TPTP Problem File: ITP093^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP093^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Kuratowski problem prob_43__5514550_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Kuratowski/prob_43__5514550_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 493 ( 113 unt; 138 typ; 0 def)
% Number of atoms : 1236 ( 234 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3205 ( 91 ~; 31 |; 122 &;2361 @)
% ( 0 <=>; 600 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 337 ( 337 >; 0 *; 0 +; 0 <<)
% Number of symbols : 108 ( 107 usr; 6 con; 0-3 aty)
% Number of variables : 1153 ( 214 ^; 904 !; 35 ?;1153 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:24.363
%------------------------------------------------------------------------------
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thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs_001tf__a_001t__Product____Type__Ounit,type,
pair_p133601421t_unit: pair_p125712459t_unit > set_Product_prod_a_a ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Nat__Onat_001t__Product____Type__Ounit,type,
pair_p1677060310t_unit: pair_p1914262621t_unit > set_nat ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
pair_p210955889t_unit: pair_p2041852168t_unit > set_Pr1986765409at_nat ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
pair_p1652294923t_unit: pair_p1593840546t_unit > set_Pr1948701895od_a_a ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
pair_p447552203t_unit: pair_p1765063010t_unit > set_Product_prod_a_a ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts_001tf__a_001t__Product____Type__Ounit,type,
pair_p1047056820t_unit: pair_p125712459t_unit > set_a ).
thf(sy_c_Pair__Digraph_Opair__wf__digraph_001t__Nat__Onat,type,
pair_p1515597646ph_nat: pair_p1914262621t_unit > $o ).
thf(sy_c_Pair__Digraph_Opair__wf__digraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
pair_p646030121od_a_a: pair_p1765063010t_unit > $o ).
thf(sy_c_Pair__Digraph_Opair__wf__digraph_001tf__a,type,
pair_p68905728raph_a: pair_p125712459t_unit > $o ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc1182842125od_a_a: set_nat > ( nat > set_Product_prod_a_a ) > set_Pr339609346od_a_a ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001tf__a,type,
product_Sigma_nat_a: set_nat > ( nat > set_a ) > set_Pr967348953_nat_a ).
thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
produc931712687_a_nat: set_Product_prod_a_a > ( product_prod_a_a > set_nat ) > set_Pr894832732_a_nat ).
thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc304751368od_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
produc1282482655_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_a ) > set_Pr1689873822_a_a_a ).
thf(sy_c_Product__Type_OSigma_001tf__a_001t__Nat__Onat,type,
product_Sigma_a_nat: set_a > ( a > set_nat ) > set_Pr548851891_a_nat ).
thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc292491723at_nat: set_a > ( a > set_Pr1986765409at_nat ) > set_Pr7688842at_nat ).
thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc520147185od_a_a: set_a > ( a > set_Product_prod_a_a ) > set_Pr681306928od_a_a ).
thf(sy_c_Product__Type_OSigma_001tf__a_001tf__a,type,
product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001_Eo,type,
produc1833107820_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
thf(sy_c_Product__Type_Oproduct_001tf__a_001tf__a,type,
product_product_a_a: set_a > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
collec453062450od_a_a: ( set_Pr1948701895od_a_a > $o ) > set_se958357159od_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: pair_p125712459t_unit ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (354)
thf(fact_0__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062,axiom,
( finite179568208od_a_a
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) ) ).
% \<open>finite (pverts G \<times> pverts G)\<close>
thf(fact_1__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062,axiom,
( ord_le1824328871od_a_a @ ( pair_p133601421t_unit @ g )
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) ) ).
% \<open>parcs G \<subseteq> pverts G \<times> pverts G\<close>
thf(fact_2_pair__fin__digraph__axioms__def,axiom,
( pair_p56914274at_nat
= ( ^ [G: pair_p2041852168t_unit] :
( ( finite772653738at_nat @ ( pair_p210955889t_unit @ G ) )
& ( finite48957584at_nat @ ( pair_p806300874t_unit @ G ) ) ) ) ) ).
% pair_fin_digraph_axioms_def
thf(fact_3_pair__fin__digraph__axioms__def,axiom,
( pair_p1906342088od_a_a
= ( ^ [G: pair_p1593840546t_unit] :
( ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G ) )
& ( finite256329232od_a_a @ ( pair_p1559300324t_unit @ G ) ) ) ) ) ).
% pair_fin_digraph_axioms_def
thf(fact_4_pair__fin__digraph__axioms__def,axiom,
( pair_p504738056od_a_a
= ( ^ [G: pair_p1765063010t_unit] :
( ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G ) )
& ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G ) ) ) ) ) ).
% pair_fin_digraph_axioms_def
thf(fact_5_pair__fin__digraph__axioms__def,axiom,
( pair_p1027063983ms_nat
= ( ^ [G: pair_p1914262621t_unit] :
( ( finite_finite_nat @ ( pair_p1677060310t_unit @ G ) )
& ( finite772653738at_nat @ ( pair_p715279805t_unit @ G ) ) ) ) ) ).
% pair_fin_digraph_axioms_def
thf(fact_6_pair__fin__digraph__axioms__def,axiom,
( pair_p1864019935ioms_a
= ( ^ [G: pair_p125712459t_unit] :
( ( finite_finite_a @ ( pair_p1047056820t_unit @ G ) )
& ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G ) ) ) ) ) ).
% pair_fin_digraph_axioms_def
thf(fact_7_pair__fin__digraph__axioms_Ointro,axiom,
! [G2: pair_p2041852168t_unit] :
( ( finite772653738at_nat @ ( pair_p210955889t_unit @ G2 ) )
=> ( ( finite48957584at_nat @ ( pair_p806300874t_unit @ G2 ) )
=> ( pair_p56914274at_nat @ G2 ) ) ) ).
% pair_fin_digraph_axioms.intro
thf(fact_8_pair__fin__digraph__axioms_Ointro,axiom,
! [G2: pair_p1593840546t_unit] :
( ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G2 ) )
=> ( ( finite256329232od_a_a @ ( pair_p1559300324t_unit @ G2 ) )
=> ( pair_p1906342088od_a_a @ G2 ) ) ) ).
% pair_fin_digraph_axioms.intro
thf(fact_9_pair__fin__digraph__axioms_Ointro,axiom,
! [G2: pair_p1765063010t_unit] :
( ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G2 ) )
=> ( ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G2 ) )
=> ( pair_p504738056od_a_a @ G2 ) ) ) ).
