: int > int,B2: set_int,I4: int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ top_top_set_int ) @ B2 )
=> ( member_int @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_438_range__subsetD,axiom,
! [F: a > a,B2: set_a,I4: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ top_top_set_a ) @ B2 )
=> ( member_a @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_439_range__subsetD,axiom,
! [F: nat > a,B2: set_a,I4: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B2 )
=> ( member_a @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_440_range__subsetD,axiom,
! [F: int > a,B2: set_a,I4: int] :
( ( ord_less_eq_set_a @ ( image_int_a @ F @ top_top_set_int ) @ B2 )
=> ( member_a @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_441_range__subsetD,axiom,
! [F: a > nat,B2: set_nat,I4: a] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ top_top_set_a ) @ B2 )
=> ( member_nat @ ( F @ I4 ) @ B2 ) ) ).
% range_subsetD
thf(fact_442_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
=> ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C2 @ A2 )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_443_atLeastLessThan__subset__iff,axiom,
! [A2: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A2 @ B ) @ ( set_or4662586982721622107an_int @ C2 @ D2 ) )
=> ( ( ord_less_eq_int @ B @ A2 )
| ( ( ord_less_eq_int @ C2 @ A2 )
& ( ord_less_eq_int @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_444_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_445_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_446_PiE__mono,axiom,
! [A: set_a,B2: a > set_a,C: a > set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_less_eq_set_a_a @ ( piE_a_a @ A @ B2 ) @ ( piE_a_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_447_PiE__mono,axiom,
! [A: set_o,B2: $o > set_a,C: $o > set_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_less_eq_set_o_a @ ( piE_o_a @ A @ B2 ) @ ( piE_o_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_448_PiE__mono,axiom,
! [A: set_int,B2: int > set_a,C: int > set_a] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le943418215940126601_int_a @ ( piE_int_a @ A @ B2 ) @ ( piE_int_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_449_PiE__mono,axiom,
! [A: set_nat,B2: nat > set_a,C: nat > set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( piE_nat_a @ A @ B2 ) @ ( piE_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_450_PiE__mono,axiom,
! [A: set_a,B2: a > set_nat,C: a > set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le1612561287239139007_a_nat @ ( piE_a_nat @ A @ B2 ) @ ( piE_a_nat @ A @ C ) ) ) ).
% PiE_mono
thf(fact_451_PiE__mono,axiom,
! [A: set_nat,B2: nat > set_nat,C: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A @ B2 ) @ ( piE_nat_nat @ A @ C ) ) ) ).
% PiE_mono
thf(fact_452_PiE__mono,axiom,
! [A: set_o,B2: $o > set_nat,C: $o > set_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le4981610546006782297_o_nat @ ( piE_o_nat @ A @ B2 ) @ ( piE_o_nat @ A @ C ) ) ) ).
% PiE_mono
thf(fact_453_PiE__mono,axiom,
! [A: set_int,B2: int > set_nat,C: int > set_nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le2023132899490853297nt_nat @ ( piE_int_nat @ A @ B2 ) @ ( piE_int_nat @ A @ C ) ) ) ).
% PiE_mono
thf(fact_454_PiE__mono,axiom,
! [A: set_nat_a,B2: ( nat > a ) > set_a,C: ( nat > a ) > set_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le3509452538356653652at_a_a @ ( piE_nat_a_a @ A @ B2 ) @ ( piE_nat_a_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_455_PiE__mono,axiom,
! [A: set_a,B2: a > set_nat_a,C: a > set_nat_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le871467723717165285_nat_a @ ( B2 @ X3 ) @ ( C @ X3 ) ) )
=> ( ord_le2508512696544544866_nat_a @ ( piE_a_nat_a @ A @ B2 ) @ ( piE_a_nat_a @ A @ C ) ) ) ).
% PiE_mono
thf(fact_456_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_457_Collect__subset,axiom,
! [A: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_458_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_459_Collect__subset,axiom,
! [A: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_460_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_461_UNIV__def,axiom,
( top_top_set_nat_a
= ( collect_nat_a
@ ^ [X: nat > a] : $true ) ) ).
% UNIV_def
thf(fact_462_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_463_UNIV__def,axiom,
( top_top_set_int
= ( collect_int
@ ^ [X: int] : $true ) ) ).
% UNIV_def
thf(fact_464_the__inv__f__f,axiom,
! [F: ( nat > a ) > nat > a,X2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ top_top_set_nat_a )
=> ( ( the_in381770415356814323_nat_a @ top_top_set_nat_a @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_465_the__inv__f__f,axiom,
! [F: nat > nat,X2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_466_inj__image__subset__iff,axiom,
! [F: a > $o,A: set_a,B2: set_a] :
( ( inj_on_a_o @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ ( image_a_o @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_467_inj__image__subset__iff,axiom,
! [F: a > int,A: set_a,B2: set_a] :
( ( inj_on_a_int @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ ( image_a_int @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_468_inj__image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B2: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_469_inj__image__subset__iff,axiom,
! [F: int > a,A: set_int,B2: set_int] :
( ( inj_on_int_a @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_a @ ( image_int_a @ F @ A ) @ ( image_int_a @ F @ B2 ) )
= ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_470_inj__image__subset__iff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_471_inj__image__subset__iff,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_472_inj__image__subset__iff,axiom,
! [F: int > nat,A: set_int,B2: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ ( image_int_nat @ F @ B2 ) )
= ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_473_inj__image__subset__iff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_474_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_475_inj__image__subset__iff,axiom,
! [F: ( nat > a ) > a,A: set_nat_a,B2: set_nat_a] :
( ( inj_on_nat_a_a @ F @ top_top_set_nat_a )
=> ( ( ord_less_eq_set_a @ ( image_nat_a_a @ F @ A ) @ ( image_nat_a_a @ F @ B2 ) )
= ( ord_le871467723717165285_nat_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_476_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_477_the__inv__into__into,axiom,
! [F: $o > a,A: set_o,X2: a,B2: set_o] :
( ( inj_on_o_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_o_a @ F @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( member_o @ ( the_inv_into_o_a @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_478_the__inv__into__into,axiom,
! [F: int > a,A: set_int,X2: a,B2: set_int] :
( ( inj_on_int_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_int_a @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( member_int @ ( the_inv_into_int_a @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_479_the__inv__into__into,axiom,
! [F: $o > nat,A: set_o,X2: nat,B2: set_o] :
( ( inj_on_o_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_o_nat @ F @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( member_o @ ( the_inv_into_o_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_480_the__inv__into__into,axiom,
! [F: int > nat,A: set_int,X2: nat,B2: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_int_nat @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( member_int @ ( the_inv_into_int_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_481_the__inv__into__into,axiom,
! [F: $o > $o,A: set_o,X2: $o,B2: set_o] :
( ( inj_on_o_o @ F @ A )
=> ( ( member_o @ X2 @ ( image_o_o @ F @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( member_o @ ( the_inv_into_o_o @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_482_the__inv__into__into,axiom,
! [F: int > $o,A: set_int,X2: $o,B2: set_int] :
( ( inj_on_int_o @ F @ A )
=> ( ( member_o @ X2 @ ( image_int_o @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( member_int @ ( the_inv_into_int_o @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_483_the__inv__into__into,axiom,
! [F: $o > int,A: set_o,X2: int,B2: set_o] :
( ( inj_on_o_int @ F @ A )
=> ( ( member_int @ X2 @ ( image_o_int @ F @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( member_o @ ( the_inv_into_o_int @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_484_the__inv__into__into,axiom,
! [F: int > int,A: set_int,X2: int,B2: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( member_int @ X2 @ ( image_int_int @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( member_int @ ( the_inv_into_int_int @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_485_the__inv__into__into,axiom,
! [F: a > a,A: set_a,X2: a,B2: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_a_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( member_a @ ( the_inv_into_a_a @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_486_the__inv__into__into,axiom,
