TPTP Problem File: SLH0509^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0018_Graph_Theory_Relations/prob_00128_004383__13343074_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1531 ( 730 unt; 246 typ; 0 def)
% Number of atoms : 3293 (1498 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 9964 ( 309 ~; 39 |; 373 &;8321 @)
% ( 0 <=>; 922 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 913 ( 913 >; 0 *; 0 +; 0 <<)
% Number of symbols : 226 ( 223 usr; 16 con; 0-4 aty)
% Number of variables : 3672 ( 490 ^;3065 !; 117 ?;3672 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:34:04.103
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
produc8857593507947890343od_a_a: $tType ).
thf(ty_n_t__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,
set_Pr8600417178894128327od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
produc3364680560414100336_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc3498347346309940967od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_Pr5845495582615845127_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc4044097585999906000od_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_se5735800977113168103od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1703568184450464039_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (223)
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001tf__a_001tf__a,type,
bNF_Gr4067040185137062718_o_a_a: set_o > ( $o > a ) > ( $o > a ) > set_Product_prod_a_a ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001tf__a,type,
bNF_Gr1149069696037075021_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > a ) > ( product_prod_a_a > a ) > set_Product_prod_a_a ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_Itf__a_J_001tf__a_001tf__a,type,
bNF_Gr6628399236160294404_a_a_a: set_set_a > ( set_a > a ) > ( set_a > a ) > set_Product_prod_a_a ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001tf__a_001tf__a,type,
bNF_Gr1766759448597441700_a_a_a: set_a > ( a > a ) > ( a > a ) > set_Product_prod_a_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
comple8421679170691845492od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple3958522678809307947_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001tf__a,type,
design6447616907850319326ment_a: set_a > set_a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001_Eo,type,
design7782887785804742939oint_o: set_o > $o > set_o ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
design3431343892158072362od_a_a: set_Product_prod_a_a > product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001t__Set__Oset_Itf__a_J,type,
design4648949625254728801_set_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001tf__a,type,
design2964366272795260673oint_a: set_a > a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001tf__a,type,
design108908007054065099oint_a: set_a > a > set_a ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite6544458595007987280od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
inj_on4851796814176604264_set_a: ( product_prod_a_a > set_a ) > set_Product_prod_a_a > $o ).
thf(sy_c_Graph__Theory__Relations_Oulgraph_Oadj__relation_001t__Nat__Onat,type,
graph_3658176268357964989on_nat: set_set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Graph__Theory__Relations_Oulgraph_Oadj__relation_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
graph_210517301254161018od_a_a: set_se5735800977113168103od_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Graph__Theory__Relations_Oulgraph_Oadj__relation_001t__Set__Oset_Itf__a_J,type,
graph_7634368646383411377_set_a: set_set_set_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Graph__Theory__Relations_Oulgraph_Oadj__relation_001tf__a,type,
graph_8122095853558514513tion_a: set_set_a > set_Product_prod_a_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
if_Product_prod_a_a: $o > product_prod_a_a > product_prod_a_a > product_prod_a_a ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
if_set_set_a: $o > set_set_a > set_set_a > set_set_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
inf_in2559554923042384936_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
inf_inf_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
inf_inf_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
inf_in8905007599844390133od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
inf_in3339382566020358357od_a_a: set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
inf_in1205276777018777868_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
sup_su3048258781599657691od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
sup_su2076012971530813682_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
bot_bo4160289986317612842_a_a_o: product_prod_a_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
bot_bot_a_a_o: a > a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bot_bo3357376287454694259od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
bot_bo777872063958040403od_a_a: set_se5735800977113168103od_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
ord_le1591150415168442102_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
ord_le1995061765932249223od_a_a: set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc4925843558922497303od_a_a: product_prod_a_a > produc3498347346309940967od_a_a > produc8857593507947890343od_a_a ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc7886510207707329367od_a_a: product_prod_a_a > product_prod_a_a > produc3498347346309940967od_a_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc7299740244201487072_set_a: set_a > produc1703568184450464039_set_a > produc3364680560414100336_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc9088192753505129239_set_a: set_a > set_a > produc1703568184450464039_set_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc431845341423274048od_a_a: a > product_prod_a_a > produc4044097585999906000od_a_a ).
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_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,
produc6436628058953941356_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc408267641121961211od_a_a: ( a > a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_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_Itf__a_Mtf__a_J_J,type,
collec1673347964119250290od_a_a: ( set_Product_prod_a_a > $o ) > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_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_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_5435475662653987220od_a_a: ( $o > product_prod_a_a ) > set_o > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__a_J,type,
image_o_set_a: ( $o > set_a ) > set_o > set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001_Eo,type,
image_9022731552424948534_a_a_o: ( product_prod_a_a > $o ) > set_Product_prod_a_a > set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_4636654165204879301od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_6690255128444368805od_a_a: ( product_prod_a_a > set_Product_prod_a_a ) > set_Product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
image_9052089385058188540_set_a: ( product_prod_a_a > set_a ) > set_Product_prod_a_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
image_3437945252899457948_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_6061375613820669477_set_a: ( set_set_a > set_a ) > set_set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_Eo,type,
image_set_a_o: ( set_a > $o ) > set_set_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
image_set_a_nat: ( set_a > nat ) > set_set_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_7677297774867312974od_a_a: ( set_a > product_prod_a_a ) > set_set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_6165024369500519726od_a_a: ( set_a > set_Product_prod_a_a ) > set_set_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_7400625782589995694od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_4421510592991446670od_a_a: ( a > set_Product_prod_a_a ) > set_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
insert914553114930139863od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
insert_set_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001tf__a,type,
undire2554140024507503526stem_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
undire4022703626023482010_adj_a: set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001_Eo,type,
undire5276842904984321022dent_o: $o > set_o > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Nat__Onat,type,
undire7858122600432113898nt_nat: nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3369688177417741453od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Set__Oset_Itf__a_J,type,
undire2320338297334612420_set_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
undire1521409233611534436dent_a: a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001_Eo,type,
undire687802054568686522dges_o: set_set_o > $o > set_set_o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001t__Nat__Onat,type,
undire4176300566717384750es_nat: set_set_nat > nat > set_set_nat ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire8905369280470868553od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001t__Set__Oset_Itf__a_J,type,
undire4631953023069350784_set_a: set_set_set_a > set_a > set_set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
undire3231912044278729248dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
undire7777452895879145676dges_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__edge_001tf__a,type,
undire6670514144573423676edge_a: product_prod_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Nat__Onat,type,
undire4970100481470743719et_nat: produc7248412053542808358at_nat > set_nat ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire2459242765783757584od_a_a: produc8857593507947890343od_a_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Set__Oset_Itf__a_J,type,
undire4638465864238448455_set_a: produc3364680560414100336_set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001tf__a,type,
undire8536760333753235943_set_a: produc4044097585999906000od_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire398746457437328754od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Set__Oset_Itf__a_J,type,
undire1186139521737116585_set_a: set_set_a > set_set_set_a > set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001tf__a,type,
undire7103218114511261257raph_a: set_a > set_set_a > set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001_Eo,type,
undire2905056519009681222raph_o: set_o > set_set_o > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Nat__Onat,type,
undire3269267262472140706ph_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4585262585102564309od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Set__Oset_Itf__a_J,type,
undire6886684016831807756_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001tf__a,type,
undire7251896706689453996raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001tf__a,type,
undire8383842906760478443ween_a: set_set_a > set_a > set_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
undire8867928226783802224gree_a: set_set_a > a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Nat__Onat,type,
undire8640779321340989627ty_nat: set_set_nat > set_nat > set_nat > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire8410861505230878716od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Set__Oset_Itf__a_J,type,
undire8927637694342045747_set_a: set_set_set_a > set_set_a > set_set_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001tf__a,type,
undire297304480579013331sity_a: set_set_a > set_a > set_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001_Eo,type,
undire5089312632164170970loop_o: set_set_o > $o > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Nat__Onat,type,
undire5005864372999571214op_nat: set_set_nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7777398424729533289od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Set__Oset_Itf__a_J,type,
undire5774735625301615776_set_a: set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
undire3617971648856834880loop_a: set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001t__Nat__Onat,type,
undire1050940535076293677ps_nat: set_set_nat > nat > set_set_nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3049230956220217098od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001t__Set__Oset_Itf__a_J,type,
undire7215034953758041409_set_a: set_set_set_a > set_a > set_set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
undire4753905205749729249oops_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001tf__a,type,
undire1270416042309875431dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001_Eo,type,
undire3044609692436228535ween_o: set_o > set_o > set_o > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Nat__Onat,type,
undire6814325412647357297en_nat: set_nat > set_nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7011261089604658374od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Set__Oset_Itf__a_J,type,
undire2578756059399487229_set_a: set_set_a > set_set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001_Eo,type,
undire3977394397555731759rtex_o: set_o > set_set_o > $o > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Nat__Onat,type,
undire5609513041723151865ex_nat: set_nat > set_set_nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3207556238582723646od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Set__Oset_Itf__a_J,type,
undire6879241558604981877_set_a: set_set_a > set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
undire8931668460104145173rtex_a: set_a > set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Nat__Onat,type,
undire643512044667278624op_nat: set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3428022325429088215od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Set__Oset_Itf__a_J,type,
undire3618949687197220622_set_a: set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
undire2905028936066782638loop_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001_Eo,type,
undire3274282514099902824hood_o: set_o > set_set_o > $o > set_o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001t__Nat__Onat,type,
undire8190396521545869824od_nat: set_nat > set_set_nat > nat > set_nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7963753511165915895od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001t__Set__Oset_Itf__a_J,type,
undire2074812191327625774_set_a: set_set_a > set_set_set_a > set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighbors__ss_001_Eo,type,
undire7951960605359180119s_ss_o: set_set_o > $o > set_o > set_o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighbors__ss_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire711355123308631142od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighbors__ss_001t__Set__Oset_Itf__a_J,type,
undire182385392924118685_set_a: set_set_set_a > set_a > set_set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighbors__ss_001tf__a,type,
undire401937927514038589s_ss_a: set_set_a > a > set_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001_Eo,type,
undire3088010775375027618_adj_o: set_set_o > $o > $o > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Nat__Onat,type,
undire1083030068171319366dj_nat: set_set_nat > nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire6135774327024169009od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Set__Oset_Itf__a_J,type,
undire3510646817838285160_set_a: set_set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
undire397441198561214472_adj_a: set_set_a > a > a > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
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,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $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,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_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_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1275)
thf(fact_0_vert__adj__sym,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ edges @ V2 @ V1 ) ) ).
% vert_adj_sym
thf(fact_1_adj__relation__def,axiom,
( ( graph_8122095853558514513tion_a @ edges )
= ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [U: a,V: a] :
( ( Uu
= ( product_Pair_a_a @ U @ V ) )
& ( undire397441198561214472_adj_a @ edges @ U @ V ) ) ) ) ).
% adj_relation_def
thf(fact_2_edge__adj__inE,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_inE
thf(fact_3_neighbors__ss__def,axiom,
! [X: a,Y: set_a] :
( ( undire401937927514038589s_ss_a @ edges @ X @ Y )
= ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y )
& ( undire397441198561214472_adj_a @ edges @ X @ Y2 ) ) ) ) ).
% neighbors_ss_def
thf(fact_4_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_5_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_6_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_7_edge__density__commute,axiom,
! [X3: set_a,Y: set_a] :
( ( undire297304480579013331sity_a @ edges @ X3 @ Y )
= ( undire297304480579013331sity_a @ edges @ Y @ X3 ) ) ).
% edge_density_commute
thf(fact_8_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_9_set__self__img__compr,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [Uu: $o] :
? [A4: $o] :
( ( Uu = A4 )
& ( member_o @ A4 @ A3 ) ) )
= A3 ) ).
% set_self_img_compr
thf(fact_10_set__self__img__compr,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: product_prod_a_a] :
( ( Uu = A4 )
& ( member1426531477525435216od_a_a @ A4 @ A3 ) ) )
= A3 ) ).
% set_self_img_compr
thf(fact_11_set__self__img__compr,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [A4: a] :
( ( Uu = A4 )
& ( member_a @ A4 @ A3 ) ) )
= A3 ) ).
% set_self_img_compr
thf(fact_12_set__self__img__compr,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [A4: set_a] :
( ( Uu = A4 )
& ( member_set_a @ A4 @ A3 ) ) )
= A3 ) ).