% pair_fin_digraph_axioms.intro
thf(fact_10_pair__fin__digraph__axioms_Ointro,axiom,
! [G2: pair_p1914262621t_unit] :
( ( finite_finite_nat @ ( pair_p1677060310t_unit @ G2 ) )
=> ( ( finite772653738at_nat @ ( pair_p715279805t_unit @ G2 ) )
=> ( pair_p1027063983ms_nat @ G2 ) ) ) ).
% pair_fin_digraph_axioms.intro
thf(fact_11_pair__fin__digraph__axioms_Ointro,axiom,
! [G2: pair_p125712459t_unit] :
( ( finite_finite_a @ ( pair_p1047056820t_unit @ G2 ) )
=> ( ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G2 ) )
=> ( pair_p1864019935ioms_a @ G2 ) ) ) ).
% pair_fin_digraph_axioms.intro
thf(fact_12_pair__fin__digraph_Opair__finite__arcs,axiom,
! [G2: pair_p1914262621t_unit] :
( ( pair_p128415500ph_nat @ G2 )
=> ( finite772653738at_nat @ ( pair_p715279805t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_arcs
thf(fact_13_pair__fin__digraph_Opair__finite__arcs,axiom,
! [G2: pair_p1765063010t_unit] :
( ( pair_p374947051od_a_a @ G2 )
=> ( finite1664988688od_a_a @ ( pair_p1783210148t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_arcs
thf(fact_14_pair__fin__digraph_Opair__finite__arcs,axiom,
! [G2: pair_p125712459t_unit] :
( ( pair_p1802376898raph_a @ G2 )
=> ( finite179568208od_a_a @ ( pair_p133601421t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_arcs
thf(fact_15_finite__subset,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( finite772653738at_nat @ B )
=> ( finite772653738at_nat @ A ) ) ) ).
% finite_subset
thf(fact_16_finite__subset,axiom,
! [A: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( ord_le456379495od_a_a @ A @ B )
=> ( ( finite1664988688od_a_a @ B )
=> ( finite1664988688od_a_a @ A ) ) ) ).
% finite_subset
thf(fact_17_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_18_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_19_finite__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B )
=> ( ( finite179568208od_a_a @ B )
=> ( finite179568208od_a_a @ A ) ) ) ).
% finite_subset
thf(fact_20_infinite__super,axiom,
! [S: set_Pr1986765409at_nat,T: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ S @ T )
=> ( ~ ( finite772653738at_nat @ S )
=> ~ ( finite772653738at_nat @ T ) ) ) ).
% infinite_super
thf(fact_21_infinite__super,axiom,
! [S: set_Pr1948701895od_a_a,T: set_Pr1948701895od_a_a] :
( ( ord_le456379495od_a_a @ S @ T )
=> ( ~ ( finite1664988688od_a_a @ S )
=> ~ ( finite1664988688od_a_a @ T ) ) ) ).
% infinite_super
thf(fact_22_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_23_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_24_infinite__super,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ S @ T )
=> ( ~ ( finite179568208od_a_a @ S )
=> ~ ( finite179568208od_a_a @ T ) ) ) ).
% infinite_super
thf(fact_25_rev__finite__subset,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B )
=> ( ( ord_le841296385at_nat @ A @ B )
=> ( finite772653738at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_26_rev__finite__subset,axiom,
! [B: set_Pr1948701895od_a_a,A: set_Pr1948701895od_a_a] :
( ( finite1664988688od_a_a @ B )
=> ( ( ord_le456379495od_a_a @ A @ B )
=> ( finite1664988688od_a_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_27_rev__finite__subset,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ B )
=> ( ( ord_le1824328871od_a_a @ A @ B )
=> ( finite179568208od_a_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_28_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_29_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_30_finite__has__maximal2,axiom,
! [A: set_se1596668135od_a_a,A2: set_Product_prod_a_a] :
( ( finite1145471536od_a_a @ A )
=> ( ( member1838126896od_a_a @ A2 @ A )
=> ? [X: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X @ A )
& ( ord_le1824328871od_a_a @ A2 @ X )
& ! [Xa: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Xa @ A )
=> ( ( ord_le1824328871od_a_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_31_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_32_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite2012248349et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_33_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ A2 @ X )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_34_finite__Collect__conjI,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
| ( finite179568208od_a_a @ ( collec645855634od_a_a @ Q ) ) )
=> ( finite179568208od_a_a
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_35_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
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_36_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
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_37_finite__Collect__conjI,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
| ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) )
=> ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_38_finite__Collect__conjI,axiom,
! [P: produc1572603623od_a_a > $o,Q: produc1572603623od_a_a > $o] :
( ( ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
| ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ Q ) ) )
=> ( finite1664988688od_a_a
@ ( collec1635618130od_a_a
@ ^ [X2: produc1572603623od_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_39_finite__Collect__disjI,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( finite179568208od_a_a
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
& ( finite179568208od_a_a @ ( collec645855634od_a_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_40_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_41_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_42_finite__Collect__disjI,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( finite772653738at_nat
@ ( collec7649004at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
& ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_43_finite__Collect__disjI,axiom,
! [P: produc1572603623od_a_a > $o,Q: produc1572603623od_a_a > $o] :
( ( finite1664988688od_a_a
@ ( collec1635618130od_a_a
@ ^ [X2: produc1572603623od_a_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
& ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_44_finite__Collect__subsets,axiom,
! [A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A )
=> ( finite1457549322at_nat
@ ( collec1606769740at_nat
@ ^ [B2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_45_finite__Collect__subsets,axiom,
! [A: set_Pr1948701895od_a_a] :
( ( finite1664988688od_a_a @ A )
=> ( finite323969008od_a_a
@ ( collec453062450od_a_a
@ ^ [B2: set_Pr1948701895od_a_a] : ( ord_le456379495od_a_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_46_finite__Collect__subsets,axiom,
