TPTP Problem File: ITP093^2.p
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%------------------------------------------------------------------------------
% File : ITP093^2 : 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 : 0.00 v7.5.0
% Syntax : Number of formulae : 338 ( 123 unt; 48 typ; 0 def)
% Number of atoms : 760 ( 210 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4141 ( 81 ~; 12 |; 64 &;3618 @)
% ( 0 <=>; 366 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 266 ( 266 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 45 usr; 4 con; 0-5 aty)
% Number of variables : 1258 ( 120 ^;1065 !; 18 ?;1258 :)
% ( 55 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:07.273
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
pair_p1731315293ph_ext: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pair__Digraph_Opair__bidirected__digraph,type,
pair_p1033978749igraph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__digraph,type,
pair_pair_digraph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__fin__digraph,type,
pair_p953238171igraph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms,type,
pair_p1373739070axioms:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__graph,type,
pair_pair_graph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__loopfree__digraph,type,
pair_p1282390076igraph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__loopfree__digraph__axioms,type,
pair_p814123999axioms:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore,type,
pair_pair_pre_more:
!>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > Z ) ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore__update,type,
pair_p1108258481update:
!>[Z: $tType,A: $tType] : ( ( Z > Z ) > ( pair_p1731315293ph_ext @ A @ Z ) > ( pair_p1731315293ph_ext @ A @ Z ) ) ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
pair_p1152105390ph_ext:
!>[A: $tType,Z: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > Z > ( pair_p1731315293ph_ext @ A @ Z ) ) ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs,type,
pair_pair_pre_parcs:
!>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts,type,
pair_pair_pre_pverts:
!>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > ( set @ A ) ) ).
thf(sy_c_Pair__Digraph_Opair__pseudo__graph,type,
pair_p389585203_graph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__sym__digraph,type,
pair_p460180511igraph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__sym__digraph__axioms,type,
pair_p477239746axioms:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Opair__wf__digraph,type,
pair_pair_wf_digraph:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
thf(sy_c_Pair__Digraph_Osubdivide,type,
pair_subdivide:
!>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > ( pair_p1731315293ph_ext @ A @ product_unit ) ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_G,type,
g: pair_p1731315293ph_ext @ a @ product_unit ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (256)
thf(fact_0_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A2: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_1__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062,axiom,
( finite_finite2 @ ( product_prod @ a @ a )
@ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
@ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) ) ).
% \<open>finite (pverts G \<times> pverts G)\<close>
thf(fact_2__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062,axiom,
( ord_less_eq @ ( set @ ( product_prod @ a @ a ) ) @ ( pair_pair_pre_parcs @ a @ product_unit @ g )
@ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
@ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) ) ).
% \<open>parcs G \<subseteq> pverts G \<times> pverts G\<close>
thf(fact_3_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ? [X_1: B] : ( P @ X @ X_1 ) )
=> ? [F: A > B] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 @ ( F @ X2 ) ) ) ) ) ).
% finite_set_choice
thf(fact_4_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_5_pair__fin__digraph__axioms__def,axiom,
! [A: $tType] :
( ( pair_p1373739070axioms @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G ) )
& ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G ) ) ) ) ) ).
% pair_fin_digraph_axioms_def
thf(fact_6_pair__fin__digraph__axioms_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( pair_p1373739070axioms @ A @ G2 ) ) ) ).
% pair_fin_digraph_axioms.intro
thf(fact_7_pair__fin__digraph_Opair__finite__arcs,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_arcs
thf(fact_8_finite__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_9_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_10_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_11_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A4 @ A3 )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less_eq @ A @ A4 @ X )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_12_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_13_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_14_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_15_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( finite_finite2 @ B @ ( B2 @ A5 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_16_pair__fin__digraph_Oaxioms_I2_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( pair_p1373739070axioms @ A @ G2 ) ) ).
% pair_fin_digraph.axioms(2)
thf(fact_17_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_18_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: B] :
( ( member @ B @ X @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A3 )
& ( R @ A6 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_19_pair__fin__digraph_Opair__fin__digraph,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( pair_p953238171igraph @ A @ G2 ) ) ).
% pair_fin_digraph.pair_fin_digraph
thf(fact_20_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_21_pair__fin__digraph_Opair__finite__verts,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ).