% set_self_img_compr
thf(fact_13_pred__equals__eq2,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ( ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_14_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_15_image2__def,axiom,
( bNF_Gr1149069696037075021_a_a_a
= ( ^ [A5: set_Product_prod_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: product_prod_a_a] :
( ( Uu
= ( product_Pair_a_a @ ( F @ A4 ) @ ( G @ A4 ) ) )
& ( member1426531477525435216od_a_a @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_16_image2__def,axiom,
( bNF_Gr6628399236160294404_a_a_a
= ( ^ [A5: set_set_a,F: set_a > a,G: set_a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: set_a] :
( ( Uu
= ( product_Pair_a_a @ ( F @ A4 ) @ ( G @ A4 ) ) )
& ( member_set_a @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_17_image2__def,axiom,
( bNF_Gr1766759448597441700_a_a_a
= ( ^ [A5: set_a,F: a > a,G: a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: a] :
( ( Uu
= ( product_Pair_a_a @ ( F @ A4 ) @ ( G @ A4 ) ) )
& ( member_a @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_18_image2__def,axiom,
( bNF_Gr4067040185137062718_o_a_a
= ( ^ [A5: set_o,F: $o > a,G: $o > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: $o] :
( ( Uu
= ( product_Pair_a_a @ ( F @ A4 ) @ ( G @ A4 ) ) )
& ( member_o @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_19_vert__adj__imp__inV,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V2 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_20_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_a] :
~ ! [A6: a,B3: a] :
( Y3
!= ( product_Pair_a_a @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_21_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_22_incident__edge__in__wf,axiom,
! [E: set_a,V3: a] :
( ( member_set_a @ E @ edges )
=> ( ( undire1521409233611534436dent_a @ V3 @ E )
=> ( member_a @ V3 @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_23_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X4: a] :
( ( member_a @ X4 @ vertices )
& ( member_a @ X4 @ E1 )
& ( member_a @ X4 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_24_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_25_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_26_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_27_comp__sgraph_Oincident__def,axiom,
undire5276842904984321022dent_o = member_o ).
% comp_sgraph.incident_def
thf(fact_28_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_29_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_30_ulgraph_Oneighbors__ss_Ocong,axiom,
undire401937927514038589s_ss_a = undire401937927514038589s_ss_a ).
% ulgraph.neighbors_ss.cong
thf(fact_31_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_32_ulgraph_Oadj__relation_Ocong,axiom,
graph_8122095853558514513tion_a = graph_8122095853558514513tion_a ).
% ulgraph.adj_relation.cong
thf(fact_33_image2__eqI,axiom,
! [B: a,F2: product_prod_a_a > a,X: product_prod_a_a,C: a,G2: product_prod_a_a > a,A3: set_Product_prod_a_a] :
( ( B
= ( F2 @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr1149069696037075021_a_a_a @ A3 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_34_image2__eqI,axiom,
! [B: a,F2: set_a > a,X: set_a,C: a,G2: set_a > a,A3: set_set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_set_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr6628399236160294404_a_a_a @ A3 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_35_image2__eqI,axiom,
! [B: a,F2: a > a,X: a,C: a,G2: a > a,A3: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr1766759448597441700_a_a_a @ A3 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_36_image2__eqI,axiom,
! [B: a,F2: $o > a,X: $o,C: a,G2: $o > a,A3: set_o] :
( ( B
= ( F2 @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr4067040185137062718_o_a_a @ A3 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_37_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_38_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A6: a,B3: a] : ( P @ ( product_Pair_a_a @ A6 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_39_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X5: a,Y4: a] :
( P2
= ( product_Pair_a_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_40_mk__triangle__set_Ocases,axiom,
! [X: produc4044097585999906000od_a_a] :
~ ! [X5: a,Y4: a,Z: a] :
( X
!= ( produc431845341423274048od_a_a @ X5 @ ( product_Pair_a_a @ Y4 @ Z ) ) ) ).
% mk_triangle_set.cases
thf(fact_41_mk__edge_Ocases,axiom,
! [X: product_prod_a_a] :
~ ! [U2: a,V4: a] :
( X
!= ( product_Pair_a_a @ U2 @ V4 ) ) ).
% mk_edge.cases
thf(fact_42_is__isolated__vertex__edge,axiom,
! [V3: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V3 )
=> ( ( member_set_a @ E @ edges )
=> ~ ( undire1521409233611534436dent_a @ V3 @ E ) ) ) ).
% is_isolated_vertex_edge
thf(fact_43_is__isolated__vertex__def,axiom,
! [V3: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V3 )
= ( ( member_a @ V3 @ vertices )
& ! [X2: a] :
( ( member_a @ X2 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X2 @ V3 ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_44_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_52_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X5: product_prod_a_a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_53_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_54_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X5: set_a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_55_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X5: $o] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_56_neighborhood__def,axiom,
! [X: a] :
( ( undire8504279938402040014hood_a @ vertices @ edges @ X )
= ( collect_a
@ ^ [V: a] :
( ( member_a @ V @ vertices )
& ( undire397441198561214472_adj_a @ edges @ X @ V ) ) ) ) ).
% neighborhood_def
thf(fact_57_has__loop__in__verts,axiom,
! [V3: a] :
( ( undire3617971648856834880loop_a @ edges @ V3 )
=> ( member_a @ V3 @ vertices ) ) ).
% has_loop_in_verts
thf(fact_58_vert__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_59_incident__edges__def,axiom,
! [V3: a] :
( ( undire3231912044278729248dges_a @ edges @ V3 )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire1521409233611534436dent_a @ V3 @ E3 ) ) ) ) ).
% incident_edges_def
thf(fact_60_edge__density__zero,axiom,
! [Y: set_a,X3: set_a] :
( ( Y = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X3 @ Y )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_61_edge__adj__def,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_def
thf(fact_62_wellformed__alt__snd,axiom,
! [X: a,Y3: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y3 @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_63_wellformed__alt__fst,axiom,
! [X: a,Y3: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_64_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_65_not__vert__adj,axiom,
! [V3: a,U3: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V3 @ U3 )
=> ~ ( member_set_a @ ( insert_a @ V3 @ ( insert_a @ U3 @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_66_vert__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_67_has__loop__def,axiom,
! [V3: a] :
( ( undire3617971648856834880loop_a @ edges @ V3 )
= ( member_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_68_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y5: set_a,E3: set_a] :
? [X2: a,Y2: a] :
( ( E3
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X6 )
& ( member_a @ Y2 @ Y5 ) ) ) ) ).
% is_edge_between_def
thf(fact_69_is__isolated__vertex__no__loop,axiom,
! [V3: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V3 )
=> ~ ( undire3617971648856834880loop_a @ edges @ V3 ) ) ).
% is_isolated_vertex_no_loop
thf(fact_70_iso__vertex__empty__neighborhood,axiom,
! [V3: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V3 )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V3 )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_71_neighborhood__incident,axiom,
! [U3: a,V3: a] :
( ( member_a @ U3 @ ( undire8504279938402040014hood_a @ vertices @ edges @ V3 ) )
= ( member_set_a @ ( insert_a @ U3 @ ( insert_a @ V3 @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V3 ) ) ) ).
% neighborhood_incident
thf(fact_72_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V3 )
= ( member_set_set_a @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_73_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V3 )
= ( member_set_nat @ ( insert_nat @ V3 @ bot_bot_set_nat ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_74_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V3 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V3 @ bot_bo3357376287454694259od_a_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_75_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V3 )
= ( member_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_76_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_77_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V5: set_a,E4: set_set_a,V6: set_a,E5: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V5 @ E4 @ V6 @ E5 )
=> ( ( undire7103218114511261257raph_a @ V6 @ E5 @ V5 @ E4 )
=> ( ( V6 = V5 )
& ( E5 = E4 ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_78_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V3 )
=> ( member1426531477525435216od_a_a @ V3 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_79_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V3 )
=> ( member_set_a @ V3 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_80_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_o,Edges: set_set_o,V3: $o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( undire5089312632164170970loop_o @ Edges @ V3 )
=> ( member_o @ V3 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_81_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( member_a @ V3 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_82_ulgraph_Oneighborhood_Ocong,axiom,
undire8504279938402040014hood_a = undire8504279938402040014hood_a ).
% ulgraph.neighborhood.cong
thf(fact_83_subgraph_Ois__subgraph__ulgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7251896706689453996raph_a @ V_G @ E_G )
=> ( undire7251896706689453996raph_a @ V_H @ E_H ) ) ) ).
% subgraph.is_subgraph_ulgraph
thf(fact_84_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_o,Edges: set_set_o,U3: $o,V3: $o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( member_o @ U3 @ ( undire3274282514099902824hood_o @ Vertices @ Edges @ V3 ) )
= ( member_set_o @ ( insert_o @ U3 @ ( insert_o @ V3 @ bot_bot_set_o ) ) @ ( undire687802054568686522dges_o @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_85_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U3: set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ U3 @ ( undire2074812191327625774_set_a @ Vertices @ Edges @ V3 ) )
= ( member_set_set_a @ ( insert_set_a @ U3 @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_86_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_nat,Edges: set_set_nat,U3: nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( member_nat @ U3 @ ( undire8190396521545869824od_nat @ Vertices @ Edges @ V3 ) )
= ( member_set_nat @ ( insert_nat @ U3 @ ( insert_nat @ V3 @ bot_bot_set_nat ) ) @ ( undire4176300566717384750es_nat @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_87_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,U3: product_prod_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ U3 @ ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V3 ) )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ U3 @ ( insert4534936382041156343od_a_a @ V3 @ bot_bo3357376287454694259od_a_a ) ) @ ( undire8905369280470868553od_a_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_88_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_a,Edges: set_set_a,U3: a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ U3 @ ( undire8504279938402040014hood_a @ Vertices @ Edges @ V3 ) )
= ( member_set_a @ ( insert_a @ U3 @ ( insert_a @ V3 @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_89_graph__system_Oincident__edges_Ocong,axiom,
undire3231912044278729248dges_a = undire3231912044278729248dges_a ).
% graph_system.incident_edges.cong
thf(fact_90_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V3 ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_91_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V3 )
=> ( ( undire2074812191327625774_set_a @ Vertices @ Edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_92_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5609513041723151865ex_nat @ Vertices @ Edges @ V3 )
=> ( ( undire8190396521545869824od_nat @ Vertices @ Edges @ V3 )
= bot_bot_set_nat ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_93_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V3 )
=> ( ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V3 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_94_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
=> ( ( undire8504279938402040014hood_a @ Vertices @ Edges @ V3 )
= bot_bot_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_95_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_96_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V3 )
= ( ( member1426531477525435216od_a_a @ V3 @ Vertices )
& ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ Vertices )
=> ~ ( undire6135774327024169009od_a_a @ Edges @ X2 @ V3 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_97_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V3 )
= ( ( member_set_a @ V3 @ Vertices )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X2 @ V3 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_98_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_o,Edges: set_set_o,V3: $o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( undire3977394397555731759rtex_o @ Vertices @ Edges @ V3 )
= ( ( member_o @ V3 @ Vertices )
& ! [X2: $o] :
( ( member_o @ X2 @ Vertices )
=> ~ ( undire3088010775375027618_adj_o @ Edges @ X2 @ V3 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_99_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
= ( ( member_a @ V3 @ Vertices )
& ! [X2: a] :
( ( member_a @ X2 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X2 @ V3 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_100_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
=> ( ( member_set_a @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V3 @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_101_ulgraph_Oneighborhood__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7963753511165915895od_a_a @ Vertices @ Edges @ X )
= ( collec3336397797384452498od_a_a
@ ^ [V: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ V @ Vertices )
& ( undire6135774327024169009od_a_a @ Edges @ X @ V ) ) ) ) ) ).
% ulgraph.neighborhood_def
thf(fact_102_ulgraph_Oneighborhood__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2074812191327625774_set_a @ Vertices @ Edges @ X )
= ( collect_set_a
@ ^ [V: set_a] :
( ( member_set_a @ V @ Vertices )
& ( undire3510646817838285160_set_a @ Edges @ X @ V ) ) ) ) ) ).
% ulgraph.neighborhood_def
thf(fact_103_ulgraph_Oneighborhood__def,axiom,
! [Vertices: set_o,Edges: set_set_o,X: $o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( undire3274282514099902824hood_o @ Vertices @ Edges @ X )
= ( collect_o
@ ^ [V: $o] :
( ( member_o @ V @ Vertices )
& ( undire3088010775375027618_adj_o @ Edges @ X @ V ) ) ) ) ) ).
% ulgraph.neighborhood_def
thf(fact_104_ulgraph_Oneighborhood__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8504279938402040014hood_a @ Vertices @ Edges @ X )
= ( collect_a
@ ^ [V: a] :
( ( member_a @ V @ Vertices )
& ( undire397441198561214472_adj_a @ Edges @ X @ V ) ) ) ) ) ).