! [F: a > nat,A: set_a,X2: nat,B2: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_a_nat @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( member_a @ ( the_inv_into_a_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_487_subset__image__iff,axiom,
! [B2: set_o,F: a > $o,A: set_a] :
( ( ord_less_eq_set_o @ B2 @ ( image_a_o @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_488_subset__image__iff,axiom,
! [B2: set_int,F: a > int,A: set_a] :
( ( ord_less_eq_set_int @ B2 @ ( image_a_int @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_489_subset__image__iff,axiom,
! [B2: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_490_subset__image__iff,axiom,
! [B2: set_a,F: int > a,A: set_int] :
( ( ord_less_eq_set_a @ B2 @ ( image_int_a @ F @ A ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A )
& ( B2
= ( image_int_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_491_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_492_subset__image__iff,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_493_subset__image__iff,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_494_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_495_subset__image__iff,axiom,
! [B2: set_a,F: ( nat > a ) > a,A: set_nat_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a_a @ F @ A ) )
= ( ? [AA: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ AA @ A )
& ( B2
= ( image_nat_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_496_subset__image__iff,axiom,
! [B2: set_nat_a,F: a > nat > a,A: set_a] :
( ( ord_le871467723717165285_nat_a @ B2 @ ( image_a_nat_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_497_image__subset__iff,axiom,
! [F: a > $o,A: set_a,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ B2 )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_o @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_498_image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_499_image__subset__iff,axiom,
! [F: a > int,A: set_a,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ B2 )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_int @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_500_image__subset__iff,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_501_image__subset__iff,axiom,
! [F: int > a,A: set_int,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_int_a @ F @ A ) @ B2 )
= ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_502_image__subset__iff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_503_image__subset__iff,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( image_nat_a_nat_a @ F @ A ) @ B2 )
= ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( member_nat_a @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_504_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_505_image__subset__iff,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_506_subset__imageE,axiom,
! [B2: set_o,F: a > $o,A: set_a] :
( ( ord_less_eq_set_o @ B2 @ ( image_a_o @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_507_subset__imageE,axiom,
! [B2: set_int,F: a > int,A: set_a] :
( ( ord_less_eq_set_int @ B2 @ ( image_a_int @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_508_subset__imageE,axiom,
! [B2: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_509_subset__imageE,axiom,
! [B2: set_a,F: int > a,A: set_int] :
( ( ord_less_eq_set_a @ B2 @ ( image_int_a @ F @ A ) )
=> ~ ! [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
=> ( B2
!= ( image_int_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_510_subset__imageE,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_511_subset__imageE,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_512_subset__imageE,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_513_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_514_subset__imageE,axiom,
! [B2: set_a,F: ( nat > a ) > a,A: set_nat_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a_a @ F @ A ) )
=> ~ ! [C3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ C3 @ A )
=> ( B2
!= ( image_nat_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_515_subset__imageE,axiom,
! [B2: set_nat_a,F: a > nat > a,A: set_a] :
( ( ord_le871467723717165285_nat_a @ B2 @ ( image_a_nat_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_516_image__subsetI,axiom,
! [A: set_a,F: a > $o,B2: set_o] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_517_image__subsetI,axiom,
! [A: set_a,F: a > int,B2: set_int] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_518_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B2: set_o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_519_image__subsetI,axiom,
! [A: set_nat,F: nat > int,B2: set_int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_520_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_521_image__subsetI,axiom,
! [A: set_o,F: $o > int,B2: set_int] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_o_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_522_image__subsetI,axiom,
! [A: set_int,F: int > $o,B2: set_o] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_int_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_523_image__subsetI,axiom,
! [A: set_int,F: int > int,B2: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_524_image__subsetI,axiom,
! [A: set_a,F: a > a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_525_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B2: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_526_image__mono,axiom,
! [A: set_int,B2: set_int,F: int > a] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ord_less_eq_set_a @ ( image_int_a @ F @ A ) @ ( image_int_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_527_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > $o] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ ( image_a_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_528_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > int] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_int @ ( image_a_int @ F @ A ) @ ( image_a_int @ F @ B2 ) ) ) ).
% image_mono
thf(fact_529_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_530_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_531_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% image_mono
thf(fact_532_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_533_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_534_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > nat > a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F @ A ) @ ( image_a_nat_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_535_image__mono,axiom,
! [A: set_nat_a,B2: set_nat_a,F: ( nat > a ) > a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ord_less_eq_set_a @ ( image_nat_a_a @ F @ A ) @ ( image_nat_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_536_in__image__by__witness,axiom,
! [A: set_o,G: $o > a,B2: set_a,F: a > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_a @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_a_o @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_537_in__image__by__witness,axiom,
! [A: set_o,G: $o > nat,B2: set_nat,F: nat > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_nat @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_nat_o @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_538_in__image__by__witness,axiom,
! [A: set_o,G: $o > $o,B2: set_o,F: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_o_o @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_539_in__image__by__witness,axiom,
! [A: set_o,G: $o > int,B2: set_int,F: int > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_int @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_o @ A @ ( image_int_o @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_540_in__image__by__witness,axiom,
! [A: set_int,G: int > a,B2: set_a,F: a > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( member_a @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_a_int @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_541_in__image__by__witness,axiom,
! [A: set_int,G: int > nat,B2: set_nat,F: nat > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( member_nat @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_nat_int @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_542_in__image__by__witness,axiom,
! [A: set_int,G: int > $o,B2: set_o,F: $o > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( member_o @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_o_int @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_543_in__image__by__witness,axiom,
! [A: set_int,G: int > int,B2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( member_int @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_int @ A @ ( image_int_int @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_544_in__image__by__witness,axiom,
! [A: set_a,G: a > a,B2: set_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( image_a_a @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_545_in__image__by__witness,axiom,
! [A: set_a,G: a > nat,B2: set_nat,F: nat > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_nat @ ( G @ X3 ) @ B2 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( image_nat_a @ F @ B2 ) ) ) ).
