TPTP Problem File: SLH0908^1.p
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%------------------------------------------------------------------------------
% 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 : Khovanskii_Theorem/0004_FiniteProduct/prob_00202_006475__13334152_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1511 ( 420 unt; 227 typ; 0 def)
% Number of atoms : 4487 (1247 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 15606 ( 424 ~; 44 |; 305 &;12354 @)
% ( 0 <=>;2479 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 2177 (2177 >; 0 *; 0 +; 0 <<)
% Number of symbols : 209 ( 206 usr; 15 con; 0-5 aty)
% Number of variables : 4515 ( 519 ^;3924 !; 72 ?;4515 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:13:33.861
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__Set__Oset_I_062_Itf__b_M_062_Itf__b_Mtf__a_J_J_J,type,
set_b_b_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__b_Mtf__a_J_Mtf__b_J_J,type,
set_b_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__b_Mtf__a_J_Mtf__a_J_J,type,
set_b_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_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mt__Nat__Onat_J_J,type,
set_b_nat: $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_It__Nat__Onat_Mtf__b_J_J,type,
set_nat_b: $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_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $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__b_Mtf__b_J_J,type,
set_b_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
set_b_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
set_a_b: $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__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (206)
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001_062_Itf__b_Mtf__a_J,type,
commut3325098377247325640fy_b_a: set_b_a > ( b > a ) > ( b > a ) > b > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001t__Nat__Onat,type,
commut810702690453168372fy_nat: set_nat > nat > nat > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001tf__a,type,
commutative_M_ify_a: set_a > a > a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001tf__b,type,
commutative_M_ify_b: set_b > b > b > b ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001t__Nat__Onat,type,
commut1028764413824576968at_nat: set_nat > ( nat > nat > nat ) > nat > ( nat > nat ) > set_nat > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001tf__a,type,
commut1549887680474846982_nat_a: set_nat > ( nat > nat > nat ) > nat > ( a > nat ) > set_a > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001tf__b,type,
commut1549887680474846983_nat_b: set_nat > ( nat > nat > nat ) > nat > ( b > nat ) > set_b > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001_062_Itf__b_Mtf__a_J,type,
commut3556868347779488380_a_b_a: set_a > ( a > a > a ) > a > ( ( b > a ) > a ) > set_b_a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001t__Nat__Onat,type,
commut6741328216151336360_a_nat: set_a > ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001tf__a,type,
commut5005951359559292710mp_a_a: set_a > ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001tf__b,type,
commut5005951359559292711mp_a_b: set_a > ( a > a > a ) > a > ( b > a ) > set_b > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__b_001_062_Itf__b_Mtf__a_J,type,
commut187354080821335165_b_b_a: set_b > ( b > b > b ) > b > ( ( b > a ) > b ) > set_b_a > b ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__b_001t__Nat__Onat,type,
commut7976772545107730857_b_nat: set_b > ( b > b > b ) > b > ( nat > b ) > set_nat > b ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__b_001tf__a,type,
commut2218495777586616677mp_b_a: set_b > ( b > b > b ) > b > ( a > b ) > set_a > b ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__b_001tf__b,type,
commut2218495777586616678mp_b_b: set_b > ( b > b > b ) > b > ( b > b ) > set_b > b ).
thf(sy_c_Finite__Set_Ocard_001_062_Itf__b_Mtf__a_J,type,
finite_card_b_a: set_b_a > 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_Ocard_001tf__b,type,
finite_card_b: set_b > nat ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on_001tf__b_001tf__a,type,
finite9173194153363770127on_b_a: set_b > ( b > a > a ) > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__b_Mtf__a_J,type,
finite_finite_b_a: set_b_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__b_J,type,
finite_finite_set_b: set_set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Finite__Set_Ofold_001_062_Itf__b_Mtf__a_J_001tf__a,type,
finite_fold_b_a_a: ( ( b > a ) > a > a ) > a > set_b_a > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001tf__a,type,
finite_fold_nat_a: ( nat > a > a ) > a > set_nat > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001tf__b,type,
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thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Nat__Onat,type,
finite_fold_a_nat: ( a > nat > nat ) > nat > set_a > nat ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001tf__a,type,
finite_fold_a_a: ( a > a > a ) > a > set_a > a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001tf__b,type,
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thf(sy_c_Finite__Set_Ofold_001tf__b_001_062_Itf__b_Mtf__a_J,type,
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thf(sy_c_Finite__Set_Ofold_001tf__b_001tf__a,type,
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thf(sy_c_Finite__Set_Ofold__graph_001t__Nat__Onat_001_062_Itf__b_Mtf__a_J,type,
finite8571984644340236092at_b_a: ( nat > ( b > a ) > b > a ) > ( b > a ) > set_nat > ( b > a ) > $o ).
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thf(sy_c_Finite__Set_Ofold__graph_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Finite__Set_Ofold__graph_001t__Nat__Onat_001tf__b,type,
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thf(sy_c_Finite__Set_Ofold__graph_001tf__a_001_062_Itf__b_Mtf__a_J,type,
finite1332093216797763036_a_b_a: ( a > ( b > a ) > b > a ) > ( b > a ) > set_a > ( b > a ) > $o ).
thf(sy_c_Finite__Set_Ofold__graph_001tf__a_001t__Nat__Onat,type,
finite5110433740378173704_a_nat: ( a > nat > nat ) > nat > set_a > nat > $o ).
thf(sy_c_Finite__Set_Ofold__graph_001tf__a_001tf__a,type,
finite7874008084079289286ph_a_a: ( a > a > a ) > a > set_a > a > $o ).
thf(sy_c_Finite__Set_Ofold__graph_001tf__a_001tf__b,type,
finite7874008084079289287ph_a_b: ( a > b > b ) > b > set_a > b > $o ).
thf(sy_c_Finite__Set_Ofold__graph_001tf__b_001_062_Itf__b_Mtf__a_J,type,
finite7185950986694385629_b_b_a: ( b > ( b > a ) > b > a ) > ( b > a ) > set_b > ( b > a ) > $o ).
thf(sy_c_Finite__Set_Ofold__graph_001tf__b_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofold__graph_001tf__b_001tf__a,type,
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thf(sy_c_Finite__Set_Ofold__graph_001tf__b_001tf__b,type,
finite5086552502106613254ph_b_b: ( b > b > b ) > b > set_b > b > $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_FuncSet_OPi_001_062_Itf__b_Mtf__a_J_001tf__a,type,
pi_b_a_a: set_b_a > ( ( b > a ) > set_a ) > set_b_a_a ).
thf(sy_c_FuncSet_OPi_001_062_Itf__b_Mtf__a_J_001tf__b,type,
pi_b_a_b: set_b_a > ( ( b > a ) > set_b ) > set_b_a_b ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
pi_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__b,type,
pi_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).
thf(sy_c_FuncSet_OPi_001tf__a_001t__Nat__Onat,type,
pi_a_nat: set_a > ( a > set_nat ) > set_a_nat ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
pi_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__b,type,
pi_a_b: set_a > ( a > set_b ) > set_a_b ).
thf(sy_c_FuncSet_OPi_001tf__b_001_062_Itf__b_Mtf__a_J,type,
pi_b_b_a: set_b > ( b > set_b_a ) > set_b_b_a ).
thf(sy_c_FuncSet_OPi_001tf__b_001t__Nat__Onat,type,
pi_b_nat: set_b > ( b > set_nat ) > set_b_nat ).
thf(sy_c_FuncSet_OPi_001tf__b_001tf__a,type,
pi_b_a: set_b > ( b > set_a ) > set_b_a ).
thf(sy_c_FuncSet_OPi_001tf__b_001tf__b,type,
pi_b_b: set_b > ( b > set_b ) > set_b_b ).
thf(sy_c_FuncSet_Oextensional_001_062_Itf__b_Mtf__a_J_001tf__a,type,
extensional_b_a_a: set_b_a > set_b_a_a ).
thf(sy_c_FuncSet_Oextensional_001t__Nat__Onat_001tf__a,type,
extensional_nat_a: set_nat > set_nat_a ).
thf(sy_c_FuncSet_Oextensional_001tf__a_001tf__a,type,
extensional_a_a: set_a > set_a_a ).
thf(sy_c_FuncSet_Oextensional_001tf__b_001tf__a,type,
extensional_b_a: set_b > set_b_a ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001tf__a,type,
restrict_a_a: ( a > a ) > set_a > a > a ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001_062_Itf__b_Mtf__a_J,type,
group_4188790030012530981id_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001t__Nat__Onat,type,
group_6791354081887936081id_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__b,type,
group_4866109990395492030noid_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Ogroup_001_062_Itf__b_Mtf__a_J,type,
group_group_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001t__Nat__Onat,type,
group_group_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__b,type,
group_group_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Omonoid_001_062_Itf__b_Mtf__a_J,type,
group_monoid_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_001t__Nat__Onat,type,
group_monoid_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__a,type,
group_monoid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__b,type,
group_monoid_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001_062_Itf__b_Mtf__a_J,type,
group_Units_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > set_b_a ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001t__Nat__Onat,type,
group_Units_nat: set_nat > ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__b,type,
group_Units_b: set_b > ( b > b > b ) > b > set_b ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_Itf__b_Mtf__a_J,type,
group_inverse_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > ( b > a ) > b > a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__b,type,
group_inverse_b: set_b > ( b > b > b ) > b > b > b ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001_062_Itf__b_Mtf__a_J,type,
group_invertible_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > ( b > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__b,type,
group_invertible_b: set_b > ( b > b > b ) > b > b > $o ).
thf(sy_c_Group__Theory_Osubgroup_001_062_Itf__b_Mtf__a_J,type,
group_subgroup_b_a: set_b_a > set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Nat__Onat,type,
group_subgroup_nat: set_nat > set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__b,type,
group_subgroup_b: set_b > set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
minus_minus_b_a_o: ( ( b > a ) > $o ) > ( ( b > a ) > $o ) > ( b > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
minus_minus_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
minus_minus_set_b_a: set_b_a > set_b_a > set_b_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_Oundefined_001_062_Itf__b_Mtf__a_J,type,
undefined_b_a: b > a ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_HOL_Oundefined_001tf__b,type,
undefined_b: b ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_If_001t__Set__Oset_Itf__b_J,type,
if_set_b: $o > set_b > set_b > set_b ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
inf_inf_b_a_o: ( ( b > a ) > $o ) > ( ( b > a ) > $o ) > ( b > a ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_M_Eo_J,type,
inf_inf_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
inf_inf_set_b_a: set_b_a > set_b_a > set_b_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
sup_sup_b_a_o: ( ( b > a ) > $o ) > ( ( b > a ) > $o ) > ( b > a ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_Eo_J,type,
sup_sup_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
sup_sup_set_b_a: set_b_a > set_b_a > set_b_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_Itf__b_Mtf__a_J_001t__Nat__Onat,type,
lattic7032157021346806415_a_nat: ( ( b > a ) > nat ) > set_b_a > b > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__b_001t__Nat__Onat,type,
lattic7575731748627795062_b_nat: ( b > nat ) > set_b > b ).
thf(sy_c_Nat_Oold_Onat_Orec__nat_001tf__a,type,
rec_nat_a: a > ( nat > a > a ) > nat > a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
bot_bot_b_a_o: ( b > a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
bot_bot_set_b_a: set_b_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
ord_less_eq_b_a_o: ( ( b > a ) > $o ) > ( ( b > a ) > $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_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $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_Itf__b_Mtf__a_J_Mtf__a_J_J,type,
ord_le4402886750609172241_b_a_a: set_b_a_a > set_b_a_a > $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_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_062_Itf__b_Mtf__a_J_J,type,
ord_less_eq_set_b_a: set_b_a > set_b_a > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Set_OCollect_001_062_Itf__b_Mtf__a_J,type,
collect_b_a: ( ( b > a ) > $o ) > set_b_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Ofilter_001_062_Itf__b_Mtf__a_J,type,
filter_b_a: ( ( b > a ) > $o ) > set_b_a > set_b_a ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001tf__a,type,
filter_a: ( a > $o ) > set_a > set_a ).
thf(sy_c_Set_Ofilter_001tf__b,type,
filter_b: ( b > $o ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001_062_Itf__b_Mtf__a_J,type,
insert_b_a: ( b > a ) > set_b_a > set_b_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__singleton_001_062_Itf__b_Mtf__a_J,type,
is_singleton_b_a: set_b_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Oremove_001_062_Itf__b_Mtf__a_J,type,
remove_b_a: ( b > a ) > set_b_a > set_b_a ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Oremove_001tf__b,type,
remove_b: b > set_b > set_b ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_member_001_062_I_062_Itf__b_Mtf__a_J_Mtf__a_J,type,
member_b_a_a: ( ( b > a ) > a ) > set_b_a_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__b_Mtf__a_J_Mtf__b_J,type,
member_b_a_b: ( ( b > a ) > b ) > set_b_a_b > $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_It__Nat__Onat_Mtf__b_J,type,
member_nat_b: ( nat > b ) > set_nat_b > $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_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__b_M_062_Itf__b_Mtf__a_J_J,type,
member_b_b_a: ( b > b > a ) > set_b_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__Nat__Onat_J,type,
member_b_nat: ( b > nat ) > set_b_nat > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > 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_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_A,type,
a2: set_b ).
thf(sy_v_M,type,
m: set_a ).
thf(sy_v_composition,type,
composition: a > a > a ).
thf(sy_v_f,type,
f: b > a ).
thf(sy_v_i,type,
i: b ).
thf(sy_v_unit,type,
unit: a ).
% Relevant facts (1276)
thf(fact_0_fin__A,axiom,
finite_finite_b @ a2 ).
% fin_A
thf(fact_1_i__in__A,axiom,
member_b @ i @ a2 ).
% i_in_A
thf(fact_2_f__Pi,axiom,
( member_b_a @ f
@ ( pi_b_a @ a2
@ ^ [Uu: b] : m ) ) ).
% f_Pi
thf(fact_3_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( composition @ X @ Y )
= ( composition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_4_left__commute,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ m )
=> ( ( member_a @ B @ m )
=> ( ( member_a @ C @ m )
=> ( ( composition @ B @ ( composition @ A @ C ) )
= ( composition @ A @ ( composition @ B @ C ) ) ) ) ) ) ).
% left_commute
thf(fact_5_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( composition @ U @ V )
= unit )
=> ( ( ( composition @ V2 @ U )
= unit )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( member_a @ V @ m )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_6_fincomp__unit__eqI,axiom,
! [A2: set_b_a,F: ( b > a ) > a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_7_fincomp__unit__eqI,axiom,
! [A2: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_8_fincomp__unit__eqI,axiom,
! [A2: set_nat,F: nat > a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_9_fincomp__unit__eqI,axiom,
! [A2: set_b,F: b > a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_10__092_060open_062fincomp_A_I_092_060lambda_062i_O_A_092_060one_062_J_A_IA_A_N_A_123i_125_J_A_061_A_092_060one_062_092_060close_062,axiom,
( ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [I: b] : unit
@ ( minus_minus_set_b @ a2 @ ( insert_b @ i @ bot_bot_set_b ) ) )
= unit ) ).
% \<open>fincomp (\<lambda>i. \<one>) (A - {i}) = \<one>\<close>
thf(fact_11_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ m @ composition @ unit ).
% commutative_monoid_axioms
thf(fact_12_fincomp__closed,axiom,
! [F: nat > a,F2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : m ) )
=> ( member_a @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_13_fincomp__closed,axiom,
! [F: a > a,F2: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : m ) )
=> ( member_a @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_14_fincomp__closed,axiom,
! [F: ( b > a ) > a,F2: set_b_a] :
( ( member_b_a_a @ F
@ ( pi_b_a_a @ F2
@ ^ [Uu: b > a] : m ) )
=> ( member_a @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_15_fincomp__closed,axiom,
! [F: b > a,F2: set_b] :
( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : m ) )
=> ( member_a @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_16_fincomp__comp,axiom,
! [F: b > a,A2: set_b,G: b > a] :
( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : m ) )
=> ( ( member_b_a @ G
@ ( pi_b_a @ A2
@ ^ [Uu: b] : m ) )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [X3: b] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 ) @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_17_fincomp__comp,axiom,
! [F: nat > a,A2: set_nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit
@ ^ [X3: nat] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_18_fincomp__comp,axiom,
! [F: a > a,A2: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit
@ ^ [X3: a] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_19_fincomp__comp,axiom,
! [F: ( b > a ) > a,A2: set_b_a,G: ( b > a ) > a] :
( ( member_b_a_a @ F
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : m ) )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : m ) )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit
@ ^ [X3: b > a] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ A2 ) @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_20_fincomp__cong_H,axiom,
! [A2: set_b,B2: set_b,G: b > a,F: b > a] :
( ( A2 = B2 )
=> ( ( member_b_a @ G
@ ( pi_b_a @ B2
@ ^ [Uu: b] : m ) )
=> ( ! [I2: b] :
( ( member_b @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_21_fincomp__cong_H,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( A2 = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : m ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_22_fincomp__cong_H,axiom,
! [A2: set_a,B2: set_a,G: a > a,F: a > a] :
( ( A2 = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : m ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_23_fincomp__cong_H,axiom,
! [A2: set_b_a,B2: set_b_a,G: ( b > a ) > a,F: ( b > a ) > a] :
( ( A2 = B2 )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ B2
@ ^ [Uu: b > a] : m ) )
=> ( ! [I2: b > a] :
( ( member_b_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ A2 )
= ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_24_associative,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ m )
=> ( ( member_a @ B @ m )
=> ( ( member_a @ C @ m )
=> ( ( composition @ ( composition @ A @ B ) @ C )
= ( composition @ A @ ( composition @ B @ C ) ) ) ) ) ) ).