% ulgraph.neighborhood_def
thf(fact_105_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_106_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V2 )
= ( member_set_nat @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_107_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_108_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_109_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a,U3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( undire3510646817838285160_set_a @ Edges @ V3 @ U3 )
=> ~ ( member_set_set_a @ ( insert_set_a @ V3 @ ( insert_set_a @ U3 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_110_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V3: nat,U3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ~ ( undire1083030068171319366dj_nat @ Edges @ V3 @ U3 )
=> ~ ( member_set_nat @ ( insert_nat @ V3 @ ( insert_nat @ U3 @ bot_bot_set_nat ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_111_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a,U3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ~ ( undire6135774327024169009od_a_a @ Edges @ V3 @ U3 )
=> ~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V3 @ ( insert4534936382041156343od_a_a @ U3 @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_112_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a,U3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire397441198561214472_adj_a @ Edges @ V3 @ U3 )
=> ~ ( member_set_a @ ( insert_a @ V3 @ ( insert_a @ U3 @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_113_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ bot_bot_set_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_114_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_nat,Edges: set_set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ~ ( member_set_nat @ bot_bot_set_nat @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_115_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ~ ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_116_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_117_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y: set_set_a,X3: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Y = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ Edges @ X3 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_118_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_nat,Edges: set_set_nat,Y: set_nat,X3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( Y = bot_bot_set_nat )
=> ( ( undire8640779321340989627ty_nat @ Edges @ X3 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_119_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( Y = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X3 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_120_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_a,Edges: set_set_a,Y: set_a,X3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Y = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ Edges @ X3 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_121_ulgraph_Overt__adj__sym,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ Edges @ V2 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_122_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ Vertices )
& ( member1426531477525435216od_a_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_123_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_124_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_o,Edges: set_set_o,V1: $o,V2: $o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( undire3088010775375027618_adj_o @ Edges @ V1 @ V2 )
=> ( ( member_o @ V1 @ Vertices )
& ( member_o @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_125_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_126_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_a,Edges: set_set_a,X3: set_a,Y: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire297304480579013331sity_a @ Edges @ X3 @ Y )
= ( undire297304480579013331sity_a @ Edges @ Y @ X3 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_127_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
= ( ( undire2320338297334612420_set_a @ V1 @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) )
& ( undire2320338297334612420_set_a @ V2 @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) )
& ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_128_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V2 )
= ( ( undire7858122600432113898nt_nat @ V1 @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) )
& ( undire7858122600432113898nt_nat @ V2 @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_129_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
= ( ( undire3369688177417741453od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( undire3369688177417741453od_a_a @ V2 @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_130_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_131_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_132_ulgraph_Oneighbors__ss__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a,Y: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire711355123308631142od_a_a @ Edges @ X @ Y )
= ( collec3336397797384452498od_a_a
@ ^ [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ Y )
& ( undire6135774327024169009od_a_a @ Edges @ X @ Y2 ) ) ) ) ) ).
% ulgraph.neighbors_ss_def
thf(fact_133_ulgraph_Oneighbors__ss__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a,Y: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire182385392924118685_set_a @ Edges @ X @ Y )
= ( collect_set_a
@ ^ [Y2: set_a] :
( ( member_set_a @ Y2 @ Y )
& ( undire3510646817838285160_set_a @ Edges @ X @ Y2 ) ) ) ) ) ).
% ulgraph.neighbors_ss_def
thf(fact_134_ulgraph_Oneighbors__ss__def,axiom,
! [Vertices: set_o,Edges: set_set_o,X: $o,Y: set_o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( undire7951960605359180119s_ss_o @ Edges @ X @ Y )
= ( collect_o
@ ^ [Y2: $o] :
( ( member_o @ Y2 @ Y )
& ( undire3088010775375027618_adj_o @ Edges @ X @ Y2 ) ) ) ) ) ).
% ulgraph.neighbors_ss_def
thf(fact_135_ulgraph_Oneighbors__ss__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire401937927514038589s_ss_a @ Edges @ X @ Y )
= ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y )
& ( undire397441198561214472_adj_a @ Edges @ X @ Y2 ) ) ) ) ) ).
% ulgraph.neighbors_ss_def
thf(fact_136_ulgraph_Oadj__relation__def,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( graph_8122095853558514513tion_a @ Edges )
= ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [U: a,V: a] :
( ( Uu
= ( product_Pair_a_a @ U @ V ) )
& ( undire397441198561214472_adj_a @ Edges @ U @ V ) ) ) ) ) ).
% ulgraph.adj_relation_def
thf(fact_137_is__loop__set__alt,axiom,
( ( collect_set_a
@ ^ [Uu: set_a] :
? [V: a] :
( ( Uu
= ( insert_a @ V @ bot_bot_set_a ) )
& ( undire3617971648856834880loop_a @ edges @ V ) ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) ) ) ).
% is_loop_set_alt
thf(fact_138_adj__relation__wf,axiom,
! [U3: a,V3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ V3 ) @ ( graph_8122095853558514513tion_a @ edges ) )
=> ( ord_less_eq_set_a @ ( insert_a @ U3 @ ( insert_a @ V3 @ bot_bot_set_a ) ) @ vertices ) ) ).
% adj_relation_wf
thf(fact_139_disjoint__insert_I2_J,axiom,
! [A3: set_o,B: $o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A3 @ ( insert_o @ B @ B4 ) ) )
= ( ~ ( member_o @ B @ A3 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_140_disjoint__insert_I2_J,axiom,
! [A3: set_a,B: a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B @ B4 ) ) )
= ( ~ ( member_a @ B @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_141_disjoint__insert_I2_J,axiom,
! [A3: set_set_a,B: set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ B @ B4 ) ) )
= ( ~ ( member_set_a @ B @ A3 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_142_disjoint__insert_I2_J,axiom,
! [A3: set_nat,B: nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ ( insert_nat @ B @ B4 ) ) )
= ( ~ ( member_nat @ B @ A3 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_143_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_a_a,B: product_prod_a_a,B4: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ B4 ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ A3 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_144_disjoint__insert_I1_J,axiom,
! [B4: set_o,A: $o,A3: set_o] :
( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A @ A3 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B4 )
& ( ( inf_inf_set_o @ B4 @ A3 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_145_disjoint__insert_I1_J,axiom,
! [B4: set_a,A: a,A3: set_a] :
( ( ( inf_inf_set_a @ B4 @ ( insert_a @ A @ A3 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ B4 @ A3 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_146_disjoint__insert_I1_J,axiom,
! [B4: set_set_a,A: set_a,A3: set_set_a] :
( ( ( inf_inf_set_set_a @ B4 @ ( insert_set_a @ A @ A3 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B4 )
& ( ( inf_inf_set_set_a @ B4 @ A3 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_147_disjoint__insert_I1_J,axiom,
! [B4: set_nat,A: nat,A3: set_nat] :
( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A @ A3 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B4 )
& ( ( inf_inf_set_nat @ B4 @ A3 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_148_disjoint__insert_I1_J,axiom,
! [B4: set_Product_prod_a_a,A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B4 @ ( insert4534936382041156343od_a_a @ A @ A3 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B4 )
& ( ( inf_in8905007599844390133od_a_a @ B4 @ A3 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_149_insert__disjoint_I2_J,axiom,
! [A: $o,A3: set_o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B4 ) )
= ( ~ ( member_o @ A @ B4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_150_insert__disjoint_I2_J,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B4 ) )
= ( ~ ( member_a @ A @ B4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_151_insert__disjoint_I2_J,axiom,
! [A: set_a,A3: set_set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ B4 ) )
= ( ~ ( member_set_a @ A @ B4 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_152_insert__disjoint_I2_J,axiom,
! [A: nat,A3: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B4 ) )
= ( ~ ( member_nat @ A @ B4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_153_insert__disjoint_I2_J,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A3 ) @ B4 ) )
= ( ~ ( member1426531477525435216od_a_a @ A @ B4 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_154_insert__disjoint_I1_J,axiom,
! [A: $o,A3: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B4 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B4 )
& ( ( inf_inf_set_o @ A3 @ B4 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_155_insert__disjoint_I1_J,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B4 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_156_insert__disjoint_I1_J,axiom,
! [A: set_a,A3: set_set_a,B4: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ B4 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B4 )
& ( ( inf_inf_set_set_a @ A3 @ B4 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_157_insert__disjoint_I1_J,axiom,
! [A: nat,A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B4 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B4 )
& ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_158_insert__disjoint_I1_J,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A3 ) @ B4 )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B4 )
& ( ( inf_in8905007599844390133od_a_a @ A3 @ B4 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_159_incident__loops__def,axiom,
! [V3: a] :
( ( undire4753905205749729249oops_a @ edges @ V3 )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( E3
= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ).
% incident_loops_def
thf(fact_160_incident__edges__neighbors__img,axiom,
! [V3: a] :
( ( undire3231912044278729248dges_a @ edges @ V3 )
= ( image_a_set_a
@ ^ [U: a] : ( insert_a @ V3 @ ( insert_a @ U @ bot_bot_set_a ) )
@ ( undire8504279938402040014hood_a @ vertices @ edges @ V3 ) ) ) ).
% incident_edges_neighbors_img
thf(fact_161_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_162_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y6: a,Z2: a] : ( Y6 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_163_singleton__conv2,axiom,
! [A: set_a] :
( ( collect_set_a
@ ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 )
@ A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv2
thf(fact_164_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_165_singleton__conv2,axiom,
! [A: product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ( ^ [Y6: product_prod_a_a,Z2: product_prod_a_a] : ( Y6 = Z2 )
@ A ) )
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singleton_conv2
thf(fact_166_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X2: $o] : ( X2 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_167_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_168_singleton__conv,axiom,
! [A: set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( X2 = A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv
thf(fact_169_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_170_singleton__conv,axiom,
! [A: product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( X2 = A ) )
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singleton_conv
thf(fact_171_Int__insert__right__if1,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_172_Int__insert__right__if1,axiom,
! [A: $o,A3: set_o,B4: set_o] :
( ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_173_Int__insert__right__if1,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_174_Int__insert__right__if1,axiom,
! [A: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_175_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_176_incident__loops__simp_I2_J,axiom,
! [V3: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V3 )
=> ( ( undire4753905205749729249oops_a @ edges @ V3 )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_177_incident__edges__empty,axiom,
! [V3: a] :
( ~ ( member_a @ V3 @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V3 )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_178_image__eqI,axiom,
! [B: a,F2: a > a,X: a,A3: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ B @ ( image_a_a @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_179_image__eqI,axiom,
! [B: $o,F2: a > $o,X: a,A3: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_o @ B @ ( image_a_o @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_180_image__eqI,axiom,
! [B: a,F2: $o > a,X: $o,A3: set_o] :
( ( B
= ( F2 @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member_a @ B @ ( image_o_a @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_181_image__eqI,axiom,
! [B: $o,F2: $o > $o,X: $o,A3: set_o] :
( ( B
= ( F2 @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member_o @ B @ ( image_o_o @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_182_image__eqI,axiom,
! [B: a,F2: set_a > a,X: set_a,A3: set_set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_a @ X @ A3 )
=> ( member_a @ B @ ( image_set_a_a @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_183_image__eqI,axiom,
! [B: $o,F2: set_a > $o,X: set_a,A3: set_set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_a @ X @ A3 )
=> ( member_o @ B @ ( image_set_a_o @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_184_image__eqI,axiom,
! [B: set_a,F2: a > set_a,X: a,A3: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_set_a @ B @ ( image_a_set_a @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_185_image__eqI,axiom,
! [B: set_a,F2: $o > set_a,X: $o,A3: set_o] :
( ( B
= ( F2 @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member_set_a @ B @ ( image_o_set_a @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_186_image__eqI,axiom,
! [B: a,F2: product_prod_a_a > a,X: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_187_image__eqI,axiom,
! [B: $o,F2: product_prod_a_a > $o,X: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member_o @ B @ ( image_9022731552424948534_a_a_o @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_188_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_189_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_190_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_191_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_192_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_193_all__not__in__conv,axiom,
! [A3: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A3 ) )
= ( A3 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_194_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_195_all__not__in__conv,axiom,
! [A3: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_196_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_197_all__not__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ! [X2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X2 @ A3 ) )
= ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_198_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_199_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_200_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_201_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_202_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_203_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_204_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_205_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_206_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_207_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_208_subsetI,axiom,
! [A3: set_o,B4: set_o] :
( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( member_o @ X5 @ B4 ) )
=> ( ord_less_eq_set_o @ A3 @ B4 ) ) ).