% pair_fin_digraph.pair_finite_verts
thf(fact_22_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A4 @ A3 )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less_eq @ A @ X @ A4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_23_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ~ ( finite_finite2 @ B @ B2 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_24_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X4: A,C2: set @ A,A3: set @ B,B2: set @ B] :
( ( member @ A @ X4 @ C2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : C2 )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : C2 ) )
= ( ord_less_eq @ ( set @ B ) @ A3 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_25_Sigma__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,C2: set @ A,B2: A > ( set @ B ),D: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X ) @ ( D @ X ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_26_subsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% subsetI
thf(fact_27_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_28_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_29_pair__pre__digraph_Oequality,axiom,
! [Z: $tType,A: $tType,R2: pair_p1731315293ph_ext @ A @ Z,R3: pair_p1731315293ph_ext @ A @ Z] :
( ( ( pair_pair_pre_pverts @ A @ Z @ R2 )
= ( pair_pair_pre_pverts @ A @ Z @ R3 ) )
=> ( ( ( pair_pair_pre_parcs @ A @ Z @ R2 )
= ( pair_pair_pre_parcs @ A @ Z @ R3 ) )
=> ( ( ( pair_pair_pre_more @ A @ Z @ R2 )
= ( pair_pair_pre_more @ A @ Z @ R3 ) )
=> ( R2 = R3 ) ) ) ) ).
% pair_pre_digraph.equality
thf(fact_30_Sigma__cong,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ A,C2: A > ( set @ B ),D: A > ( set @ B )] :
( ( A3 = B2 )
=> ( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( product_Sigma @ A @ B @ A3 @ C2 )
= ( product_Sigma @ A @ B @ B2 @ D ) ) ) ) ).
% Sigma_cong
thf(fact_31_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X4: A,C2: set @ A,A3: set @ B,B2: set @ B] :
( ( member @ A @ X4 @ C2 )
=> ( ( ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : C2 )
= ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : C2 ) )
= ( A3 = B2 ) ) ) ).
% Times_eq_cancel2
thf(fact_32_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_33_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B3: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A2 )
@ ^ [X3: A] : ( member @ A @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_34_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_35_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X4: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_36_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X4: A,Q: A > $o] :
( ( P @ X4 )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_37_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_38_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_39_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ B4 )
=> ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_40_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: A,B6: A] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_41_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_42_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_43_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_44_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C3 )
=> ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G3: A > B] :
( ! [X: A] :
( ( F2 @ X )
= ( G3 @ X ) )
=> ( F2 = G3 ) ) ).
% ext
thf(fact_49_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C3: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C3 )
=> ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_50_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_51_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X4: A] :
( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_52_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y2: A,Z3: A] :
( ( ( ord_less_eq @ A @ X4 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_53_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C3 )
=> ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% order.trans
thf(fact_54_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% le_cases
thf(fact_55_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y2: A] :
( ( X4 = Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% eq_refl
thf(fact_56_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% linear
thf(fact_57_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ) ).
% antisym
thf(fact_58_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_59_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F2: A > B,C3: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F2 @ B4 )
= C3 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F2: B > A,B4: B,C3: B] :
( ( A4
= ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C3 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_61_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C3 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_62_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B4: B,C3: B] :
( ( ord_less_eq @ A @ A4 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C3 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_63_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G4: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_64_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G3: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G3 @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G3 ) ) ) ).
% le_funI
thf(fact_65_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G3: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G3 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G3 @ X4 ) ) ) ) ).
% le_funE
thf(fact_66_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G3: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G3 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G3 @ X4 ) ) ) ) ).
% le_funD
thf(fact_67_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_68_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z2: set @ A] : Y = Z2 )
= ( ^ [A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_69_subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_70_Collect__mono,axiom,
! [A: $tType,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_71_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_72_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B3: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A2 )
=> ( member @ A @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_73_equalityD2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_74_equalityD1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_75_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B3: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_76_equalityE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_77_subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C3 @ A3 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_78_in__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_79_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_80_pair__pre__digraph_Osurjective,axiom,
! [Z: $tType,A: $tType,R2: pair_p1731315293ph_ext @ A @ Z] :
( R2
= ( pair_p1152105390ph_ext @ A @ Z @ ( pair_pair_pre_pverts @ A @ Z @ R2 ) @ ( pair_pair_pre_parcs @ A @ Z @ R2 ) @ ( pair_pair_pre_more @ A @ Z @ R2 ) ) ) ).