% in_image_by_witness
thf(fact_546_all__subset__image,axiom,
! [F: a > $o,A: set_a,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_a_o @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_547_all__subset__image,axiom,
! [F: a > int,A: set_a,P: set_int > $o] :
( ( ! [B3: set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_a_int @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_int @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_548_all__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ! [B3: set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_int @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_549_all__subset__image,axiom,
! [F: int > a,A: set_int,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_int_a @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_int] :
( ( ord_less_eq_set_int @ B3 @ A )
=> ( P @ ( image_int_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_550_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_551_all__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_552_all__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_553_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_554_all__subset__image,axiom,
! [F: ( nat > a ) > a,A: set_nat_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_nat_a_a @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B3 @ A )
=> ( P @ ( image_nat_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_555_all__subset__image,axiom,
! [F: a > nat > a,A: set_a,P: set_nat_a > $o] :
( ( ! [B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B3 @ ( image_a_nat_a @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_nat_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_556_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_557_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_558_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_559_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_560_image__int__atLeastLessThan,axiom,
! [A2: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_561_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_562_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_563_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_564_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_565_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_566_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J3: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_567_range__eqI,axiom,
! [B: a,F: a > a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_568_range__eqI,axiom,
! [B: nat,F: a > nat,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_569_range__eqI,axiom,
! [B: $o,F: a > $o,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_a_o @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_570_range__eqI,axiom,
! [B: int,F: a > int,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_a_int @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_571_range__eqI,axiom,
! [B: a,F: nat > a,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_572_range__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_573_range__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_574_range__eqI,axiom,
! [B: int,F: nat > int,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_575_range__eqI,axiom,
! [B: a,F: int > a,X2: int] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_int_a @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_576_range__eqI,axiom,
! [B: nat,F: int > nat,X2: int] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_577_rangeI,axiom,
! [F: a > a,X2: a] : ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_578_rangeI,axiom,
! [F: a > nat,X2: a] : ( member_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_579_rangeI,axiom,
! [F: a > $o,X2: a] : ( member_o @ ( F @ X2 ) @ ( image_a_o @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_580_rangeI,axiom,
! [F: a > int,X2: a] : ( member_int @ ( F @ X2 ) @ ( image_a_int @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_581_rangeI,axiom,
! [F: nat > a,X2: nat] : ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_582_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_583_rangeI,axiom,
! [F: nat > $o,X2: nat] : ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_584_rangeI,axiom,
! [F: nat > int,X2: nat] : ( member_int @ ( F @ X2 ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_585_rangeI,axiom,
! [F: int > a,X2: int] : ( member_a @ ( F @ X2 ) @ ( image_int_a @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_586_rangeI,axiom,
! [F: int > nat,X2: int] : ( member_nat @ ( F @ X2 ) @ ( image_int_nat @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_587_surj__def,axiom,
! [F: a > $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
= ( ! [Y3: $o] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_588_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y3: a] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_589_surj__def,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_590_surj__def,axiom,
! [F: a > int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
= ( ! [Y3: int] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_591_surj__def,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ! [Y3: a] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_592_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_593_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y3: int] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_594_surj__def,axiom,
! [F: int > a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
= ( ! [Y3: a] :
? [X: int] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_595_surj__def,axiom,
! [F: int > nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: int] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_596_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y3: int] :
? [X: int] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_597_surjI,axiom,
! [G: a > $o,F: $o > a] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_o @ G @ top_top_set_a )
= top_top_set_o ) ) ).
% surjI
thf(fact_598_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_599_surjI,axiom,
! [G: a > nat,F: nat > a] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_600_surjI,axiom,
! [G: a > int,F: int > a] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_int @ G @ top_top_set_a )
= top_top_set_int ) ) ).
% surjI
thf(fact_601_surjI,axiom,
! [G: nat > a,F: a > nat] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a ) ) ).
% surjI
thf(fact_602_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_603_surjI,axiom,
! [G: nat > int,F: int > nat] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_604_surjI,axiom,
! [G: int > a,F: a > int] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_a @ G @ top_top_set_int )
= top_top_set_a ) ) ).
% surjI
thf(fact_605_surjI,axiom,
! [G: int > nat,F: nat > int] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_nat @ G @ top_top_set_int )
= top_top_set_nat ) ) ).
% surjI
thf(fact_606_surjI,axiom,
! [G: int > int,F: int > int] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_int @ G @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_607_surjE,axiom,
! [F: a > $o,Y2: $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ~ ! [X3: a] :
( Y2
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_608_surjE,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X3: a] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_609_surjE,axiom,
! [F: a > nat,Y2: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ~ ! [X3: a] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_610_surjE,axiom,
! [F: a > int,Y2: int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ~ ! [X3: a] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_611_surjE,axiom,
! [F: nat > a,Y2: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ~ ! [X3: nat] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_612_surjE,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_613_surjE,axiom,
! [F: nat > int,Y2: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X3: nat] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_614_surjE,axiom,
! [F: int > a,Y2: a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ~ ! [X3: int] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_615_surjE,axiom,
! [F: int > nat,Y2: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ~ ! [X3: int] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_616_surjE,axiom,
! [F: int > int,Y2: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X3: int] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_617_surjD,axiom,
! [F: a > $o,Y2: $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ? [X3: a] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_618_surjD,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X3: a] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_619_surjD,axiom,
! [F: a > nat,Y2: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ? [X3: a] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_620_surjD,axiom,
! [F: a > int,Y2: int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ? [X3: a] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_621_surjD,axiom,
! [F: nat > a,Y2: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_622_surjD,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_623_surjD,axiom,
! [F: nat > int,Y2: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_624_surjD,axiom,
! [F: int > a,Y2: a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ? [X3: int] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_625_surjD,axiom,
! [F: int > nat,Y2: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ? [X3: int] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_626_surjD,axiom,
! [F: int > int,Y2: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ? [X3: int] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_627_inj__on__subset,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,B2: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ A )
=> ( inj_on_nat_a_nat_a @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_628_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( inj_on_nat_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_629_subset__inj__on,axiom,
! [F: ( nat > a ) > nat > a,B2: set_nat_a,A: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( inj_on_nat_a_nat_a @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_630_subset__inj__on,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_631_injD,axiom,
! [F: ( nat > a ) > nat > a,X2: nat > a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ top_top_set_nat_a )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% injD
thf(fact_632_injD,axiom,
! [F: nat > nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% injD
thf(fact_633_injI,axiom,
! [F: ( nat > a ) > nat > a] :
( ! [X3: nat > a,Y4: nat > a] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on_nat_a_nat_a @ F @ top_top_set_nat_a ) ) ).
% injI
thf(fact_634_injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_635_inj__eq,axiom,
! [F: ( nat > a ) > nat > a,X2: nat > a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ top_top_set_nat_a )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% inj_eq
thf(fact_636_inj__eq,axiom,
! [F: nat > nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% inj_eq
thf(fact_637_inj__def,axiom,
! [F: ( nat > a ) > nat > a] :
( ( inj_on_nat_a_nat_a @ F @ top_top_set_nat_a )
= ( ! [X: nat > a,Y3: nat > a] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ) ).
% inj_def
thf(fact_638_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X: nat,Y3: nat] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ) ).