% subsetI
thf(fact_209_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_a @ X5 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_210_subsetI,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ( member_set_a @ X5 @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_211_subsetI,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( member1426531477525435216od_a_a @ X5 @ B4 ) )
=> ( ord_le746702958409616551od_a_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_212_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_213_subset__antisym,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_214_subset__antisym,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ( ord_le746702958409616551od_a_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_215_insertCI,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B4 )
=> ( A = B ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_216_insertCI,axiom,
! [A: set_a,B4: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B4 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_217_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_218_insertCI,axiom,
! [A: $o,B4: set_o,B: $o] :
( ( ~ ( member_o @ A @ B4 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% insertCI
thf(fact_219_insert__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A3 ) )
= ( ( A = B )
| ( member1426531477525435216od_a_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_220_insert__iff,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_221_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_222_insert__iff,axiom,
! [A: $o,B: $o,A3: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A3 ) )
= ( ( A = B )
| ( member_o @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_223_insert__absorb2,axiom,
! [X: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
= ( insert_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_224_insert__absorb2,axiom,
! [X: set_a,A3: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A3 ) )
= ( insert_set_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_225_IntI,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( ( member1426531477525435216od_a_a @ C @ B4 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_226_IntI,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ A3 )
=> ( ( member_o @ C @ B4 )
=> ( member_o @ C @ ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_227_IntI,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_228_IntI,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_229_Int__iff,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) )
= ( ( member1426531477525435216od_a_a @ C @ A3 )
& ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_230_Int__iff,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B4 ) )
= ( ( member_o @ C @ A3 )
& ( member_o @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_231_Int__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_232_Int__iff,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B4 ) )
= ( ( member_set_a @ C @ A3 )
& ( member_set_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_233_image__ident,axiom,
! [Y: set_Product_prod_a_a] :
( ( image_4636654165204879301od_a_a
@ ^ [X2: product_prod_a_a] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_234_incident__loops__simp_I1_J,axiom,
! [V3: a] :
( ( undire3617971648856834880loop_a @ edges @ V3 )
=> ( ( undire4753905205749729249oops_a @ edges @ V3 )
= ( insert_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_235_image__empty,axiom,
! [F2: a > a] :
( ( image_a_a @ F2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_236_image__empty,axiom,
! [F2: a > nat] :
( ( image_a_nat @ F2 @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_237_image__empty,axiom,
! [F2: nat > a] :
( ( image_nat_a @ F2 @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_238_image__empty,axiom,
! [F2: nat > nat] :
( ( image_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_239_image__empty,axiom,
! [F2: a > set_a] :
( ( image_a_set_a @ F2 @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_240_image__empty,axiom,
! [F2: set_a > a] :
( ( image_set_a_a @ F2 @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_241_image__empty,axiom,
! [F2: set_a > nat] :
( ( image_set_a_nat @ F2 @ bot_bot_set_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_242_image__empty,axiom,
! [F2: nat > set_a] :
( ( image_nat_set_a @ F2 @ bot_bot_set_nat )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_243_image__empty,axiom,
! [F2: a > set_set_a] :
( ( image_a_set_set_a @ F2 @ bot_bot_set_a )
= bot_bo3380559777022489994_set_a ) ).
% image_empty
thf(fact_244_image__empty,axiom,
! [F2: a > product_prod_a_a] :
( ( image_7400625782589995694od_a_a @ F2 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% image_empty
thf(fact_245_empty__is__image,axiom,
! [F2: a > a,A3: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_246_empty__is__image,axiom,
! [F2: nat > a,A3: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_247_empty__is__image,axiom,
! [F2: a > nat,A3: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_248_empty__is__image,axiom,
! [F2: nat > nat,A3: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_249_empty__is__image,axiom,
! [F2: set_a > a,A3: set_set_a] :
( ( bot_bot_set_a
= ( image_set_a_a @ F2 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_250_empty__is__image,axiom,
! [F2: a > set_a,A3: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_251_empty__is__image,axiom,
! [F2: nat > set_a,A3: set_nat] :
( ( bot_bot_set_set_a
= ( image_nat_set_a @ F2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_252_empty__is__image,axiom,
! [F2: set_a > nat,A3: set_set_a] :
( ( bot_bot_set_nat
= ( image_set_a_nat @ F2 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_253_empty__is__image,axiom,
! [F2: a > set_set_a,A3: set_a] :
( ( bot_bo3380559777022489994_set_a
= ( image_a_set_set_a @ F2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_254_empty__is__image,axiom,
! [F2: product_prod_a_a > a,A3: set_Product_prod_a_a] :
( ( bot_bot_set_a
= ( image_3437945252899457948_a_a_a @ F2 @ A3 ) )
= ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% empty_is_image
thf(fact_255_image__is__empty,axiom,
! [F2: a > a,A3: set_a] :
( ( ( image_a_a @ F2 @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_256_image__is__empty,axiom,
! [F2: nat > a,A3: set_nat] :
( ( ( image_nat_a @ F2 @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_257_image__is__empty,axiom,
! [F2: a > nat,A3: set_a] :
( ( ( image_a_nat @ F2 @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_258_image__is__empty,axiom,
! [F2: nat > nat,A3: set_nat] :
( ( ( image_nat_nat @ F2 @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_259_image__is__empty,axiom,
! [F2: set_a > a,A3: set_set_a] :
( ( ( image_set_a_a @ F2 @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_260_image__is__empty,axiom,
! [F2: a > set_a,A3: set_a] :
( ( ( image_a_set_a @ F2 @ A3 )
= bot_bot_set_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_261_image__is__empty,axiom,
! [F2: nat > set_a,A3: set_nat] :
( ( ( image_nat_set_a @ F2 @ A3 )
= bot_bot_set_set_a )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_262_image__is__empty,axiom,
! [F2: set_a > nat,A3: set_set_a] :
( ( ( image_set_a_nat @ F2 @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_263_image__is__empty,axiom,
! [F2: a > set_set_a,A3: set_a] :
( ( ( image_a_set_set_a @ F2 @ A3 )
= bot_bo3380559777022489994_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_264_image__is__empty,axiom,
! [F2: product_prod_a_a > a,A3: set_Product_prod_a_a] :
( ( ( image_3437945252899457948_a_a_a @ F2 @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% image_is_empty
thf(fact_265_subset__empty,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_266_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_267_subset__empty,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_268_subset__empty,axiom,
! [A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ bot_bo3357376287454694259od_a_a )
= ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_269_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_270_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_271_empty__subsetI,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A3 ) ).
% empty_subsetI
thf(fact_272_empty__subsetI,axiom,
! [A3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A3 ) ).
% empty_subsetI
thf(fact_273_image__insert,axiom,
! [F2: product_prod_a_a > product_prod_a_a,A: product_prod_a_a,B4: set_Product_prod_a_a] :
( ( image_4636654165204879301od_a_a @ F2 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( insert4534936382041156343od_a_a @ ( F2 @ A ) @ ( image_4636654165204879301od_a_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_274_image__insert,axiom,
! [F2: product_prod_a_a > set_a,A: product_prod_a_a,B4: set_Product_prod_a_a] :
( ( image_9052089385058188540_set_a @ F2 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( insert_set_a @ ( F2 @ A ) @ ( image_9052089385058188540_set_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_275_image__insert,axiom,
! [F2: a > set_set_a,A: a,B4: set_a] :
( ( image_a_set_set_a @ F2 @ ( insert_a @ A @ B4 ) )
= ( insert_set_set_a @ ( F2 @ A ) @ ( image_a_set_set_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_276_image__insert,axiom,
! [F2: a > a,A: a,B4: set_a] :
( ( image_a_a @ F2 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ ( F2 @ A ) @ ( image_a_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_277_image__insert,axiom,
! [F2: a > set_a,A: a,B4: set_a] :
( ( image_a_set_a @ F2 @ ( insert_a @ A @ B4 ) )
= ( insert_set_a @ ( F2 @ A ) @ ( image_a_set_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_278_image__insert,axiom,
! [F2: set_a > set_Product_prod_a_a,A: set_a,B4: set_set_a] :
( ( image_6165024369500519726od_a_a @ F2 @ ( insert_set_a @ A @ B4 ) )
= ( insert914553114930139863od_a_a @ ( F2 @ A ) @ ( image_6165024369500519726od_a_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_279_image__insert,axiom,
! [F2: set_a > a,A: set_a,B4: set_set_a] :
( ( image_set_a_a @ F2 @ ( insert_set_a @ A @ B4 ) )
= ( insert_a @ ( F2 @ A ) @ ( image_set_a_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_280_image__insert,axiom,
! [F2: set_a > set_a,A: set_a,B4: set_set_a] :
( ( image_set_a_set_a @ F2 @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ ( F2 @ A ) @ ( image_set_a_set_a @ F2 @ B4 ) ) ) ).
% image_insert
thf(fact_281_insert__image,axiom,
! [X: a,A3: set_a,F2: a > a] :
( ( member_a @ X @ A3 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A3 ) )
= ( image_a_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_282_insert__image,axiom,
! [X: $o,A3: set_o,F2: $o > a] :
( ( member_o @ X @ A3 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_o_a @ F2 @ A3 ) )
= ( image_o_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_283_insert__image,axiom,
! [X: set_a,A3: set_set_a,F2: set_a > a] :
( ( member_set_a @ X @ A3 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_set_a_a @ F2 @ A3 ) )
= ( image_set_a_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_284_insert__image,axiom,
! [X: a,A3: set_a,F2: a > set_a] :
( ( member_a @ X @ A3 )
=> ( ( insert_set_a @ ( F2 @ X ) @ ( image_a_set_a @ F2 @ A3 ) )
= ( image_a_set_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_285_insert__image,axiom,
! [X: $o,A3: set_o,F2: $o > set_a] :
( ( member_o @ X @ A3 )
=> ( ( insert_set_a @ ( F2 @ X ) @ ( image_o_set_a @ F2 @ A3 ) )
= ( image_o_set_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_286_insert__image,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,F2: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) )
= ( image_3437945252899457948_a_a_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_287_insert__image,axiom,
! [X: set_a,A3: set_set_a,F2: set_a > set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( insert_set_a @ ( F2 @ X ) @ ( image_set_a_set_a @ F2 @ A3 ) )
= ( image_set_a_set_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_288_insert__image,axiom,
! [X: a,A3: set_a,F2: a > set_set_a] :
( ( member_a @ X @ A3 )
=> ( ( insert_set_set_a @ ( F2 @ X ) @ ( image_a_set_set_a @ F2 @ A3 ) )
= ( image_a_set_set_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_289_insert__image,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,F2: product_prod_a_a > set_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( insert_set_a @ ( F2 @ X ) @ ( image_9052089385058188540_set_a @ F2 @ A3 ) )
= ( image_9052089385058188540_set_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_290_insert__image,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( insert4534936382041156343od_a_a @ ( F2 @ X ) @ ( image_4636654165204879301od_a_a @ F2 @ A3 ) )
= ( image_4636654165204879301od_a_a @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_291_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_292_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_293_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_294_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_295_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_296_insert__subset,axiom,
! [X: $o,A3: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A3 ) @ B4 )
= ( ( member_o @ X @ B4 )
& ( ord_less_eq_set_o @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_297_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_298_insert__subset,axiom,
! [X: set_a,A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A3 ) @ B4 )
= ( ( member_set_a @ X @ B4 )
& ( ord_le3724670747650509150_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_299_insert__subset,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A3 ) @ B4 )
= ( ( member1426531477525435216od_a_a @ X @ B4 )
& ( ord_le746702958409616551od_a_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_300_Int__subset__iff,axiom,
! [C2: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( ( ord_less_eq_set_a @ C2 @ A3 )
& ( ord_less_eq_set_a @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_301_Int__subset__iff,axiom,
! [C2: set_set_a,A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B4 ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A3 )
& ( ord_le3724670747650509150_set_a @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_302_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A3 )
& ( ord_le746702958409616551od_a_a @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_303_Int__insert__left__if0,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B4 ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_304_Int__insert__left__if0,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( inf_inf_set_o @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_305_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_306_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_307_Int__insert__left__if1,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B4 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_308_Int__insert__left__if1,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_309_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_310_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B4: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_311_insert__inter__insert,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_312_insert__inter__insert,axiom,
! [A: set_a,A3: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_313_Int__insert__right__if0,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_314_Int__insert__right__if0,axiom,
! [A: $o,A3: set_o,B4: set_o] :
( ~ ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B4 ) )
= ( inf_inf_set_o @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_315_Int__insert__right__if0,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_316_Int__insert__right__if0,axiom,
! [A: set_a,A3: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B4 ) )
= ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_317_singleton__insert__inj__eq_H,axiom,
! [A: nat,A3: set_nat,B: nat] :
( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_318_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_319_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A3: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A3 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_320_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A3 )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_321_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A3: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_322_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_323_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A3: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_324_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_a,A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_325_image__Int__subset,axiom,
! [F2: a > set_set_a,A3: set_a,B4: set_a] : ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F2 @ ( inf_inf_set_a @ A3 @ B4 ) ) @ ( inf_in1205276777018777868_set_a @ ( image_a_set_set_a @ F2 @ A3 ) @ ( image_a_set_set_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_326_image__Int__subset,axiom,
! [F2: set_a > set_Product_prod_a_a,A3: set_set_a,B4: set_set_a] : ( ord_le1995061765932249223od_a_a @ ( image_6165024369500519726od_a_a @ F2 @ ( inf_inf_set_set_a @ A3 @ B4 ) ) @ ( inf_in3339382566020358357od_a_a @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) @ ( image_6165024369500519726od_a_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_327_image__Int__subset,axiom,
! [F2: a > a,A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F2 @ ( inf_inf_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F2 @ A3 ) @ ( image_a_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_328_image__Int__subset,axiom,
! [F2: set_a > a,A3: set_set_a,B4: set_set_a] : ( ord_less_eq_set_a @ ( image_set_a_a @ F2 @ ( inf_inf_set_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( image_set_a_a @ F2 @ A3 ) @ ( image_set_a_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_329_image__Int__subset,axiom,
! [F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] : ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ F2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) @ ( inf_inf_set_set_a @ ( image_9052089385058188540_set_a @ F2 @ A3 ) @ ( image_9052089385058188540_set_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_330_image__Int__subset,axiom,
! [F2: a > set_a,A3: set_a,B4: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ ( inf_inf_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F2 @ A3 ) @ ( image_a_set_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_331_image__Int__subset,axiom,
! [F2: set_a > set_a,A3: set_set_a,B4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F2 @ ( inf_inf_set_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_set_a @ ( image_set_a_set_a @ F2 @ A3 ) @ ( image_set_a_set_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_332_image__Int__subset,axiom,
! [F2: product_prod_a_a > product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ F2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_4636654165204879301od_a_a @ F2 @ A3 ) @ ( image_4636654165204879301od_a_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_333_image__Int__subset,axiom,
! [F2: a > product_prod_a_a,A3: set_a,B4: set_a] : ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F2 @ ( inf_inf_set_a @ A3 @ B4 ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_7400625782589995694od_a_a @ F2 @ A3 ) @ ( image_7400625782589995694od_a_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_334_image__Int__subset,axiom,
! [F2: set_a > product_prod_a_a,A3: set_set_a,B4: set_set_a] : ( ord_le746702958409616551od_a_a @ ( image_7677297774867312974od_a_a @ F2 @ ( inf_inf_set_set_a @ A3 @ B4 ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_7677297774867312974od_a_a @ F2 @ A3 ) @ ( image_7677297774867312974od_a_a @ F2 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_335_imageE,axiom,
! [B: a,F2: a > a,A3: set_a] :
( ( member_a @ B @ ( image_a_a @ F2 @ A3 ) )
=> ~ ! [X5: a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_336_imageE,axiom,
! [B: a,F2: $o > a,A3: set_o] :
( ( member_a @ B @ ( image_o_a @ F2 @ A3 ) )
=> ~ ! [X5: $o] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_o @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_337_imageE,axiom,
! [B: $o,F2: a > $o,A3: set_a] :
( ( member_o @ B @ ( image_a_o @ F2 @ A3 ) )
=> ~ ! [X5: a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_338_imageE,axiom,
! [B: $o,F2: $o > $o,A3: set_o] :
( ( member_o @ B @ ( image_o_o @ F2 @ A3 ) )
=> ~ ! [X5: $o] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_o @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_339_imageE,axiom,
! [B: set_a,F2: a > set_a,A3: set_a] :
( ( member_set_a @ B @ ( image_a_set_a @ F2 @ A3 ) )
=> ~ ! [X5: a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_340_imageE,axiom,
! [B: set_a,F2: $o > set_a,A3: set_o] :
( ( member_set_a @ B @ ( image_o_set_a @ F2 @ A3 ) )
=> ~ ! [X5: $o] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_o @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_341_imageE,axiom,
! [B: a,F2: set_a > a,A3: set_set_a] :
( ( member_a @ B @ ( image_set_a_a @ F2 @ A3 ) )
=> ~ ! [X5: set_a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_set_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_342_imageE,axiom,
! [B: $o,F2: set_a > $o,A3: set_set_a] :
( ( member_o @ B @ ( image_set_a_o @ F2 @ A3 ) )
=> ~ ! [X5: set_a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_set_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_343_imageE,axiom,
! [B: set_set_a,F2: a > set_set_a,A3: set_a] :
( ( member_set_set_a @ B @ ( image_a_set_set_a @ F2 @ A3 ) )
=> ~ ! [X5: a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_344_imageE,axiom,
! [B: product_prod_a_a,F2: a > product_prod_a_a,A3: set_a] :
( ( member1426531477525435216od_a_a @ B @ ( image_7400625782589995694od_a_a @ F2 @ A3 ) )
=> ~ ! [X5: a] :
( ( B
= ( F2 @ X5 ) )
=> ~ ( member_a @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_345_imageI,axiom,
! [X: a,A3: set_a,F2: a > a] :
( ( member_a @ X @ A3 )
=> ( member_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_346_imageI,axiom,
! [X: a,A3: set_a,F2: a > $o] :
( ( member_a @ X @ A3 )
=> ( member_o @ ( F2 @ X ) @ ( image_a_o @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_347_imageI,axiom,
! [X: $o,A3: set_o,F2: $o > a] :
( ( member_o @ X @ A3 )
=> ( member_a @ ( F2 @ X ) @ ( image_o_a @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_348_imageI,axiom,
! [X: $o,A3: set_o,F2: $o > $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ ( F2 @ X ) @ ( image_o_o @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_349_imageI,axiom,
! [X: set_a,A3: set_set_a,F2: set_a > a] :
( ( member_set_a @ X @ A3 )
=> ( member_a @ ( F2 @ X ) @ ( image_set_a_a @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_350_imageI,axiom,
! [X: set_a,A3: set_set_a,F2: set_a > $o] :
( ( member_set_a @ X @ A3 )
=> ( member_o @ ( F2 @ X ) @ ( image_set_a_o @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_351_imageI,axiom,
! [X: a,A3: set_a,F2: a > set_a] :
( ( member_a @ X @ A3 )
=> ( member_set_a @ ( F2 @ X ) @ ( image_a_set_a @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_352_imageI,axiom,
! [X: $o,A3: set_o,F2: $o > set_a] :
( ( member_o @ X @ A3 )
=> ( member_set_a @ ( F2 @ X ) @ ( image_o_set_a @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_353_imageI,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,F2: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member_a @ ( F2 @ X ) @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_354_imageI,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,F2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member_o @ ( F2 @ X ) @ ( image_9022731552424948534_a_a_o @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_355_in__mono,axiom,
! [A3: set_o,B4: set_o,X: $o] :
( ( ord_less_eq_set_o @ A3 @ B4 )
=> ( ( member_o @ X @ A3 )
=> ( member_o @ X @ B4 ) ) ) ).
% in_mono
thf(fact_356_in__mono,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_357_in__mono,axiom,
! [A3: set_set_a,B4: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_358_in__mono,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_359_subsetD,axiom,
! [A3: set_o,B4: set_o,C: $o] :
( ( ord_less_eq_set_o @ A3 @ B4 )
=> ( ( member_o @ C @ A3 )
=> ( member_o @ C @ B4 ) ) ) ).
% subsetD
thf(fact_360_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_361_subsetD,axiom,
! [A3: set_set_a,B4: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_362_subsetD,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_363_equalityE,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_364_equalityE,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ~ ( ord_le3724670747650509150_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_365_equalityE,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( A3 = B4 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ~ ( ord_le746702958409616551od_a_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_366_image__iff,axiom,
! [Z3: set_set_a,F2: a > set_set_a,A3: set_a] :
( ( member_set_set_a @ Z3 @ ( image_a_set_set_a @ F2 @ A3 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( Z3
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_367_image__iff,axiom,
! [Z3: set_Product_prod_a_a,F2: set_a > set_Product_prod_a_a,A3: set_set_a] :
( ( member1816616512716248880od_a_a @ Z3 @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( Z3
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_368_image__iff,axiom,
! [Z3: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z3 @ ( image_4636654165204879301od_a_a @ F2 @ A3 ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( Z3
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_369_image__iff,axiom,
! [Z3: set_a,F2: a > set_a,A3: set_a] :
( ( member_set_a @ Z3 @ ( image_a_set_a @ F2 @ A3 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( Z3
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_370_image__iff,axiom,
! [Z3: set_a,F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a] :
( ( member_set_a @ Z3 @ ( image_9052089385058188540_set_a @ F2 @ A3 ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( Z3
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_371_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A5 )
=> ( member_o @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_372_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_373_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_374_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A5 )
=> ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_375_bex__imageD,axiom,
! [F2: a > set_a,A3: set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_set_a @ F2 @ A3 ) )
& ( P @ X4 ) )
=> ? [X5: a] :
( ( member_a @ X5 @ A3 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_376_bex__imageD,axiom,
! [F2: a > set_set_a,A3: set_a,P: set_set_a > $o] :
( ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ ( image_a_set_set_a @ F2 @ A3 ) )
& ( P @ X4 ) )
=> ? [X5: a] :
( ( member_a @ X5 @ A3 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_377_bex__imageD,axiom,
! [F2: set_a > set_Product_prod_a_a,A3: set_set_a,P: set_Product_prod_a_a > $o] :
( ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) )
& ( P @ X4 ) )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_378_bex__imageD,axiom,
! [F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_9052089385058188540_set_a @ F2 @ A3 ) )
& ( P @ X4 ) )
=> ? [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_379_bex__imageD,axiom,
! [F2: product_prod_a_a > product_prod_a_a,A3: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( image_4636654165204879301od_a_a @ F2 @ A3 ) )
& ( P @ X4 ) )
=> ? [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_380_equalityD1,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_381_equalityD1,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( A3 = B4 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_382_equalityD1,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( A3 = B4 )
=> ( ord_le746702958409616551od_a_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_383_equalityD2,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_384_equalityD2,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( A3 = B4 )
=> ( ord_le3724670747650509150_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_385_equalityD2,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( A3 = B4 )
=> ( ord_le746702958409616551od_a_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_386_image__cong,axiom,
! [M: set_Product_prod_a_a,N: set_Product_prod_a_a,F2: product_prod_a_a > set_a,G2: product_prod_a_a > set_a] :
( ( M = N )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ N )
=> ( ( F2 @ X5 )
= ( G2 @ X5 ) ) )
=> ( ( image_9052089385058188540_set_a @ F2 @ M )
= ( image_9052089385058188540_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_387_image__cong,axiom,
! [M: set_Product_prod_a_a,N: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,G2: product_prod_a_a > product_prod_a_a] :
( ( M = N )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ N )
=> ( ( F2 @ X5 )
= ( G2 @ X5 ) ) )
=> ( ( image_4636654165204879301od_a_a @ F2 @ M )
= ( image_4636654165204879301od_a_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_388_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F2: set_a > set_Product_prod_a_a,G2: set_a > set_Product_prod_a_a] :
( ( M = N )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ N )
=> ( ( F2 @ X5 )
= ( G2 @ X5 ) ) )
=> ( ( image_6165024369500519726od_a_a @ F2 @ M )
= ( image_6165024369500519726od_a_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_389_image__cong,axiom,
! [M: set_a,N: set_a,F2: a > set_a,G2: a > set_a] :
( ( M = N )
=> ( ! [X5: a] :
( ( member_a @ X5 @ N )
=> ( ( F2 @ X5 )
= ( G2 @ X5 ) ) )
=> ( ( image_a_set_a @ F2 @ M )
= ( image_a_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_390_image__cong,axiom,
! [M: set_a,N: set_a,F2: a > set_set_a,G2: a > set_set_a] :
( ( M = N )
=> ( ! [X5: a] :
( ( member_a @ X5 @ N )
=> ( ( F2 @ X5 )
= ( G2 @ X5 ) ) )
=> ( ( image_a_set_set_a @ F2 @ M )
= ( image_a_set_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_391_image__mono,axiom,
! [A3: set_a,B4: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A3 ) @ ( image_a_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_392_image__mono,axiom,
! [A3: set_a,B4: set_a,F2: a > set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A3 ) @ ( image_a_set_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_393_image__mono,axiom,
! [A3: set_set_a,B4: set_set_a,F2: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F2 @ A3 ) @ ( image_set_a_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_394_image__mono,axiom,
! [A3: set_a,B4: set_a,F2: a > set_set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F2 @ A3 ) @ ( image_a_set_set_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_395_image__mono,axiom,
! [A3: set_a,B4: set_a,F2: a > product_prod_a_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F2 @ A3 ) @ ( image_7400625782589995694od_a_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_396_image__mono,axiom,
! [A3: set_set_a,B4: set_set_a,F2: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F2 @ A3 ) @ ( image_set_a_set_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_397_image__mono,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,F2: product_prod_a_a > a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) @ ( image_3437945252899457948_a_a_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_398_image__mono,axiom,
! [A3: set_set_a,B4: set_set_a,F2: set_a > product_prod_a_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_le746702958409616551od_a_a @ ( image_7677297774867312974od_a_a @ F2 @ A3 ) @ ( image_7677297774867312974od_a_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_399_image__mono,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,F2: product_prod_a_a > set_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ F2 @ A3 ) @ ( image_9052089385058188540_set_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_400_image__mono,axiom,
! [A3: set_set_a,B4: set_set_a,F2: set_a > set_Product_prod_a_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_le1995061765932249223od_a_a @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) @ ( image_6165024369500519726od_a_a @ F2 @ B4 ) ) ) ).