% associative
thf(fact_25_composition__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ m )
=> ( ( member_a @ B @ m )
=> ( member_a @ ( composition @ A @ B ) @ m ) ) ) ).
% composition_closed
thf(fact_26_unit__closed,axiom,
member_a @ unit @ m ).
% unit_closed
thf(fact_27_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ m )
=> ( ( composition @ unit @ A )
= A ) ) ).
% left_unit
thf(fact_28_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ m )
=> ( ( composition @ A @ unit )
= A ) ) ).
% right_unit
thf(fact_29__092_060open_062_092_060lbrakk_062finite_A_IA_A_N_A_123i_125_J_059_Ai_A_092_060notin_062_AA_A_N_A_123i_125_059_A_I_092_060lambda_062j_O_Aif_Ai_A_061_Aj_Athen_Af_Aj_Aelse_A_092_060one_062_J_A_092_060in_062_AA_A_N_A_123i_125_A_092_060rightarrow_062_AM_059_A_Iif_Ai_A_061_Ai_Athen_Af_Ai_Aelse_A_092_060one_062_J_A_092_060in_062_AM_092_060rbrakk_062_A_092_060Longrightarrow_062_Afincomp_A_I_092_060lambda_062j_O_Aif_Ai_A_061_Aj_Athen_Af_Aj_Aelse_A_092_060one_062_J_A_Iinsert_Ai_A_IA_A_N_A_123i_125_J_J_A_061_A_Iif_Ai_A_061_Ai_Athen_Af_Ai_Aelse_A_092_060one_062_J_A_092_060cdot_062_Afincomp_A_I_092_060lambda_062j_O_Aif_Ai_A_061_Aj_Athen_Af_Aj_Aelse_A_092_060one_062_J_A_IA_A_N_A_123i_125_J_092_060close_062,axiom,
( ( finite_finite_b @ ( minus_minus_set_b @ a2 @ ( insert_b @ i @ bot_bot_set_b ) ) )
=> ( ~ ( member_b @ i @ ( minus_minus_set_b @ a2 @ ( insert_b @ i @ bot_bot_set_b ) ) )
=> ( ( member_b_a
@ ^ [J: b] : ( if_a @ ( i = J ) @ ( f @ J ) @ unit )
@ ( pi_b_a @ ( minus_minus_set_b @ a2 @ ( insert_b @ i @ bot_bot_set_b ) )
@ ^ [Uu: b] : m ) )
=> ( ( member_a @ ( f @ i ) @ m )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [J: b] : ( if_a @ ( i = J ) @ ( f @ J ) @ unit )
@ ( insert_b @ i @ ( minus_minus_set_b @ a2 @ ( insert_b @ i @ bot_bot_set_b ) ) ) )
= ( composition @ ( if_a @ $true @ ( f @ i ) @ unit )
@ ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [J: b] : ( if_a @ ( i = J ) @ ( f @ J ) @ unit )
@ ( minus_minus_set_b @ a2 @ ( insert_b @ i @ bot_bot_set_b ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>finite (A - {i}); i \<notin> A - {i}; (\<lambda>j. if i = j then f j else \<one>) \<in> A - {i} \<rightarrow> M; (if i = i then f i else \<one>) \<in> M\<rbrakk> \<Longrightarrow> fincomp (\<lambda>j. if i = j then f j else \<one>) (insert i (A - {i})) = (if i = i then f i else \<one>) \<cdot> fincomp (\<lambda>j. if i = j then f j else \<one>) (A - {i})\<close>
thf(fact_30_fincomp__unit,axiom,
! [A2: set_b] :
( ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [I: b] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_31_fincomp__unit,axiom,
! [A2: set_nat] :
( ( commut6741328216151336360_a_nat @ m @ composition @ unit
@ ^ [I: nat] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_32_fincomp__unit,axiom,
! [A2: set_a] :
( ( commut5005951359559292710mp_a_a @ m @ composition @ unit
@ ^ [I: a] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_33_fincomp__unit,axiom,
! [A2: set_b_a] :
( ( commut3556868347779488380_a_b_a @ m @ composition @ unit
@ ^ [I: b > a] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_34_M__ify__def,axiom,
! [X: a] :
( ( ( member_a @ X @ m )
=> ( ( commutative_M_ify_a @ m @ unit @ X )
= X ) )
& ( ~ ( member_a @ X @ m )
=> ( ( commutative_M_ify_a @ m @ unit @ X )
= unit ) ) ) ).
% M_ify_def
thf(fact_35_commutative__monoid_Ofincomp_Ocong,axiom,
commut5005951359559292711mp_a_b = commut5005951359559292711mp_a_b ).
% commutative_monoid.fincomp.cong
thf(fact_36_commutative__monoid_Ofincomp_Ocong,axiom,
commut6741328216151336360_a_nat = commut6741328216151336360_a_nat ).
% commutative_monoid.fincomp.cong
thf(fact_37_commutative__monoid_Ofincomp_Ocong,axiom,
commut5005951359559292710mp_a_a = commut5005951359559292710mp_a_a ).
% commutative_monoid.fincomp.cong
thf(fact_38_commutative__monoid_Ofincomp_Ocong,axiom,
commut3556868347779488380_a_b_a = commut3556868347779488380_a_b_a ).
% commutative_monoid.fincomp.cong
thf(fact_39_fincomp__empty,axiom,
! [F: b > a] :
( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ bot_bot_set_b )
= unit ) ).
% fincomp_empty
thf(fact_40_fincomp__empty,axiom,
! [F: nat > a] :
( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ bot_bot_set_nat )
= unit ) ).
% fincomp_empty
thf(fact_41_fincomp__empty,axiom,
! [F: a > a] :
( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ bot_bot_set_a )
= unit ) ).
% fincomp_empty
thf(fact_42_fincomp__empty,axiom,
! [F: ( b > a ) > a] :
( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ bot_bot_set_b_a )
= unit ) ).
% fincomp_empty
thf(fact_43_fincomp__infinite,axiom,
! [A2: set_b,F: b > a] :
( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_44_fincomp__infinite,axiom,
! [A2: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_45_fincomp__infinite,axiom,
! [A2: set_a,F: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_46_fincomp__infinite,axiom,
! [A2: set_b_a,F: ( b > a ) > a] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_47_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( composition @ U @ V2 )
= unit )
=> ( ( ( composition @ V2 @ U )
= unit )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_48_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ! [V3: a] :
( ( ( ( composition @ U @ V3 )
= unit )
& ( ( composition @ V3 @ U )
= unit ) )
=> ~ ( member_a @ V3 @ m ) )
=> ~ ( member_a @ U @ m ) ) ) ).
% invertibleE
thf(fact_49_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ m )
=> ( ( group_invertible_a @ m @ composition @ unit @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ m )
& ( ( composition @ U @ X3 )
= unit )
& ( ( composition @ X3 @ U )
= unit ) ) ) ) ) ).
% invertible_def
thf(fact_50_unit__invertible,axiom,
group_invertible_a @ m @ composition @ unit @ unit ).
% unit_invertible
thf(fact_51_monoid__axioms,axiom,
group_monoid_a @ m @ composition @ unit ).
% monoid_axioms
thf(fact_52_fincomp__insert,axiom,
! [F2: set_b,A: b,F: b > a] :
( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ ( insert_b @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_53_fincomp__insert,axiom,
! [F2: set_nat,A: nat,F: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( insert_nat @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_54_fincomp__insert,axiom,
! [F2: set_a,A: a,F: a > a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ ( insert_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_55_fincomp__insert,axiom,
! [F2: set_b_a,A: b > a,F: ( b > a ) > a] :
( ( finite_finite_b_a @ F2 )
=> ( ~ ( member_b_a @ A @ F2 )
=> ( ( member_b_a_a @ F
@ ( pi_b_a_a @ F2
@ ^ [Uu: b > a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ ( insert_b_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_56_inverse__unit,axiom,
( ( group_inverse_a @ m @ composition @ unit @ unit )
= unit ) ).
% inverse_unit
thf(fact_57_fincomp__const,axiom,
! [A: a,A2: set_b] :
( ( member_a @ A @ m )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [X3: b] : A
@ A2 )
= ( rec_nat_a @ unit
@ ^ [U2: nat] : ( composition @ A )
@ ( finite_card_b @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_58_fincomp__const,axiom,
! [A: a,A2: set_nat] :
( ( member_a @ A @ m )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit
@ ^ [X3: nat] : A
@ A2 )
= ( rec_nat_a @ unit
@ ^ [U2: nat] : ( composition @ A )
@ ( finite_card_nat @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_59_fincomp__const,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ m )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit
@ ^ [X3: a] : A
@ A2 )
= ( rec_nat_a @ unit
@ ^ [U2: nat] : ( composition @ A )
@ ( finite_card_a @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_60_fincomp__const,axiom,
! [A: a,A2: set_b_a] :
( ( member_a @ A @ m )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit
@ ^ [X3: b > a] : A
@ A2 )
= ( rec_nat_a @ unit
@ ^ [U2: nat] : ( composition @ A )
@ ( finite_card_b_a @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_61_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( group_invertible_a @ m @ composition @ unit @ Y )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( group_invertible_a @ m @ composition @ unit @ ( composition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_62_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( composition @ U @ V2 )
= unit )
=> ( ( ( composition @ V2 @ U )
= unit )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( group_invertible_a @ m @ composition @ unit @ U ) ) ) ) ) ).
% invertibleI
thf(fact_63_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( composition @ U @ ( composition @ ( group_inverse_a @ m @ composition @ unit @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_64_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( composition @ ( group_inverse_a @ m @ composition @ unit @ U ) @ ( composition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_65_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( group_invertible_a @ m @ composition @ unit @ Y )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ ( composition @ X @ Y ) )
= ( composition @ ( group_inverse_a @ m @ composition @ unit @ Y ) @ ( group_inverse_a @ m @ composition @ unit @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_66_invertible__right__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( member_a @ Z @ m )
=> ( ( ( composition @ Y @ X )
= ( composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_67_invertible__left__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( member_a @ Z @ m )
=> ( ( ( composition @ X @ Y )
= ( composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_68_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( composition @ U @ ( group_inverse_a @ m @ composition @ unit @ U ) )
= unit ) ) ) ).
% invertible_right_inverse
thf(fact_69_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( composition @ ( group_inverse_a @ m @ composition @ unit @ U ) @ U )
= unit ) ) ) ).
% invertible_left_inverse
thf(fact_70_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( group_invertible_a @ m @ composition @ unit @ ( group_inverse_a @ m @ composition @ unit @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_71_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ ( group_inverse_a @ m @ composition @ unit @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_72_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( member_a @ ( group_inverse_a @ m @ composition @ unit @ U ) @ m ) ) ) ).
% invertible_inverse_closed
thf(fact_73_commutative__monoid_OM__ify__def,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( ( member_b @ X @ M )
=> ( ( commutative_M_ify_b @ M @ Unit @ X )
= X ) )
& ( ~ ( member_b @ X @ M )
=> ( ( commutative_M_ify_b @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_74_commutative__monoid_OM__ify__def,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a] :
( ( group_4188790030012530981id_b_a @ M @ Composition @ Unit )
=> ( ( ( member_b_a @ X @ M )
=> ( ( commut3325098377247325640fy_b_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_b_a @ X @ M )
=> ( ( commut3325098377247325640fy_b_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_75_commutative__monoid_OM__ify__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( ( member_nat @ X @ M )
=> ( ( commut810702690453168372fy_nat @ M @ Unit @ X )
= X ) )
& ( ~ ( member_nat @ X @ M )
=> ( ( commut810702690453168372fy_nat @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_76_commutative__monoid_OM__ify__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( member_a @ X @ M )
=> ( ( commutative_M_ify_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_a @ X @ M )
=> ( ( commutative_M_ify_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_77_commutative__monoid_Oleft__commute,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b,B: b,C: b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( member_b @ B @ M )
=> ( ( member_b @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_78_commutative__monoid_Oleft__commute,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a,B: b > a,C: b > a] :
( ( group_4188790030012530981id_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( member_b_a @ B @ M )
=> ( ( member_b_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_79_commutative__monoid_Oleft__commute,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat,C: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( member_nat @ B @ M )
=> ( ( member_nat @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_80_commutative__monoid_Oleft__commute,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B: a,C: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( ( member_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_81_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A: b > a,P: ( b > a ) > $o] :
( ( member_b_a @ A @ ( collect_b_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A2: set_b_a] :
( ( collect_b_a
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_90_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_91_commutative__monoid_Ofincomp__const,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit
@ ^ [X3: b] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_b @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_92_commutative__monoid_Ofincomp__const,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit
@ ^ [X3: nat] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_93_commutative__monoid_Ofincomp__const,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit
@ ^ [X3: a] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_a @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_94_commutative__monoid_Ofincomp__const,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_b_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit
@ ^ [X3: b > a] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_b_a @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_95_commutative__monoid_OM__ify_Ocong,axiom,
commutative_M_ify_a = commutative_M_ify_a ).
% commutative_monoid.M_ify.cong
thf(fact_96_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_a,A: a,F: a > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_b @ F
@ ( pi_a_b @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut2218495777586616677mp_b_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut2218495777586616677mp_b_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_97_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_a,A: a,F: a > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_nat @ F
@ ( pi_a_nat @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut1549887680474846982_nat_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1549887680474846982_nat_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_98_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_b,A: b,F: b > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_b @ F
@ ( pi_b_b @ F2
@ ^ [Uu: b] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut2218495777586616678mp_b_b @ M @ Composition @ Unit @ F @ ( insert_b @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut2218495777586616678mp_b_b @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_99_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_b,A: b,F: b > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_nat @ F
@ ( pi_b_nat @ F2
@ ^ [Uu: b] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut1549887680474846983_nat_b @ M @ Composition @ Unit @ F @ ( insert_b @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1549887680474846983_nat_b @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_100_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_nat,A: nat,F: nat > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_b @ F
@ ( pi_nat_b @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut7976772545107730857_b_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut7976772545107730857_b_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_101_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_nat,A: nat,F: nat > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_nat @ F
@ ( pi_nat_nat @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut1028764413824576968at_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1028764413824576968at_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_102_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F2: set_b,A: b,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : M ) )
=> ( ( member_a @ ( F @ A ) @ M )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ ( insert_b @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_103_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F2: set_nat,A: nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_a @ ( F @ A ) @ M )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_104_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F2: set_a,A: a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_a @ ( F @ A ) @ M )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_105_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_b_a,A: b > a,F: ( b > a ) > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_b_a @ F2 )
=> ( ~ ( member_b_a @ A @ F2 )
=> ( ( member_b_a_b @ F
@ ( pi_b_a_b @ F2
@ ^ [Uu: b > a] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut187354080821335165_b_b_a @ M @ Composition @ Unit @ F @ ( insert_b_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut187354080821335165_b_b_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_106_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_107_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_108_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_109_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a,F: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_b_a @ A2 )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_110_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ bot_bot_set_b )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_111_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ bot_bot_set_nat )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_112_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ bot_bot_set_a )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_113_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ bot_bot_set_b_a )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_114_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,F2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : M ) )
=> ( member_a @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_115_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,F2: set_nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M ) )
=> ( member_a @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_116_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a,F2: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : M ) )
=> ( member_a @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_117_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( b > a ) > a,F2: set_b_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a_a @ F
@ ( pi_b_a_a @ F2
@ ^ [Uu: b > a] : M ) )
=> ( member_a @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_118_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,B2: set_b,G: b > a,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_b_a @ G
@ ( pi_b_a @ B2
@ ^ [Uu: b] : M ) )
=> ( ! [I2: b] :
( ( member_b @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_119_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_120_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_121_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a,B2: set_b_a,G: ( b > a ) > a,F: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ B2
@ ^ [Uu: b > a] : M ) )
=> ( ! [I2: b > a] :
( ( member_b_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ A2 )
= ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_122_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,A2: set_b,G: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : M ) )
=> ( ( member_b_a @ G
@ ( pi_b_a @ A2
@ ^ [Uu: b] : M ) )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit
@ ^ [X3: b] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 ) @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_123_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,A2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit
@ ^ [X3: nat] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_124_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a,A2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit
@ ^ [X3: a] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_125_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( b > a ) > a,A2: set_b_a,G: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a_a @ F
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : M ) )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : M ) )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit
@ ^ [X3: b > a] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ A2 ) @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_126_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_127_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_128_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_129_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a,F: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_130_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit
@ ^ [I: b] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_131_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit
@ ^ [I: nat] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_132_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit
@ ^ [I: a] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_133_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit
@ ^ [I: b > a] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_134_Units__def,axiom,
( ( group_Units_a @ m @ composition @ unit )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ m )
& ( group_invertible_a @ m @ composition @ unit @ U2 ) ) ) ) ).
% Units_def
thf(fact_135_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( member_a @ U @ ( group_Units_a @ m @ composition @ unit ) ) ) ) ).
% mem_UnitsI
thf(fact_136_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ m @ composition @ unit ) )
=> ( ( group_invertible_a @ m @ composition @ unit @ U )
& ( member_a @ U @ m ) ) ) ).
% mem_UnitsD
thf(fact_137_comp__fun__commute__onI,axiom,
! [F: b > a,F2: set_b] :
( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : m ) )
=> ( finite9173194153363770127on_b_a @ F2
@ ^ [X3: b,Y2: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y2 ) ) ) ) ).