% pair_pre_digraph.surjective
thf(fact_81_member__product,axiom,
! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( product_product @ A @ B @ A3 @ B2 ) )
= ( member @ ( product_prod @ A @ B ) @ X4
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) ) ).
% member_product
thf(fact_82_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A2: set @ A,B3: set @ B] :
( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B3 ) ) ) ).
% Product_Type.product_def
thf(fact_83_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A3: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_84_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_85_Collect__restrict,axiom,
! [A: $tType,X5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_86_prop__restrict,axiom,
! [A: $tType,X4: A,Z4: set @ A,X5: set @ A,P: A > $o] :
( ( member @ A @ X4 @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_87_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_88_pair__pre__digraph_Oext__inject,axiom,
! [A: $tType,Z: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z,Pverts2: set @ A,Parcs2: set @ ( product_prod @ A @ A ),More2: Z] :
( ( ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More )
= ( pair_p1152105390ph_ext @ A @ Z @ Pverts2 @ Parcs2 @ More2 ) )
= ( ( Pverts = Pverts2 )
& ( Parcs = Parcs2 )
& ( More = More2 ) ) ) ).
% pair_pre_digraph.ext_inject
thf(fact_89_pair__pre__digraph_Oinduct__scheme,axiom,
! [Z: $tType,A: $tType,P: ( pair_p1731315293ph_ext @ A @ Z ) > $o,R2: pair_p1731315293ph_ext @ A @ Z] :
( ! [Pverts3: set @ A,Parcs3: set @ ( product_prod @ A @ A ),More3: Z] : ( P @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts3 @ Parcs3 @ More3 ) )
=> ( P @ R2 ) ) ).
% pair_pre_digraph.induct_scheme
thf(fact_90_pair__pre__digraph_Ocases__scheme,axiom,
! [A: $tType,Z: $tType,R2: pair_p1731315293ph_ext @ A @ Z] :
~ ! [Pverts3: set @ A,Parcs3: set @ ( product_prod @ A @ A ),More3: Z] :
( R2
!= ( pair_p1152105390ph_ext @ A @ Z @ Pverts3 @ Parcs3 @ More3 ) ) ).
% pair_pre_digraph.cases_scheme
thf(fact_91_pair__pre__digraph_Oselect__convs_I2_J,axiom,
! [Z: $tType,A: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
( ( pair_pair_pre_parcs @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
= Parcs ) ).
% pair_pre_digraph.select_convs(2)
thf(fact_92_pair__pre__digraph_Oselect__convs_I1_J,axiom,
! [Z: $tType,A: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
( ( pair_pair_pre_pverts @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
= Pverts ) ).
% pair_pre_digraph.select_convs(1)
thf(fact_93_pair__pre__digraph_Oselect__convs_I3_J,axiom,
! [A: $tType,Z: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
( ( pair_pair_pre_more @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
= More ) ).
% pair_pre_digraph.select_convs(3)
thf(fact_94_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A2: set @ A] :
( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
& ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_95_pair__fin__digraph__def,axiom,
! [A: $tType] :
( ( pair_p953238171igraph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_wf_digraph @ A @ G )
& ( pair_p1373739070axioms @ A @ G ) ) ) ) ).
% pair_fin_digraph_def
thf(fact_96_pair__fin__digraph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_wf_digraph @ A @ G2 )
=> ( ( pair_p1373739070axioms @ A @ G2 )
=> ( pair_p953238171igraph @ A @ G2 ) ) ) ).
% pair_fin_digraph.intro
thf(fact_97_pair__pre__digraph_Ounfold__congs_I3_J,axiom,
! [A: $tType,A7: $tType,R2: pair_p1731315293ph_ext @ A7 @ A,R3: pair_p1731315293ph_ext @ A7 @ A,V: A,F2: A > A,F4: A > A] :
( ( R2 = R3 )
=> ( ( ( pair_pair_pre_more @ A7 @ A @ R3 )
= V )
=> ( ! [V2: A] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F4 @ V2 ) ) )
=> ( ( pair_p1108258481update @ A @ A7 @ F2 @ R2 )
= ( pair_p1108258481update @ A @ A7 @ F4 @ R3 ) ) ) ) ) ).