% image_mono
thf(fact_401_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A5 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_402_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_403_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_404_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A5 )
=> ( member1426531477525435216od_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_405_ball__imageD,axiom,
! [F2: a > set_a,A3: set_a,P: set_a > $o] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( image_a_set_a @ F2 @ A3 ) )
=> ( P @ X5 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_406_ball__imageD,axiom,
! [F2: a > set_set_a,A3: set_a,P: set_set_a > $o] :
( ! [X5: set_set_a] :
( ( member_set_set_a @ X5 @ ( image_a_set_set_a @ F2 @ A3 ) )
=> ( P @ X5 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_407_ball__imageD,axiom,
! [F2: set_a > set_Product_prod_a_a,A3: set_set_a,P: set_Product_prod_a_a > $o] :
( ! [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) )
=> ( P @ X5 ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_408_ball__imageD,axiom,
! [F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a,P: set_a > $o] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( image_9052089385058188540_set_a @ F2 @ A3 ) )
=> ( P @ X5 ) )
=> ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A3 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_409_ball__imageD,axiom,
! [F2: product_prod_a_a > product_prod_a_a,A3: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( image_4636654165204879301od_a_a @ F2 @ A3 ) )
=> ( P @ X5 ) )
=> ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A3 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_410_image__image,axiom,
! [F2: a > set_a,G2: a > a,A3: set_a] :
( ( image_a_set_a @ F2 @ ( image_a_a @ G2 @ A3 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_411_image__image,axiom,
! [F2: set_a > set_a,G2: a > set_a,A3: set_a] :
( ( image_set_a_set_a @ F2 @ ( image_a_set_a @ G2 @ A3 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_412_image__image,axiom,
! [F2: a > set_set_a,G2: a > a,A3: set_a] :
( ( image_a_set_set_a @ F2 @ ( image_a_a @ G2 @ A3 ) )
= ( image_a_set_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_413_image__image,axiom,
! [F2: set_a > set_set_a,G2: a > set_a,A3: set_a] :
( ( image_4955109552351689957_set_a @ F2 @ ( image_a_set_a @ G2 @ A3 ) )
= ( image_a_set_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_414_image__image,axiom,
! [F2: set_set_a > set_a,G2: a > set_set_a,A3: set_a] :
( ( image_6061375613820669477_set_a @ F2 @ ( image_a_set_set_a @ G2 @ A3 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_415_image__image,axiom,
! [F2: a > set_a,G2: product_prod_a_a > a,A3: set_Product_prod_a_a] :
( ( image_a_set_a @ F2 @ ( image_3437945252899457948_a_a_a @ G2 @ A3 ) )
= ( image_9052089385058188540_set_a
@ ^ [X2: product_prod_a_a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_416_image__image,axiom,
! [F2: product_prod_a_a > set_a,G2: a > product_prod_a_a,A3: set_a] :
( ( image_9052089385058188540_set_a @ F2 @ ( image_7400625782589995694od_a_a @ G2 @ A3 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_417_image__image,axiom,
! [F2: set_set_a > set_set_a,G2: a > set_set_a,A3: set_a] :
( ( image_1042221919965026181_set_a @ F2 @ ( image_a_set_set_a @ G2 @ A3 ) )
= ( image_a_set_set_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_418_image__image,axiom,
! [F2: set_a > set_a,G2: product_prod_a_a > set_a,A3: set_Product_prod_a_a] :
( ( image_set_a_set_a @ F2 @ ( image_9052089385058188540_set_a @ G2 @ A3 ) )
= ( image_9052089385058188540_set_a
@ ^ [X2: product_prod_a_a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_419_image__image,axiom,
! [F2: set_a > set_Product_prod_a_a,G2: a > set_a,A3: set_a] :
( ( image_6165024369500519726od_a_a @ F2 @ ( image_a_set_a @ G2 @ A3 ) )
= ( image_4421510592991446670od_a_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_420_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_421_subset__refl,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_422_subset__refl,axiom,
! [A3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_423_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X5: $o] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_424_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_425_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X5: set_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_426_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X5: product_prod_a_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_427_subset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_428_subset__trans,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_429_subset__trans,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ( ord_le746702958409616551od_a_a @ B4 @ C2 )
=> ( ord_le746702958409616551od_a_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_430_image__subsetI,axiom,
! [A3: set_a,F2: a > $o,B4: set_o] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_o @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_431_image__subsetI,axiom,
! [A3: set_o,F2: $o > $o,B4: set_o] :
( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( member_o @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_432_image__subsetI,axiom,
! [A3: set_a,F2: a > a,B4: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_a @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_433_image__subsetI,axiom,
! [A3: set_o,F2: $o > a,B4: set_a] :
( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( member_a @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_434_image__subsetI,axiom,
! [A3: set_set_a,F2: set_a > $o,B4: set_o] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ( member_o @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_set_a_o @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_435_image__subsetI,axiom,
! [A3: set_set_a,F2: set_a > a,B4: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ( member_a @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_436_image__subsetI,axiom,
! [A3: set_a,F2: a > set_a,B4: set_set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_set_a @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_437_image__subsetI,axiom,
! [A3: set_o,F2: $o > set_a,B4: set_set_a] :
( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( member_set_a @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_o_set_a @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_438_image__subsetI,axiom,
! [A3: set_Product_prod_a_a,F2: product_prod_a_a > $o,B4: set_o] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( member_o @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_9022731552424948534_a_a_o @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_439_image__subsetI,axiom,
! [A3: set_a,F2: a > set_set_a,B4: set_set_set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_set_set_a @ ( F2 @ X5 ) @ B4 ) )
=> ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F2 @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_440_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: a,F2: a > a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_a_a @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_441_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: $o,F2: a > $o] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_o @ B @ ( image_a_o @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_442_rev__image__eqI,axiom,
! [X: $o,A3: set_o,B: a,F2: $o > a] :
( ( member_o @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_o_a @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_443_rev__image__eqI,axiom,
! [X: $o,A3: set_o,B: $o,F2: $o > $o] :
( ( member_o @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_o @ B @ ( image_o_o @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_444_rev__image__eqI,axiom,
! [X: set_a,A3: set_set_a,B: a,F2: set_a > a] :
( ( member_set_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_445_rev__image__eqI,axiom,
! [X: set_a,A3: set_set_a,B: $o,F2: set_a > $o] :
( ( member_set_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_o @ B @ ( image_set_a_o @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_446_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: set_a,F2: a > set_a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_447_rev__image__eqI,axiom,
! [X: $o,A3: set_o,B: set_a,F2: $o > set_a] :
( ( member_o @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_a @ B @ ( image_o_set_a @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_448_rev__image__eqI,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B: a,F2: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_449_rev__image__eqI,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B: $o,F2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_o @ B @ ( image_9022731552424948534_a_a_o @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_450_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_451_set__eq__subset,axiom,
( ( ^ [Y6: set_set_a,Z2: set_set_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_452_set__eq__subset,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y6 = Z2 ) )
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_453_subset__imageE,axiom,
! [B4: set_a,F2: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F2 @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B4
!= ( image_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_454_subset__imageE,axiom,
! [B4: set_a,F2: set_a > a,A3: set_set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_set_a_a @ F2 @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B4
!= ( image_set_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_455_subset__imageE,axiom,
! [B4: set_set_a,F2: a > set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F2 @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B4
!= ( image_a_set_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_456_subset__imageE,axiom,
! [B4: set_set_set_a,F2: a > set_set_a,A3: set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ ( image_a_set_set_a @ F2 @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B4
!= ( image_a_set_set_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_457_subset__imageE,axiom,
! [B4: set_a,F2: product_prod_a_a > a,A3: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) )
=> ~ ! [C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C3 @ A3 )
=> ( B4
!= ( image_3437945252899457948_a_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_458_subset__imageE,axiom,
! [B4: set_set_a,F2: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F2 @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B4
!= ( image_set_a_set_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_459_subset__imageE,axiom,
! [B4: set_Product_prod_a_a,F2: a > product_prod_a_a,A3: set_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ ( image_7400625782589995694od_a_a @ F2 @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B4
!= ( image_7400625782589995694od_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_460_subset__imageE,axiom,
! [B4: set_set_a,F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_9052089385058188540_set_a @ F2 @ A3 ) )
=> ~ ! [C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C3 @ A3 )
=> ( B4
!= ( image_9052089385058188540_set_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_461_subset__imageE,axiom,
! [B4: set_Product_prod_a_a,F2: set_a > product_prod_a_a,A3: set_set_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ ( image_7677297774867312974od_a_a @ F2 @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B4
!= ( image_7677297774867312974od_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_462_subset__imageE,axiom,
! [B4: set_se5735800977113168103od_a_a,F2: set_a > set_Product_prod_a_a,A3: set_set_a] :
( ( ord_le1995061765932249223od_a_a @ B4 @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B4
!= ( image_6165024369500519726od_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_463_Collect__subset,axiom,
! [A3: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_464_Collect__subset,axiom,
! [A3: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_465_Collect__subset,axiom,
! [A3: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_466_Collect__subset,axiom,
! [A3: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_467_Compr__image__eq,axiom,
! [F2: a > a,A3: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_468_Compr__image__eq,axiom,
! [F2: $o > a,A3: set_o,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_o_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_o_a @ F2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_469_Compr__image__eq,axiom,
! [F2: a > $o,A3: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_a_o @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_a_o @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_470_Compr__image__eq,axiom,
! [F2: $o > $o,A3: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_471_Compr__image__eq,axiom,
! [F2: set_a > a,A3: set_set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_set_a_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_set_a_a @ F2
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_472_Compr__image__eq,axiom,
! [F2: a > set_a,A3: set_a,P: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_a_set_a @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_473_Compr__image__eq,axiom,
! [F2: $o > set_a,A3: set_o,P: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ ( image_o_set_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_o_set_a @ F2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_474_Compr__image__eq,axiom,
! [F2: set_a > $o,A3: set_set_a,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_set_a_o @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_set_a_o @ F2
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_475_Compr__image__eq,axiom,
! [F2: a > set_set_a,A3: set_a,P: set_set_a > $o] :
( ( collect_set_set_a
@ ^ [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( image_a_set_set_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_a_set_set_a @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_476_Compr__image__eq,axiom,
! [F2: a > product_prod_a_a,A3: set_a,P: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( image_7400625782589995694od_a_a @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image_7400625782589995694od_a_a @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_477_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_478_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_479_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_480_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X2: product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_481_image__subset__iff,axiom,
! [F2: a > set_set_a,A3: set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F2 @ A3 ) @ B4 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_set_set_a @ ( F2 @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_482_image__subset__iff,axiom,
! [F2: set_a > set_Product_prod_a_a,A3: set_set_a,B4: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) @ B4 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member1816616512716248880od_a_a @ ( F2 @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_483_image__subset__iff,axiom,
! [F2: a > set_a,A3: set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A3 ) @ B4 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_set_a @ ( F2 @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_484_image__subset__iff,axiom,
! [F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ F2 @ A3 ) @ B4 )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( member_set_a @ ( F2 @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_485_image__subset__iff,axiom,
! [F2: product_prod_a_a > product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ F2 @ A3 ) @ B4 )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( member1426531477525435216od_a_a @ ( F2 @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_486_subset__image__iff,axiom,
! [B4: set_a,F2: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F2 @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_487_subset__image__iff,axiom,
! [B4: set_a,F2: set_a > a,A3: set_set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_set_a_a @ F2 @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B4
= ( image_set_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_488_subset__image__iff,axiom,
! [B4: set_set_a,F2: a > set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F2 @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_a_set_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_489_subset__image__iff,axiom,
! [B4: set_set_set_a,F2: a > set_set_a,A3: set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ ( image_a_set_set_a @ F2 @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_a_set_set_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_490_subset__image__iff,axiom,
! [B4: set_a,F2: product_prod_a_a > a,A3: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_3437945252899457948_a_a_a @ F2 @ A3 ) )
= ( ? [AA: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ AA @ A3 )
& ( B4
= ( image_3437945252899457948_a_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_491_subset__image__iff,axiom,
! [B4: set_set_a,F2: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F2 @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B4
= ( image_set_a_set_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_492_subset__image__iff,axiom,
! [B4: set_Product_prod_a_a,F2: a > product_prod_a_a,A3: set_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ ( image_7400625782589995694od_a_a @ F2 @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_7400625782589995694od_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_493_subset__image__iff,axiom,
! [B4: set_set_a,F2: product_prod_a_a > set_a,A3: set_Product_prod_a_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_9052089385058188540_set_a @ F2 @ A3 ) )
= ( ? [AA: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ AA @ A3 )
& ( B4
= ( image_9052089385058188540_set_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_494_subset__image__iff,axiom,
! [B4: set_Product_prod_a_a,F2: set_a > product_prod_a_a,A3: set_set_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ ( image_7677297774867312974od_a_a @ F2 @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B4
= ( image_7677297774867312974od_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_495_subset__image__iff,axiom,
! [B4: set_se5735800977113168103od_a_a,F2: set_a > set_Product_prod_a_a,A3: set_set_a] :
( ( ord_le1995061765932249223od_a_a @ B4 @ ( image_6165024369500519726od_a_a @ F2 @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B4
= ( image_6165024369500519726od_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_496_ulgraph_Oincident__loops_Ocong,axiom,
undire4753905205749729249oops_a = undire4753905205749729249oops_a ).