! [A: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ A )
=> ( finite1145471536od_a_a
@ ( collec183727474od_a_a
@ ^ [B2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_47_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite2012248349et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_48_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_49_finite__SigmaI,axiom,
! [A: set_a,B: a > set_nat] :
( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( finite_finite_nat @ ( B @ A3 ) ) )
=> ( finite1743148308_a_nat @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_50_finite__SigmaI,axiom,
! [A: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( finite_finite_a @ ( B @ A3 ) ) )
=> ( finite1808550458_nat_a @ ( product_Sigma_nat_a @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_51_finite__SigmaI,axiom,
! [A: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( finite_finite_nat @ ( B @ A3 ) ) )
=> ( finite772653738at_nat @ ( produc45129834at_nat @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_52_finite__SigmaI,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( finite_finite_a @ ( B @ A3 ) ) )
=> ( finite179568208od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_53_finite__SigmaI,axiom,
! [A: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
( ( finite179568208od_a_a @ A )
=> ( ! [A3: product_prod_a_a] :
( ( member449909584od_a_a @ A3 @ A )
=> ( finite_finite_a @ ( B @ A3 ) ) )
=> ( finite1919032935_a_a_a @ ( produc1282482655_a_a_a @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_54_finite__SigmaI,axiom,
! [A: set_Product_prod_a_a,B: product_prod_a_a > set_nat] :
( ( finite179568208od_a_a @ A )
=> ( ! [A3: product_prod_a_a] :
( ( member449909584od_a_a @ A3 @ A )
=> ( finite_finite_nat @ ( B @ A3 ) ) )
=> ( finite1837575485_a_nat @ ( produc931712687_a_nat @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_55_finite__SigmaI,axiom,
! [A: set_a,B: a > set_Product_prod_a_a] :
( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( finite179568208od_a_a @ ( B @ A3 ) ) )
=> ( finite676513017od_a_a @ ( produc520147185od_a_a @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_56_finite__SigmaI,axiom,
! [A: set_a,B: a > set_Pr1986765409at_nat] :
( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( finite772653738at_nat @ ( B @ A3 ) ) )
=> ( finite942416723at_nat @ ( produc292491723at_nat @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_57_finite__SigmaI,axiom,
! [A: set_nat,B: nat > set_Product_prod_a_a] :
( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( finite179568208od_a_a @ ( B @ A3 ) ) )
=> ( finite1297454819od_a_a @ ( produc1182842125od_a_a @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_58_finite__SigmaI,axiom,
! [A: set_nat,B: nat > set_Pr1986765409at_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( finite772653738at_nat @ ( B @ A3 ) ) )
=> ( finite277291581at_nat @ ( produc894163943at_nat @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_59_pair__fin__digraph_Oaxioms_I2_J,axiom,
! [G2: pair_p125712459t_unit] :
( ( pair_p1802376898raph_a @ G2 )
=> ( pair_p1864019935ioms_a @ G2 ) ) ).
% pair_fin_digraph.axioms(2)
thf(fact_60_pair__fin__digraph_Oaxioms_I2_J,axiom,
! [G2: pair_p1914262621t_unit] :
( ( pair_p128415500ph_nat @ G2 )
=> ( pair_p1027063983ms_nat @ G2 ) ) ).
% pair_fin_digraph.axioms(2)
thf(fact_61_pair__fin__digraph_Oaxioms_I2_J,axiom,
! [G2: pair_p1765063010t_unit] :
( ( pair_p374947051od_a_a @ G2 )
=> ( pair_p504738056od_a_a @ G2 ) ) ).
% pair_fin_digraph.axioms(2)
thf(fact_62_not__finite__existsD,axiom,
! [P: product_prod_a_a > $o] :
( ~ ( finite179568208od_a_a @ ( collec645855634od_a_a @ P ) )
=> ? [X_1: product_prod_a_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_63_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_64_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_65_not__finite__existsD,axiom,
! [P: product_prod_nat_nat > $o] :
( ~ ( finite772653738at_nat @ ( collec7649004at_nat @ P ) )
=> ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_66_not__finite__existsD,axiom,
! [P: produc1572603623od_a_a > $o] :
( ~ ( finite1664988688od_a_a @ ( collec1635618130od_a_a @ P ) )
=> ? [X_1: produc1572603623od_a_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_67_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_68_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_69_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_70_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_71_pigeonhole__infinite__rel,axiom,
! [A: set_Product_prod_a_a,B: set_a,R: product_prod_a_a > a > $o] :
( ~ ( finite179568208od_a_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite179568208od_a_a
@ ( collec645855634od_a_a
@ ^ [A4: product_prod_a_a] :
( ( member449909584od_a_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_72_pigeonhole__infinite__rel,axiom,
! [A: set_Product_prod_a_a,B: set_nat,R: product_prod_a_a > nat > $o] :
( ~ ( finite179568208od_a_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite179568208od_a_a
@ ( collec645855634od_a_a
@ ^ [A4: product_prod_a_a] :
( ( member449909584od_a_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_73_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_Product_prod_a_a,R: a > product_prod_a_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite179568208od_a_a @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_74_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_Pr1986765409at_nat,R: a > product_prod_nat_nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite772653738at_nat @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: product_prod_nat_nat] :
( ( member701585322at_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: product_prod_nat_nat] :
( ( member701585322at_nat @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_75_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_Product_prod_a_a,R: nat > product_prod_a_a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite179568208od_a_a @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_76_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_Pr1986765409at_nat,R: nat > product_prod_nat_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite772653738at_nat @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: product_prod_nat_nat] :
( ( member701585322at_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: product_prod_nat_nat] :
( ( member701585322at_nat @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_77_pair__fin__digraph_Opair__fin__digraph,axiom,
! [G2: pair_p125712459t_unit] :
( ( pair_p1802376898raph_a @ G2 )
=> ( pair_p1802376898raph_a @ G2 ) ) ).