% pair_pre_digraph.unfold_congs(3)
thf(fact_98_pair__pre__digraph_Ofold__congs_I3_J,axiom,
! [A: $tType,A7: $tType,R2: pair_p1731315293ph_ext @ A7 @ A,R3: pair_p1731315293ph_ext @ A7 @ A,V: A,F2: A > A,F4: A > A] :
( ( R2 = R3 )
=> ( ( ( pair_pair_pre_more @ A7 @ A @ R3 )
= V )
=> ( ! [V2: A] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F4 @ V2 ) ) )
=> ( ( pair_p1108258481update @ A @ A7 @ F2 @ R2 )
= ( pair_p1108258481update @ A @ A7 @ F4 @ R3 ) ) ) ) ) ).
% pair_pre_digraph.fold_congs(3)
thf(fact_99_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A3 )
& ( finite_finite2 @ B @ B2 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_100_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_101_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_102_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_103_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_104_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_105_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_106_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_107_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_108_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_109_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_110_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% empty_def
thf(fact_111_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I: set @ A,X5: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I @ X5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ I )
=> ( ( X5 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_112_empty__in__Fpow,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ).
% empty_in_Fpow
thf(fact_113_Fpow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_Fpow @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_114_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_115_equals0D,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A3 ) ) ).
% equals0D
thf(fact_116_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_117_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_118_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_119_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
=> ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_120_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
= ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_121_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_122_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_123_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_124_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X: A] :
~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_125_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B,C2: set @ A,D: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 )
= ( product_Sigma @ A @ B @ C2
@ ^ [Uu: A] : D ) )
= ( ( ( A3 = C2 )
& ( B2 = D ) )
| ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C2
= ( bot_bot @ ( set @ A ) ) )
| ( D
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_126_Fpow__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% Fpow_mono
thf(fact_127_pair__fin__digraph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% pair_fin_digraph.axioms(1)
thf(fact_128_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_129_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_130_pair__pre__digraph_Oupdate__convs_I3_J,axiom,
! [A: $tType,Z: $tType,More2: Z > Z,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
( ( pair_p1108258481update @ Z @ A @ More2 @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
= ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ ( More2 @ More ) ) ) ).
% pair_pre_digraph.update_convs(3)
thf(fact_131_times__subset__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,C2: set @ B,B2: set @ A,D: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C2 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : D ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( C2
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
& ( ord_less_eq @ ( set @ B ) @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_132_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ( B2 @ X3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( finite_finite2 @ B @ ( B2 @ A5 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% finite_SigmaI2
thf(fact_133_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_cartesian_productD1
thf(fact_134_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_cartesian_productD2
thf(fact_135_finite__transitivity__chain,axiom,
! [A: $tType,A3: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X: A] :
~ ( R @ X @ X )
=> ( ! [X: A,Y4: A,Z5: A] :
( ( R @ X @ Y4 )
=> ( ( R @ Y4 @ Z5 )
=> ( R @ X @ Z5 ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( R @ X @ Y5 ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_136_pair__wf__digraph_Opair__wf__digraph__subdivide,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
( ( pair_pair_wf_digraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( pair_pair_wf_digraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% pair_wf_digraph.pair_wf_digraph_subdivide
thf(fact_137_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A2: set @ A] :
( A2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_138_pair__wf__digraph_Oin__arcsD2,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
( ( pair_pair_wf_digraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( member @ A @ V3 @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ) ).
% pair_wf_digraph.in_arcsD2
thf(fact_139_pair__wf__digraph_Oin__arcsD1,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
( ( pair_pair_wf_digraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( member @ A @ U @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ) ).
% pair_wf_digraph.in_arcsD1
thf(fact_140_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B4: B,A8: A,B7: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= ( product_Pair @ A @ B @ A8 @ B7 ) )
= ( ( A4 = A8 )
& ( B4 = B7 ) ) ) ).