% inj_def
thf(fact_639_range__composition,axiom,
! [F: $o > $o,G: a > $o] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_o_o @ F @ ( image_a_o @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_640_range__composition,axiom,
! [F: nat > $o,G: a > nat] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_nat_o @ F @ ( image_a_nat @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_641_range__composition,axiom,
! [F: int > $o,G: a > int] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_int_o @ F @ ( image_a_int @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_642_range__composition,axiom,
! [F: a > $o,G: a > a] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_a_o @ F @ ( image_a_a @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_643_range__composition,axiom,
! [F: $o > nat,G: a > $o] :
( ( image_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_o_nat @ F @ ( image_a_o @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_644_range__composition,axiom,
! [F: int > nat,G: a > int] :
( ( image_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_int_nat @ F @ ( image_a_int @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_645_range__composition,axiom,
! [F: nat > nat,G: a > nat] :
( ( image_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_nat_nat @ F @ ( image_a_nat @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_646_range__composition,axiom,
! [F: a > nat,G: a > a] :
( ( image_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_a_nat @ F @ ( image_a_a @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_647_range__composition,axiom,
! [F: $o > int,G: a > $o] :
( ( image_a_int
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_o_int @ F @ ( image_a_o @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_648_range__composition,axiom,
! [F: int > int,G: a > int] :
( ( image_a_int
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_int_int @ F @ ( image_a_int @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_649_rangeE,axiom,
! [B: a,F: a > a] :
( ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) )
=> ~ ! [X3: a] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_650_rangeE,axiom,
! [B: nat,F: a > nat] :
( ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) )
=> ~ ! [X3: a] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_651_rangeE,axiom,
! [B: $o,F: a > $o] :
( ( member_o @ B @ ( image_a_o @ F @ top_top_set_a ) )
=> ~ ! [X3: a] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_652_rangeE,axiom,
! [B: int,F: a > int] :
( ( member_int @ B @ ( image_a_int @ F @ top_top_set_a ) )
=> ~ ! [X3: a] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_653_rangeE,axiom,
! [B: a,F: nat > a] :
( ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_654_rangeE,axiom,
! [B: nat,F: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_655_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_656_rangeE,axiom,
! [B: int,F: nat > int] :
( ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_657_rangeE,axiom,
! [B: a,F: int > a] :
( ( member_a @ B @ ( image_int_a @ F @ top_top_set_int ) )
=> ~ ! [X3: int] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_658_rangeE,axiom,
! [B: nat,F: int > nat] :
( ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) )
=> ~ ! [X3: int] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_659_sorted__list__of__set_Oinj__on,axiom,
( inj_on_nat_nat
@ ^ [X: nat] : X
@ top_top_set_nat ) ).
% sorted_list_of_set.inj_on
thf(fact_660_sorted__list__of__set_Oinj__on,axiom,
( inj_on_int_int
@ ^ [X: int] : X
@ top_top_set_int ) ).
% sorted_list_of_set.inj_on
thf(fact_661_the__inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_662_the__inv__into__f__f,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,X2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( member_nat_a @ X2 @ A )
=> ( ( the_in381770415356814323_nat_a @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_663_the__inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X2 )
= Y2 )
=> ( ( member_nat @ X2 @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ Y2 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_664_the__inv__into__f__eq,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,X2: nat > a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( ( F @ X2 )
= Y2 )
=> ( ( member_nat_a @ X2 @ A )
=> ( ( the_in381770415356814323_nat_a @ A @ F @ Y2 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_665_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_666_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_667_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_668_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_669_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_670_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_671_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_672_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_673_inj__on__image__eq__iff,axiom,
! [F: int > a,C: set_int,A: set_int,B2: set_int] :
( ( inj_on_int_a @ F @ C )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( ord_less_eq_set_int @ B2 @ C )
=> ( ( ( image_int_a @ F @ A )
= ( image_int_a @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_674_inj__on__image__eq__iff,axiom,
! [F: a > $o,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_o @ F @ C )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( ( image_a_o @ F @ A )
= ( image_a_o @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_675_inj__on__image__eq__iff,axiom,
! [F: a > nat,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ C )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_676_inj__on__image__eq__iff,axiom,
! [F: a > int,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_int @ F @ C )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( ( image_a_int @ F @ A )
= ( image_a_int @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_677_inj__on__image__eq__iff,axiom,
! [F: a > a,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_678_inj__on__image__eq__iff,axiom,
! [F: ( nat > a ) > nat > a,C: set_nat_a,A: set_nat_a,B2: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ C )
=> ( ( ord_le871467723717165285_nat_a @ A @ C )
=> ( ( ord_le871467723717165285_nat_a @ B2 @ C )
=> ( ( ( image_nat_a_nat_a @ F @ A )
= ( image_nat_a_nat_a @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_679_inj__on__image__eq__iff,axiom,
! [F: nat > a,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_a @ F @ A )
= ( image_nat_a @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_680_inj__on__image__eq__iff,axiom,
! [F: nat > int,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_int @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_int @ F @ A )
= ( image_nat_int @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_681_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_682_inj__on__image__mem__iff,axiom,
! [F: $o > a,B2: set_o,A2: $o,A: set_o] :
( ( inj_on_o_a @ F @ B2 )
=> ( ( member_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_o_a @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_683_inj__on__image__mem__iff,axiom,
! [F: $o > nat,B2: set_o,A2: $o,A: set_o] :
( ( inj_on_o_nat @ F @ B2 )
=> ( ( member_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_o_nat @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_684_inj__on__image__mem__iff,axiom,
! [F: $o > $o,B2: set_o,A2: $o,A: set_o] :
( ( inj_on_o_o @ F @ B2 )
=> ( ( member_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ ( F @ A2 ) @ ( image_o_o @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_685_inj__on__image__mem__iff,axiom,
! [F: $o > int,B2: set_o,A2: $o,A: set_o] :
( ( inj_on_o_int @ F @ B2 )
=> ( ( member_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_int @ ( F @ A2 ) @ ( image_o_int @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_686_inj__on__image__mem__iff,axiom,
! [F: int > a,B2: set_int,A2: int,A: set_int] :
( ( inj_on_int_a @ F @ B2 )
=> ( ( member_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_int_a @ F @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_687_inj__on__image__mem__iff,axiom,
! [F: int > nat,B2: set_int,A2: int,A: set_int] :
( ( inj_on_int_nat @ F @ B2 )
=> ( ( member_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_int_nat @ F @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_688_inj__on__image__mem__iff,axiom,
! [F: int > $o,B2: set_int,A2: int,A: set_int] :
( ( inj_on_int_o @ F @ B2 )
=> ( ( member_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_o @ ( F @ A2 ) @ ( image_int_o @ F @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_689_inj__on__image__mem__iff,axiom,
! [F: int > int,B2: set_int,A2: int,A: set_int] :
( ( inj_on_int_int @ F @ B2 )
=> ( ( member_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ ( F @ A2 ) @ ( image_int_int @ F @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_690_inj__on__image__mem__iff,axiom,
! [F: a > a,B2: set_a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( member_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_691_inj__on__image__mem__iff,axiom,
! [F: a > nat,B2: set_a,A2: a,A: set_a] :
( ( inj_on_a_nat @ F @ B2 )