% pair_fin_digraph.pair_fin_digraph
thf(fact_78_finite__cartesian__product,axiom,
! [A: set_a,B: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite1743148308_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_79_finite__cartesian__product,axiom,
! [A: set_nat,B: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( finite1808550458_nat_a
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_80_finite__cartesian__product,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite772653738at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_81_finite__cartesian__product,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite179568208od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_82_finite__cartesian__product,axiom,
! [A: set_Product_prod_a_a,B: set_a] :
( ( finite179568208od_a_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite1919032935_a_a_a
@ ( produc1282482655_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_83_finite__cartesian__product,axiom,
! [A: set_Product_prod_a_a,B: set_nat] :
( ( finite179568208od_a_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite1837575485_a_nat
@ ( produc931712687_a_nat @ A
@ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_84_finite__cartesian__product,axiom,
! [A: set_a,B: set_Product_prod_a_a] :
( ( finite_finite_a @ A )
=> ( ( finite179568208od_a_a @ B )
=> ( finite676513017od_a_a
@ ( produc520147185od_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_85_finite__cartesian__product,axiom,
! [A: set_a,B: set_Pr1986765409at_nat] :
( ( finite_finite_a @ A )
=> ( ( finite772653738at_nat @ B )
=> ( finite942416723at_nat
@ ( produc292491723at_nat @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_86_finite__cartesian__product,axiom,
! [A: set_nat,B: set_Product_prod_a_a] :
( ( finite_finite_nat @ A )
=> ( ( finite179568208od_a_a @ B )
=> ( finite1297454819od_a_a
@ ( produc1182842125od_a_a @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_87_finite__cartesian__product,axiom,
! [A: set_nat,B: set_Pr1986765409at_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite772653738at_nat @ B )
=> ( finite277291581at_nat
@ ( produc894163943at_nat @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_88_pair__fin__digraph_Opair__finite__verts,axiom,
! [G2: pair_p2041852168t_unit] :
( ( pair_p752841413at_nat @ G2 )
=> ( finite772653738at_nat @ ( pair_p210955889t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_verts
thf(fact_89_pair__fin__digraph_Opair__finite__verts,axiom,
! [G2: pair_p1593840546t_unit] :
( ( pair_p159207083od_a_a @ G2 )
=> ( finite1664988688od_a_a @ ( pair_p1652294923t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_verts
thf(fact_90_pair__fin__digraph_Opair__finite__verts,axiom,
! [G2: pair_p125712459t_unit] :
( ( pair_p1802376898raph_a @ G2 )
=> ( finite_finite_a @ ( pair_p1047056820t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_verts
thf(fact_91_pair__fin__digraph_Opair__finite__verts,axiom,
! [G2: pair_p1914262621t_unit] :
( ( pair_p128415500ph_nat @ G2 )
=> ( finite_finite_nat @ ( pair_p1677060310t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_verts
thf(fact_92_pair__fin__digraph_Opair__finite__verts,axiom,
! [G2: pair_p1765063010t_unit] :
( ( pair_p374947051od_a_a @ G2 )
=> ( finite179568208od_a_a @ ( pair_p447552203t_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_verts
thf(fact_93_finite__has__minimal2,axiom,
! [A: set_se1596668135od_a_a,A2: set_Product_prod_a_a] :
( ( finite1145471536od_a_a @ A )
=> ( ( member1838126896od_a_a @ A2 @ A )
=> ? [X: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X @ A )
& ( ord_le1824328871od_a_a @ X @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Xa @ A )
=> ( ( ord_le1824328871od_a_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_94_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_95_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite2012248349et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_96_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_97_infinite__cartesian__product,axiom,
! [A: set_a,B: set_nat] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite_finite_nat @ B )
=> ~ ( finite1743148308_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_98_infinite__cartesian__product,axiom,
! [A: set_nat,B: set_a] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite_finite_a @ B )
=> ~ ( finite1808550458_nat_a
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_99_infinite__cartesian__product,axiom,
! [A: set_nat,B: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite_finite_nat @ B )
=> ~ ( finite772653738at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_100_infinite__cartesian__product,axiom,
! [A: set_a,B: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite_finite_a @ B )
=> ~ ( finite179568208od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_101_infinite__cartesian__product,axiom,
! [A: set_Product_prod_a_a,B: set_a] :
( ~ ( finite179568208od_a_a @ A )
=> ( ~ ( finite_finite_a @ B )
=> ~ ( finite1919032935_a_a_a
@ ( produc1282482655_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_102_infinite__cartesian__product,axiom,
! [A: set_Product_prod_a_a,B: set_nat] :
( ~ ( finite179568208od_a_a @ A )
=> ( ~ ( finite_finite_nat @ B )
=> ~ ( finite1837575485_a_nat
@ ( produc931712687_a_nat @ A
@ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_103_infinite__cartesian__product,axiom,
! [A: set_a,B: set_Product_prod_a_a] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite179568208od_a_a @ B )
=> ~ ( finite676513017od_a_a
@ ( produc520147185od_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_104_infinite__cartesian__product,axiom,
! [A: set_a,B: set_Pr1986765409at_nat] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite772653738at_nat @ B )
=> ~ ( finite942416723at_nat
@ ( produc292491723at_nat @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_105_infinite__cartesian__product,axiom,
! [A: set_nat,B: set_Product_prod_a_a] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite179568208od_a_a @ B )
=> ~ ( finite1297454819od_a_a
@ ( produc1182842125od_a_a @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_106_infinite__cartesian__product,axiom,
! [A: set_nat,B: set_Pr1986765409at_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ~ ( finite772653738at_nat @ B )
=> ~ ( finite277291581at_nat
@ ( produc894163943at_nat @ A
@ ^ [Uu: nat] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_107_Times__subset__cancel2,axiom,
! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ C )
=> ( ( ord_le456379495od_a_a
@ ( produc304751368od_a_a @ A
@ ^ [Uu: product_prod_a_a] : C )
@ ( produc304751368od_a_a @ B
@ ^ [Uu: product_prod_a_a] : C ) )
= ( ord_le1824328871od_a_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_108_Times__subset__cancel2,axiom,