% ulgraph.incident_loops.cong
thf(fact_497_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_498_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_499_insert__mono,axiom,
! [C2: set_Product_prod_a_a,D: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ D )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A @ C2 ) @ ( insert4534936382041156343od_a_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_500_subset__insert,axiom,
! [X: $o,A3: set_o,B4: set_o] :
( ~ ( member_o @ X @ A3 )
=> ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X @ B4 ) )
= ( ord_less_eq_set_o @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_501_subset__insert,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_502_subset__insert,axiom,
! [X: set_a,A3: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ B4 ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_503_subset__insert,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X @ B4 ) )
= ( ord_le746702958409616551od_a_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_504_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_505_subset__insertI,axiom,
! [B4: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B4 @ ( insert_set_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_506_subset__insertI,axiom,
! [B4: set_Product_prod_a_a,A: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B4 @ ( insert4534936382041156343od_a_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_507_subset__insertI2,axiom,
! [A3: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_508_subset__insertI2,axiom,
! [A3: set_set_a,B4: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_509_subset__insertI2,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_510_subset__CollectI,axiom,
! [B4: set_o,A3: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B4 @ A3 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ B4 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_511_subset__CollectI,axiom,
! [B4: set_a,A3: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ B4 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_512_subset__CollectI,axiom,
! [B4: set_set_a,A3: set_set_a,Q: set_a > $o,P: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ B4 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_513_subset__CollectI,axiom,
! [B4: set_Product_prod_a_a,A3: set_Product_prod_a_a,Q: product_prod_a_a > $o,P: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ B4 @ A3 )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ B4 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_514_subset__Collect__iff,axiom,
! [B4: set_o,A3: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B4 @ A3 )
=> ( ( ord_less_eq_set_o @ B4
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_515_subset__Collect__iff,axiom,
! [B4: set_a,A3: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( ord_less_eq_set_a @ B4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_516_subset__Collect__iff,axiom,
! [B4: set_set_a,A3: set_set_a,P: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ B4
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_517_subset__Collect__iff,axiom,
! [B4: set_Product_prod_a_a,A3: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ B4 @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ B4
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_518_Int__mono,axiom,
! [A3: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_519_Int__mono,axiom,
! [A3: set_set_a,C2: set_set_a,B4: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B4 ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_520_Int__mono,axiom,
! [A3: set_Product_prod_a_a,C2: set_Product_prod_a_a,B4: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B4 @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_521_Int__lower1,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_522_Int__lower1,axiom,
! [A3: set_set_a,B4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_523_Int__lower1,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_524_Int__lower2,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_525_Int__lower2,axiom,
! [A3: set_set_a,B4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_526_Int__lower2,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_527_Int__absorb1,axiom,
! [B4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_528_Int__absorb1,axiom,
! [B4: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_529_Int__absorb1,axiom,
! [B4: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_530_Int__absorb2,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_531_Int__absorb2,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( inf_inf_set_set_a @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_532_Int__absorb2,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_533_Int__greatest,axiom,
! [C2: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_534_Int__greatest,axiom,
! [C2: set_set_a,A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B4 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_535_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B4 )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_536_Int__Collect__mono,axiom,
! [A3: set_o,B4: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A3 @ B4 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B4 @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_537_Int__Collect__mono,axiom,
! [A3: set_a,B4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_538_Int__Collect__mono,axiom,
! [A3: set_set_a,B4: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B4 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_539_Int__Collect__mono,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B4 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_540_setcompr__eq__image,axiom,
! [F2: a > a,P: a > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_541_setcompr__eq__image,axiom,
! [F2: $o > a,P: $o > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: $o] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_o_a @ F2 @ ( collect_o @ P ) ) ) ).
% setcompr_eq_image
thf(fact_542_setcompr__eq__image,axiom,
! [F2: a > $o,P: a > $o] :
( ( collect_o
@ ^ [Uu: $o] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_a_o @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_543_setcompr__eq__image,axiom,
! [F2: $o > $o,P: $o > $o] :
( ( collect_o
@ ^ [Uu: $o] :
? [X2: $o] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F2 @ ( collect_o @ P ) ) ) ).
% setcompr_eq_image
thf(fact_544_setcompr__eq__image,axiom,
! [F2: set_a > a,P: set_a > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: set_a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_set_a_a @ F2 @ ( collect_set_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_545_setcompr__eq__image,axiom,
! [F2: a > set_a,P: a > $o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_a_set_a @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_546_setcompr__eq__image,axiom,
! [F2: $o > set_a,P: $o > $o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: $o] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_o_set_a @ F2 @ ( collect_o @ P ) ) ) ).
% setcompr_eq_image
thf(fact_547_setcompr__eq__image,axiom,
! [F2: set_a > $o,P: set_a > $o] :
( ( collect_o
@ ^ [Uu: $o] :
? [X2: set_a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_set_a_o @ F2 @ ( collect_set_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_548_setcompr__eq__image,axiom,
! [F2: a > set_set_a,P: a > $o] :
( ( collect_set_set_a
@ ^ [Uu: set_set_a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_a_set_set_a @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_549_setcompr__eq__image,axiom,
! [F2: a > product_prod_a_a,P: a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image_7400625782589995694od_a_a @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_550_Setcompr__eq__image,axiom,
! [F2: a > a,A3: set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_a @ X2 @ A3 ) ) )
= ( image_a_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_551_Setcompr__eq__image,axiom,
! [F2: $o > a,A3: set_o] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: $o] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_o @ X2 @ A3 ) ) )
= ( image_o_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_552_Setcompr__eq__image,axiom,
! [F2: a > $o,A3: set_a] :
( ( collect_o
@ ^ [Uu: $o] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_a @ X2 @ A3 ) ) )
= ( image_a_o @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_553_Setcompr__eq__image,axiom,
! [F2: $o > $o,A3: set_o] :
( ( collect_o
@ ^ [Uu: $o] :
? [X2: $o] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_o @ X2 @ A3 ) ) )
= ( image_o_o @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_554_Setcompr__eq__image,axiom,
! [F2: set_a > a,A3: set_set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: set_a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_set_a @ X2 @ A3 ) ) )
= ( image_set_a_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_555_Setcompr__eq__image,axiom,
! [F2: a > set_a,A3: set_a] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_a @ X2 @ A3 ) ) )
= ( image_a_set_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_556_Setcompr__eq__image,axiom,
! [F2: $o > set_a,A3: set_o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: $o] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_o @ X2 @ A3 ) ) )
= ( image_o_set_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_557_Setcompr__eq__image,axiom,
! [F2: set_a > $o,A3: set_set_a] :
( ( collect_o
@ ^ [Uu: $o] :
? [X2: set_a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_set_a @ X2 @ A3 ) ) )
= ( image_set_a_o @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_558_Setcompr__eq__image,axiom,
! [F2: a > set_set_a,A3: set_a] :
( ( collect_set_set_a
@ ^ [Uu: set_set_a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_a @ X2 @ A3 ) ) )
= ( image_a_set_set_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_559_Setcompr__eq__image,axiom,
! [F2: a > product_prod_a_a,A3: set_a] :
( ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [X2: a] :
( ( Uu
= ( F2 @ X2 ) )
& ( member_a @ X2 @ A3 ) ) )
= ( image_7400625782589995694od_a_a @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_560_subset__singleton__iff,axiom,
! [X3: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X3 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X3 = bot_bot_set_nat )
| ( X3
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_561_subset__singleton__iff,axiom,
! [X3: set_a,A: a] :
( ( ord_less_eq_set_a @ X3 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X3 = bot_bot_set_a )
| ( X3
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_562_subset__singleton__iff,axiom,
! [X3: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X3 = bot_bot_set_set_a )
| ( X3
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_563_subset__singleton__iff,axiom,
! [X3: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( ( X3 = bot_bo3357376287454694259od_a_a )
| ( X3
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_564_subset__singletonD,axiom,
! [A3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A3 = bot_bot_set_nat )
| ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_565_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_566_subset__singletonD,axiom,
! [A3: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A3 = bot_bot_set_set_a )
| ( A3
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_567_subset__singletonD,axiom,
! [A3: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A3 = bot_bo3357376287454694259od_a_a )
| ( A3
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_568_ulgraph_Oincident__loops__simp_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( ( undire4753905205749729249oops_a @ Edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_loops_simp(2)
thf(fact_569_subgraph_Overts__ss,axiom,
! [V_H: set_set_a,E_H: set_set_set_a,V_G: set_set_a,E_G: set_set_set_a] :
( ( undire1186139521737116585_set_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_570_subgraph_Overts__ss,axiom,
! [V_H: set_Product_prod_a_a,E_H: set_se5735800977113168103od_a_a,V_G: set_Product_prod_a_a,E_G: set_se5735800977113168103od_a_a] :
( ( undire398746457437328754od_a_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le746702958409616551od_a_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_571_subgraph_Overts__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_less_eq_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_572_image__constant__conv,axiom,
! [A3: set_a,C: a] :
( ( ( A3 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A3 )
= bot_bot_set_a ) )
& ( ( A3 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_573_image__constant__conv,axiom,
! [A3: set_a,C: nat] :
( ( ( A3 = bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X2: a] : C
@ A3 )
= bot_bot_set_nat ) )
& ( ( A3 != bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X2: a] : C
@ A3 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_574_image__constant__conv,axiom,
! [A3: set_nat,C: a] :
( ( ( A3 = bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X2: nat] : C
@ A3 )
= bot_bot_set_a ) )
& ( ( A3 != bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X2: nat] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_575_image__constant__conv,axiom,
! [A3: set_nat,C: nat] :
( ( ( A3 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C
@ A3 )
= bot_bot_set_nat ) )
& ( ( A3 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C
@ A3 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_576_image__constant__conv,axiom,
! [A3: set_a,C: set_a] :
( ( ( A3 = bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X2: a] : C
@ A3 )
= bot_bot_set_set_a ) )
& ( ( A3 != bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X2: a] : C
@ A3 )
= ( insert_set_a @ C @ bot_bot_set_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_577_image__constant__conv,axiom,
! [A3: set_set_a,C: a] :
( ( ( A3 = bot_bot_set_set_a )
=> ( ( image_set_a_a
@ ^ [X2: set_a] : C
@ A3 )
= bot_bot_set_a ) )
& ( ( A3 != bot_bot_set_set_a )
=> ( ( image_set_a_a
@ ^ [X2: set_a] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_578_image__constant__conv,axiom,
! [A3: set_set_a,C: nat] :
( ( ( A3 = bot_bot_set_set_a )
=> ( ( image_set_a_nat
@ ^ [X2: set_a] : C
@ A3 )
= bot_bot_set_nat ) )
& ( ( A3 != bot_bot_set_set_a )
=> ( ( image_set_a_nat
@ ^ [X2: set_a] : C
@ A3 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_579_image__constant__conv,axiom,
! [A3: set_nat,C: set_a] :
( ( ( A3 = bot_bot_set_nat )
=> ( ( image_nat_set_a
@ ^ [X2: nat] : C
@ A3 )
= bot_bot_set_set_a ) )
& ( ( A3 != bot_bot_set_nat )
=> ( ( image_nat_set_a
@ ^ [X2: nat] : C
@ A3 )
= ( insert_set_a @ C @ bot_bot_set_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_580_image__constant__conv,axiom,
! [A3: set_a,C: set_set_a] :
( ( ( A3 = bot_bot_set_a )
=> ( ( image_a_set_set_a
@ ^ [X2: a] : C
@ A3 )
= bot_bo3380559777022489994_set_a ) )
& ( ( A3 != bot_bot_set_a )
=> ( ( image_a_set_set_a
@ ^ [X2: a] : C
@ A3 )
= ( insert_set_set_a @ C @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_581_image__constant__conv,axiom,
! [A3: set_a,C: product_prod_a_a] :
( ( ( A3 = bot_bot_set_a )
=> ( ( image_7400625782589995694od_a_a
@ ^ [X2: a] : C
@ A3 )
= bot_bo3357376287454694259od_a_a ) )
& ( ( A3 != bot_bot_set_a )
=> ( ( image_7400625782589995694od_a_a
@ ^ [X2: a] : C
@ A3 )
= ( insert4534936382041156343od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% image_constant_conv
thf(fact_582_image__constant,axiom,
! [X: a,A3: set_a,C: a] :
( ( member_a @ X @ A3 )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_583_image__constant,axiom,
! [X: $o,A3: set_o,C: a] :
( ( member_o @ X @ A3 )
=> ( ( image_o_a
@ ^ [X2: $o] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_584_image__constant,axiom,
! [X: a,A3: set_a,C: nat] :
( ( member_a @ X @ A3 )
=> ( ( image_a_nat
@ ^ [X2: a] : C
@ A3 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_585_image__constant,axiom,
! [X: $o,A3: set_o,C: nat] :
( ( member_o @ X @ A3 )
=> ( ( image_o_nat
@ ^ [X2: $o] : C
@ A3 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_586_image__constant,axiom,
! [X: set_a,A3: set_set_a,C: a] :
( ( member_set_a @ X @ A3 )
=> ( ( image_set_a_a
@ ^ [X2: set_a] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_587_image__constant,axiom,
! [X: a,A3: set_a,C: set_a] :
( ( member_a @ X @ A3 )
=> ( ( image_a_set_a
@ ^ [X2: a] : C
@ A3 )
= ( insert_set_a @ C @ bot_bot_set_set_a ) ) ) ).
% image_constant
thf(fact_588_image__constant,axiom,
! [X: $o,A3: set_o,C: set_a] :
( ( member_o @ X @ A3 )
=> ( ( image_o_set_a
@ ^ [X2: $o] : C
@ A3 )
= ( insert_set_a @ C @ bot_bot_set_set_a ) ) ) ).