% comp_fun_commute_onI
thf(fact_138_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_139_finite__Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) ) )
= ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_140_insert__Diff__single,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( insert_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_141_fincomp__def,axiom,
! [A2: set_b,F: b > a] :
( ( ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_b_a
@ ^ [X3: b,Y2: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y2 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_142_fincomp__def,axiom,
! [A2: set_nat,F: nat > a] :
( ( ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_nat_a
@ ^ [X3: nat,Y2: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y2 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_143_fincomp__def,axiom,
! [A2: set_a,F: a > a] :
( ( ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_a_a
@ ^ [X3: a,Y2: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y2 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_144_fincomp__def,axiom,
! [A2: set_b_a,F: ( b > a ) > a] :
( ( ( finite_finite_b_a @ A2 )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_b_a_a
@ ^ [X3: b > a,Y2: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y2 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_b_a @ A2 )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_145_Pi__eq__empty,axiom,
! [A2: set_b,B2: b > set_a] :
( ( ( pi_b_a @ A2 @ B2 )
= bot_bot_set_b_a )
= ( ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ( ( B2 @ X3 )
= bot_bot_set_a ) ) ) ) ).
% Pi_eq_empty
thf(fact_146_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_147_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_148_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 )
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_149_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_150_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_151_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X3: b] : ( X3 = A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_152_Pi__split__insert__domain,axiom,
! [X: b > b,I3: b,I4: set_b,X4: b > set_b] :
( ( member_b_b @ X @ ( pi_b_b @ ( insert_b @ I3 @ I4 ) @ X4 ) )
= ( ( member_b_b @ X @ ( pi_b_b @ I4 @ X4 ) )
& ( member_b @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_153_Pi__split__insert__domain,axiom,
! [X: b > b > a,I3: b,I4: set_b,X4: b > set_b_a] :
( ( member_b_b_a @ X @ ( pi_b_b_a @ ( insert_b @ I3 @ I4 ) @ X4 ) )
= ( ( member_b_b_a @ X @ ( pi_b_b_a @ I4 @ X4 ) )
& ( member_b_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_154_Pi__split__insert__domain,axiom,
! [X: b > nat,I3: b,I4: set_b,X4: b > set_nat] :
( ( member_b_nat @ X @ ( pi_b_nat @ ( insert_b @ I3 @ I4 ) @ X4 ) )
= ( ( member_b_nat @ X @ ( pi_b_nat @ I4 @ X4 ) )
& ( member_nat @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_155_Pi__split__insert__domain,axiom,
! [X: b > a,I3: b,I4: set_b,X4: b > set_a] :
( ( member_b_a @ X @ ( pi_b_a @ ( insert_b @ I3 @ I4 ) @ X4 ) )
= ( ( member_b_a @ X @ ( pi_b_a @ I4 @ X4 ) )
& ( member_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_156_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_157_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_158_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_159_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_160_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_161_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_162_all__not__in__conv,axiom,
! [A2: set_b_a] :
( ( ! [X3: b > a] :
~ ( member_b_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_b_a ) ) ).
% all_not_in_conv
thf(fact_163_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_164_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_165_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_166_empty__iff,axiom,
! [C: b > a] :
~ ( member_b_a @ C @ bot_bot_set_b_a ) ).
% empty_iff
thf(fact_167_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_168_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_169_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_170_insert__absorb2,axiom,
! [X: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
= ( insert_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_171_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_172_insert__iff,axiom,
! [A: b > a,B: b > a,A2: set_b_a] :
( ( member_b_a @ A @ ( insert_b_a @ B @ A2 ) )
= ( ( A = B )
| ( member_b_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_173_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_174_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_175_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_176_insertCI,axiom,
! [A: b > a,B2: set_b_a,B: b > a] :
( ( ~ ( member_b_a @ A @ B2 )
=> ( A = B ) )
=> ( member_b_a @ A @ ( insert_b_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_177_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_178_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_179_Diff__idemp,axiom,
! [A2: set_b,B2: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_180_Diff__iff,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
= ( ( member_b_a @ C @ A2 )
& ~ ( member_b_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_181_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_182_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_183_Diff__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_184_DiffI,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ A2 )
=> ( ~ ( member_b_a @ C @ B2 )
=> ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_185_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_186_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_187_DiffI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_188_Pi__I,axiom,
! [A2: set_b,F: b > b,B2: b > set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_189_Pi__I,axiom,
! [A2: set_b,F: b > a,B2: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_190_Pi__I,axiom,
! [A2: set_b,F: b > nat,B2: b > set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_191_Pi__I,axiom,
! [A2: set_a,F: a > b,B2: a > set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_192_Pi__I,axiom,
! [A2: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_193_Pi__I,axiom,
! [A2: set_a,F: a > nat,B2: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_194_Pi__I,axiom,
! [A2: set_nat,F: nat > b,B2: nat > set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_195_Pi__I,axiom,
! [A2: set_nat,F: nat > a,B2: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_196_Pi__I,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_197_Pi__I,axiom,
! [A2: set_b,F: b > b > a,B2: b > set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b_a @ F @ ( pi_b_b_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_198_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_199_finite__Collect__conjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( ( finite_finite_b @ ( collect_b @ P ) )
| ( finite_finite_b @ ( collect_b @ Q ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X3: b] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_200_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_201_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_202_finite__Collect__disjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X3: b] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_b @ ( collect_b @ P ) )
& ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_203_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_204_singletonI,axiom,
! [A: b > a] : ( member_b_a @ A @ ( insert_b_a @ A @ bot_bot_set_b_a ) ) ).
% singletonI
thf(fact_205_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_206_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_207_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_208_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_209_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_210_Diff__cancel,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ A2 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_211_empty__Diff,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A2 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_212_Diff__empty,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Diff_empty
thf(fact_213_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_214_finite__Diff2,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_215_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_216_finite__Diff,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_217_insert__Diff1,axiom,
! [X: b > a,B2: set_b_a,A2: set_b_a] :
( ( member_b_a @ X @ B2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_218_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_219_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_220_insert__Diff1,axiom,
! [X: b,B2: set_b,A2: set_b] :
( ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_221_Diff__insert0,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ B2 ) )
= ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_222_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_223_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_224_Diff__insert0,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_225_fold__closed__eq,axiom,
! [A2: set_b,B2: set_b,F: b > b > b,G: b > b > b,Z: b] :
( ! [A3: b,B3: b] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: b] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite_fold_b_b @ F @ Z @ A2 )
= ( finite_fold_b_b @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_226_fold__closed__eq,axiom,
! [A2: set_b,B2: set_a,F: b > a > a,G: b > a > a,Z: a] :
( ! [A3: b,B3: a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite_fold_b_a @ F @ Z @ A2 )
= ( finite_fold_b_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_227_fold__closed__eq,axiom,
! [A2: set_b,B2: set_nat,F: b > nat > nat,G: b > nat > nat,Z: nat] :
( ! [A3: b,B3: nat] :
( ( member_b @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: nat] :
( ( member_b @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite_fold_b_nat @ F @ Z @ A2 )
= ( finite_fold_b_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_228_fold__closed__eq,axiom,
! [A2: set_a,B2: set_b,F: a > b > b,G: a > b > b,Z: b] :
( ! [A3: a,B3: b] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: b] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite_fold_a_b @ F @ Z @ A2 )
= ( finite_fold_a_b @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_229_fold__closed__eq,axiom,
! [A2: set_a,B2: set_a,F: a > a > a,G: a > a > a,Z: a] :
( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite_fold_a_a @ F @ Z @ A2 )
= ( finite_fold_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_230_fold__closed__eq,axiom,
! [A2: set_a,B2: set_nat,F: a > nat > nat,G: a > nat > nat,Z: nat] :
( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite_fold_a_nat @ F @ Z @ A2 )
= ( finite_fold_a_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_231_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_b,F: nat > b > b,G: nat > b > b,Z: b] :
( ! [A3: nat,B3: b] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: b] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite_fold_nat_b @ F @ Z @ A2 )
= ( finite_fold_nat_b @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_232_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_a,F: nat > a > a,G: nat > a > a,Z: a] :
( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite_fold_nat_a @ F @ Z @ A2 )
= ( finite_fold_nat_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_233_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z: nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite_fold_nat_nat @ F @ Z @ A2 )
= ( finite_fold_nat_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_234_fold__closed__eq,axiom,
! [A2: set_b,B2: set_b_a,F: b > ( b > a ) > b > a,G: b > ( b > a ) > b > a,Z: b > a] :
( ! [A3: b,B3: b > a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: b > a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( member_b_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b_a @ Z @ B2 )
=> ( ( finite_fold_b_b_a @ F @ Z @ A2 )
= ( finite_fold_b_b_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_235_ex__in__conv,axiom,
! [A2: set_b_a] :
( ( ? [X3: b > a] : ( member_b_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_b_a ) ) ).
% ex_in_conv
thf(fact_236_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_237_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_238_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_239_equals0I,axiom,
! [A2: set_b_a] :
( ! [Y4: b > a] :
~ ( member_b_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_b_a ) ) ).
% equals0I
thf(fact_240_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_241_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_242_equals0I,axiom,
! [A2: set_b] :
( ! [Y4: b] :
~ ( member_b @ Y4 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_243_equals0D,axiom,
! [A2: set_b_a,A: b > a] :
( ( A2 = bot_bot_set_b_a )
=> ~ ( member_b_a @ A @ A2 ) ) ).
% equals0D
thf(fact_244_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_245_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_246_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_247_emptyE,axiom,
! [A: b > a] :
~ ( member_b_a @ A @ bot_bot_set_b_a ) ).
% emptyE
thf(fact_248_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_249_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_250_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_251_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B4: set_b] :
( ( A2
= ( insert_b @ A @ B4 ) )
& ~ ( member_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_252_mk__disjoint__insert,axiom,
! [A: b > a,A2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ? [B4: set_b_a] :
( ( A2
= ( insert_b_a @ A @ B4 ) )
& ~ ( member_b_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_253_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_254_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_255_insert__commute,axiom,
! [X: b,Y: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A2 ) )
= ( insert_b @ Y @ ( insert_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_256_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_b] :
( ( A2
= ( insert_b @ B @ C2 ) )
& ~ ( member_b @ B @ C2 )
& ( B2
= ( insert_b @ A @ C2 ) )
& ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_257_insert__eq__iff,axiom,
! [A: b > a,A2: set_b_a,B: b > a,B2: set_b_a] :
( ~ ( member_b_a @ A @ A2 )
=> ( ~ ( member_b_a @ B @ B2 )
=> ( ( ( insert_b_a @ A @ A2 )
= ( insert_b_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_b_a] :
( ( A2
= ( insert_b_a @ B @ C2 ) )
& ~ ( member_b_a @ B @ C2 )
& ( B2
= ( insert_b_a @ A @ C2 ) )
& ~ ( member_b_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_258_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_a] :
( ( A2
= ( insert_a @ B @ C2 ) )
& ~ ( member_a @ B @ C2 )
& ( B2
= ( insert_a @ A @ C2 ) )
& ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_259_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_nat] :
( ( A2
= ( insert_nat @ B @ C2 ) )
& ~ ( member_nat @ B @ C2 )
& ( B2
= ( insert_nat @ A @ C2 ) )
& ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_260_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_261_insert__absorb,axiom,
! [A: b > a,A2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ( ( insert_b_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_262_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_263_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_264_insert__ident,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ A2 )
= ( insert_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_265_insert__ident,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ~ ( member_b_a @ X @ B2 )
=> ( ( ( insert_b_a @ X @ A2 )
= ( insert_b_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_266_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_267_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_268_Set_Oset__insert,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ~ ! [B4: set_b] :
( ( A2
= ( insert_b @ X @ B4 ) )
=> ( member_b @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_269_Set_Oset__insert,axiom,
! [X: b > a,A2: set_b_a] :
( ( member_b_a @ X @ A2 )
=> ~ ! [B4: set_b_a] :
( ( A2
= ( insert_b_a @ X @ B4 ) )
=> ( member_b_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_270_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_271_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X @ B4 ) )
=> ( member_nat @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_272_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_273_insertI2,axiom,
! [A: b > a,B2: set_b_a,B: b > a] :
( ( member_b_a @ A @ B2 )
=> ( member_b_a @ A @ ( insert_b_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_274_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_275_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_276_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_277_insertI1,axiom,
! [A: b > a,B2: set_b_a] : ( member_b_a @ A @ ( insert_b_a @ A @ B2 ) ) ).
% insertI1
thf(fact_278_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_279_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_280_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_281_insertE,axiom,
! [A: b > a,B: b > a,A2: set_b_a] :
( ( member_b_a @ A @ ( insert_b_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_282_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_283_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_284_DiffD2,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
=> ~ ( member_b_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_285_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_286_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_287_DiffD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( member_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_288_DiffD1,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
=> ( member_b_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_289_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_290_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_291_DiffD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_292_DiffE,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
=> ~ ( ( member_b_a @ C @ A2 )
=> ( member_b_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_293_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_294_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_295_DiffE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_296_Pi__cong,axiom,
! [A2: set_b,F: b > a,G: b > a,B2: b > set_a] :
( ! [W: b] :
( ( member_b @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) )
= ( member_b_a @ G @ ( pi_b_a @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_297_Pi__mem,axiom,
! [F: b > b,A2: set_b,B2: b > set_b,X: b] :
( ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_298_Pi__mem,axiom,
! [F: b > nat,A2: set_b,B2: b > set_nat,X: b] :
( ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_299_Pi__mem,axiom,
! [F: a > b,A2: set_a,B2: a > set_b,X: a] :
( ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_300_Pi__mem,axiom,
! [F: a > a,A2: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_301_Pi__mem,axiom,
! [F: a > nat,A2: set_a,B2: a > set_nat,X: a] :
( ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_302_Pi__mem,axiom,
! [F: nat > b,A2: set_nat,B2: nat > set_b,X: nat] :
( ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_303_Pi__mem,axiom,
! [F: nat > a,A2: set_nat,B2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_304_Pi__mem,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_305_Pi__mem,axiom,
! [F: b > a,A2: set_b,B2: b > set_a,X: b] :
( ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_306_Pi__mem,axiom,
! [F: b > b > a,A2: set_b,B2: b > set_b_a,X: b] :
( ( member_b_b_a @ F @ ( pi_b_b_a @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_307_Pi__iff,axiom,
! [F: b > a,I4: set_b,X4: b > set_a] :
( ( member_b_a @ F @ ( pi_b_a @ I4 @ X4 ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ I4 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_308_Pi__I_H,axiom,
! [A2: set_b,F: b > b,B2: b > set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_309_Pi__I_H,axiom,
! [A2: set_b,F: b > a,B2: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_310_Pi__I_H,axiom,
! [A2: set_b,F: b > nat,B2: b > set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_311_Pi__I_H,axiom,
! [A2: set_a,F: a > b,B2: a > set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_312_Pi__I_H,axiom,
! [A2: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_313_Pi__I_H,axiom,
! [A2: set_a,F: a > nat,B2: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_314_Pi__I_H,axiom,
! [A2: set_nat,F: nat > b,B2: nat > set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_315_Pi__I_H,axiom,
! [A2: set_nat,F: nat > a,B2: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_316_Pi__I_H,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_317_Pi__I_H,axiom,
! [A2: set_b,F: b > b > a,B2: b > set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b_a @ F @ ( pi_b_b_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_318_PiE,axiom,
! [F: b > b,A2: set_b,B2: b > set_b,X: b] :
( ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b @ X @ A2 ) ) ) ).
% PiE
thf(fact_319_PiE,axiom,
! [F: a > b,A2: set_a,B2: a > set_b,X: a] :
( ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_320_PiE,axiom,
! [F: nat > b,A2: set_nat,B2: nat > set_b,X: nat] :
( ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_321_PiE,axiom,
! [F: a > a,A2: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_322_PiE,axiom,
! [F: nat > a,A2: set_nat,B2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_323_PiE,axiom,
! [F: b > nat,A2: set_b,B2: b > set_nat,X: b] :
( ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b @ X @ A2 ) ) ) ).
% PiE
thf(fact_324_PiE,axiom,
! [F: a > nat,A2: set_a,B2: a > set_nat,X: a] :
( ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_325_PiE,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_326_PiE,axiom,
! [F: b > a,A2: set_b,B2: b > set_a,X: b] :
( ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b @ X @ A2 ) ) ) ).
% PiE
thf(fact_327_PiE,axiom,
! [F: ( b > a ) > b,A2: set_b_a,B2: ( b > a ) > set_b,X: b > a] :
( ( member_b_a_b @ F @ ( pi_b_a_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_328_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_329_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_330_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X3: b] : $false ) ) ).
% empty_def
thf(fact_331_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_332_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_333_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_334_pigeonhole__infinite__rel,axiom,
! [A2: set_b_a,B2: set_b,R: ( b > a ) > b > $o] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_b_a
@ ( collect_b_a
@ ^ [A4: b > a] :
( ( member_b_a @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_335_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_b,R: a > b > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_336_pigeonhole__infinite__rel,axiom,
! [A2: set_b_a,B2: set_nat,R: ( b > a ) > nat > $o] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_b_a
@ ( collect_b_a
@ ^ [A4: b > a] :
( ( member_b_a @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_337_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_338_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_b,R: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_339_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_nat,R: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_340_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_b,R: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_341_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_342_insert__compr,axiom,
( insert_b
= ( ^ [A4: b,B5: set_b] :
( collect_b
@ ^ [X3: b] :
( ( X3 = A4 )
| ( member_b @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_343_insert__compr,axiom,
( insert_b_a
= ( ^ [A4: b > a,B5: set_b_a] :
( collect_b_a
@ ^ [X3: b > a] :
( ( X3 = A4 )
| ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_344_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A4 )
| ( member_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_345_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A4 )
| ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_346_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U2: b] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_347_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_348_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_349_set__diff__eq,axiom,
( minus_minus_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A5 )
& ~ ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_350_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ~ ( member_a @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_351_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A5 )
& ~ ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_352_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A5 )
& ~ ( member_b @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_353_commutative__monoid_Ocomp__fun__commute__onI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,F2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : M ) )
=> ( finite9173194153363770127on_b_a @ F2
@ ^ [X3: b,Y2: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y2 ) ) ) ) ) ).