! [X3: nat,C: set_nat,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member_nat @ X3 @ C )
=> ( ( ord_le492294332_a_nat
@ ( produc931712687_a_nat @ A
@ ^ [Uu: product_prod_a_a] : C )
@ ( produc931712687_a_nat @ B
@ ^ [Uu: product_prod_a_a] : C ) )
= ( ord_le1824328871od_a_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_109_Times__subset__cancel2,axiom,
! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_nat,B: set_nat] :
( ( member449909584od_a_a @ X3 @ C )
=> ( ( ord_le2084554594od_a_a
@ ( produc1182842125od_a_a @ A
@ ^ [Uu: nat] : C )
@ ( produc1182842125od_a_a @ B
@ ^ [Uu: nat] : C ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_110_Times__subset__cancel2,axiom,
! [X3: nat,C: set_nat,A: set_nat,B: set_nat] :
( ( member_nat @ X3 @ C )
=> ( ( ord_le841296385at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : C )
@ ( produc45129834at_nat @ B
@ ^ [Uu: nat] : C ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_111_Times__subset__cancel2,axiom,
! [X3: product_prod_a_a,C: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( member449909584od_a_a @ X3 @ C )
=> ( ( ord_le1816232656od_a_a
@ ( produc520147185od_a_a @ A
@ ^ [Uu: a] : C )
@ ( produc520147185od_a_a @ B
@ ^ [Uu: a] : C ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_112_Times__subset__cancel2,axiom,
! [X3: nat,C: set_nat,A: set_a,B: set_a] :
( ( member_nat @ X3 @ C )
=> ( ( ord_le2073555219_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : C )
@ ( product_Sigma_a_nat @ B
@ ^ [Uu: a] : C ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_113_Times__subset__cancel2,axiom,
! [X3: a,C: set_a,A: set_a,B: set_a] :
( ( member_a @ X3 @ C )
=> ( ( ord_le1824328871od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C )
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : C ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_114_Sigma__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a,D: product_prod_a_a > set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ C )
=> ( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ A )
=> ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le456379495od_a_a @ ( produc304751368od_a_a @ A @ B ) @ ( produc304751368od_a_a @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_115_Sigma__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_nat,D: product_prod_a_a > set_nat] :
( ( ord_le1824328871od_a_a @ A @ C )
=> ( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le492294332_a_nat @ ( produc931712687_a_nat @ A @ B ) @ ( produc931712687_a_nat @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_116_Sigma__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: product_prod_a_a > set_a,D: product_prod_a_a > set_a] :
( ( ord_le1824328871od_a_a @ A @ C )
=> ( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le677315902_a_a_a @ ( produc1282482655_a_a_a @ A @ B ) @ ( produc1282482655_a_a_a @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_117_Sigma__mono,axiom,
! [A: set_nat,C: set_nat,B: nat > set_Product_prod_a_a,D: nat > set_Product_prod_a_a] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le2084554594od_a_a @ ( produc1182842125od_a_a @ A @ B ) @ ( produc1182842125od_a_a @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_118_Sigma__mono,axiom,
! [A: set_nat,C: set_nat,B: nat > set_nat,D: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le841296385at_nat @ ( produc45129834at_nat @ A @ B ) @ ( produc45129834at_nat @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_119_Sigma__mono,axiom,
! [A: set_nat,C: set_nat,B: nat > set_a,D: nat > set_a] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le344568633_nat_a @ ( product_Sigma_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_120_Sigma__mono,axiom,
! [A: set_a,C: set_a,B: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le1824328871od_a_a @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le1816232656od_a_a @ ( produc520147185od_a_a @ A @ B ) @ ( produc520147185od_a_a @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_121_Sigma__mono,axiom,
! [A: set_a,C: set_a,B: a > set_nat,D: a > set_nat] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le2073555219_a_nat @ ( product_Sigma_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_122_Sigma__mono,axiom,
! [A: set_a,C: set_a,B: a > set_a,D: a > set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ ( D @ X ) ) )
=> ( ord_le1824328871od_a_a @ ( product_Sigma_a_a @ A @ B ) @ ( product_Sigma_a_a @ C @ D ) ) ) ) ).
% Sigma_mono
thf(fact_123_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ A )
=> ( member449909584od_a_a @ X @ B ) )
=> ( ord_le1824328871od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_124_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_125_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_126_subset__antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B )
=> ( ( ord_le1824328871od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_127_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_128_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_129_order__refl,axiom,
! [X3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_130_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_131_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_132_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_133_pair__pre__digraph_Oequality,axiom,
! [R2: pair_p125712459t_unit,R3: pair_p125712459t_unit] :
( ( ( pair_p1047056820t_unit @ R2 )
= ( pair_p1047056820t_unit @ R3 ) )
=> ( ( ( pair_p133601421t_unit @ R2 )
= ( pair_p133601421t_unit @ R3 ) )
=> ( ( ( pair_p1896222615t_unit @ R2 )
= ( pair_p1896222615t_unit @ R3 ) )
=> ( R2 = R3 ) ) ) ) ).
% pair_pre_digraph.equality
thf(fact_134_pair__pre__digraph_Oequality,axiom,
! [R2: pair_p1914262621t_unit,R3: pair_p1914262621t_unit] :
( ( ( pair_p1677060310t_unit @ R2 )
= ( pair_p1677060310t_unit @ R3 ) )
=> ( ( ( pair_p715279805t_unit @ R2 )
= ( pair_p715279805t_unit @ R3 ) )
=> ( ( ( pair_p69470259t_unit @ R2 )
= ( pair_p69470259t_unit @ R3 ) )
=> ( R2 = R3 ) ) ) ) ).
% pair_pre_digraph.equality
thf(fact_135_pair__pre__digraph_Oequality,axiom,
! [R2: pair_p1765063010t_unit,R3: pair_p1765063010t_unit] :
( ( ( pair_p447552203t_unit @ R2 )
= ( pair_p447552203t_unit @ R3 ) )
=> ( ( ( pair_p1783210148t_unit @ R2 )
= ( pair_p1783210148t_unit @ R3 ) )
=> ( ( ( pair_p1984658862t_unit @ R2 )
= ( pair_p1984658862t_unit @ R3 ) )
=> ( R2 = R3 ) ) ) ) ).
% pair_pre_digraph.equality
thf(fact_136_Sigma__cong,axiom,
! [A: set_a,B: set_a,C: a > set_a,D: a > set_a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( product_Sigma_a_a @ A @ C )
= ( product_Sigma_a_a @ B @ D ) ) ) ) ).
% Sigma_cong
thf(fact_137_Times__eq__cancel2,axiom,
! [X3: a,C: set_a,A: set_a,B: set_a] :
( ( member_a @ X3 @ C )
=> ( ( ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C )
= ( product_Sigma_a_a @ B
@ ^ [Uu: a] : C ) )
= ( A = B ) ) ) ).