=> ( ( member_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_a_nat @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_692_linorder__injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_693_range__ex1__eq,axiom,
! [F: a > a,B: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) )
= ( ? [X: a] :
( ( B
= ( F @ X ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_694_range__ex1__eq,axiom,
! [F: a > nat,B: nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) )
= ( ? [X: a] :
( ( B
= ( F @ X ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_695_range__ex1__eq,axiom,
! [F: a > $o,B: $o] :
( ( inj_on_a_o @ F @ top_top_set_a )
=> ( ( member_o @ B @ ( image_a_o @ F @ top_top_set_a ) )
= ( ? [X: a] :
( ( B
= ( F @ X ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_696_range__ex1__eq,axiom,
! [F: a > int,B: int] :
( ( inj_on_a_int @ F @ top_top_set_a )
=> ( ( member_int @ B @ ( image_a_int @ F @ top_top_set_a ) )
= ( ? [X: a] :
( ( B
= ( F @ X ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_697_range__ex1__eq,axiom,
! [F: nat > a,B: a] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_698_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_699_range__ex1__eq,axiom,
! [F: nat > $o,B: $o] :
( ( inj_on_nat_o @ F @ top_top_set_nat )
=> ( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_700_range__ex1__eq,axiom,
! [F: nat > int,B: int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_701_range__ex1__eq,axiom,
! [F: int > a,B: a] :
( ( inj_on_int_a @ F @ top_top_set_int )
=> ( ( member_a @ B @ ( image_int_a @ F @ top_top_set_int ) )
= ( ? [X: int] :
( ( B
= ( F @ X ) )
& ! [Y3: int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_702_range__ex1__eq,axiom,
! [F: int > nat,B: nat] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) )
= ( ? [X: int] :
( ( B
= ( F @ X ) )
& ! [Y3: int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_703_inj__image__eq__iff,axiom,
! [F: a > $o,A: set_a,B2: set_a] :
( ( inj_on_a_o @ F @ top_top_set_a )
=> ( ( ( image_a_o @ F @ A )
= ( image_a_o @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_704_inj__image__eq__iff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_705_inj__image__eq__iff,axiom,
! [F: a > int,A: set_a,B2: set_a] :
( ( inj_on_a_int @ F @ top_top_set_a )
=> ( ( ( image_a_int @ F @ A )
= ( image_a_int @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_706_inj__image__eq__iff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_707_inj__image__eq__iff,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,B2: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ top_top_set_nat_a )
=> ( ( ( image_nat_a_nat_a @ F @ A )
= ( image_nat_a_nat_a @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_708_inj__image__eq__iff,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( ( image_nat_a @ F @ A )
= ( image_nat_a @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_709_inj__image__eq__iff,axiom,
! [F: nat > int,A: set_nat,B2: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ( image_nat_int @ F @ A )
= ( image_nat_int @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_710_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_711_inj__image__eq__iff,axiom,
! [F: int > a,A: set_int,B2: set_int] :
( ( inj_on_int_a @ F @ top_top_set_int )
=> ( ( ( image_int_a @ F @ A )
= ( image_int_a @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_712_inj__image__mem__iff,axiom,
! [F: a > a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_713_inj__image__mem__iff,axiom,
! [F: $o > a,A2: $o,A: set_o] :
( ( inj_on_o_a @ F @ top_top_set_o )
=> ( ( member_a @ ( F @ A2 ) @ ( image_o_a @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_714_inj__image__mem__iff,axiom,
! [F: a > nat,A2: a,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_a_nat @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_715_inj__image__mem__iff,axiom,
! [F: $o > nat,A2: $o,A: set_o] :
( ( inj_on_o_nat @ F @ top_top_set_o )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_o_nat @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_716_inj__image__mem__iff,axiom,
! [F: a > $o,A2: a,A: set_a] :
( ( inj_on_a_o @ F @ top_top_set_a )
=> ( ( member_o @ ( F @ A2 ) @ ( image_a_o @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_717_inj__image__mem__iff,axiom,
! [F: $o > $o,A2: $o,A: set_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( member_o @ ( F @ A2 ) @ ( image_o_o @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_718_inj__image__mem__iff,axiom,
! [F: a > int,A2: a,A: set_a] :
( ( inj_on_a_int @ F @ top_top_set_a )
=> ( ( member_int @ ( F @ A2 ) @ ( image_a_int @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_719_inj__image__mem__iff,axiom,
! [F: $o > int,A2: $o,A: set_o] :
( ( inj_on_o_int @ F @ top_top_set_o )
=> ( ( member_int @ ( F @ A2 ) @ ( image_o_int @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_720_inj__image__mem__iff,axiom,
! [F: nat > a,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ ( F @ A2 ) @ ( image_nat_a @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_721_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_722_f__the__inv__into__f,axiom,
! [F: nat > a,A: set_nat,Y2: a] :
( ( inj_on_nat_a @ F @ A )
=> ( ( member_a @ Y2 @ ( image_nat_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_a @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_723_f__the__inv__into__f,axiom,
! [F: int > a,A: set_int,Y2: a] :
( ( inj_on_int_a @ F @ A )
=> ( ( member_a @ Y2 @ ( image_int_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_int_a @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_724_f__the__inv__into__f,axiom,
! [F: a > a,A: set_a,Y2: a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ Y2 @ ( image_a_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_a @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_725_f__the__inv__into__f,axiom,
! [F: a > nat,A: set_a,Y2: nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( member_nat @ Y2 @ ( image_a_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_nat @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_726_f__the__inv__into__f,axiom,
! [F: nat > nat,A: set_nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ Y2 @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_727_f__the__inv__into__f,axiom,
! [F: a > $o,A: set_a,Y2: $o] :
( ( inj_on_a_o @ F @ A )
=> ( ( member_o @ Y2 @ ( image_a_o @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_o @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_728_f__the__inv__into__f,axiom,
! [F: nat > int,A: set_nat,Y2: int] :
( ( inj_on_nat_int @ F @ A )
=> ( ( member_int @ Y2 @ ( image_nat_int @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_int @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_729_f__the__inv__into__f,axiom,
! [F: a > int,A: set_a,Y2: int] :
( ( inj_on_a_int @ F @ A )
=> ( ( member_int @ Y2 @ ( image_a_int @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_int @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_730_f__the__inv__into__f,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a,Y2: nat > a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( ( member_nat_a @ Y2 @ ( image_nat_a_nat_a @ F @ A ) )
=> ( ( F @ ( the_in381770415356814323_nat_a @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_731_inj__on__the__inv__into,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( inj_on_a_nat @ ( the_inv_into_nat_a @ A @ F ) @ ( image_nat_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_732_inj__on__the__inv__into,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( inj_on_int_nat @ ( the_inv_into_nat_int @ A @ F ) @ ( image_nat_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_733_inj__on__the__inv__into,axiom,
! [F: int > a,A: set_int] :
( ( inj_on_int_a @ F @ A )
=> ( inj_on_a_int @ ( the_inv_into_int_a @ A @ F ) @ ( image_int_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_734_inj__on__the__inv__into,axiom,
! [F: a > $o,A: set_a] :
( ( inj_on_a_o @ F @ A )
=> ( inj_on_o_a @ ( the_inv_into_a_o @ A @ F ) @ ( image_a_o @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_735_inj__on__the__inv__into,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_nat_a @ ( the_inv_into_a_nat @ A @ F ) @ ( image_a_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_736_inj__on__the__inv__into,axiom,
! [F: a > int,A: set_a] :
( ( inj_on_a_int @ F @ A )
=> ( inj_on_int_a @ ( the_inv_into_a_int @ A @ F ) @ ( image_a_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_737_inj__on__the__inv__into,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( inj_on_a_a @ ( the_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_738_inj__on__the__inv__into,axiom,
! [F: ( nat > a ) > nat > a,A: set_nat_a] :
( ( inj_on_nat_a_nat_a @ F @ A )
=> ( inj_on_nat_a_nat_a @ ( the_in381770415356814323_nat_a @ A @ F ) @ ( image_nat_a_nat_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_739_inj__on__the__inv__into,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_740_inj__fun,axiom,
! [F: ( nat > a ) > a] :
( ( inj_on_nat_a_a @ F @ top_top_set_nat_a )
=> ( inj_on_nat_a_nat_a
@ ^ [X: nat > a,Y3: nat] : ( F @ X )
@ top_top_set_nat_a ) ) ).