% commutative_monoid.comp_fun_commute_onI
thf(fact_354_funcsetI,axiom,
! [A2: set_b,F: b > b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( member_b_b @ F
@ ( pi_b_b @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_355_funcsetI,axiom,
! [A2: set_b,F: b > a,B2: set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_356_funcsetI,axiom,
! [A2: set_b,F: b > nat,B2: set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_b_nat @ F
@ ( pi_b_nat @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_357_funcsetI,axiom,
! [A2: set_a,F: a > b,B2: set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( member_a_b @ F
@ ( pi_a_b @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_358_funcsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_359_funcsetI,axiom,
! [A2: set_a,F: a > nat,B2: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_a_nat @ F
@ ( pi_a_nat @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_360_funcsetI,axiom,
! [A2: set_nat,F: nat > b,B2: set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_b @ F
@ ( pi_nat_b @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_361_funcsetI,axiom,
! [A2: set_nat,F: nat > a,B2: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_362_funcsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_nat @ F
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_363_funcsetI,axiom,
! [A2: set_b,F: b > b > a,B2: set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ B2 ) )
=> ( member_b_b_a @ F
@ ( pi_b_b_a @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_364_funcset__mem,axiom,
! [F: b > b,A2: set_b,B2: set_b,X: b] :
( ( member_b_b @ F
@ ( pi_b_b @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_365_funcset__mem,axiom,
! [F: b > nat,A2: set_b,B2: set_nat,X: b] :
( ( member_b_nat @ F
@ ( pi_b_nat @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_366_funcset__mem,axiom,
! [F: a > b,A2: set_a,B2: set_b,X: a] :
( ( member_a_b @ F
@ ( pi_a_b @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_367_funcset__mem,axiom,
! [F: a > a,A2: set_a,B2: set_a,X: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_368_funcset__mem,axiom,
! [F: a > nat,A2: set_a,B2: set_nat,X: a] :
( ( member_a_nat @ F
@ ( pi_a_nat @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_369_funcset__mem,axiom,
! [F: nat > b,A2: set_nat,B2: set_b,X: nat] :
( ( member_nat_b @ F
@ ( pi_nat_b @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_370_funcset__mem,axiom,
! [F: nat > a,A2: set_nat,B2: set_a,X: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_371_funcset__mem,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,X: nat] :
( ( member_nat_nat @ F
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_372_funcset__mem,axiom,
! [F: b > a,A2: set_b,B2: set_a,X: b] :
( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_373_funcset__mem,axiom,
! [F: b > b > a,A2: set_b,B2: set_b_a,X: b] :
( ( member_b_b_a @ F
@ ( pi_b_b_a @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_374_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_b_a
@ ^ [X3: b,Y2: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y2 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_375_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_nat_a
@ ^ [X3: nat,Y2: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y2 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_376_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_a_a
@ ^ [X3: a,Y2: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y2 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_377_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a,F: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_b_a @ A2 )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_b_a_a
@ ^ [X3: b > a,Y2: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y2 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_b_a @ A2 )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_378_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_379_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_380_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_381_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_382_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_383_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_384_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_385_singleton__iff,axiom,
! [B: b > a,A: b > a] :
( ( member_b_a @ B @ ( insert_b_a @ A @ bot_bot_set_b_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_386_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_387_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_388_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_389_singletonD,axiom,
! [B: b > a,A: b > a] :
( ( member_b_a @ B @ ( insert_b_a @ A @ bot_bot_set_b_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_390_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_391_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_392_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_393_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_394_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_395_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_396_Diff__infinite__finite,axiom,
! [T: set_b,S: set_b] :
( ( finite_finite_b @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_397_insert__Diff__if,axiom,
! [X: b > a,B2: set_b_a,A2: set_b_a] :
( ( ( member_b_a @ X @ B2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_b_a @ A2 @ B2 ) ) )
& ( ~ ( member_b_a @ X @ B2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( insert_b_a @ X @ ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_398_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_399_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_400_insert__Diff__if,axiom,
! [X: b,B2: set_b,A2: set_b] :
( ( ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) )
& ( ~ ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( insert_b @ X @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_401_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_402_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_403_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_404_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_405_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_406_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_407_infinite__finite__induct,axiom,
! [P: set_b_a > $o,A2: set_b_a] :
( ! [A6: set_b_a] :
( ~ ( finite_finite_b_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [X2: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ~ ( member_b_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_408_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_409_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_410_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A6: set_b] :
( ~ ( finite_finite_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X2: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_411_finite__ne__induct,axiom,
! [F2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( F2 != bot_bot_set_b_a )
=> ( ! [X2: b > a] : ( P @ ( insert_b_a @ X2 @ bot_bot_set_b_a ) )
=> ( ! [X2: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ( F3 != bot_bot_set_b_a )
=> ( ~ ( member_b_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_412_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_413_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_414_finite__ne__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( F2 != bot_bot_set_b )
=> ( ! [X2: b] : ( P @ ( insert_b @ X2 @ bot_bot_set_b ) )
=> ( ! [X2: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ~ ( member_b @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_415_finite__induct,axiom,
! [F2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [X2: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ~ ( member_b_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_416_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_417_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_418_finite__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X2: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_419_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A5: set_nat,B6: nat] :
( ( A4
= ( insert_nat @ B6 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_420_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A4: set_b] :
( ( A4 = bot_bot_set_b )
| ? [A5: set_b,B6: b] :
( ( A4
= ( insert_b @ B6 @ A5 ) )
& ( finite_finite_b @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_421_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_422_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A6: set_b] :
( ? [A3: b] :
( A
= ( insert_b @ A3 @ A6 ) )
=> ~ ( finite_finite_b @ A6 ) ) ) ) ).
% finite.cases
thf(fact_423_Diff__insert__absorb,axiom,
! [X: b > a,A2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ ( insert_b_a @ X @ bot_bot_set_b_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_424_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_425_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_426_Diff__insert__absorb,axiom,
! [X: b,A2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ ( insert_b @ X @ bot_bot_set_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_427_Diff__insert2,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_428_insert__Diff,axiom,
! [A: b > a,A2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ( ( insert_b_a @ A @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ A @ bot_bot_set_b_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_429_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_430_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_431_insert__Diff,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_432_Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_433_finite__empty__induct,axiom,
! [A2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: b > a,A6: set_b_a] :
( ( finite_finite_b_a @ A6 )
=> ( ( member_b_a @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_b_a @ A6 @ ( insert_b_a @ A3 @ bot_bot_set_b_a ) ) ) ) ) )
=> ( P @ bot_bot_set_b_a ) ) ) ) ).
% finite_empty_induct
thf(fact_434_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_435_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_436_finite__empty__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: b,A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( member_b @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ) ) )
=> ( P @ bot_bot_set_b ) ) ) ) ).
% finite_empty_induct
thf(fact_437_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A6: set_nat] :
( ( X4 @ A6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_438_infinite__coinduct,axiom,
! [X4: set_b > $o,A2: set_b] :
( ( X4 @ A2 )
=> ( ! [A6: set_b] :
( ( X4 @ A6 )
=> ? [X5: b] :
( ( member_b @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_b @ A6 @ ( insert_b @ X5 @ bot_bot_set_b ) ) )
| ~ ( finite_finite_b @ ( minus_minus_set_b @ A6 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) ) ) )
=> ~ ( finite_finite_b @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_439_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_440_infinite__remove,axiom,
! [S: set_b,A: b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% infinite_remove
thf(fact_441_monoid_OUnits__def,axiom,
! [M: set_b,Composition: b > b > b,Unit: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_Units_b @ M @ Composition @ Unit )
= ( collect_b
@ ^ [U2: b] :
( ( member_b @ U2 @ M )
& ( group_invertible_b @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_442_monoid_OUnits__def,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_Units_b_a @ M @ Composition @ Unit )
= ( collect_b_a
@ ^ [U2: b > a] :
( ( member_b_a @ U2 @ M )
& ( group_invertible_b_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_443_monoid_OUnits__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_Units_nat @ M @ Composition @ Unit )
= ( collect_nat
@ ^ [U2: nat] :
( ( member_nat @ U2 @ M )
& ( group_invertible_nat @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_444_monoid_OUnits__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_Units_a @ M @ Composition @ Unit )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ M )
& ( group_invertible_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_445_monoid_Omem__UnitsI,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( member_b @ U @ ( group_Units_b @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_446_monoid_Omem__UnitsI,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( member_b_a @ U @ ( group_Units_b_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_447_monoid_Omem__UnitsI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( member_nat @ U @ ( group_Units_nat @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_448_monoid_Omem__UnitsI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_449_monoid_Omem__UnitsD,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ U @ ( group_Units_b @ M @ Composition @ Unit ) )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
& ( member_b @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_450_monoid_Omem__UnitsD,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ U @ ( group_Units_b_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
& ( member_b_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_451_monoid_Omem__UnitsD,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ ( group_Units_nat @ M @ Composition @ Unit ) )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
& ( member_nat @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_452_monoid_Omem__UnitsD,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
& ( member_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_453_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( group_invertible_b @ M @ Composition @ Unit @ ( group_inverse_b @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_454_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_455_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_456_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_457_monoid_Oinverse__composition__commute,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ Y )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ Y ) @ ( group_inverse_b @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_458_monoid_Oinverse__composition__commute,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ Y )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_b_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_459_monoid_Oinverse__composition__commute,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ Y )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ Y ) @ ( group_inverse_nat @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_460_monoid_Oinverse__composition__commute,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_461_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ ( group_inverse_b @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_462_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_463_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_464_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_465_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_466_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_467_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_468_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_469_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( member_b @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_470_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( member_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_471_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_472_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_473_monoid_Oinvertible__right__inverse,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_b @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_474_monoid_Oinvertible__right__inverse,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_475_monoid_Oinvertible__right__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_476_monoid_Oinvertible__right__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_477_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_478_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_479_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_480_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_481_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_b,Composition: b > b > b,Unit: b] :
( ! [A3: b,B3: b] :
( ( member_b @ A3 @ M )
=> ( ( member_b @ B3 @ M )
=> ( member_b @ ( Composition @ A3 @ B3 ) @ M ) ) )
=> ( ( member_b @ Unit @ M )
=> ( ! [A3: b,B3: b,C3: b] :
( ( member_b @ A3 @ M )
=> ( ( member_b @ B3 @ M )
=> ( ( member_b @ C3 @ M )
=> ( ( Composition @ ( Composition @ A3 @ B3 ) @ C3 )
= ( Composition @ A3 @ ( Composition @ B3 @ C3 ) ) ) ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ M )
=> ( ( Composition @ Unit @ A3 )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ M )
=> ( ( Composition @ A3 @ Unit )
= A3 ) )
=> ( group_monoid_b @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_482_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ! [A3: b > a,B3: b > a] :
( ( member_b_a @ A3 @ M )
=> ( ( member_b_a @ B3 @ M )
=> ( member_b_a @ ( Composition @ A3 @ B3 ) @ M ) ) )
=> ( ( member_b_a @ Unit @ M )
=> ( ! [A3: b > a,B3: b > a,C3: b > a] :
( ( member_b_a @ A3 @ M )
=> ( ( member_b_a @ B3 @ M )
=> ( ( member_b_a @ C3 @ M )
=> ( ( Composition @ ( Composition @ A3 @ B3 ) @ C3 )
= ( Composition @ A3 @ ( Composition @ B3 @ C3 ) ) ) ) ) )
=> ( ! [A3: b > a] :
( ( member_b_a @ A3 @ M )
=> ( ( Composition @ Unit @ A3 )
= A3 ) )
=> ( ! [A3: b > a] :
( ( member_b_a @ A3 @ M )
=> ( ( Composition @ A3 @ Unit )
= A3 ) )
=> ( group_monoid_b_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_483_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ M )
=> ( ( member_nat @ B3 @ M )
=> ( member_nat @ ( Composition @ A3 @ B3 ) @ M ) ) )
=> ( ( member_nat @ Unit @ M )
=> ( ! [A3: nat,B3: nat,C3: nat] :
( ( member_nat @ A3 @ M )
=> ( ( member_nat @ B3 @ M )
=> ( ( member_nat @ C3 @ M )
=> ( ( Composition @ ( Composition @ A3 @ B3 ) @ C3 )
= ( Composition @ A3 @ ( Composition @ B3 @ C3 ) ) ) ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ M )
=> ( ( Composition @ Unit @ A3 )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ M )
=> ( ( Composition @ A3 @ Unit )
= A3 ) )
=> ( group_monoid_nat @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_484_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ! [A3: a,B3: a] :
( ( member_a @ A3 @ M )
=> ( ( member_a @ B3 @ M )
=> ( member_a @ ( Composition @ A3 @ B3 ) @ M ) ) )
=> ( ( member_a @ Unit @ M )
=> ( ! [A3: a,B3: a,C3: a] :
( ( member_a @ A3 @ M )
=> ( ( member_a @ B3 @ M )
=> ( ( member_a @ C3 @ M )
=> ( ( Composition @ ( Composition @ A3 @ B3 ) @ C3 )
= ( Composition @ A3 @ ( Composition @ B3 @ C3 ) ) ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ M )
=> ( ( Composition @ Unit @ A3 )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ M )
=> ( ( Composition @ A3 @ Unit )
= A3 ) )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_485_monoid_Oleft__unit,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_486_monoid_Oleft__unit,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_487_monoid_Oleft__unit,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_488_monoid_Oleft__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_489_monoid_Oright__unit,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_490_monoid_Oright__unit,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_491_monoid_Oright__unit,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_492_monoid_Oright__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_493_monoid_Oassociative,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b,B: b,C: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( member_b @ B @ M )
=> ( ( member_b @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_494_monoid_Oassociative,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a,B: b > a,C: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( member_b_a @ B @ M )
=> ( ( member_b_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_495_monoid_Oassociative,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat,C: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( member_nat @ B @ M )
=> ( ( member_nat @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_496_monoid_Oassociative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B: a,C: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( ( member_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_497_monoid_Ounit__closed,axiom,
! [M: set_b,Composition: b > b > b,Unit: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( member_b @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_498_monoid_Ounit__closed,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( member_b_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_499_monoid_Ounit__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( member_nat @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_500_monoid_Ounit__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( member_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_501_monoid_Oinverse__unique,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( member_b @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_502_monoid_Oinverse__unique,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( member_b_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_503_monoid_Oinverse__unique,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( member_nat @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_504_monoid_Oinverse__unique,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( member_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_505_monoid_Ocomposition__closed,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b,B: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( member_b @ B @ M )
=> ( member_b @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_506_monoid_Ocomposition__closed,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a,B: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( member_b_a @ B @ M )
=> ( member_b_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_507_monoid_Ocomposition__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( member_nat @ B @ M )
=> ( member_nat @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_508_monoid_Ocomposition__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( member_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_509_Group__Theory_Omonoid__def,axiom,
( group_monoid_b
= ( ^ [M2: set_b,Composition2: b > b > b,Unit2: b] :
( ! [A4: b,B6: b] :
( ( member_b @ A4 @ M2 )
=> ( ( member_b @ B6 @ M2 )
=> ( member_b @ ( Composition2 @ A4 @ B6 ) @ M2 ) ) )
& ( member_b @ Unit2 @ M2 )
& ! [A4: b,B6: b,C4: b] :
( ( member_b @ A4 @ M2 )
=> ( ( member_b @ B6 @ M2 )
=> ( ( member_b @ C4 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A4 @ B6 ) @ C4 )
= ( Composition2 @ A4 @ ( Composition2 @ B6 @ C4 ) ) ) ) ) )
& ! [A4: b] :
( ( member_b @ A4 @ M2 )
=> ( ( Composition2 @ Unit2 @ A4 )
= A4 ) )
& ! [A4: b] :
( ( member_b @ A4 @ M2 )
=> ( ( Composition2 @ A4 @ Unit2 )
= A4 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_510_Group__Theory_Omonoid__def,axiom,
( group_monoid_b_a
= ( ^ [M2: set_b_a,Composition2: ( b > a ) > ( b > a ) > b > a,Unit2: b > a] :
( ! [A4: b > a,B6: b > a] :
( ( member_b_a @ A4 @ M2 )
=> ( ( member_b_a @ B6 @ M2 )
=> ( member_b_a @ ( Composition2 @ A4 @ B6 ) @ M2 ) ) )
& ( member_b_a @ Unit2 @ M2 )
& ! [A4: b > a,B6: b > a,C4: b > a] :
( ( member_b_a @ A4 @ M2 )
=> ( ( member_b_a @ B6 @ M2 )
=> ( ( member_b_a @ C4 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A4 @ B6 ) @ C4 )
= ( Composition2 @ A4 @ ( Composition2 @ B6 @ C4 ) ) ) ) ) )
& ! [A4: b > a] :
( ( member_b_a @ A4 @ M2 )
=> ( ( Composition2 @ Unit2 @ A4 )
= A4 ) )
& ! [A4: b > a] :
( ( member_b_a @ A4 @ M2 )
=> ( ( Composition2 @ A4 @ Unit2 )
= A4 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_511_Group__Theory_Omonoid__def,axiom,
( group_monoid_nat
= ( ^ [M2: set_nat,Composition2: nat > nat > nat,Unit2: nat] :
( ! [A4: nat,B6: nat] :
( ( member_nat @ A4 @ M2 )
=> ( ( member_nat @ B6 @ M2 )
=> ( member_nat @ ( Composition2 @ A4 @ B6 ) @ M2 ) ) )
& ( member_nat @ Unit2 @ M2 )
& ! [A4: nat,B6: nat,C4: nat] :
( ( member_nat @ A4 @ M2 )
=> ( ( member_nat @ B6 @ M2 )
=> ( ( member_nat @ C4 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A4 @ B6 ) @ C4 )
= ( Composition2 @ A4 @ ( Composition2 @ B6 @ C4 ) ) ) ) ) )
& ! [A4: nat] :
( ( member_nat @ A4 @ M2 )
=> ( ( Composition2 @ Unit2 @ A4 )
= A4 ) )
& ! [A4: nat] :
( ( member_nat @ A4 @ M2 )
=> ( ( Composition2 @ A4 @ Unit2 )
= A4 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_512_Group__Theory_Omonoid__def,axiom,
( group_monoid_a
= ( ^ [M2: set_a,Composition2: a > a > a,Unit2: a] :
( ! [A4: a,B6: a] :
( ( member_a @ A4 @ M2 )
=> ( ( member_a @ B6 @ M2 )
=> ( member_a @ ( Composition2 @ A4 @ B6 ) @ M2 ) ) )
& ( member_a @ Unit2 @ M2 )
& ! [A4: a,B6: a,C4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( member_a @ B6 @ M2 )
=> ( ( member_a @ C4 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A4 @ B6 ) @ C4 )
= ( Composition2 @ A4 @ ( Composition2 @ B6 @ C4 ) ) ) ) ) )
& ! [A4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( Composition2 @ Unit2 @ A4 )
= A4 ) )
& ! [A4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( Composition2 @ A4 @ Unit2 )
= A4 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_513_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_514_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_515_commutative__monoid_Ocommutative,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_516_commutative__monoid_Ocommutative,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a] :
( ( group_4188790030012530981id_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_517_commutative__monoid_Ocommutative,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_518_commutative__monoid_Ocommutative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_519_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_520_monoid_OinvertibleE,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ! [V3: b] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_b @ V3 @ M ) )
=> ~ ( member_b @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_521_monoid_OinvertibleE,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: b > a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_b_a @ V3 @ M ) )
=> ~ ( member_b_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_522_monoid_OinvertibleE,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ! [V3: nat] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_nat @ V3 @ M ) )
=> ~ ( member_nat @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_523_monoid_OinvertibleE,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_a @ V3 @ M ) )
=> ~ ( member_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_524_monoid_OinvertibleI,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( group_invertible_b @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_525_monoid_OinvertibleI,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_526_monoid_OinvertibleI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_527_monoid_OinvertibleI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_528_monoid_Oinvertible__def,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ U @ M )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
= ( ? [X3: b] :
( ( member_b @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_529_monoid_Oinvertible__def,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
= ( ? [X3: b > a] :
( ( member_b_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_530_monoid_Oinvertible__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ M )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_531_monoid_Oinvertible__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ M )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_532_monoid_Ounit__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_invertible_a @ M @ Composition @ Unit @ Unit ) ) ).