% Times_eq_cancel2
thf(fact_138_Collect__subset,axiom,
! [A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le1824328871od_a_a
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_139_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_140_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_141_less__eq__set__def,axiom,
( ord_le1824328871od_a_a
= ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ord_le1347718902_a_a_o
@ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ A5 )
@ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_142_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_143_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B2: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_144_dual__order_Oantisym,axiom,
! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ B3 @ A2 )
=> ( ( ord_le1824328871od_a_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_145_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_146_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ B4 @ A4 )
& ( ord_le1824328871od_a_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_149_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_150_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_dual__order_Otrans,axiom,
! [B3: set_Product_prod_a_a,A2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ B3 @ A2 )
=> ( ( ord_le1824328871od_a_a @ C2 @ B3 )
=> ( ord_le1824328871od_a_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_153_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_154_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B3 )
=> ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_155_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_156_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: nat,B5: nat] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_157_dual__order_Orefl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_158_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_159_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_160_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_161_order__trans,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X3 @ Y2 )
=> ( ( ord_le1824328871od_a_a @ Y2 @ Z2 )
=> ( ord_le1824328871od_a_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_162_order__trans,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_163_order__trans,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_164_order__trans,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_165_order__class_Oorder_Oantisym,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B3 )
=> ( ( ord_le1824328871od_a_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_166_order__class_Oorder_Oantisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_167_order__class_Oorder_Oantisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_168_order__class_Oorder_Oantisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_169_ord__le__eq__trans,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_170_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_171_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_172_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_173_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_174_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_175_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_177_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X: product_prod_a_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec645855634od_a_a @ P )
= ( collec645855634od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_178_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_179_ord__eq__le__trans,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( A2 = B3 )
=> ( ( ord_le1824328871od_a_a @ B3 @ C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_180_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C2: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_181_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C2: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_182_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_183_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A4 @ B4 )
& ( ord_le1824328871od_a_a @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_184_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_185_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_186_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_187_antisym__conv,axiom,
! [Y2: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ Y2 @ X3 )
=> ( ( ord_le1824328871od_a_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv
thf(fact_188_antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv
thf(fact_189_antisym__conv,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv
thf(fact_190_antisym__conv,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv
thf(fact_191_le__cases3,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_192_order_Otrans,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B3 )
=> ( ( ord_le1824328871od_a_a @ B3 @ C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_193_order_Otrans,axiom,
! [A2: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_194_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_195_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_196_le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% le_cases
thf(fact_197_eq__refl,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( X3 = Y2 )
=> ( ord_le1824328871od_a_a @ X3 @ Y2 ) ) ).
% eq_refl
thf(fact_198_eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% eq_refl
thf(fact_199_eq__refl,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).
% eq_refl
thf(fact_200_eq__refl,axiom,
! [X3: set_a,Y2: set_a] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% eq_refl
thf(fact_201_linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linear
thf(fact_202_antisym,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X3 @ Y2 )
=> ( ( ord_le1824328871od_a_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% antisym
thf(fact_203_antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% antisym
thf(fact_204_antisym,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% antisym
thf(fact_205_antisym,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% antisym
thf(fact_206_eq__iff,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X2 @ Y3 )
& ( ord_le1824328871od_a_a @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_207_eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_208_eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_209_eq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_210_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_211_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_212_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_213_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_214_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_215_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_216_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_217_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_218_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_219_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
( ( ord_le1824328871od_a_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_220_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_221_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_222_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C2: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_223_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_224_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C2: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_225_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C2: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_226_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_nat > set_a,B3: set_nat,C2: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_227_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_a > set_nat,B3: set_a,C2: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_228_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_229_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Product_prod_a_a > nat,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le1824328871od_a_a @ B3 @ C2 )
=> ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_230_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_231_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_232_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_233_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_234_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_235_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_236_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_237_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_238_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_239_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
( ( ord_le1824328871od_a_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_240_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_241_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_243_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_244_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_245_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_246_order__subst1,axiom,
! [A2: set_nat,F: set_a > set_nat,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_247_order__subst1,axiom,
! [A2: set_a,F: set_nat > set_a,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_248_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_249_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B3: nat,C2: nat] :
( ( ord_le1824328871od_a_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le1824328871od_a_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1824328871od_a_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_250_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) )
= ( ! [X2: product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_251_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_252_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_253_set__eq__subset,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A5 @ B2 )
& ( ord_le1824328871od_a_a @ B2 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_254_set__eq__subset,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [A5: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_255_set__eq__subset,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [A5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_256_subset__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B )
=> ( ( ord_le1824328871od_a_a @ B @ C )
=> ( ord_le1824328871od_a_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_257_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_258_subset__trans,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 ) ) ) ).
% subset_trans
thf(fact_259_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X: product_prod_a_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_260_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_261_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_262_subset__refl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A @ A ) ).
% subset_refl
thf(fact_263_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_264_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_265_subset__iff,axiom,
( ord_le1824328871od_a_a
= ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [T2: product_prod_a_a] :
( ( member449909584od_a_a @ T2 @ A5 )
=> ( member449909584od_a_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_266_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_267_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B2: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_268_equalityD2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ( ord_le1824328871od_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_269_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_270_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_271_equalityD1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ( ord_le1824328871od_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_272_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_273_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_274_subset__eq,axiom,
( ord_le1824328871od_a_a
= ( ^ [A5: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A5 )
=> ( member449909584od_a_a @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_275_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_276_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B2: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_277_equalityE,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ~ ( ( ord_le1824328871od_a_a @ A @ B )
=> ~ ( ord_le1824328871od_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_278_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_279_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_280_subsetD,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B )
=> ( ( member449909584od_a_a @ C2 @ A )
=> ( member449909584od_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_281_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_282_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_283_in__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B )
=> ( ( member449909584od_a_a @ X3 @ A )
=> ( member449909584od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_284_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_285_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_286_pred__subset__eq,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le1347718902_a_a_o
@ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ R )
@ ^ [X2: product_prod_a_a] : ( member449909584od_a_a @ X2 @ S ) )
= ( ord_le1824328871od_a_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_287_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_288_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_289_pair__pre__digraph_Osurjective,axiom,
! [R2: pair_p125712459t_unit] :
( R2
= ( pair_p1621517565t_unit @ ( pair_p1047056820t_unit @ R2 ) @ ( pair_p133601421t_unit @ R2 ) @ ( pair_p1896222615t_unit @ R2 ) ) ) ).
% pair_pre_digraph.surjective
thf(fact_290_pair__pre__digraph_Osurjective,axiom,
! [R2: pair_p1914262621t_unit] :
( R2
= ( pair_p1167410509t_unit @ ( pair_p1677060310t_unit @ R2 ) @ ( pair_p715279805t_unit @ R2 ) @ ( pair_p69470259t_unit @ R2 ) ) ) ).
% pair_pre_digraph.surjective
thf(fact_291_pair__pre__digraph_Osurjective,axiom,
! [R2: pair_p1765063010t_unit] :
( R2
= ( pair_p398687508t_unit @ ( pair_p447552203t_unit @ R2 ) @ ( pair_p1783210148t_unit @ R2 ) @ ( pair_p1984658862t_unit @ R2 ) ) ) ).
% pair_pre_digraph.surjective
thf(fact_292_member__product,axiom,
! [X3: product_prod_a_a,A: set_a,B: set_a] :
( ( member449909584od_a_a @ X3 @ ( product_product_a_a @ A @ B ) )
= ( member449909584od_a_a @ X3
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) ) ) ).