% inj_fun
thf(fact_741_subset__image__inj,axiom,
! [S2: set_o,F: a > $o,T: set_a] :
( ( ord_less_eq_set_o @ S2 @ ( image_a_o @ F @ T ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T )
& ( inj_on_a_o @ F @ U2 )
& ( S2
= ( image_a_o @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_742_subset__image__inj,axiom,
! [S2: set_int,F: a > int,T: set_a] :
( ( ord_less_eq_set_int @ S2 @ ( image_a_int @ F @ T ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T )
& ( inj_on_a_int @ F @ U2 )
& ( S2
= ( image_a_int @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_743_subset__image__inj,axiom,
! [S2: set_int,F: nat > int,T: set_nat] :
( ( ord_less_eq_set_int @ S2 @ ( image_nat_int @ F @ T ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T )
& ( inj_on_nat_int @ F @ U2 )
& ( S2
= ( image_nat_int @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_744_subset__image__inj,axiom,
! [S2: set_a,F: int > a,T: set_int] :
( ( ord_less_eq_set_a @ S2 @ ( image_int_a @ F @ T ) )
= ( ? [U2: set_int] :
( ( ord_less_eq_set_int @ U2 @ T )
& ( inj_on_int_a @ F @ U2 )
& ( S2
= ( image_int_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_745_subset__image__inj,axiom,
! [S2: set_a,F: a > a,T: set_a] :
( ( ord_less_eq_set_a @ S2 @ ( image_a_a @ F @ T ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T )
& ( inj_on_a_a @ F @ U2 )
& ( S2
= ( image_a_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_746_subset__image__inj,axiom,
! [S2: set_a,F: nat > a,T: set_nat] :
( ( ord_less_eq_set_a @ S2 @ ( image_nat_a @ F @ T ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T )
& ( inj_on_nat_a @ F @ U2 )
& ( S2
= ( image_nat_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_747_subset__image__inj,axiom,
! [S2: set_nat,F: a > nat,T: set_a] :
( ( ord_less_eq_set_nat @ S2 @ ( image_a_nat @ F @ T ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T )
& ( inj_on_a_nat @ F @ U2 )
& ( S2
= ( image_a_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_748_subset__image__inj,axiom,
! [S2: set_nat,F: nat > nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( image_nat_nat @ F @ T ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T )
& ( inj_on_nat_nat @ F @ U2 )
& ( S2
= ( image_nat_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_749_subset__image__inj,axiom,
! [S2: set_a,F: ( nat > a ) > a,T: set_nat_a] :
( ( ord_less_eq_set_a @ S2 @ ( image_nat_a_a @ F @ T ) )
= ( ? [U2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ U2 @ T )
& ( inj_on_nat_a_a @ F @ U2 )
& ( S2
= ( image_nat_a_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_750_subset__image__inj,axiom,
! [S2: set_nat_a,F: a > nat > a,T: set_a] :
( ( ord_le871467723717165285_nat_a @ S2 @ ( image_a_nat_a @ F @ T ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T )
& ( inj_on_a_nat_a @ F @ U2 )
& ( S2
= ( image_a_nat_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_751_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_752_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_753_iso__tuple__UNIV__I,axiom,
! [X2: nat > a] : ( member_nat_a @ X2 @ top_top_set_nat_a ) ).
% iso_tuple_UNIV_I
thf(fact_754_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_755_iso__tuple__UNIV__I,axiom,
! [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_756_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_757_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_758_order__refl,axiom,
! [X2: set_nat_a] : ( ord_le871467723717165285_nat_a @ X2 @ X2 ) ).
% order_refl
thf(fact_759_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_760_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_761_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_762_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_763_dual__order_Orefl,axiom,
! [A2: set_nat_a] : ( ord_le871467723717165285_nat_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_764_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_765_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_766_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > $o,B2: set_o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_767_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > int,B2: set_int] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_a_int @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_768_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o,B2: set_o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_769_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > int,B2: set_int] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_770_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > $o,B2: set_o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_int_o @ F @ ( collect_int @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_771_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > int,B2: set_int] :
( ! [X3: int] :
( ( P @ X3 )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ ( collect_int @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_772_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B2: set_a] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_773_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B2: set_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_774_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > a,B2: set_a] :
( ! [X3: int] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_int_a @ F @ ( collect_int @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_775_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > nat,B2: set_nat] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_776_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_777_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_778_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_779_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_780_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_781_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_782_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_783_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_784_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_785_top__set__def,axiom,
( top_top_set_nat_a
= ( collect_nat_a @ top_top_nat_a_o ) ) ).