% monoid.unit_invertible
thf(fact_533_monoid_Ocomposition__invertible,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ Y )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( group_invertible_b @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_534_monoid_Ocomposition__invertible,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ Y )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_535_monoid_Ocomposition__invertible,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ Y )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_536_monoid_Ocomposition__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_537_monoid_Oinvertible__left__cancel,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b,Z: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( member_b @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_538_monoid_Oinvertible__left__cancel,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a,Z: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( member_b_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_539_monoid_Oinvertible__left__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_540_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_541_monoid_Oinvertible__right__cancel,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b,Z: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( member_b @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_542_monoid_Oinvertible__right__cancel,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a,Z: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( member_b_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_543_monoid_Oinvertible__right__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_544_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_545_monoid_Oinverse__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ Unit )
= Unit ) ) ).
% monoid.inverse_unit
thf(fact_546_monoid_Oinverse__equality,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_547_monoid_Oinverse__equality,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_548_monoid_Oinverse__equality,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_549_monoid_Oinverse__equality,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_550_commutative__monoid_Oaxioms_I1_J,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ).
% commutative_monoid.axioms(1)
thf(fact_551_monoid_Oinvertible__left__inverse,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_552_monoid_Oinvertible__left__inverse,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_553_monoid_Oinvertible__left__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_554_monoid_Oinvertible__left__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_555_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ m @ composition @ unit ) @ composition @ unit ).
% group_of_Units
thf(fact_556_the__elem__eq,axiom,
! [X: b] :
( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
= X ) ).
% the_elem_eq
thf(fact_557_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_558_is__singletonI,axiom,
! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_559_fincomp__Un__disjoint,axiom,
! [A2: set_b,B2: set_b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( member_b_a @ G
@ ( pi_b_a @ A2
@ ^ [Uu: b] : m ) )
=> ( ( member_b_a @ G
@ ( pi_b_a @ B2
@ ^ [Uu: b] : m ) )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( composition @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ A2 ) @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_560_fincomp__Un__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : m ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_561_fincomp__Un__disjoint,axiom,
! [A2: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : m ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_562_fincomp__Un__disjoint,axiom,
! [A2: set_b_a,B2: set_b_a,G: ( b > a ) > a] :
( ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b_a @ B2 )
=> ( ( ( inf_inf_set_b_a @ A2 @ B2 )
= bot_bot_set_b_a )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : m ) )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ B2
@ ^ [Uu: b > a] : m ) )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ ( sup_sup_set_b_a @ A2 @ B2 ) )
= ( composition @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ A2 ) @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_563_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_564_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_565_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_566_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_567_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_568_subsetI,axiom,
! [A2: set_b_a,B2: set_b_a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( member_b_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_569_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_570_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_571_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_572_IntI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ( member_b @ C @ B2 )
=> ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_573_IntI,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ A2 )
=> ( ( member_b_a @ C @ B2 )
=> ( member_b_a @ C @ ( inf_inf_set_b_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_574_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_575_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_576_Int__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ( member_b @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_577_Int__iff,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( inf_inf_set_b_a @ A2 @ B2 ) )
= ( ( member_b_a @ C @ A2 )
& ( member_b_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_578_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_579_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_580_UnCI,axiom,
! [C: b,B2: set_b,A2: set_b] :
( ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ A2 ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_581_UnCI,axiom,
! [C: b > a,B2: set_b_a,A2: set_b_a] :
( ( ~ ( member_b_a @ C @ B2 )
=> ( member_b_a @ C @ A2 ) )
=> ( member_b_a @ C @ ( sup_sup_set_b_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_582_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_583_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_584_Un__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
| ( member_b @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_585_Un__iff,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( sup_sup_set_b_a @ A2 @ B2 ) )
= ( ( member_b_a @ C @ A2 )
| ( member_b_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_586_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_587_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_588_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_589_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_590_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_591_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_592_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_593_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_594_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_595_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_596_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_597_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_598_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_599_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_600_inf__bot__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_601_inf__bot__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_602_sup__bot__left,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X )
= X ) ).
% sup_bot_left
thf(fact_603_sup__bot__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% sup_bot_right
thf(fact_604_bot__eq__sup__iff,axiom,
! [X: set_b,Y: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X @ Y ) )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_605_sup__eq__bot__iff,axiom,
! [X: set_b,Y: set_b] :
( ( ( sup_sup_set_b @ X @ Y )
= bot_bot_set_b )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_606_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( sup_sup_set_b @ A @ B )
= bot_bot_set_b )
= ( ( A = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_607_sup__bot_Oleft__neutral,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_608_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_b,B: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A @ B ) )
= ( ( A = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_609_sup__bot_Oright__neutral,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ bot_bot_set_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_610_insert__subset,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( ( member_b @ X @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_611_insert__subset,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ( ord_less_eq_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( ( member_b_a @ X @ B2 )
& ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_612_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_613_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_614_finite__Int,axiom,
! [F2: set_b,G2: set_b] :
( ( ( finite_finite_b @ F2 )
| ( finite_finite_b @ G2 ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_615_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_616_Un__empty,axiom,
! [A2: set_b,B2: set_b] :
( ( ( sup_sup_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ( A2 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_617_Int__subset__iff,axiom,
! [C5: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C5 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C5 @ A2 )
& ( ord_less_eq_set_a @ C5 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_618_finite__Un,axiom,
! [F2: set_b,G2: set_b] :
( ( finite_finite_b @ ( sup_sup_set_b @ F2 @ G2 ) )
= ( ( finite_finite_b @ F2 )
& ( finite_finite_b @ G2 ) ) ) ).
% finite_Un
thf(fact_619_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_620_Int__insert__right__if1,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_621_Int__insert__right__if1,axiom,
! [A: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ( ( inf_inf_set_b_a @ A2 @ ( insert_b_a @ A @ B2 ) )
= ( insert_b_a @ A @ ( inf_inf_set_b_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_622_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_623_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_624_Int__insert__right__if0,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_625_Int__insert__right__if0,axiom,
! [A: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ A @ A2 )
=> ( ( inf_inf_set_b_a @ A2 @ ( insert_b_a @ A @ B2 ) )
= ( inf_inf_set_b_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_626_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_627_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_628_insert__inter__insert,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_629_Int__insert__left__if1,axiom,
! [A: b,C5: set_b,B2: set_b] :
( ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C5 )
= ( insert_b @ A @ ( inf_inf_set_b @ B2 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_630_Int__insert__left__if1,axiom,
! [A: b > a,C5: set_b_a,B2: set_b_a] :
( ( member_b_a @ A @ C5 )
=> ( ( inf_inf_set_b_a @ ( insert_b_a @ A @ B2 ) @ C5 )
= ( insert_b_a @ A @ ( inf_inf_set_b_a @ B2 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_631_Int__insert__left__if1,axiom,
! [A: a,C5: set_a,B2: set_a] :
( ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C5 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_632_Int__insert__left__if1,axiom,
! [A: nat,C5: set_nat,B2: set_nat] :
( ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_633_Int__insert__left__if0,axiom,
! [A: b,C5: set_b,B2: set_b] :
( ~ ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C5 )
= ( inf_inf_set_b @ B2 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_634_Int__insert__left__if0,axiom,
! [A: b > a,C5: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ A @ C5 )
=> ( ( inf_inf_set_b_a @ ( insert_b_a @ A @ B2 ) @ C5 )
= ( inf_inf_set_b_a @ B2 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_635_Int__insert__left__if0,axiom,
! [A: a,C5: set_a,B2: set_a] :
( ~ ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C5 )
= ( inf_inf_set_a @ B2 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_636_Int__insert__left__if0,axiom,
! [A: nat,C5: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
= ( inf_inf_set_nat @ B2 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_637_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C5 )
= ( ( ord_less_eq_set_a @ A2 @ C5 )
& ( ord_less_eq_set_a @ B2 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_638_Un__insert__right,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( sup_sup_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_639_Un__insert__left,axiom,
! [A: b,B2: set_b,C5: set_b] :
( ( sup_sup_set_b @ ( insert_b @ A @ B2 ) @ C5 )
= ( insert_b @ A @ ( sup_sup_set_b @ B2 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_640_Un__Diff__cancel2,axiom,
! [B2: set_b,A2: set_b] :
( ( sup_sup_set_b @ ( minus_minus_set_b @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_b @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_641_Un__Diff__cancel,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ A2 @ ( minus_minus_set_b @ B2 @ A2 ) )
= ( sup_sup_set_b @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_642_Pi__split__domain,axiom,
! [X: b > a,I4: set_b,J2: set_b,X4: b > set_a] :
( ( member_b_a @ X @ ( pi_b_a @ ( sup_sup_set_b @ I4 @ J2 ) @ X4 ) )
= ( ( member_b_a @ X @ ( pi_b_a @ I4 @ X4 ) )
& ( member_b_a @ X @ ( pi_b_a @ J2 @ X4 ) ) ) ) ).
% Pi_split_domain
thf(fact_643_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B5: set_b] : ( ord_less_eq_set_b @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_644_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_645_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B5: set_a] : ( ord_less_eq_set_a @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_646_funcset__Int__left,axiom,
! [F: b > a,A2: set_b,C5: set_a,B2: set_b] :
( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : C5 ) )
=> ( ( member_b_a @ F
@ ( pi_b_a @ B2
@ ^ [Uu: b] : C5 ) )
=> ( member_b_a @ F
@ ( pi_b_a @ ( inf_inf_set_b @ A2 @ B2 )
@ ^ [Uu: b] : C5 ) ) ) ) ).
% funcset_Int_left
thf(fact_647_funcset__Un__left,axiom,
! [F: b > a,A2: set_b,B2: set_b,C5: set_a] :
( ( member_b_a @ F
@ ( pi_b_a @ ( sup_sup_set_b @ A2 @ B2 )
@ ^ [Uu: b] : C5 ) )
= ( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : C5 ) )
& ( member_b_a @ F
@ ( pi_b_a @ B2
@ ^ [Uu: b] : C5 ) ) ) ) ).
% funcset_Un_left
thf(fact_648_fincomp__Un__Int,axiom,
! [A2: set_b,B2: set_b,G: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( member_b_a @ G
@ ( pi_b_a @ A2
@ ^ [Uu: b] : m ) )
=> ( ( member_b_a @ G
@ ( pi_b_a @ B2
@ ^ [Uu: b] : m ) )
=> ( ( composition @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ ( sup_sup_set_b @ A2 @ B2 ) ) @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ ( inf_inf_set_b @ A2 @ B2 ) ) )
= ( composition @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ A2 ) @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_649_fincomp__Un__Int,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : m ) )
=> ( ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_650_fincomp__Un__Int,axiom,
! [A2: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : m ) )
=> ( ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ ( inf_inf_set_a @ A2 @ B2 ) ) )
= ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_651_fincomp__Un__Int,axiom,
! [A2: set_b_a,B2: set_b_a,G: ( b > a ) > a] :
( ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b_a @ B2 )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : m ) )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ B2
@ ^ [Uu: b > a] : m ) )
=> ( ( composition @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ ( sup_sup_set_b_a @ A2 @ B2 ) ) @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ ( inf_inf_set_b_a @ A2 @ B2 ) ) )
= ( composition @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ A2 ) @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_652_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_653_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_654_singleton__insert__inj__eq,axiom,
! [B: b,A: b,A2: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_655_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_656_insert__disjoint_I1_J,axiom,
! [A: b > a,A2: set_b_a,B2: set_b_a] :
( ( ( inf_inf_set_b_a @ ( insert_b_a @ A @ A2 ) @ B2 )
= bot_bot_set_b_a )
= ( ~ ( member_b_a @ A @ B2 )
& ( ( inf_inf_set_b_a @ A2 @ B2 )
= bot_bot_set_b_a ) ) ) ).
% insert_disjoint(1)
thf(fact_657_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_658_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_659_insert__disjoint_I1_J,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B2 )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B2 )
& ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_660_insert__disjoint_I2_J,axiom,
! [A: b > a,A2: set_b_a,B2: set_b_a] :
( ( bot_bot_set_b_a
= ( inf_inf_set_b_a @ ( insert_b_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_b_a @ A @ B2 )
& ( bot_bot_set_b_a
= ( inf_inf_set_b_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_661_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_662_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_663_insert__disjoint_I2_J,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B2 ) )
= ( ~ ( member_b @ A @ B2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_664_disjoint__insert_I1_J,axiom,
! [B2: set_b_a,A: b > a,A2: set_b_a] :
( ( ( inf_inf_set_b_a @ B2 @ ( insert_b_a @ A @ A2 ) )
= bot_bot_set_b_a )
= ( ~ ( member_b_a @ A @ B2 )
& ( ( inf_inf_set_b_a @ B2 @ A2 )
= bot_bot_set_b_a ) ) ) ).
% disjoint_insert(1)
thf(fact_665_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_666_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_667_disjoint__insert_I1_J,axiom,
! [B2: set_b,A: b,A2: set_b] :
( ( ( inf_inf_set_b @ B2 @ ( insert_b @ A @ A2 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B2 )
& ( ( inf_inf_set_b @ B2 @ A2 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_668_disjoint__insert_I2_J,axiom,
! [A2: set_b_a,B: b > a,B2: set_b_a] :
( ( bot_bot_set_b_a
= ( inf_inf_set_b_a @ A2 @ ( insert_b_a @ B @ B2 ) ) )
= ( ~ ( member_b_a @ B @ A2 )
& ( bot_bot_set_b_a
= ( inf_inf_set_b_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_669_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_670_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_671_disjoint__insert_I2_J,axiom,
! [A2: set_b,B: b,B2: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ ( insert_b @ B @ B2 ) ) )
= ( ~ ( member_b @ B @ A2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_672_Diff__eq__empty__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( minus_minus_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_673_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_674_Diff__disjoint,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ A2 @ ( minus_minus_set_b @ B2 @ A2 ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_675_Un__Diff__Int,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_676_Int__Diff__Un,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_677_Diff__Int,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( minus_minus_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C5 ) )
= ( sup_sup_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ A2 @ C5 ) ) ) ).
% Diff_Int
thf(fact_678_Diff__Un,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( minus_minus_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C5 ) )
= ( inf_inf_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ A2 @ C5 ) ) ) ).