% old.prod.inject
thf(fact_141_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_142_SigmaI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B4: B,B2: A > ( set @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ B @ B4 @ ( B2 @ A4 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_143_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
= ( ( member @ A @ A4 @ A3 )
& ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_144_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_145_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_146_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_147_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_148_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
~ ! [A5: A,B6: B] :
( Y2
!= ( product_Pair @ A @ B @ A5 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_149_prod__induct7,axiom,
! [G5: $tType,F5: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) )] :
( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5,G6: G5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G5 ) @ E3 @ ( product_Pair @ F5 @ G5 @ F @ G6 ) ) ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct7
thf(fact_150_prod__induct6,axiom,
! [F5: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) )] :
( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F5 ) @ D3 @ ( product_Pair @ E2 @ F5 @ E3 @ F ) ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct6
thf(fact_151_prod__induct5,axiom,
! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct5
thf(fact_152_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A5: A,B6: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B6 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct4
thf(fact_153_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B6: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_154_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F5: $tType,G5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) )] :
~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5,G6: G5] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G5 ) @ E3 @ ( product_Pair @ F5 @ G5 @ F @ G6 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_155_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) )] :
~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F5 ) @ D3 @ ( product_Pair @ E2 @ F5 @ E3 @ F ) ) ) ) ) ) ).
% prod_cases6
thf(fact_156_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_157_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A5: A,B6: B,C4: C,D3: D2] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B6 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_158_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B6: B,C4: C] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) ) ).
% prod_cases3
thf(fact_159_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B4: B,A8: A,B7: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ~ ( ( A4 = A8 )
=> ( B4 != B7 ) ) ) ).
% Pair_inject
thf(fact_160_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_161_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% surj_pair
thf(fact_162_subdivide_Oinduct,axiom,
! [A: $tType,P: ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > $o,A0: pair_p1731315293ph_ext @ A @ product_unit,A1: product_prod @ A @ A,A22: A] :
( ! [G7: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V2: A,X_12: A] : ( P @ G7 @ ( product_Pair @ A @ A @ U2 @ V2 ) @ X_12 )
=> ( P @ A0 @ A1 @ A22 ) ) ).
% subdivide.induct
thf(fact_163_subdivide_Ocases,axiom,
! [A: $tType,X4: product_prod @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A )] :
~ ! [G7: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V2: A,W2: A] :
( X4
!= ( product_Pair @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A ) @ G7 @ ( product_Pair @ ( product_prod @ A @ A ) @ A @ ( product_Pair @ A @ A @ U2 @ V2 ) @ W2 ) ) ) ).
% subdivide.cases
thf(fact_164_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_165_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_166_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A3 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B2 @ X ) )
=> ( C3
!= ( product_Pair @ A @ B @ X @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_167_SigmaD1,axiom,
! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ( member @ A @ A4 @ A3 ) ) ).
% SigmaD1
thf(fact_168_SigmaD2,axiom,
! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ).
% SigmaD2
thf(fact_169_SigmaE2,axiom,
! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ~ ( ( member @ A @ A4 @ A3 )
=> ~ ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ) ).
% SigmaE2
thf(fact_170_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% subrelI
thf(fact_171_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_172_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R4 )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_173_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B4: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( F1 @ A4 @ B4 ) ) ).
% old.prod.rec
thf(fact_174_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) ) ).
% less_by_empty
thf(fact_175_calculation,axiom,
( ( finite_finite2 @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g ) )
& ( ( finite_card @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g ) )
= n )
& ( ( pair_pair_pre_parcs @ a @ product_unit @ g )
= ( collect @ ( product_prod @ a @ a )
@ ( product_case_prod @ a @ a @ $o
@ ^ [U3: a,V4: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ U3 @ V4 )
@ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
@ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) )
& ( U3 != V4 ) ) ) ) ) ) ).
% calculation
thf(fact_176_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X: A,Y4: B] :
( ( P @ X @ Y4 )
=> ( Q @ X @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_177_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( P
& ( Q @ A6 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_178_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A4: B,B4: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A4 @ B4 ) )
= ( F2 @ A4 @ B4 ) ) ).
% case_prod_conv
thf(fact_179_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B6: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( C3 @ A5 @ B6 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_180_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B4: B] :
( ( F2 @ A4 @ B4 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ).
% case_prodI
thf(fact_181_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( ( P @ A6 )
& ( Q @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_182_bot2E,axiom,
! [A: $tType,B: $tType,X4: A,Y2: B] :
~ ( bot_bot @ ( A > B > $o ) @ X4 @ Y2 ) ).
% bot2E
thf(fact_183_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_184_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B4: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( F2 @ A4 @ B4 ) ) ).
% case_prodD
thf(fact_185_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
=> ~ ! [X: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X @ Y4 ) )
=> ~ ( C3 @ X @ Y4 ) ) ) ).