% member_product
thf(fact_293_Product__Type_Oproduct__def,axiom,
( product_product_a_a
= ( ^ [A5: set_a,B2: set_a] :
( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B2 ) ) ) ).
% Product_Type.product_def
thf(fact_294_subset__Collect__iff,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ B @ A )
=> ( ( ord_le1824328871od_a_a @ B
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_295_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_296_subset__Collect__iff,axiom,
! [B: set_a,A: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_297_subset__CollectI,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,Q: product_prod_a_a > $o,P: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ B @ A )
=> ( ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le1824328871od_a_a
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_298_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_299_subset__CollectI,axiom,
! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_300_Collect__restrict,axiom,
! [X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le1824328871od_a_a
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_301_Collect__restrict,axiom,
! [X4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_302_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_303_prop__restrict,axiom,
! [X3: product_prod_a_a,Z3: set_Product_prod_a_a,X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ X3 @ Z3 )
=> ( ( ord_le1824328871od_a_a @ Z3
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_304_prop__restrict,axiom,
! [X3: nat,Z3: set_nat,X4: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_305_prop__restrict,axiom,
! [X3: a,Z3: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X3 @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_306_conj__subset__def,axiom,
! [A: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ A
@ ( collec645855634od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le1824328871od_a_a @ A @ ( collec645855634od_a_a @ P ) )
& ( ord_le1824328871od_a_a @ A @ ( collec645855634od_a_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_307_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_308_conj__subset__def,axiom,
! [A: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_309_pair__pre__digraph_Oselect__convs_I2_J,axiom,
! [Pverts: set_a,Parcs: set_Product_prod_a_a,More: product_unit] :
( ( pair_p133601421t_unit @ ( pair_p1621517565t_unit @ Pverts @ Parcs @ More ) )
= Parcs ) ).
% pair_pre_digraph.select_convs(2)
thf(fact_310_pair__pre__digraph_Oselect__convs_I2_J,axiom,
! [Pverts: set_nat,Parcs: set_Pr1986765409at_nat,More: product_unit] :
( ( pair_p715279805t_unit @ ( pair_p1167410509t_unit @ Pverts @ Parcs @ More ) )
= Parcs ) ).
% pair_pre_digraph.select_convs(2)
thf(fact_311_pair__pre__digraph_Oselect__convs_I2_J,axiom,
! [Pverts: set_Product_prod_a_a,Parcs: set_Pr1948701895od_a_a,More: product_unit] :
( ( pair_p1783210148t_unit @ ( pair_p398687508t_unit @ Pverts @ Parcs @ More ) )
= Parcs ) ).
% pair_pre_digraph.select_convs(2)
thf(fact_312_pair__pre__digraph_Oselect__convs_I1_J,axiom,
! [Pverts: set_a,Parcs: set_Product_prod_a_a,More: product_unit] :
( ( pair_p1047056820t_unit @ ( pair_p1621517565t_unit @ Pverts @ Parcs @ More ) )
= Pverts ) ).
% pair_pre_digraph.select_convs(1)
thf(fact_313_pair__pre__digraph_Oselect__convs_I1_J,axiom,
! [Pverts: set_nat,Parcs: set_Pr1986765409at_nat,More: product_unit] :
( ( pair_p1677060310t_unit @ ( pair_p1167410509t_unit @ Pverts @ Parcs @ More ) )
= Pverts ) ).
% pair_pre_digraph.select_convs(1)
thf(fact_314_pair__pre__digraph_Oselect__convs_I1_J,axiom,
! [Pverts: set_Product_prod_a_a,Parcs: set_Pr1948701895od_a_a,More: product_unit] :
( ( pair_p447552203t_unit @ ( pair_p398687508t_unit @ Pverts @ Parcs @ More ) )
= Pverts ) ).
% pair_pre_digraph.select_convs(1)
thf(fact_315_Fpow__def,axiom,
( finite361944167at_nat
= ( ^ [A5: set_Pr1986765409at_nat] :
( collec1606769740at_nat
@ ^ [X5: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X5 @ A5 )
& ( finite772653738at_nat @ X5 ) ) ) ) ) ).
% Fpow_def
thf(fact_316_Fpow__def,axiom,
( finite702915405od_a_a
= ( ^ [A5: set_Pr1948701895od_a_a] :
( collec453062450od_a_a
@ ^ [X5: set_Pr1948701895od_a_a] :
( ( ord_le456379495od_a_a @ X5 @ A5 )
& ( finite1664988688od_a_a @ X5 ) ) ) ) ) ).
% Fpow_def
thf(fact_317_Fpow__def,axiom,
( finite351630733od_a_a
= ( ^ [A5: set_Product_prod_a_a] :
( collec183727474od_a_a
@ ^ [X5: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X5 @ A5 )
& ( finite179568208od_a_a @ X5 ) ) ) ) ) ).
% Fpow_def
thf(fact_318_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A5: set_nat] :
( collect_set_nat
@ ^ [X5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ A5 )
& ( finite_finite_nat @ X5 ) ) ) ) ) ).
% Fpow_def
thf(fact_319_Fpow__def,axiom,
( finite_Fpow_a
= ( ^ [A5: set_a] :
( collect_set_a
@ ^ [X5: set_a] :
( ( ord_less_eq_set_a @ X5 @ A5 )
& ( finite_finite_a @ X5 ) ) ) ) ) ).
% Fpow_def
thf(fact_320_pair__fin__digraph__def,axiom,
( pair_p1802376898raph_a
= ( ^ [G: pair_p125712459t_unit] :
( ( pair_p68905728raph_a @ G )
& ( pair_p1864019935ioms_a @ G ) ) ) ) ).
% pair_fin_digraph_def
thf(fact_321_pair__fin__digraph__def,axiom,
( pair_p128415500ph_nat
= ( ^ [G: pair_p1914262621t_unit] :
( ( pair_p1515597646ph_nat @ G )
& ( pair_p1027063983ms_nat @ G ) ) ) ) ).
% pair_fin_digraph_def
thf(fact_322_pair__fin__digraph__def,axiom,
( pair_p374947051od_a_a
= ( ^ [G: pair_p1765063010t_unit] :
( ( pair_p646030121od_a_a @ G )
& ( pair_p504738056od_a_a @ G ) ) ) ) ).