% Diff_Un
thf(fact_679_Diff__subset__conv,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ C5 )
= ( ord_less_eq_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_680_Diff__subset__conv,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C5 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_681_Diff__partition,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( sup_sup_set_b @ A2 @ ( minus_minus_set_b @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_682_Diff__partition,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_683_Pi__anti__mono,axiom,
! [A7: set_b,A2: set_b,B2: b > set_a] :
( ( ord_less_eq_set_b @ A7 @ A2 )
=> ( ord_less_eq_set_b_a @ ( pi_b_a @ A2 @ B2 ) @ ( pi_b_a @ A7 @ B2 ) ) ) ).
% Pi_anti_mono
thf(fact_684_Pi__mono,axiom,
! [A2: set_b,B2: b > set_a,C5: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_less_eq_set_b_a @ ( pi_b_a @ A2 @ B2 ) @ ( pi_b_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_685_Pi__mono,axiom,
! [A2: set_b_a,B2: ( b > a ) > set_a,C5: ( b > a ) > set_a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_le4402886750609172241_b_a_a @ ( pi_b_a_a @ A2 @ B2 ) @ ( pi_b_a_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_686_Pi__mono,axiom,
! [A2: set_a,B2: a > set_a,C5: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A2 @ B2 ) @ ( pi_a_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_687_Pi__mono,axiom,
! [A2: set_nat,B2: nat > set_a,C5: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( pi_nat_a @ A2 @ B2 ) @ ( pi_nat_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_688_Un__def,axiom,
( sup_sup_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A5 )
| ( member_b @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_689_Un__def,axiom,
( sup_sup_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A5 )
| ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_690_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
| ( member_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_691_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A5 )
| ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_692_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_693_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_694_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A5 )
& ( member_b @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_695_Int__def,axiom,
( inf_inf_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A5 )
& ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_696_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( member_a @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_697_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A5 )
& ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_698_Int__Collect,axiom,
! [X: b,A2: set_b,P: b > $o] :
( ( member_b @ X @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) )
= ( ( member_b @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_699_Int__Collect,axiom,
! [X: b > a,A2: set_b_a,P: ( b > a ) > $o] :
( ( member_b_a @ X @ ( inf_inf_set_b_a @ A2 @ ( collect_b_a @ P ) ) )
= ( ( member_b_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_700_Int__Collect,axiom,
! [X: a,A2: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_701_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_702_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_703_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_704_bot_Oextremum,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% bot.extremum
thf(fact_705_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_706_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_707_bot_Oextremum__unique,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_708_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_709_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_710_bot_Oextremum__uniqueI,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
=> ( A = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_711_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_712_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_713_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_714_Collect__subset,axiom,
! [A2: set_b_a,P: ( b > a ) > $o] :
( ord_less_eq_set_b_a
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_715_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_716_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_717_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_718_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_719_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_720_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_721_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_724_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_725_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_726_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_727_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_728_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_729_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_730_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_731_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_732_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_733_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_734_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_735_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_736_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_737_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_738_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_739_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A4 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_740_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ A4 @ B6 )
& ( ord_less_eq_nat @ B6 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_741_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_742_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_743_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_744_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_745_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_746_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_747_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B6: set_a] :
( ( sup_sup_set_a @ A4 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_748_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
( ( sup_sup_nat @ A4 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_749_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B6 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_750_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B6 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_751_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_752_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_753_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_754_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_755_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_756_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_757_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B6: set_a] :
( ( inf_inf_set_a @ A4 @ B6 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_758_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
( ( inf_inf_nat @ A4 @ B6 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_759_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_760_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_761_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_762_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_763_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_764_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_765_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_766_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_767_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_768_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_769_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B6: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_770_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
( A4
= ( inf_inf_nat @ A4 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_771_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_772_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_773_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_774_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_775_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_776_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_777_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_778_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_779_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_780_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_781_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_782_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_783_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_784_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_785_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_786_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_787_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_788_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_789_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_790_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_791_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_792_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_793_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_794_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_795_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_796_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_797_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_798_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_799_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_800_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_801_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_802_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_803_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_804_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_805_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_806_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( sup_sup_nat @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_807_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_808_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_809_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_810_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_811_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_812_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_813_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_814_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_815_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_816_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_817_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_818_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_819_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_820_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A4 )
& ( ord_less_eq_nat @ A4 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_821_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_822_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_823_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_824_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_825_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_826_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_827_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_828_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_829_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_830_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_831_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_832_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_833_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_834_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_835_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_836_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_837_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_838_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_839_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_840_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_841_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_842_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_843_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_844_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_845_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_846_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_847_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_848_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_849_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_850_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_851_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_852_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_853_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_854_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_855_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_856_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_857_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_858_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_859_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_860_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_861_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_862_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_863_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_864_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_865_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_866_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_867_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_868_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_869_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_870_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_871_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_872_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_873_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_874_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_875_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_876_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_877_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_878_UnE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% UnE
thf(fact_879_UnE,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( sup_sup_set_b_a @ A2 @ B2 ) )
=> ( ~ ( member_b_a @ C @ A2 )
=> ( member_b_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_880_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_881_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_882_IntE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ~ ( member_b @ C @ B2 ) ) ) ).
% IntE
thf(fact_883_IntE,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( inf_inf_set_b_a @ A2 @ B2 ) )
=> ~ ( ( member_b_a @ C @ A2 )
=> ~ ( member_b_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_884_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_885_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_886_UnI1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_887_UnI1,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ A2 )
=> ( member_b_a @ C @ ( sup_sup_set_b_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_888_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_889_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_890_UnI2,axiom,
! [C: b,B2: set_b,A2: set_b] :
( ( member_b @ C @ B2 )
=> ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_891_UnI2,axiom,
! [C: b > a,B2: set_b_a,A2: set_b_a] :
( ( member_b_a @ C @ B2 )
=> ( member_b_a @ C @ ( sup_sup_set_b_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_892_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_893_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_894_IntD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% IntD1
thf(fact_895_IntD1,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( inf_inf_set_b_a @ A2 @ B2 ) )
=> ( member_b_a @ C @ A2 ) ) ).
% IntD1
thf(fact_896_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_897_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_898_IntD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ B2 ) ) ).
% IntD2
thf(fact_899_IntD2,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( inf_inf_set_b_a @ A2 @ B2 ) )
=> ( member_b_a @ C @ B2 ) ) ).
% IntD2
thf(fact_900_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_901_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_902_Un__mono,axiom,
! [A2: set_a,C5: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C5 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_903_in__mono,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_904_in__mono,axiom,
! [A2: set_b_a,B2: set_b_a,X: b > a] :
( ( ord_less_eq_set_b_a @ A2 @ B2 )
=> ( ( member_b_a @ X @ A2 )
=> ( member_b_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_905_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_906_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_907_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_908_subsetD,axiom,
! [A2: set_b_a,B2: set_b_a,C: b > a] :
( ( ord_less_eq_set_b_a @ A2 @ B2 )
=> ( ( member_b_a @ C @ A2 )
=> ( member_b_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_909_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_910_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_911_Int__mono,axiom,
! [A2: set_a,C5: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C5 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_912_Un__least,axiom,
! [A2: set_a,C5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( ord_less_eq_set_a @ B2 @ C5 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C5 ) ) ) ).
% Un_least
thf(fact_913_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_914_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_915_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_916_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B5: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ( member_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_917_subset__eq,axiom,
( ord_less_eq_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
! [X3: b > a] :
( ( member_b_a @ X3 @ A5 )
=> ( member_b_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_918_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_919_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_920_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_921_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_922_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_923_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_924_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_925_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_926_subset__UnE,axiom,
! [C5: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C5 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A2 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B2 )
=> ( C5
!= ( sup_sup_set_a @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_927_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B5: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A5 )
=> ( member_b @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_928_subset__iff,axiom,
( ord_less_eq_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
! [T2: b > a] :
( ( member_b_a @ T2 @ A5 )
=> ( member_b_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_929_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_930_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_931_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_932_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_933_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_934_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_935_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_936_Int__greatest,axiom,
! [C5: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( ( ord_less_eq_set_a @ C5 @ B2 )
=> ( ord_less_eq_set_a @ C5 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_937_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_938_subset__trans,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C5 )
=> ( ord_less_eq_set_a @ A2 @ C5 ) ) ) ).
% subset_trans
thf(fact_939_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_940_Un__Int__assoc__eq,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C5 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C5 ) ) )
= ( ord_less_eq_set_a @ C5 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_941_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_942_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_943_Int__Collect__mono,axiom,
! [A2: set_b,B2: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B2 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_944_Int__Collect__mono,axiom,
! [A2: set_b_a,B2: set_b_a,P: ( b > a ) > $o,Q: ( b > a ) > $o] :
( ( ord_less_eq_set_b_a @ A2 @ B2 )
=> ( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_b_a @ ( inf_inf_set_b_a @ A2 @ ( collect_b_a @ P ) ) @ ( inf_inf_set_b_a @ B2 @ ( collect_b_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_945_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_946_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_947_Pi__Int,axiom,
! [I4: set_b,E: b > set_a,F2: b > set_a] :
( ( inf_inf_set_b_a @ ( pi_b_a @ I4 @ E ) @ ( pi_b_a @ I4 @ F2 ) )
= ( pi_b_a @ I4
@ ^ [I: b] : ( inf_inf_set_a @ ( E @ I ) @ ( F2 @ I ) ) ) ) ).
% Pi_Int
thf(fact_948_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_949_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_950_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_951_Un__empty__right,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Un_empty_right
thf(fact_952_Un__empty__left,axiom,
! [B2: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_953_infinite__Un,axiom,
! [S: set_b,T: set_b] :
( ( ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T ) ) )
= ( ~ ( finite_finite_b @ S )
| ~ ( finite_finite_b @ T ) ) ) ).
% infinite_Un
thf(fact_954_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_955_Un__infinite,axiom,
! [S: set_b,T: set_b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T ) ) ) ).
% Un_infinite
thf(fact_956_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_957_finite__UnI,axiom,
! [F2: set_b,G2: set_b] :
( ( finite_finite_b @ F2 )
=> ( ( finite_finite_b @ G2 )
=> ( finite_finite_b @ ( sup_sup_set_b @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_958_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_959_disjoint__iff__not__equal,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ( X3 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_960_Int__empty__right,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_961_Int__empty__left,axiom,
! [B2: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B2 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_962_disjoint__iff,axiom,
! [A2: set_b_a,B2: set_b_a] :
( ( ( inf_inf_set_b_a @ A2 @ B2 )
= bot_bot_set_b_a )
= ( ! [X3: b > a] :
( ( member_b_a @ X3 @ A2 )
=> ~ ( member_b_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_963_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_964_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_965_disjoint__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ~ ( member_b @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_966_Int__emptyI,axiom,
! [A2: set_b_a,B2: set_b_a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ~ ( member_b_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_b_a @ A2 @ B2 )
= bot_bot_set_b_a ) ) ).
% Int_emptyI
thf(fact_967_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_968_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_969_Int__emptyI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ~ ( member_b @ X2 @ B2 ) )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_970_Int__insert__right,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B2 ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_971_Int__insert__right,axiom,
! [A: b > a,A2: set_b_a,B2: set_b_a] :
( ( ( member_b_a @ A @ A2 )
=> ( ( inf_inf_set_b_a @ A2 @ ( insert_b_a @ A @ B2 ) )
= ( insert_b_a @ A @ ( inf_inf_set_b_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_b_a @ A @ A2 )
=> ( ( inf_inf_set_b_a @ A2 @ ( insert_b_a @ A @ B2 ) )
= ( inf_inf_set_b_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_972_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_973_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_974_Int__insert__left,axiom,
! [A: b,C5: set_b,B2: set_b] :
( ( ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C5 )
= ( insert_b @ A @ ( inf_inf_set_b @ B2 @ C5 ) ) ) )
& ( ~ ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C5 )
= ( inf_inf_set_b @ B2 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_975_Int__insert__left,axiom,
! [A: b > a,C5: set_b_a,B2: set_b_a] :
( ( ( member_b_a @ A @ C5 )
=> ( ( inf_inf_set_b_a @ ( insert_b_a @ A @ B2 ) @ C5 )
= ( insert_b_a @ A @ ( inf_inf_set_b_a @ B2 @ C5 ) ) ) )
& ( ~ ( member_b_a @ A @ C5 )
=> ( ( inf_inf_set_b_a @ ( insert_b_a @ A @ B2 ) @ C5 )
= ( inf_inf_set_b_a @ B2 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_976_Int__insert__left,axiom,
! [A: a,C5: set_a,B2: set_a] :
( ( ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C5 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C5 ) ) ) )
& ( ~ ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C5 )
= ( inf_inf_set_a @ B2 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_977_Int__insert__left,axiom,
! [A: nat,C5: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C5 ) ) ) )
& ( ~ ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
= ( inf_inf_set_nat @ B2 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_978_Un__Diff,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( minus_minus_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ C5 )
= ( sup_sup_set_b @ ( minus_minus_set_b @ A2 @ C5 ) @ ( minus_minus_set_b @ B2 @ C5 ) ) ) ).
% Un_Diff
thf(fact_979_Diff__Int__distrib2,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( inf_inf_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ C5 )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C5 ) @ ( inf_inf_set_b @ B2 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_980_Diff__Int__distrib,axiom,
! [C5: set_b,A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ C5 @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ C5 @ A2 ) @ ( inf_inf_set_b @ C5 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_981_Diff__Diff__Int,axiom,
! [A2: set_b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_982_Diff__Int2,axiom,
! [A2: set_b,C5: set_b,B2: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C5 ) @ ( inf_inf_set_b @ B2 @ C5 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C5 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_983_Int__Diff,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C5 )
= ( inf_inf_set_b @ A2 @ ( minus_minus_set_b @ B2 @ C5 ) ) ) ).
% Int_Diff
thf(fact_984_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_985_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_986_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_987_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_988_rev__finite__subset,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_989_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_990_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_991_infinite__super,axiom,
! [S: set_b,T: set_b] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T ) ) ) ).
% infinite_super
thf(fact_992_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_993_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_994_finite__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_995_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_996_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_997_subset__insertI2,axiom,
! [A2: set_b,B2: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_998_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_999_subset__insertI,axiom,
! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1000_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1001_subset__insert,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1002_subset__insert,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ( ord_less_eq_set_b_a @ A2 @ ( insert_b_a @ X @ B2 ) )
= ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1003_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1004_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1005_insert__mono,axiom,
! [C5: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C5 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C5 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1006_insert__mono,axiom,
! [C5: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C5 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C5 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1007_double__diff,axiom,
! [A2: set_b,B2: set_b,C5: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C5 )
=> ( ( minus_minus_set_b @ B2 @ ( minus_minus_set_b @ C5 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1008_double__diff,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C5 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C5 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1009_Diff__subset,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1010_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1011_Diff__mono,axiom,
! [A2: set_b,C5: set_b,D2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C5 )
=> ( ( ord_less_eq_set_b @ D2 @ B2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ C5 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1012_Diff__mono,axiom,
! [A2: set_a,C5: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C5 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1013_minus__set__def,axiom,
( minus_minus_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( collect_b_a
@ ( minus_minus_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A5 )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1014_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1015_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1016_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A5 )
@ ^ [X3: b] : ( member_b @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1017_card__Diff__subset__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1018_card__Diff__subset__Int,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1019_Group__Theory_Ogroup_Oaxioms_I1_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( group_monoid_a @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group.axioms(1)
thf(fact_1020_group_Oinvertible,axiom,
! [G2: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_group_b @ G2 @ Composition @ Unit )
=> ( ( member_b @ U @ G2 )
=> ( group_invertible_b @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_1021_group_Oinvertible,axiom,
! [G2: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_group_b_a @ G2 @ Composition @ Unit )
=> ( ( member_b_a @ U @ G2 )
=> ( group_invertible_b_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_1022_group_Oinvertible,axiom,
! [G2: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_group_nat @ G2 @ Composition @ Unit )
=> ( ( member_nat @ U @ G2 )
=> ( group_invertible_nat @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_1023_group_Oinvertible,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( member_a @ U @ G2 )
=> ( group_invertible_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_1024_insert__def,axiom,
( insert_b
= ( ^ [A4: b] :
( sup_sup_set_b
@ ( collect_b
@ ^ [X3: b] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1025_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X3: a] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1026_insert__def,axiom,
( insert_nat
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1027_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1028_card__Diff__subset,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1029_card__Diff__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1030_singleton__Un__iff,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ( ( insert_b @ X @ bot_bot_set_b )
= ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_b )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) )
| ( ( A2
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2 = bot_bot_set_b ) )
| ( ( A2
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1031_Un__singleton__iff,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ( sup_sup_set_b @ A2 @ B2 )
= ( insert_b @ X @ bot_bot_set_b ) )
= ( ( ( A2 = bot_bot_set_b )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) )
| ( ( A2
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2 = bot_bot_set_b ) )
| ( ( A2
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1032_insert__is__Un,axiom,
( insert_b
= ( ^ [A4: b] : ( sup_sup_set_b @ ( insert_b @ A4 @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_1033_union__fold__insert,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= ( finite_fold_b_set_b @ insert_b @ B2 @ A2 ) ) ) ).
% union_fold_insert
thf(fact_1034_union__fold__insert,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= ( finite5529483035118572448et_nat @ insert_nat @ B2 @ A2 ) ) ) ).
% union_fold_insert
thf(fact_1035_Int__Diff__disjoint,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ A2 @ B2 ) )
= bot_bot_set_b ) ).