% case_prodE
thf(fact_186_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z3: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z3 ) )
=> ~ ! [X: B,Y4: C] :
( ( Z3
= ( product_Pair @ B @ C @ X @ Y4 ) )
=> ~ ( Q @ ( P @ X @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_187_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( F2 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_188_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X: A,Y4: B] :
( ( F2 @ X @ Y4 )
= ( G3 @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_189_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_190_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F2: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
( ! [X: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X @ Y4 )
= Q2 )
=> ( ( F2 @ X @ Y4 )
= ( G3 @ X @ Y4 ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P2 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_191_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X4: A,Y2: B,Q: A > B > $o] :
( ( P @ X4 @ Y2 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X4 @ Y2 ) ) ) ).
% rev_predicate2D
thf(fact_192_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X4: A,Y2: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X4 @ Y2 )
=> ( Q @ X4 @ Y2 ) ) ) ).
% predicate2D
thf(fact_193_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( P @ X3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_194_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_195_prod_Ocase__distrib,axiom,
! [C: $tType,D2: $tType,B: $tType,A: $tType,H: C > D2,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D2
@ ^ [X12: A,X23: B] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_196_card__subset__eq,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B2 ) )
=> ( A3 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_197_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B8: set @ A] :
( ( finite_finite2 @ A @ B8 )
& ( ( finite_card @ A @ B8 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_198_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F6: set @ A,C2: nat] :
( ! [G7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G7 @ F6 )
=> ( ( finite_finite2 @ A @ G7 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G7 ) @ C2 ) ) )
=> ( ( finite_finite2 @ A @ F6 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F6 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_199_card__seteq,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B2 ) ) ) ) ).
% card_seteq
thf(fact_200_card__mono,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% card_mono
thf(fact_201_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
=> ( ( ( finite_card @ A @ T4 )
= N )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_202_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_203_card__Ex__subset,axiom,
! [A: $tType,K: nat,M: set @ A] :
( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ M ) )
=> ? [N2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ N2 @ M )
& ( ( finite_card @ A @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_204_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z3: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B6: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B6 ) )
=> ( member @ C @ Z3 @ ( C3 @ A5 @ B6 ) ) )
=> ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_205_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),A4: B,B4: C] :
( ( member @ A @ Z3 @ ( C3 @ A4 @ B4 ) )
=> ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% mem_case_prodI
thf(fact_206_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X4: C] :
( ! [A5: A,B6: B] :
( ( ( product_Pair @ A @ B @ A5 @ B6 )
= P2 )
=> ( C3 @ A5 @ B6 @ X4 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X4 ) ) ).
% case_prodI2'
thf(fact_207_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_208_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A4: A,B4: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A4 @ B4 ) @ C3 )
=> ( R @ A4 @ B4 @ C3 ) ) ).
% case_prodD'
thf(fact_209_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z3 )
=> ~ ! [X: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X @ Y4 ) )
=> ~ ( C3 @ X @ Y4 @ Z3 ) ) ) ).
% case_prodE'
thf(fact_210_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
=> ~ ! [X: B,Y4: C] :
( ( P2
= ( product_Pair @ B @ C @ X @ Y4 ) )
=> ~ ( member @ A @ Z3 @ ( C3 @ X @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_211_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N4: set @ nat] :
? [M2: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N4 )
=> ( ord_less_eq @ nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_212_infinite__nat__iff__unbounded__le,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
& ( member @ nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_213_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M: nat] :
( ( P @ X4 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ X @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_214_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( F2 @ N5 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F2 @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_215_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B2: set @ A,A3: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ? [B9: A] :
( ( member @ A @ B9 @ B2 )
& ( R2 @ A5 @ B9 ) ) )
=> ( ! [A12: B,A23: B,B6: A] :
( ( member @ B @ A12 @ A3 )
=> ( ( member @ B @ A23 @ A3 )
=> ( ( member @ A @ B6 @ B2 )
=> ( ( R2 @ A12 @ B6 )
=> ( ( R2 @ A23 @ B6 )
=> ( A12 = A23 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_216_pair__pseudo__graph_Opair__pseudo__graph__subdivide,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
( ( pair_p389585203_graph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( pair_p389585203_graph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% pair_pseudo_graph.pair_pseudo_graph_subdivide
thf(fact_217_pair__graph_Opair__graph__subdivide,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
( ( pair_pair_graph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( pair_pair_graph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% pair_graph.pair_graph_subdivide
thf(fact_218_pair__graph_Oaxioms_I2_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_graph @ A @ G2 )
=> ( pair_p389585203_graph @ A @ G2 ) ) ).