% top_set_def
thf(fact_786_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_787_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_788_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_789_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B3: set_int] :
( ord_less_eq_int_o
@ ^ [X: int] : ( member_int @ X @ A4 )
@ ^ [X: int] : ( member_int @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_790_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_791_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A4 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_792_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_793_order__antisym__conv,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_794_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_795_order__antisym__conv,axiom,
! [Y2: set_nat_a,X2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ Y2 @ X2 )
=> ( ( ord_le871467723717165285_nat_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_796_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_797_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_798_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_799_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_800_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_801_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_802_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_803_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_804_ord__le__eq__subst,axiom,
! [A2: set_a,B: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_805_ord__le__eq__subst,axiom,
! [A2: set_a,B: set_a,F: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_806_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_807_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_808_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_809_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_810_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_811_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_812_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_813_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_814_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B: set_a,C2: set_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_815_ord__eq__le__subst,axiom,
! [A2: int,F: set_a > int,B: set_a,C2: set_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_816_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_817_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_818_ord__eq__le__subst,axiom,
! [A2: set_a,F: int > set_a,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_819_ord__eq__le__subst,axiom,
! [A2: set_nat,F: int > set_nat,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_820_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_821_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_822_order__eq__refl,axiom,
! [X2: set_a,Y2: set_a] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_823_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_824_order__eq__refl,axiom,
! [X2: set_nat_a,Y2: set_nat_a] :
( ( X2 = Y2 )
=> ( ord_le871467723717165285_nat_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_825_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_826_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_827_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_828_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_829_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_830_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_831_order__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_832_order__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_833_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_834_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_835_order__subst2,axiom,
! [A2: int,B: int,F: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_836_order__subst2,axiom,
! [A2: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_837_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_838_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_839_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_840_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_841_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_842_order__subst1,axiom,
! [A2: set_a,F: int > set_a,B: int,C2: int] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_843_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_844_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_845_order__subst1,axiom,
! [A2: int,F: set_a > int,B: set_a,C2: set_a] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_846_order__subst1,axiom,
! [A2: int,F: set_nat > int,B: set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_847_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_848_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_849_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat_a,Z2: set_nat_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_nat_a,B4: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A5 @ B4 )
& ( ord_le871467723717165285_nat_a @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_850_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z2: int] : ( Y6 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_851_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_852_antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_853_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_854_antisym,axiom,
! [A2: set_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B )
=> ( ( ord_le871467723717165285_nat_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_855_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_856_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_857_dual__order_Otrans,axiom,
! [B: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_858_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_859_dual__order_Otrans,axiom,
! [B: set_nat_a,A2: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ C2 @ B )
=> ( ord_le871467723717165285_nat_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_860_dual__order_Otrans,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_861_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_862_dual__order_Oantisym,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_863_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_864_dual__order_Oantisym,axiom,
! [B: set_nat_a,A2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_865_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_866_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_867_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_868_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_869_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat_a,Z2: set_nat_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_nat_a,B4: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ B4 @ A5 )
& ( ord_le871467723717165285_nat_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_870_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z2: int] : ( Y6 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_871_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_872_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: nat,B5: nat] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_873_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: int,B5: int] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_874_order__trans,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_875_order__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_876_order__trans,axiom,
! [X2: set_nat_a,Y2: set_nat_a,Z: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X2 @ Y2 )
=> ( ( ord_le871467723717165285_nat_a @ Y2 @ Z )
=> ( ord_le871467723717165285_nat_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_877_order__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_878_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_879_order_Otrans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_880_order_Otrans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_881_order_Otrans,axiom,
! [A2: set_nat_a,B: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B )
=> ( ( ord_le871467723717165285_nat_a @ B @ C2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_882_order_Otrans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_883_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_884_order__antisym,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_885_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_886_order__antisym,axiom,
! [X2: set_nat_a,Y2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X2 @ Y2 )
=> ( ( ord_le871467723717165285_nat_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_887_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_888_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_889_ord__le__eq__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_890_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_891_ord__le__eq__trans,axiom,
! [A2: set_nat_a,B: set_nat_a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_892_ord__le__eq__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_893_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_894_ord__eq__le__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( A2 = B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_895_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_896_ord__eq__le__trans,axiom,
! [A2: set_nat_a,B: set_nat_a,C2: set_nat_a] :
( ( A2 = B )
=> ( ( ord_le871467723717165285_nat_a @ B @ C2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_897_ord__eq__le__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_898_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_899_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_900_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_901_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat_a,Z2: set_nat_a] : ( Y6 = Z2 ) )
= ( ^ [X: set_nat_a,Y3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X @ Y3 )
& ( ord_le871467723717165285_nat_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_902_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z2: int] : ( Y6 = Z2 ) )
= ( ^ [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
& ( ord_less_eq_int @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_903_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_904_le__cases3,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_905_le__cases3,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_906_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_907_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_908_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_909_order__less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_910_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_911_order__less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_912_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_913_order__less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_914_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_915_linorder__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_916_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_917_order__less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_918_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_919_order__less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_920_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_921_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_922_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_923_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_924_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_925_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_926_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_927_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_928_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_929_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_930_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_931_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_932_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_933_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_934_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_935_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_936_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_937_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_938_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_939_order__less__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_940_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_941_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_942_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_943_linorder__neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_944_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_945_order__less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_946_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_947_linorder__neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_948_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_949_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_950_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_951_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_952_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_953_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_954_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_955_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_956_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_957_order_Ostrict__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_958_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B5: nat] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_959_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B5: int] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_960_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P2: nat > $o] :
? [N4: nat] :
( ( P2 @ N4 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ~ ( P2 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_961_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_962_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_963_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_964_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_965_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_966_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_967_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_968_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_969_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_970_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_971_ord__less__eq__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_972_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_973_ord__eq__less__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_974_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_975_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_976_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_977_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_978_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_979_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_980_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_981_subset__eq__atLeast0__lessThan__card,axiom,
! [N2: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N2 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_982_Collect__restrict,axiom,
! [X6: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_983_Collect__restrict,axiom,
! [X6: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_984_Collect__restrict,axiom,
! [X6: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_985_Collect__restrict,axiom,
! [X6: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_986_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_987_prop__restrict,axiom,
! [X2: $o,Z3: set_o,X6: set_o,P: $o > $o] :
( ( member_o @ X2 @ Z3 )
=> ( ( ord_less_eq_set_o @ Z3
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_988_prop__restrict,axiom,
! [X2: int,Z3: set_int,X6: set_int,P: int > $o] :
( ( member_int @ X2 @ Z3 )
=> ( ( ord_less_eq_set_int @ Z3
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_989_prop__restrict,axiom,
! [X2: a,Z3: set_a,X6: set_a,P: a > $o] :
( ( member_a @ X2 @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_990_prop__restrict,axiom,
! [X2: nat > a,Z3: set_nat_a,X6: set_nat_a,P: ( nat > a ) > $o] :
( ( member_nat_a @ X2 @ Z3 )
=> ( ( ord_le871467723717165285_nat_a @ Z3
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_991_prop__restrict,axiom,
! [X2: nat,Z3: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_992_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_set_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_993_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_994_order__le__imp__less__or__eq,axiom,
! [X2: set_nat_a,Y2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X2 @ Y2 )
=> ( ( ord_less_set_nat_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_995_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_996_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_997_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_998_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_999_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1000_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > set_a,C2: set_a] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1001_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1002_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1003_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat_a,C2: set_nat_a] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le871467723717165285_nat_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1004_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > set_nat_a,C2: set_nat_a] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_le871467723717165285_nat_a @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1005_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1006_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1007_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1008_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1009_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1010_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1011_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1012_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1013_order__less__le__subst1,axiom,
! [A2: nat,F: set_a > nat,B: set_a,C2: set_a] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1014_order__less__le__subst1,axiom,
! [A2: int,F: set_a > int,B: set_a,C2: set_a] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1015_order__less__le__subst1,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C2: nat] :
( ( ord_less_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1016_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1017_order__less__le__subst1,axiom,
! [A2: set_a,F: int > set_a,B: int,C2: int] :
( ( ord_less_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1018_order__less__le__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B: int,C2: int] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1019_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1020_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1021_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1022_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1023_order__le__less__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1024_order__le__less__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1025_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1026_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1027_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1028_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1029_order__le__less__subst1,axiom,