% Int_Diff_disjoint
thf(fact_1036_Diff__triv,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( minus_minus_set_b @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1037_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1038_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1039_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1040_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1041_subset__singleton__iff,axiom,
! [X4: set_b,A: b] :
( ( ord_less_eq_set_b @ X4 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X4 = bot_bot_set_b )
| ( X4
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_1042_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1043_subset__singletonD,axiom,
! [A2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_1044_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1045_infinite__arbitrarily__large,axiom,
! [A2: set_b,N: nat] :
( ~ ( finite_finite_b @ A2 )
=> ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ( finite_card_b @ B4 )
= N )
& ( ord_less_eq_set_b @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1046_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ( finite_card_nat @ B4 )
= N )
& ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1047_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ( finite_card_a @ B4 )
= N )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1048_card__subset__eq,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ( finite_card_b @ A2 )
= ( finite_card_b @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1049_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1050_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1051_subset__Diff__insert,axiom,
! [A2: set_b_a,B2: set_b_a,X: b > a,C5: set_b_a] :
( ( ord_less_eq_set_b_a @ A2 @ ( minus_minus_set_b_a @ B2 @ ( insert_b_a @ X @ C5 ) ) )
= ( ( ord_less_eq_set_b_a @ A2 @ ( minus_minus_set_b_a @ B2 @ C5 ) )
& ~ ( member_b_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1052_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C5 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C5 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1053_subset__Diff__insert,axiom,
! [A2: set_b,B2: set_b,X: b,C5: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ ( insert_b @ X @ C5 ) ) )
= ( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ C5 ) )
& ~ ( member_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1054_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C5 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C5 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1055_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,B2: set_b,G: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( member_b_a @ G
@ ( pi_b_a @ A2
@ ^ [Uu: b] : M ) )
=> ( ( member_b_a @ G
@ ( pi_b_a @ B2
@ ^ [Uu: b] : M ) )
=> ( ( Composition @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ ( sup_sup_set_b @ A2 @ B2 ) ) @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ ( inf_inf_set_b @ A2 @ B2 ) ) )
= ( Composition @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ A2 ) @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1056_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M ) )
=> ( ( Composition @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( Composition @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1057_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M ) )
=> ( ( Composition @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ ( inf_inf_set_a @ A2 @ B2 ) ) )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1058_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a,B2: set_b_a,G: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b_a @ B2 )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : M ) )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ B2
@ ^ [Uu: b > a] : M ) )
=> ( ( Composition @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ ( sup_sup_set_b_a @ A2 @ B2 ) ) @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ ( inf_inf_set_b_a @ A2 @ B2 ) ) )
= ( Composition @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ A2 ) @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1059_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A5: set_b] :
( A5
= ( insert_b @ ( the_elem_b @ A5 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1060_monoid_Oinverse__undefined,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ~ ( member_b @ U @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ U )
= undefined_b ) ) ) ).
% monoid.inverse_undefined
thf(fact_1061_monoid_Oinverse__undefined,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ~ ( member_b_a @ U @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ U )
= undefined_b_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_1062_monoid_Oinverse__undefined,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ~ ( member_nat @ U @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= undefined_nat ) ) ) ).
% monoid.inverse_undefined
thf(fact_1063_monoid_Oinverse__undefined,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ~ ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= undefined_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_1064_is__singletonI_H,axiom,
! [A2: set_b_a] :
( ( A2 != bot_bot_set_b_a )
=> ( ! [X2: b > a,Y4: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ( member_b_a @ Y4 @ A2 )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_b_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1065_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1066_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y4: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1067_is__singletonI_H,axiom,
! [A2: set_b] :
( ( A2 != bot_bot_set_b )
=> ( ! [X2: b,Y4: b] :
( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y4 @ A2 )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1068_commutative__monoid_Ofuncset__Un__left,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,A2: set_b,B2: set_b,C5: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ ( sup_sup_set_b @ A2 @ B2 )
@ ^ [Uu: b] : C5 ) )
= ( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : C5 ) )
& ( member_b_a @ F
@ ( pi_b_a @ B2
@ ^ [Uu: b] : C5 ) ) ) ) ) ).
% commutative_monoid.funcset_Un_left
thf(fact_1069_commutative__monoid_Ofuncset__Int__left,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,A2: set_b,C5: set_a,B2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : C5 ) )
=> ( ( member_b_a @ F
@ ( pi_b_a @ B2
@ ^ [Uu: b] : C5 ) )
=> ( member_b_a @ F
@ ( pi_b_a @ ( inf_inf_set_b @ A2 @ B2 )
@ ^ [Uu: b] : C5 ) ) ) ) ) ).
% commutative_monoid.funcset_Int_left
thf(fact_1070_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,B2: set_b,G: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( member_b_a @ G
@ ( pi_b_a @ A2
@ ^ [Uu: b] : M ) )
=> ( ( member_b_a @ G
@ ( pi_b_a @ B2
@ ^ [Uu: b] : M ) )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( Composition @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ A2 ) @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1071_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( Composition @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1072_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1073_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b_a,B2: set_b_a,G: ( b > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b_a @ B2 )
=> ( ( ( inf_inf_set_b_a @ A2 @ B2 )
= bot_bot_set_b_a )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ A2
@ ^ [Uu: b > a] : M ) )
=> ( ( member_b_a_a @ G
@ ( pi_b_a_a @ B2
@ ^ [Uu: b > a] : M ) )
=> ( ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ ( sup_sup_set_b_a @ A2 @ B2 ) )
= ( Composition @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ A2 ) @ ( commut3556868347779488380_a_b_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1074_monoid_Ogroup__of__Units,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_group_a @ ( group_Units_a @ M @ Composition @ Unit ) @ Composition @ Unit ) ) ).
% monoid.group_of_Units
thf(fact_1075_finite__subset__induct_H,axiom,
! [F2: set_b_a,A2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( ord_less_eq_set_b_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [A3: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ( member_b_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_b_a @ F3 @ A2 )
=> ( ~ ( member_b_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1076_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1077_finite__subset__induct_H,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A3 @ A2 )
=> ( ( ord_less_eq_set_b @ F3 @ A2 )
=> ( ~ ( member_b @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1078_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1079_finite__subset__induct,axiom,
! [F2: set_b_a,A2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( ord_less_eq_set_b_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [A3: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ( member_b_a @ A3 @ A2 )
=> ( ~ ( member_b_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1080_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1081_finite__subset__induct,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A3 @ A2 )
=> ( ~ ( member_b @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1082_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1083_Diff__single__insert,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1084_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1085_subset__insert__iff,axiom,
! [A2: set_b_a,X: b > a,B2: set_b_a] :
( ( ord_less_eq_set_b_a @ A2 @ ( insert_b_a @ X @ B2 ) )
= ( ( ( member_b_a @ X @ A2 )
=> ( ord_less_eq_set_b_a @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ bot_bot_set_b_a ) ) @ B2 ) )
& ( ~ ( member_b_a @ X @ A2 )
=> ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1086_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1087_subset__insert__iff,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ( ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1088_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1089_finite__remove__induct,axiom,
! [B2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ B2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [A6: set_b_a] :
( ( finite_finite_b_a @ A6 )
=> ( ( A6 != bot_bot_set_b_a )
=> ( ( ord_less_eq_set_b_a @ A6 @ B2 )
=> ( ! [X5: b > a] :
( ( member_b_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_b_a @ A6 @ ( insert_b_a @ X5 @ bot_bot_set_b_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1090_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1091_finite__remove__induct,axiom,
! [B2: set_b,P: set_b > $o] :
( ( finite_finite_b @ B2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( A6 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A6 @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1092_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1093_remove__induct,axiom,
! [P: set_b_a > $o,B2: set_b_a] :
( ( P @ bot_bot_set_b_a )
=> ( ( ~ ( finite_finite_b_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_b_a] :
( ( finite_finite_b_a @ A6 )
=> ( ( A6 != bot_bot_set_b_a )
=> ( ( ord_less_eq_set_b_a @ A6 @ B2 )
=> ( ! [X5: b > a] :
( ( member_b_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_b_a @ A6 @ ( insert_b_a @ X5 @ bot_bot_set_b_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1094_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1095_remove__induct,axiom,
! [P: set_b > $o,B2: set_b] :
( ( P @ bot_bot_set_b )
=> ( ( ~ ( finite_finite_b @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( A6 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A6 @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1096_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1097_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A5: set_b] :
? [X3: b] :
( A5
= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_1098_is__singletonE,axiom,
! [A2: set_b] :
( ( is_singleton_b @ A2 )
=> ~ ! [X2: b] :
( A2
!= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_1099_subgroupI,axiom,
! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ m )
=> ( ( member_a @ unit @ G2 )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G2 )
=> ( ( member_a @ H @ G2 )
=> ( member_a @ ( composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( group_invertible_a @ m @ composition @ unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( member_a @ ( group_inverse_a @ m @ composition @ unit @ G3 ) @ G2 ) )
=> ( group_subgroup_a @ G2 @ m @ composition @ unit ) ) ) ) ) ) ).
% subgroupI
thf(fact_1100_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_1101_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_1102_finite__ranking__induct,axiom,
! [S: set_b_a,P: set_b_a > $o,F: ( b > a ) > nat] :
( ( finite_finite_b_a @ S )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [X2: b > a,S2: set_b_a] :
( ( finite_finite_b_a @ S2 )
=> ( ! [Y5: b > a] :
( ( member_b_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_b_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1103_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1104_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1105_finite__ranking__induct,axiom,
! [S: set_b,P: set_b > $o,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X2: b,S2: set_b] :
( ( finite_finite_b @ S2 )
=> ( ! [Y5: b] :
( ( member_b @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_b @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1106_diff__shunt__var,axiom,
! [X: set_b,Y: set_b] :
( ( ( minus_minus_set_b @ X @ Y )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1107_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1108_subgroup__transitive,axiom,
! [K: set_a,H2: set_a,Composition: a > a > a,Unit: a,G2: set_a] :
( ( group_subgroup_a @ K @ H2 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G2 @ Composition @ Unit )
=> ( group_subgroup_a @ K @ G2 @ Composition @ Unit ) ) ) ).
% subgroup_transitive
thf(fact_1109_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A5 )
@ ^ [X3: b] : ( member_b @ X3 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_1110_inf__set__def,axiom,
( inf_inf_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( collect_b_a
@ ( inf_inf_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A5 )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_1111_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_1112_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_1113_sup__set__def,axiom,
( sup_sup_set_b
= ( ^ [A5: set_b,B5: set_b] :
( collect_b
@ ( sup_sup_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A5 )
@ ^ [X3: b] : ( member_b @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1114_sup__set__def,axiom,
( sup_sup_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( collect_b_a
@ ( sup_sup_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A5 )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1115_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1116_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1117_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B5: set_b] :
( ord_less_eq_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A5 )
@ ^ [X3: b] : ( member_b @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1118_less__eq__set__def,axiom,
( ord_less_eq_set_b_a
= ( ^ [A5: set_b_a,B5: set_b_a] :
( ord_less_eq_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A5 )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1119_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1120_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1121_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_b,M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_subgroup_b @ G2 @ M @ Composition @ Unit )
=> ( ( member_b @ U @ G2 )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ U )
= ( group_inverse_b @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_1122_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_b_a,M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_b_a @ U @ G2 )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ U )
= ( group_inverse_b_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_1123_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_subgroup_nat @ G2 @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ G2 )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= ( group_inverse_nat @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_1124_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_a @ U @ G2 )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= ( group_inverse_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_1125_subgroup_Oaxioms_I2_J,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( group_group_a @ G2 @ Composition @ Unit ) ) ).
% subgroup.axioms(2)
thf(fact_1126_card__insert__le,axiom,
! [A2: set_b,X: b] : ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ ( insert_b @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1127_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_b,M: set_b,Composition: b > b > b,Unit: b,X: b] :
( ( group_subgroup_b @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ ( group_inverse_b @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_b @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_1128_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_b_a,M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a] :
( ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_b_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_1129_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_subgroup_nat @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_nat @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_1130_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_1131_card__mono,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B2 ) ) ) ) ).
% card_mono
thf(fact_1132_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_1133_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_1134_card__seteq,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ B2 ) @ ( finite_card_b @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1135_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1136_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1137_exists__subset__between,axiom,
! [A2: set_b,N: nat,C5: set_b] :
( ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_b @ C5 ) )
=> ( ( ord_less_eq_set_b @ A2 @ C5 )
=> ( ( finite_finite_b @ C5 )
=> ? [B4: set_b] :
( ( ord_less_eq_set_b @ A2 @ B4 )
& ( ord_less_eq_set_b @ B4 @ C5 )
& ( ( finite_card_b @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1138_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C5: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C5 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C5 )
=> ( ( finite_finite_nat @ C5 )
=> ? [B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ C5 )
& ( ( finite_card_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1139_exists__subset__between,axiom,
! [A2: set_a,N: nat,C5: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C5 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( finite_finite_a @ C5 )
=> ? [B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ C5 )
& ( ( finite_card_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1140_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_b] :
( ( ord_less_eq_nat @ N @ ( finite_card_b @ S ) )
=> ~ ! [T3: set_b] :
( ( ord_less_eq_set_b @ T3 @ S )
=> ( ( ( finite_card_b @ T3 )
= N )
=> ~ ( finite_finite_b @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1141_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1142_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1143_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_b,C5: nat] :
( ! [G4: set_b] :
( ( ord_less_eq_set_b @ G4 @ F2 )
=> ( ( finite_finite_b @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_b @ G4 ) @ C5 ) ) )
=> ( ( finite_finite_b @ F2 )
& ( ord_less_eq_nat @ ( finite_card_b @ F2 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1144_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C5: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F2 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C5 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1145_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C5: nat] :
( ! [G4: set_a] :
( ( ord_less_eq_set_a @ G4 @ F2 )
=> ( ( finite_finite_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G4 ) @ C5 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1146_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1147_card__le__sym__Diff,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_b @ ( minus_minus_set_b @ A2 @ B2 ) ) @ ( finite_card_b @ ( minus_minus_set_b @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1148_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1149_diff__card__le__card__Diff,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B2 ) ) @ ( finite_card_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1150_card__Diff1__le,axiom,
! [A2: set_b,X: b] : ( ord_less_eq_nat @ ( finite_card_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) ) @ ( finite_card_b @ A2 ) ) ).
% card_Diff1_le
thf(fact_1151_monoid_OsubgroupI,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,G2: set_b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_b @ G2 @ M )
=> ( ( member_b @ Unit @ G2 )
=> ( ! [G3: b,H: b] :
( ( member_b @ G3 @ G2 )
=> ( ( member_b @ H @ G2 )
=> ( member_b @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: b] :
( ( member_b @ G3 @ G2 )
=> ( group_invertible_b @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: b] :
( ( member_b @ G3 @ G2 )
=> ( member_b @ ( group_inverse_b @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_b @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1152_monoid_OsubgroupI,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,G2: set_b_a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_b_a @ G2 @ M )
=> ( ( member_b_a @ Unit @ G2 )
=> ( ! [G3: b > a,H: b > a] :
( ( member_b_a @ G3 @ G2 )
=> ( ( member_b_a @ H @ G2 )
=> ( member_b_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: b > a] :
( ( member_b_a @ G3 @ G2 )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: b > a] :
( ( member_b_a @ G3 @ G2 )
=> ( member_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1153_monoid_OsubgroupI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,G2: set_nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ G2 @ M )
=> ( ( member_nat @ Unit @ G2 )
=> ( ! [G3: nat,H: nat] :
( ( member_nat @ G3 @ G2 )
=> ( ( member_nat @ H @ G2 )
=> ( member_nat @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G2 )
=> ( group_invertible_nat @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G2 )
=> ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_nat @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1154_monoid_OsubgroupI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,G2: set_a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G2 @ M )
=> ( ( member_a @ Unit @ G2 )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G2 )
=> ( ( member_a @ H @ G2 )
=> ( member_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( group_invertible_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1155_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1156_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1157_Set__filter__fold,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( filter_nat @ P @ A2 )
= ( finite5529483035118572448et_nat
@ ^ [X3: nat,A9: set_nat] : ( if_set_nat @ ( P @ X3 ) @ ( insert_nat @ X3 @ A9 ) @ A9 )
@ bot_bot_set_nat
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_1158_Set__filter__fold,axiom,
! [A2: set_b,P: b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( filter_b @ P @ A2 )
= ( finite_fold_b_set_b
@ ^ [X3: b,A9: set_b] : ( if_set_b @ ( P @ X3 ) @ ( insert_b @ X3 @ A9 ) @ A9 )
@ bot_bot_set_b
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_1159_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1160_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1161_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1162_bot__empty__eq,axiom,
( bot_bot_b_a_o
= ( ^ [X3: b > a] : ( member_b_a @ X3 @ bot_bot_set_b_a ) ) ) ).