% pair_graph.axioms(2)
thf(fact_219_pair__graph_Opair__graph,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_graph @ A @ G2 )
=> ( pair_pair_graph @ A @ G2 ) ) ).
% pair_graph.pair_graph
thf(fact_220_pair__pseudo__graph_Opair__pseudo__graph,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p389585203_graph @ A @ G2 )
=> ( pair_p389585203_graph @ A @ G2 ) ) ).
% pair_pseudo_graph.pair_pseudo_graph
thf(fact_221_pair__pseudo__graph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p389585203_graph @ A @ G2 )
=> ( pair_p953238171igraph @ A @ G2 ) ) ).
% pair_pseudo_graph.axioms(1)
thf(fact_222_case__prod__app,axiom,
! [A: $tType,D2: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D2 > A ) )
= ( ^ [F3: B > C > D2 > A,X3: product_prod @ B @ C,Y3: D2] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R4: C] : ( F3 @ L @ R4 @ Y3 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_223_pair__bidirected__digraph_Opair__bidirected__digraph__subdivide,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
( ( pair_p1033978749igraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( pair_p1033978749igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% pair_bidirected_digraph.pair_bidirected_digraph_subdivide
thf(fact_224_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X4: A,Y2: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X4 @ Y2 )
=> ( Q @ X4 @ Y2 ) ) ) ) ).
% predicate2D_conj
thf(fact_225_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_226_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y: A,Z2: A] : Y = Z2
@ ^ [A6: A,B5: A] :
( ( P @ A6 @ B5 )
| ( A6 = B5 ) ) ) ).
% eq_subset
thf(fact_227_pair__loopfree__digraph_Opair__loopfree__digraph__subdivide,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
( ( pair_p1282390076igraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( pair_p1282390076igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% pair_loopfree_digraph.pair_loopfree_digraph_subdivide
thf(fact_228_pair__graph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_digraph @ A @ G2 )
=> ( ( pair_p389585203_graph @ A @ G2 )
=> ( pair_pair_graph @ A @ G2 ) ) ) ).
% pair_graph.intro
thf(fact_229_pair__bidirected__digraph_Oaxioms_I2_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p1033978749igraph @ A @ G2 )
=> ( pair_p1282390076igraph @ A @ G2 ) ) ).
% pair_bidirected_digraph.axioms(2)
thf(fact_230_pair__loopfree__digraph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p1282390076igraph @ A @ G2 )
=> ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% pair_loopfree_digraph.axioms(1)
thf(fact_231_pair__digraph_Opair__wf__digraph,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_digraph @ A @ G2 )
=> ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% pair_digraph.pair_wf_digraph
thf(fact_232_pair__digraph_Oaxioms_I2_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_digraph @ A @ G2 )
=> ( pair_p1282390076igraph @ A @ G2 ) ) ).
% pair_digraph.axioms(2)
thf(fact_233_pair__digraph_Opair__digraph,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_digraph @ A @ G2 )
=> ( pair_pair_digraph @ A @ G2 ) ) ).
% pair_digraph.pair_digraph
thf(fact_234_pair__digraph__def,axiom,
! [A: $tType] :
( ( pair_pair_digraph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G )
& ( pair_p1282390076igraph @ A @ G ) ) ) ) ).
% pair_digraph_def
thf(fact_235_pair__digraph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( ( pair_p1282390076igraph @ A @ G2 )
=> ( pair_pair_digraph @ A @ G2 ) ) ) ).
% pair_digraph.intro
thf(fact_236_pair__digraph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_digraph @ A @ G2 )
=> ( pair_p953238171igraph @ A @ G2 ) ) ).
% pair_digraph.axioms(1)
thf(fact_237_pair__graph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_graph @ A @ G2 )
=> ( pair_pair_digraph @ A @ G2 ) ) ).
% pair_graph.axioms(1)
thf(fact_238_pair__loopfree__digraph_Ono__loops_H,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
( ( pair_p1282390076igraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( U != V3 ) ) ) ).
% pair_loopfree_digraph.no_loops'
thf(fact_239_pair__graph__def,axiom,
! [A: $tType] :
( ( pair_pair_graph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_digraph @ A @ G )
& ( pair_p389585203_graph @ A @ G ) ) ) ) ).