% pair_fin_digraph_def
thf(fact_323_pair__fin__digraph_Ointro,axiom,
! [G2: pair_p125712459t_unit] :
( ( pair_p68905728raph_a @ G2 )
=> ( ( pair_p1864019935ioms_a @ G2 )
=> ( pair_p1802376898raph_a @ G2 ) ) ) ).
% pair_fin_digraph.intro
thf(fact_324_pair__fin__digraph_Ointro,axiom,
! [G2: pair_p1914262621t_unit] :
( ( pair_p1515597646ph_nat @ G2 )
=> ( ( pair_p1027063983ms_nat @ G2 )
=> ( pair_p128415500ph_nat @ G2 ) ) ) ).
% pair_fin_digraph.intro
thf(fact_325_pair__fin__digraph_Ointro,axiom,
! [G2: pair_p1765063010t_unit] :
( ( pair_p646030121od_a_a @ G2 )
=> ( ( pair_p504738056od_a_a @ G2 )
=> ( pair_p374947051od_a_a @ G2 ) ) ) ).
% pair_fin_digraph.intro
thf(fact_326_finite__cartesian__product__iff,axiom,
! [A: set_a,B: set_nat] :
( ( finite1743148308_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B ) )
= ( ( A = bot_bot_set_a )
| ( B = bot_bot_set_nat )
| ( ( finite_finite_a @ A )
& ( finite_finite_nat @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_327_finite__cartesian__product__iff,axiom,
! [A: set_nat,B: set_a] :
( ( finite1808550458_nat_a
@ ( product_Sigma_nat_a @ A
@ ^ [Uu: nat] : B ) )
= ( ( A = bot_bot_set_nat )
| ( B = bot_bot_set_a )
| ( ( finite_finite_nat @ A )
& ( finite_finite_a @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_328_finite__cartesian__product__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite772653738at_nat
@ ( produc45129834at_nat @ A
@ ^ [Uu: nat] : B ) )
= ( ( A = bot_bot_set_nat )
| ( B = bot_bot_set_nat )
| ( ( finite_finite_nat @ A )
& ( finite_finite_nat @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_329_finite__cartesian__product__iff,axiom,
! [A: set_a,B: set_a] :
( ( finite179568208od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) )
= ( ( A = bot_bot_set_a )
| ( B = bot_bot_set_a )
| ( ( finite_finite_a @ A )
& ( finite_finite_a @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_330_finite__cartesian__product__iff,axiom,
! [A: set_Product_prod_a_a,B: set_a] :
( ( finite1919032935_a_a_a
@ ( produc1282482655_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : B ) )
= ( ( A = bot_bo2131659635od_a_a )
| ( B = bot_bot_set_a )
| ( ( finite179568208od_a_a @ A )
& ( finite_finite_a @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_331_finite__cartesian__product__iff,axiom,
! [A: set_Product_prod_a_a,B: set_nat] :
( ( finite1837575485_a_nat
@ ( produc931712687_a_nat @ A
@ ^ [Uu: product_prod_a_a] : B ) )
= ( ( A = bot_bo2131659635od_a_a )
| ( B = bot_bot_set_nat )
| ( ( finite179568208od_a_a @ A )
& ( finite_finite_nat @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_332_finite__cartesian__product__iff,axiom,
! [A: set_a,B: set_Product_prod_a_a] :
( ( finite676513017od_a_a
@ ( produc520147185od_a_a @ A
@ ^ [Uu: a] : B ) )
= ( ( A = bot_bot_set_a )
| ( B = bot_bo2131659635od_a_a )
| ( ( finite_finite_a @ A )
& ( finite179568208od_a_a @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_333_finite__cartesian__product__iff,axiom,
! [A: set_a,B: set_Pr1986765409at_nat] :
( ( finite942416723at_nat
@ ( produc292491723at_nat @ A
@ ^ [Uu: a] : B ) )
= ( ( A = bot_bot_set_a )
| ( B = bot_bo2130386637at_nat )
| ( ( finite_finite_a @ A )
& ( finite772653738at_nat @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_334_finite__cartesian__product__iff,axiom,
! [A: set_nat,B: set_Product_prod_a_a] :
( ( finite1297454819od_a_a
@ ( produc1182842125od_a_a @ A
@ ^ [Uu: nat] : B ) )
= ( ( A = bot_bot_set_nat )
| ( B = bot_bo2131659635od_a_a )
| ( ( finite_finite_nat @ A )
& ( finite179568208od_a_a @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_335_finite__cartesian__product__iff,axiom,
! [A: set_nat,B: set_Pr1986765409at_nat] :
( ( finite277291581at_nat
@ ( produc894163943at_nat @ A
@ ^ [Uu: nat] : B ) )
= ( ( A = bot_bot_set_nat )
| ( B = bot_bo2130386637at_nat )
| ( ( finite_finite_nat @ A )
& ( finite772653738at_nat @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_336_empty__iff,axiom,
! [C2: product_prod_a_a] :
~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% empty_iff
thf(fact_337_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_338_all__not__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ! [X2: product_prod_a_a] :
~ ( member449909584od_a_a @ X2 @ A ) )
= ( A = bot_bo2131659635od_a_a ) ) ).
% all_not_in_conv
thf(fact_339_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_340_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec645855634od_a_a @ P )
= bot_bo2131659635od_a_a )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_341_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_342_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo2131659635od_a_a
= ( collec645855634od_a_a @ P ) )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_343_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_344_empty__subsetI,axiom,
! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A ) ).
% empty_subsetI
thf(fact_345_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_346_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_347_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_348_calculation,axiom,
( ( finite_finite_a @ ( pair_p1047056820t_unit @ g ) )
& ( ( finite_card_a @ ( pair_p1047056820t_unit @ g ) )
= n )
& ( ( pair_p133601421t_unit @ g )
= ( collec645855634od_a_a
@ ( produc1833107820_a_a_o
@ ^ [U: a,V: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ U @ V )
@ ( product_Sigma_a_a @ ( pair_p1047056820t_unit @ g )
@ ^ [Uu: a] : ( pair_p1047056820t_unit @ g ) ) )
& ( U != V ) ) ) ) ) ) ).
% calculation
thf(fact_349_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_350_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_351_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_352_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_353_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U2 ) ) ) ) ).
% finite_less_ub
% Conjectures (1)
thf(conj_0,conjecture,
finite179568208od_a_a @ ( pair_p133601421t_unit @ g ) ).
%------------------------------------------------------------------------------