! [A2: set_a,F: nat > set_a,B: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1030_order__le__less__subst1,axiom,
! [A2: set_a,F: int > set_a,B: int,C2: int] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1031_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1032_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1033_order__le__less__subst1,axiom,
! [A2: set_nat_a,F: nat > set_nat_a,B: nat,C2: nat] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1034_order__le__less__subst1,axiom,
! [A2: set_nat_a,F: int > set_nat_a,B: int,C2: int] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1035_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1036_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1037_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1038_order__le__less__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B: int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1039_order__less__le__trans,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1040_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1041_order__less__le__trans,axiom,
! [X2: set_nat_a,Y2: set_nat_a,Z: set_nat_a] :
( ( ord_less_set_nat_a @ X2 @ Y2 )
=> ( ( ord_le871467723717165285_nat_a @ Y2 @ Z )
=> ( ord_less_set_nat_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1042_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1043_order__less__le__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1044_order__le__less__trans,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1045_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1046_order__le__less__trans,axiom,
! [X2: set_nat_a,Y2: set_nat_a,Z: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X2 @ Y2 )
=> ( ( ord_less_set_nat_a @ Y2 @ Z )
=> ( ord_less_set_nat_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1047_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1048_order__le__less__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1049_order__neq__le__trans,axiom,
! [A2: set_a,B: set_a] :
( ( A2 != B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1050_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1051_order__neq__le__trans,axiom,
! [A2: set_nat_a,B: set_nat_a] :
( ( A2 != B )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B )
=> ( ord_less_set_nat_a @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1052_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1053_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1054_order__le__neq__trans,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1055_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1056_order__le__neq__trans,axiom,
! [A2: set_nat_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat_a @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1057_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1058_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1059_order__less__imp__le,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1060_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1061_order__less__imp__le,axiom,
! [X2: set_nat_a,Y2: set_nat_a] :
( ( ord_less_set_nat_a @ X2 @ Y2 )
=> ( ord_le871467723717165285_nat_a @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1062_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1063_order__less__imp__le,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1064_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_1065_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_1066_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_1067_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_1068_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1069_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1070_order__less__le,axiom,
( ord_less_set_nat_a
= ( ^ [X: set_nat_a,Y3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1071_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1072_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1073_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_set_a @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1074_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1075_order__le__less,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [X: set_nat_a,Y3: set_nat_a] :
( ( ord_less_set_nat_a @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1076_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1077_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1078_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1079_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1080_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_a,A2: set_nat_a] :
( ( ord_less_set_nat_a @ B @ A2 )
=> ( ord_le871467723717165285_nat_a @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1081_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1082_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1083_order_Ostrict__implies__order,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1084_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1085_order_Ostrict__implies__order,axiom,
! [A2: set_nat_a,B: set_nat_a] :
( ( ord_less_set_nat_a @ A2 @ B )
=> ( ord_le871467723717165285_nat_a @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1086_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1087_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1088_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1089_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1090_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1091_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1092_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1093_verit__la__generic,axiom,
! [A2: int,X2: int] :
( ( ord_less_eq_int @ A2 @ X2 )
| ( A2 = X2 )
| ( ord_less_eq_int @ X2 @ A2 ) ) ).
% verit_la_generic
thf(fact_1094_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1095_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1096_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1097_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1098_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
@ ^ [X: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1099_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1100_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1101_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_1102_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1103_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1104_finite__interval__int1,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A2 @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_1105_finite__interval__int4,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A2 @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_1106_finite__interval__int3,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A2 @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1107_finite__interval__int2,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A2 @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1108_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1109_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1110_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1111_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1112_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1113_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1114_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1115_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1116_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I4: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( P @ K )
& ( ord_less_nat @ K @ I4 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1117_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1118_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1119_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_1120_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1121_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1122_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1123_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1124_subset__eq__atLeast0__lessThan__finite,axiom,
! [N2: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1125_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1126_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_1127_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1128_unbounded__k__infinite,axiom,
! [K2: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K2 @ M5 )
=> ? [N6: nat] :
( ( ord_less_nat @ M5 @ N6 )
& ( member_nat @ N6 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1129_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1130_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1131_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1132_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1133_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1134_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1135_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1136_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1137_image__Suc__atLeastLessThan,axiom,
! [I4: nat,J3: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I4 @ J3 ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I4 ) @ ( suc @ J3 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1138_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1139_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1140_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1141_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1142_inj__Suc,axiom,
! [N2: set_nat] : ( inj_on_nat_nat @ suc @ N2 ) ).
% inj_Suc
thf(fact_1143_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1144_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1145_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1146_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1147_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1148_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1149_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1150_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1151_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1152_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1153_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1154_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1155_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1156_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1157_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1158_strict__inc__induct,axiom,
! [I4: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_1159_less__Suc__induct,axiom,
! [I4: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K3 )
=> ( ( P @ I2 @ J )
=> ( ( P @ J @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_1160_less__trans__Suc,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( suc @ I4 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1161_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1162_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1163_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1164_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1165_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1166_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1167_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1168_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1169_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1170_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1171_Suc__lessE,axiom,
! [I4: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K2 )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( K2
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_1172_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1173_Nat_OlessE,axiom,
! [I4: nat,K2: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( ( K2
!= ( suc @ I4 ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( K2
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_1174_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1175_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1176_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1177_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1178_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1179_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1180_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1181_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1182_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1183_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1184_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1185_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_1186_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1187_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1188_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1189_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1190_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1191_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1192_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1193_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1194_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1195_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1196_inc__induct,axiom,
! [I4: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( P @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_1197_dec__induct,axiom,
! [I4: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( P @ I4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_1198_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1199_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1200_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1201_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1202_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1203_card__less,axiom,
! [M2: set_nat,I4: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I4 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1204_card__less__Suc,axiom,
! [M2: set_nat,I4: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M2 )
& ( ord_less_nat @ K @ I4 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I4 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1205_card__less__Suc2,axiom,
! [M2: set_nat,I4: nat] :
( ~ ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M2 )
& ( ord_less_nat @ K @ I4 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I4 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1206_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1207_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1208_negD,axiom,
! [X2: int] :
( ( ord_less_int @ X2 @ zero_zero_int )
=> ? [N3: nat] :
( X2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1209_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1210_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1211_infinite__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ? [R2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
& ! [N6: nat] : ( member_nat @ ( R2 @ N6 ) @ S2 ) ) ) ).
% infinite_enumerate
thf(fact_1212_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1213_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1214_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1215_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1216_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_1217_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1218_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1219_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1220_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1221_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1222_less__imp__add__positive,axiom,
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I4 @ K3 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_1223_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1224_less__add__Suc1,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1225_less__add__Suc2,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_1226_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
? [K: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M3 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1227_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1228_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1229_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1230_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1231_nat__arith_Osuc1,axiom,
! [A: nat,K2: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K2 @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K2 @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1232_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
? [K: nat] :
( N4
= ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_1233_trans__le__add2,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_1234_trans__le__add1,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_1235_add__le__mono1,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1236_add__le__mono,axiom,
! [I4: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1237_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1238_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1239_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1240_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1241_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1242_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1243_add__lessD1,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K2 )
=> ( ord_less_nat @ I4 @ K2 ) ) ).
% add_lessD1
thf(fact_1244_add__less__mono,axiom,
! [I4: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1245_not__add__less1,axiom,
! [I4: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ I4 ) ).
% not_add_less1
thf(fact_1246_not__add__less2,axiom,
! [J3: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_1247_add__less__mono1,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1248_trans__less__add1,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_1249_trans__less__add2,axiom,
! [I4: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_1250_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1251_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1252_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1253_subset__card__intvl__is__intvl,axiom,
! [A: set_nat,K2: nat] :
( ( ord_less_eq_set_nat @ A @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A ) ) ) )
=> ( A
= ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1254_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1255_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1256_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1257_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1258_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1259_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1260_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1261_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( finite_card_nat_a
@ ( image_nat_a_nat_a @ f
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I: nat] : b ) ) )
= ( finite_card_nat_a
@ ( piE_nat_a @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n )
@ ^ [I: nat] : b ) ) ) ).
%------------------------------------------------------------------------------