% pair_graph_def
thf(fact_240_pair__loopfree__digraph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_wf_digraph @ A @ G2 )
=> ( ( pair_p814123999axioms @ A @ G2 )
=> ( pair_p1282390076igraph @ A @ G2 ) ) ) ).
% pair_loopfree_digraph.intro
thf(fact_241_pair__loopfree__digraph__def,axiom,
! [A: $tType] :
( ( pair_p1282390076igraph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_wf_digraph @ A @ G )
& ( pair_p814123999axioms @ A @ G ) ) ) ) ).
% pair_loopfree_digraph_def
thf(fact_242_pair__loopfree__digraph_Oaxioms_I2_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p1282390076igraph @ A @ G2 )
=> ( pair_p814123999axioms @ A @ G2 ) ) ).
% pair_loopfree_digraph.axioms(2)
thf(fact_243_pair__sym__digraph_Opair__sym__digraph__subdivide,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
( ( pair_p460180511igraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
=> ( pair_p460180511igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% pair_sym_digraph.pair_sym_digraph_subdivide
thf(fact_244_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_245_pair__sym__digraph_Oarcs__symmetric,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,A4: A,B4: A] :
( ( pair_p460180511igraph @ A @ G2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) ) ) ) ).
% pair_sym_digraph.arcs_symmetric
thf(fact_246_pair__bidirected__digraph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p1033978749igraph @ A @ G2 )
=> ( pair_p460180511igraph @ A @ G2 ) ) ).
% pair_bidirected_digraph.axioms(1)
thf(fact_247_pair__sym__digraph_Oaxioms_I1_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p460180511igraph @ A @ G2 )
=> ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% pair_sym_digraph.axioms(1)
thf(fact_248_pair__pseudo__graph_Oaxioms_I2_J,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p389585203_graph @ A @ G2 )
=> ( pair_p460180511igraph @ A @ G2 ) ) ).
% pair_pseudo_graph.axioms(2)
thf(fact_249_pair__bidirected__digraph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p460180511igraph @ A @ G2 )
=> ( ( pair_p1282390076igraph @ A @ G2 )
=> ( pair_p1033978749igraph @ A @ G2 ) ) ) ).
% pair_bidirected_digraph.intro
thf(fact_250_pair__bidirected__digraph__def,axiom,
! [A: $tType] :
( ( pair_p1033978749igraph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p460180511igraph @ A @ G )
& ( pair_p1282390076igraph @ A @ G ) ) ) ) ).
% pair_bidirected_digraph_def
thf(fact_251_pair__pseudo__graph__def,axiom,
! [A: $tType] :
( ( pair_p389585203_graph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G )
& ( pair_p460180511igraph @ A @ G ) ) ) ) ).
% pair_pseudo_graph_def
thf(fact_252_pair__pseudo__graph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_p953238171igraph @ A @ G2 )
=> ( ( pair_p460180511igraph @ A @ G2 )
=> ( pair_p389585203_graph @ A @ G2 ) ) ) ).
% pair_pseudo_graph.intro
thf(fact_253_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A4: B,B4: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A4 @ B4 ) )
= ( C3 @ A4 @ B4 ) ) ).
% internal_case_prod_conv
thf(fact_254_pair__sym__digraph_Ointro,axiom,
! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_wf_digraph @ A @ G2 )
=> ( ( pair_p477239746axioms @ A @ G2 )
=> ( pair_p460180511igraph @ A @ G2 ) ) ) ).
% pair_sym_digraph.intro
thf(fact_255_pair__sym__digraph__def,axiom,
! [A: $tType] :
( ( pair_p460180511igraph @ A )
= ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
( ( pair_pair_wf_digraph @ A @ G )
& ( pair_p477239746axioms @ A @ G ) ) ) ) ).
% pair_sym_digraph_def
% Type constructors (33)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A7 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A7 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A7: $tType,A9: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A7 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A7 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A7 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A7 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( finite_finite @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
bot @ $o ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_19,axiom,
! [A7: $tType,A9: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( product_prod @ A7 @ A9 ) ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_20,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_21,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_22,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_23,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_24,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_25,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_26,axiom,
bot @ product_unit ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite2 @ ( product_prod @ a @ a ) @ ( pair_pair_pre_parcs @ a @ product_unit @ g ) ).
%------------------------------------------------------------------------------