% bot_empty_eq
thf(fact_1163_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1164_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1165_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X3: b] : ( member_b @ X3 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_1166_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1167_member__filter,axiom,
! [X: b,P: b > $o,A2: set_b] :
( ( member_b @ X @ ( filter_b @ P @ A2 ) )
= ( ( member_b @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_1168_member__filter,axiom,
! [X: b > a,P: ( b > a ) > $o,A2: set_b_a] :
( ( member_b_a @ X @ ( filter_b_a @ P @ A2 ) )
= ( ( member_b_a @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_1169_member__filter,axiom,
! [X: a,P: a > $o,A2: set_a] :
( ( member_a @ X @ ( filter_a @ P @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_1170_member__filter,axiom,
! [X: nat,P: nat > $o,A2: set_nat] :
( ( member_nat @ X @ ( filter_nat @ P @ A2 ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_1171_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 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1172_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_1173_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_1174_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_1175_le__trans,axiom,
! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_1176_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1177_Set_Ofilter__def,axiom,
( filter_b
= ( ^ [P2: b > $o,A5: set_b] :
( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A5 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1178_Set_Ofilter__def,axiom,
( filter_b_a
= ( ^ [P2: ( b > a ) > $o,A5: set_b_a] :
( collect_b_a
@ ^ [A4: b > a] :
( ( member_b_a @ A4 @ A5 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1179_Set_Ofilter__def,axiom,
( filter_a
= ( ^ [P2: a > $o,A5: set_a] :
( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A5 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1180_Set_Ofilter__def,axiom,
( filter_nat
= ( ^ [P2: nat > $o,A5: set_nat] :
( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A5 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1181_finite__filter,axiom,
! [S: set_b,P: b > $o] :
( ( finite_finite_b @ S )
=> ( finite_finite_b @ ( filter_b @ P @ S ) ) ) ).
% finite_filter
thf(fact_1182_finite__filter,axiom,
! [S: set_nat,P: nat > $o] :
( ( finite_finite_nat @ S )
=> ( finite_finite_nat @ ( filter_nat @ P @ S ) ) ) ).
% finite_filter
thf(fact_1183_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1184_le__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1185_diff__commute,axiom,
! [I3: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_1186_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1187_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1188_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_1189_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1190_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1191_pred__subset__eq,axiom,
! [R: set_b,S: set_b] :
( ( ord_less_eq_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R )
@ ^ [X3: b] : ( member_b @ X3 @ S ) )
= ( ord_less_eq_set_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1192_pred__subset__eq,axiom,
! [R: set_b_a,S: set_b_a] :
( ( ord_less_eq_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ R )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ S ) )
= ( ord_less_eq_set_b_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1193_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1194_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1195_inter__Set__filter,axiom,
! [B2: set_b_a,A2: set_b_a] :
( ( finite_finite_b_a @ B2 )
=> ( ( inf_inf_set_b_a @ A2 @ B2 )
= ( filter_b_a
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A2 )
@ B2 ) ) ) ).
% inter_Set_filter
thf(fact_1196_inter__Set__filter,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= ( filter_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ B2 ) ) ) ).
% inter_Set_filter
thf(fact_1197_inter__Set__filter,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= ( filter_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 )
@ B2 ) ) ) ).
% inter_Set_filter
thf(fact_1198_inter__Set__filter,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= ( filter_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ B2 ) ) ) ).
% inter_Set_filter
thf(fact_1199_inf__Int__eq,axiom,
! [R: set_b,S: set_b] :
( ( inf_inf_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R )
@ ^ [X3: b] : ( member_b @ X3 @ S ) )
= ( ^ [X3: b] : ( member_b @ X3 @ ( inf_inf_set_b @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1200_inf__Int__eq,axiom,
! [R: set_b_a,S: set_b_a] :
( ( inf_inf_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ R )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ S ) )
= ( ^ [X3: b > a] : ( member_b_a @ X3 @ ( inf_inf_set_b_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1201_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1202_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1203_sup__Un__eq,axiom,
! [R: set_b,S: set_b] :
( ( sup_sup_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R )
@ ^ [X3: b] : ( member_b @ X3 @ S ) )
= ( ^ [X3: b] : ( member_b @ X3 @ ( sup_sup_set_b @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1204_sup__Un__eq,axiom,
! [R: set_b_a,S: set_b_a] :
( ( sup_sup_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ R )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ S ) )
= ( ^ [X3: b > a] : ( member_b_a @ X3 @ ( sup_sup_set_b_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1205_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1206_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1207_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_b,R2: b > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: b,A22: b,B3: a] :
( ( member_b @ A1 @ A2 )
=> ( ( member_b @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1208_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1209_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1210_card__le__if__inj__on__rel,axiom,
! [B2: set_b,A2: set_b,R2: b > b > $o] :
( ( finite_finite_b @ B2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ A2 )
=> ? [B8: b] :
( ( member_b @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: b,A22: b,B3: b] :
( ( member_b @ A1 @ A2 )
=> ( ( member_b @ A22 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1211_card__le__if__inj__on__rel,axiom,
! [B2: set_b,A2: set_a,R2: a > b > $o] :
( ( finite_finite_b @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B8: b] :
( ( member_b @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B3: b] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_b @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1212_card__le__if__inj__on__rel,axiom,
! [B2: set_b,A2: set_nat,R2: nat > b > $o] :
( ( finite_finite_b @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B8: b] :
( ( member_b @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: b] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_b @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1213_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_b,R2: b > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: b,A22: b,B3: nat] :
( ( member_b @ A1 @ A2 )
=> ( ( member_b @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1214_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1215_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1216_card__le__if__inj__on__rel,axiom,
! [B2: set_b_a,A2: set_b,R2: b > ( b > a ) > $o] :
( ( finite_finite_b_a @ B2 )
=> ( ! [A3: b] :
( ( member_b @ A3 @ A2 )
=> ? [B8: b > a] :
( ( member_b_a @ B8 @ B2 )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: b,A22: b,B3: b > a] :
( ( member_b @ A1 @ A2 )
=> ( ( member_b @ A22 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( ( R2 @ A1 @ B3 )
=> ( ( R2 @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1217_arg__min__least,axiom,
! [S: set_b_a,Y: b > a,F: ( b > a ) > nat] :
( ( finite_finite_b_a @ S )
=> ( ( S != bot_bot_set_b_a )
=> ( ( member_b_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7032157021346806415_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1218_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1219_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1220_arg__min__least,axiom,
! [S: set_b,Y: b,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( S != bot_bot_set_b )
=> ( ( member_b @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7575731748627795062_b_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1221_insert__subsetI,axiom,
! [X: b,A2: set_b,X4: set_b] :
( ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ X4 @ A2 )
=> ( ord_less_eq_set_b @ ( insert_b @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1222_insert__subsetI,axiom,
! [X: b > a,A2: set_b_a,X4: set_b_a] :
( ( member_b_a @ X @ A2 )
=> ( ( ord_less_eq_set_b_a @ X4 @ A2 )
=> ( ord_less_eq_set_b_a @ ( insert_b_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1223_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X4: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1224_insert__subsetI,axiom,
! [X: a,A2: set_a,X4: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1225_subset__emptyI,axiom,
! [A2: set_b_a] :
( ! [X2: b > a] :
~ ( member_b_a @ X2 @ A2 )
=> ( ord_less_eq_set_b_a @ A2 @ bot_bot_set_b_a ) ) ).
% subset_emptyI
thf(fact_1226_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1227_subset__emptyI,axiom,
! [A2: set_b] :
( ! [X2: b] :
~ ( member_b @ X2 @ A2 )
=> ( ord_less_eq_set_b @ A2 @ bot_bot_set_b ) ) ).
% subset_emptyI
thf(fact_1228_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1229_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1230_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1231_prop__restrict,axiom,
! [X: b,Z4: set_b,X4: set_b,P: b > $o] :
( ( member_b @ X @ Z4 )
=> ( ( ord_less_eq_set_b @ Z4
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1232_prop__restrict,axiom,
! [X: b > a,Z4: set_b_a,X4: set_b_a,P: ( b > a ) > $o] :
( ( member_b_a @ X @ Z4 )
=> ( ( ord_less_eq_set_b_a @ Z4
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1233_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X4: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1234_prop__restrict,axiom,
! [X: a,Z4: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1235_Collect__restrict,axiom,
! [X4: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1236_Collect__restrict,axiom,
! [X4: set_b_a,P: ( b > a ) > $o] :
( ord_less_eq_set_b_a
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1237_Collect__restrict,axiom,
! [X4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1238_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1239_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1240_minus__fold__remove,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( minus_minus_set_nat @ B2 @ A2 )
= ( finite5529483035118572448et_nat @ remove_nat @ B2 @ A2 ) ) ) ).
% minus_fold_remove
thf(fact_1241_minus__fold__remove,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( minus_minus_set_b @ B2 @ A2 )
= ( finite_fold_b_set_b @ remove_b @ B2 @ A2 ) ) ) ).
% minus_fold_remove
thf(fact_1242_member__remove,axiom,
! [X: b,Y: b,A2: set_b] :
( ( member_b @ X @ ( remove_b @ Y @ A2 ) )
= ( ( member_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1243_member__remove,axiom,
! [X: b > a,Y: b > a,A2: set_b_a] :
( ( member_b_a @ X @ ( remove_b_a @ Y @ A2 ) )
= ( ( member_b_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1244_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1245_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1246_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_1247_remove__def,axiom,
( remove_b
= ( ^ [X3: b,A5: set_b] : ( minus_minus_set_b @ A5 @ ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% remove_def
thf(fact_1248_extensional__empty,axiom,
( ( extensional_b_a @ bot_bot_set_b )
= ( insert_b_a
@ ^ [X3: b] : undefined_a
@ bot_bot_set_b_a ) ) ).
% extensional_empty
thf(fact_1249_extensional__insert__cancel,axiom,
! [A: b > a,I4: set_b,I3: b] :
( ( member_b_a @ A @ ( extensional_b_a @ I4 ) )
=> ( member_b_a @ A @ ( extensional_b_a @ ( insert_b @ I3 @ I4 ) ) ) ) ).
% extensional_insert_cancel
thf(fact_1250_extensional__subset,axiom,
! [F: b > a,A2: set_b,B2: set_b] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( member_b_a @ F @ ( extensional_b_a @ B2 ) ) ) ) ).
% extensional_subset
thf(fact_1251_extensionalityI,axiom,
! [F: b > a,A2: set_b,G: b > a] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ( member_b_a @ G @ ( extensional_b_a @ A2 ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( F = G ) ) ) ) ).
% extensionalityI
thf(fact_1252_extensional__arb,axiom,
! [F: ( b > a ) > a,A2: set_b_a,X: b > a] :
( ( member_b_a_a @ F @ ( extensional_b_a_a @ A2 ) )
=> ( ~ ( member_b_a @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_1253_extensional__arb,axiom,
! [F: a > a,A2: set_a,X: a] :
( ( member_a_a @ F @ ( extensional_a_a @ A2 ) )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_1254_extensional__arb,axiom,
! [F: nat > a,A2: set_nat,X: nat] :
( ( member_nat_a @ F @ ( extensional_nat_a @ A2 ) )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_1255_extensional__arb,axiom,
! [F: b > a,A2: set_b,X: b] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ~ ( member_b @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_1256_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_b,F: b > b > b,G: b > b > b,Z: b] :
( ! [A3: b,B3: b] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: b] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite5086552502106613254ph_b_b @ F @ Z @ A2 )
= ( finite5086552502106613254ph_b_b @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1257_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_a,F: b > a > a,G: b > a > a,Z: a] :
( ! [A3: b,B3: a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite5086552502106613253ph_b_a @ F @ Z @ A2 )
= ( finite5086552502106613253ph_b_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1258_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_nat,F: b > nat > nat,G: b > nat > nat,Z: nat] :
( ! [A3: b,B3: nat] :
( ( member_b @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: nat] :
( ( member_b @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite6345878069334568201_b_nat @ F @ Z @ A2 )
= ( finite6345878069334568201_b_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1259_fold__graph__closed__eq,axiom,
! [A2: set_a,B2: set_b,F: a > b > b,G: a > b > b,Z: b] :
( ! [A3: a,B3: b] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: b] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite7874008084079289287ph_a_b @ F @ Z @ A2 )
= ( finite7874008084079289287ph_a_b @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1260_fold__graph__closed__eq,axiom,
! [A2: set_a,B2: set_a,F: a > a > a,G: a > a > a,Z: a] :
( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite7874008084079289286ph_a_a @ F @ Z @ A2 )
= ( finite7874008084079289286ph_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1261_fold__graph__closed__eq,axiom,
! [A2: set_a,B2: set_nat,F: a > nat > nat,G: a > nat > nat,Z: nat] :
( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite5110433740378173704_a_nat @ F @ Z @ A2 )
= ( finite5110433740378173704_a_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1262_fold__graph__closed__eq,axiom,
! [A2: set_nat,B2: set_b,F: nat > b > b,G: nat > b > b,Z: b] :
( ! [A3: nat,B3: b] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: b] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite9142365241556460135_nat_b @ F @ Z @ A2 )
= ( finite9142365241556460135_nat_b @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1263_fold__graph__closed__eq,axiom,
! [A2: set_nat,B2: set_a,F: nat > a > a,G: nat > a > a,Z: a] :
( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite9142365241556460134_nat_a @ F @ Z @ A2 )
= ( finite9142365241556460134_nat_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1264_fold__graph__closed__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z: nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite1441398328259824232at_nat @ F @ Z @ A2 )
= ( finite1441398328259824232at_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1265_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_b_a,F: b > ( b > a ) > b > a,G: b > ( b > a ) > b > a,Z: b > a] :
( ! [A3: b,B3: b > a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: b,B3: b > a] :
( ( member_b @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( member_b_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b_a @ Z @ B2 )
=> ( ( finite7185950986694385629_b_b_a @ F @ Z @ A2 )
= ( finite7185950986694385629_b_b_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1266_fold__graph__closed__lemma,axiom,
! [G: a > b > b,Z: b,A2: set_a,X: b,B2: set_b,F: a > b > b] :
( ( finite7874008084079289287ph_a_b @ G @ Z @ A2 @ X )
=> ( ! [A3: a,B3: b] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: b] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite7874008084079289287ph_a_b @ F @ Z @ A2 @ X )
& ( member_b @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1267_fold__graph__closed__lemma,axiom,
! [G: a > ( b > a ) > b > a,Z: b > a,A2: set_a,X: b > a,B2: set_b_a,F: a > ( b > a ) > b > a] :
( ( finite1332093216797763036_a_b_a @ G @ Z @ A2 @ X )
=> ( ! [A3: a,B3: b > a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: b > a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( member_b_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b_a @ Z @ B2 )
=> ( ( finite1332093216797763036_a_b_a @ F @ Z @ A2 @ X )
& ( member_b_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1268_fold__graph__closed__lemma,axiom,
! [G: a > a > a,Z: a,A2: set_a,X: a,B2: set_a,F: a > a > a] :
( ( finite7874008084079289286ph_a_a @ G @ Z @ A2 @ X )
=> ( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite7874008084079289286ph_a_a @ F @ Z @ A2 @ X )
& ( member_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1269_fold__graph__closed__lemma,axiom,
! [G: a > nat > nat,Z: nat,A2: set_a,X: nat,B2: set_nat,F: a > nat > nat] :
( ( finite5110433740378173704_a_nat @ G @ Z @ A2 @ X )
=> ( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite5110433740378173704_a_nat @ F @ Z @ A2 @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1270_fold__graph__closed__lemma,axiom,
! [G: nat > b > b,Z: b,A2: set_nat,X: b,B2: set_b,F: nat > b > b] :
( ( finite9142365241556460135_nat_b @ G @ Z @ A2 @ X )
=> ( ! [A3: nat,B3: b] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: b] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b @ B3 @ B2 )
=> ( member_b @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b @ Z @ B2 )
=> ( ( finite9142365241556460135_nat_b @ F @ Z @ A2 @ X )
& ( member_b @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1271_fold__graph__closed__lemma,axiom,
! [G: nat > ( b > a ) > b > a,Z: b > a,A2: set_nat,X: b > a,B2: set_b_a,F: nat > ( b > a ) > b > a] :
( ( finite8571984644340236092at_b_a @ G @ Z @ A2 @ X )
=> ( ! [A3: nat,B3: b > a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: b > a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_b_a @ B3 @ B2 )
=> ( member_b_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_b_a @ Z @ B2 )
=> ( ( finite8571984644340236092at_b_a @ F @ Z @ A2 @ X )
& ( member_b_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1272_fold__graph__closed__lemma,axiom,
! [G: nat > a > a,Z: a,A2: set_nat,X: a,B2: set_a,F: nat > a > a] :
( ( finite9142365241556460134_nat_a @ G @ Z @ A2 @ X )
=> ( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite9142365241556460134_nat_a @ F @ Z @ A2 @ X )
& ( member_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1273_fold__graph__closed__lemma,axiom,
! [G: nat > nat > nat,Z: nat,A2: set_nat,X: nat,B2: set_nat,F: nat > nat > nat] :
( ( finite1441398328259824232at_nat @ G @ Z @ A2 @ X )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite1441398328259824232at_nat @ F @ Z @ A2 @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1274_inverse__def,axiom,
( ( group_inverse_a @ m @ composition @ unit )
= ( restrict_a_a
@ ^ [U2: a] :
( the_a
@ ^ [V4: a] :
( ( member_a @ V4 @ m )
& ( ( composition @ U2 @ V4 )
= unit )
& ( ( composition @ V4 @ U2 )
= unit ) ) )
@ m ) ) ).
% inverse_def
thf(fact_1275_inj__on__diff__nat,axiom,
! [N5: set_nat,K2: nat] :
( ! [N4: nat] :
( ( member_nat @ N4 @ N5 )
=> ( ord_less_eq_nat @ K2 @ N4 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K2 )
@ N5 ) ) ).
% inj_on_diff_nat
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X: set_b,Y: set_b] :
( ( if_set_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X: set_b,Y: set_b] :
( ( if_set_b @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( commut5005951359559292711mp_a_b @ m @ composition @ unit
@ ^ [J: b] : ( if_a @ ( i = J ) @ ( f @ J ) @ unit )
@ a2 )
= ( f @ i ) ) ).
%------------------------------------------------------------------------------