% image_constant
thf(fact_589_image__constant,axiom,
! [X: set_a,A3: set_set_a,C: nat] :
( ( member_set_a @ X @ A3 )
=> ( ( image_set_a_nat
@ ^ [X2: set_a] : C
@ A3 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_590_image__constant,axiom,
! [X: a,A3: set_a,C: set_set_a] :
( ( member_a @ X @ A3 )
=> ( ( image_a_set_set_a
@ ^ [X2: a] : C
@ A3 )
= ( insert_set_set_a @ C @ bot_bo3380559777022489994_set_a ) ) ) ).
% image_constant
thf(fact_591_image__constant,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ( image_3437945252899457948_a_a_a
@ ^ [X2: product_prod_a_a] : C
@ A3 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_592_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_593_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_594_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_595_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_596_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_597_equals0D,axiom,
! [A3: set_o,A: $o] :
( ( A3 = bot_bot_set_o )
=> ~ ( member_o @ A @ A3 ) ) ).
% equals0D
thf(fact_598_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_599_equals0D,axiom,
! [A3: set_set_a,A: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_600_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_601_equals0D,axiom,
! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A3 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A3 ) ) ).
% equals0D
thf(fact_602_equals0I,axiom,
! [A3: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A3 )
=> ( A3 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_603_equals0I,axiom,
! [A3: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_604_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_605_equals0I,axiom,
! [A3: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_606_equals0I,axiom,
! [A3: set_Product_prod_a_a] :
( ! [Y4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y4 @ A3 )
=> ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_607_ex__in__conv,axiom,
! [A3: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A3 ) )
= ( A3 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_608_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_609_ex__in__conv,axiom,
! [A3: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A3 ) )
= ( A3 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_610_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_611_ex__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ? [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A3 ) )
= ( A3 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_612_insertE,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member1426531477525435216od_a_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_613_insertE,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_614_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_615_insertE,axiom,
! [A: $o,B: $o,A3: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A3 ) )
=> ( ( A = ~ B )
=> ( member_o @ A @ A3 ) ) ) ).
% insertE
thf(fact_616_insertI1,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B4 ) ) ).
% insertI1
thf(fact_617_insertI1,axiom,
! [A: set_a,B4: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B4 ) ) ).
% insertI1
thf(fact_618_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_619_insertI1,axiom,
! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).
% insertI1
thf(fact_620_insertI2,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ B4 )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_621_insertI2,axiom,
! [A: set_a,B4: set_set_a,B: set_a] :
( ( member_set_a @ A @ B4 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_622_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_623_insertI2,axiom,
! [A: $o,B4: set_o,B: $o] :
( ( member_o @ A @ B4 )
=> ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% insertI2
thf(fact_624_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ~ ! [B6: set_Product_prod_a_a] :
( ( A3
= ( insert4534936382041156343od_a_a @ X @ B6 ) )
=> ( member1426531477525435216od_a_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_625_Set_Oset__insert,axiom,
! [X: set_a,A3: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ~ ! [B6: set_set_a] :
( ( A3
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_626_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_627_Set_Oset__insert,axiom,
! [X: $o,A3: set_o] :
( ( member_o @ X @ A3 )
=> ~ ! [B6: set_o] :
( ( A3
= ( insert_o @ X @ B6 ) )
=> ( member_o @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_628_insert__ident,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A3 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B4 )
=> ( ( ( insert4534936382041156343od_a_a @ X @ A3 )
= ( insert4534936382041156343od_a_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_629_insert__ident,axiom,
! [X: set_a,A3: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ~ ( member_set_a @ X @ B4 )
=> ( ( ( insert_set_a @ X @ A3 )
= ( insert_set_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_630_insert__ident,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B4 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_631_insert__ident,axiom,
! [X: $o,A3: set_o,B4: set_o] :
( ~ ( member_o @ X @ A3 )
=> ( ~ ( member_o @ X @ B4 )
=> ( ( ( insert_o @ X @ A3 )
= ( insert_o @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_632_insert__absorb,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A3 )
=> ( ( insert4534936382041156343od_a_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_633_insert__absorb,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( insert_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_634_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_635_insert__absorb,axiom,
! [A: $o,A3: set_o] :
( ( member_o @ A @ A3 )
=> ( ( insert_o @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_636_insert__eq__iff,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A3 )
=> ( ~ ( member1426531477525435216od_a_a @ B @ B4 )
=> ( ( ( insert4534936382041156343od_a_a @ A @ A3 )
= ( insert4534936382041156343od_a_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C4: set_Product_prod_a_a] :
( ( A3
= ( insert4534936382041156343od_a_a @ B @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ B @ C4 )
& ( B4
= ( insert4534936382041156343od_a_a @ A @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_637_insert__eq__iff,axiom,
! [A: set_a,A3: set_set_a,B: set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ~ ( member_set_a @ B @ B4 )
=> ( ( ( insert_set_a @ A @ A3 )
= ( insert_set_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A3
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B4
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_638_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A3
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B4
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_639_insert__eq__iff,axiom,
! [A: $o,A3: set_o,B: $o,B4: set_o] :
( ~ ( member_o @ A @ A3 )
=> ( ~ ( member_o @ B @ B4 )
=> ( ( ( insert_o @ A @ A3 )
= ( insert_o @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A = ~ B )
=> ? [C4: set_o] :
( ( A3
= ( insert_o @ B @ C4 ) )
& ~ ( member_o @ B @ C4 )
& ( B4
= ( insert_o @ A @ C4 ) )
& ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_640_insert__commute,axiom,
! [X: a,Y3: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ Y3 @ A3 ) )
= ( insert_a @ Y3 @ ( insert_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_641_insert__commute,axiom,
! [X: set_a,Y3: set_a,A3: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y3 @ A3 ) )
= ( insert_set_a @ Y3 @ ( insert_set_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_642_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A3 )
=> ? [B6: set_Product_prod_a_a] :
( ( A3
= ( insert4534936382041156343od_a_a @ A @ B6 ) )
& ~ ( member1426531477525435216od_a_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_643_mk__disjoint__insert,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ? [B6: set_set_a] :
( ( A3
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_644_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_645_mk__disjoint__insert,axiom,
! [A: $o,A3: set_o] :
( ( member_o @ A @ A3 )
=> ? [B6: set_o] :
( ( A3
= ( insert_o @ A @ B6 ) )
& ~ ( member_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_646_IntE,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A3 )
=> ~ ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_647_IntE,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B4 ) )
=> ~ ( ( member_o @ C @ A3 )
=> ~ ( member_o @ C @ B4 ) ) ) ).
% IntE
thf(fact_648_IntE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ~ ( member_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_649_IntE,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B4 ) )
=> ~ ( ( member_set_a @ C @ A3 )
=> ~ ( member_set_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_650_IntD1,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) )
=> ( member1426531477525435216od_a_a @ C @ A3 ) ) ).
% IntD1
thf(fact_651_IntD1,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B4 ) )
=> ( member_o @ C @ A3 ) ) ).
% IntD1
thf(fact_652_IntD1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ A3 ) ) ).
% IntD1
thf(fact_653_IntD1,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B4 ) )
=> ( member_set_a @ C @ A3 ) ) ).
% IntD1
thf(fact_654_IntD2,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ).
% IntD2
thf(fact_655_IntD2,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B4 ) )
=> ( member_o @ C @ B4 ) ) ).
% IntD2
thf(fact_656_IntD2,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ B4 ) ) ).
% IntD2
thf(fact_657_IntD2,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B4 ) )
=> ( member_set_a @ C @ B4 ) ) ).
% IntD2
thf(fact_658_Int__assoc,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_659_Int__assoc,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_660_Int__absorb,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_661_Int__absorb,axiom,
! [A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_662_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_663_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_664_Int__left__absorb,axiom,
! [A3: set_a,B4: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( inf_inf_set_a @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_665_Int__left__absorb,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ A3 @ B4 ) )
= ( inf_inf_set_set_a @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_666_Int__left__commute,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_667_Int__left__commute,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B4 @ C2 ) )
= ( inf_inf_set_set_a @ B4 @ ( inf_inf_set_set_a @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_668_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X2: $o] : $false ) ) ).
% empty_def
thf(fact_669_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_670_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X2: set_a] : $false ) ) ).
% empty_def
thf(fact_671_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_672_empty__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : $false ) ) ).
% empty_def
thf(fact_673_insert__compr,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A4: product_prod_a_a,B5: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( X2 = A4 )
| ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_674_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( X2 = A4 )
| ( member_a @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_675_insert__compr,axiom,
( insert_set_a
= ( ^ [A4: set_a,B5: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( X2 = A4 )
| ( member_set_a @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_676_insert__compr,axiom,
( insert_o
= ( ^ [A4: $o,B5: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( X2 = A4 )
| ( member_o @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_677_insert__Collect,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( insert4534936382041156343od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( collec3336397797384452498od_a_a
@ ^ [U: product_prod_a_a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_678_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_679_insert__Collect,axiom,
! [A: set_a,P: set_a > $o] :
( ( insert_set_a @ A @ ( collect_set_a @ P ) )
= ( collect_set_a
@ ^ [U: set_a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_680_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_681_Int__def,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A5 )
& ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_682_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A5: set_o,B5: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A5 )
& ( member_o @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_683_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
& ( member_a @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_684_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
& ( member_set_a @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_685_Int__Collect,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A3 @ ( collec3336397797384452498od_a_a @ P ) ) )
= ( ( member1426531477525435216od_a_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_686_Int__Collect,axiom,
! [X: $o,A3: set_o,P: $o > $o] :
( ( member_o @ X @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
= ( ( member_o @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_687_Int__Collect,axiom,
! [X: a,A3: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_688_Int__Collect,axiom,
! [X: set_a,A3: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_689_Collect__conj__eq,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_in8905007599844390133od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_690_Collect__conj__eq,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_691_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_692_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_693_ulgraph_Oincident__loops__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7215034953758041409_set_a @ Edges @ V3 )
= ( collect_set_set_a
@ ^ [E3: set_set_a] :
( ( member_set_set_a @ E3 @ Edges )
& ( E3
= ( insert_set_a @ V3 @ bot_bot_set_set_a ) ) ) ) ) ) ).
% ulgraph.incident_loops_def
thf(fact_694_ulgraph_Oincident__loops__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1050940535076293677ps_nat @ Edges @ V3 )
= ( collect_set_nat
@ ^ [E3: set_nat] :
( ( member_set_nat @ E3 @ Edges )
& ( E3
= ( insert_nat @ V3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% ulgraph.incident_loops_def
thf(fact_695_ulgraph_Oincident__loops__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3049230956220217098od_a_a @ Edges @ V3 )
= ( collec1673347964119250290od_a_a
@ ^ [E3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E3 @ Edges )
& ( E3
= ( insert4534936382041156343od_a_a @ V3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% ulgraph.incident_loops_def
thf(fact_696_ulgraph_Oincident__loops__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire4753905205749729249oops_a @ Edges @ V3 )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ Edges )
& ( E3
= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ) ).
% ulgraph.incident_loops_def
thf(fact_697_ulgraph_Oincident__edges__neighbors__img,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4631953023069350784_set_a @ Edges @ V3 )
= ( image_4955109552351689957_set_a
@ ^ [U: set_a] : ( insert_set_a @ V3 @ ( insert_set_a @ U @ bot_bot_set_set_a ) )
@ ( undire2074812191327625774_set_a @ Vertices @ Edges @ V3 ) ) ) ) ).
% ulgraph.incident_edges_neighbors_img
thf(fact_698_ulgraph_Oincident__edges__neighbors__img,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire4176300566717384750es_nat @ Edges @ V3 )
= ( image_nat_set_nat
@ ^ [U: nat] : ( insert_nat @ V3 @ ( insert_nat @ U @ bot_bot_set_nat ) )
@ ( undire8190396521545869824od_nat @ Vertices @ Edges @ V3 ) ) ) ) ).
% ulgraph.incident_edges_neighbors_img
thf(fact_699_ulgraph_Oincident__edges__neighbors__img,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire8905369280470868553od_a_a @ Edges @ V3 )
= ( image_6690255128444368805od_a_a
@ ^ [U: product_prod_a_a] : ( insert4534936382041156343od_a_a @ V3 @ ( insert4534936382041156343od_a_a @ U @ bot_bo3357376287454694259od_a_a ) )
@ ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V3 ) ) ) ) ).
% ulgraph.incident_edges_neighbors_img
thf(fact_700_ulgraph_Oincident__edges__neighbors__img,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3231912044278729248dges_a @ Edges @ V3 )
= ( image_a_set_a
@ ^ [U: a] : ( insert_a @ V3 @ ( insert_a @ U @ bot_bot_set_a ) )
@ ( undire8504279938402040014hood_a @ Vertices @ Edges @ V3 ) ) ) ) ).
% ulgraph.incident_edges_neighbors_img
thf(fact_701_ulgraph_Oadj__relation__wf,axiom,
! [Vertices: set_nat,Edges: set_set_nat,U3: nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U3 @ V3 ) @ ( graph_3658176268357964989on_nat @ Edges ) )
=> ( ord_less_eq_set_nat @ ( insert_nat @ U3 @ ( insert_nat @ V3 @ bot_bot_set_nat ) ) @ Vertices ) ) ) ).
% ulgraph.adj_relation_wf
thf(fact_702_ulgraph_Oadj__relation__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U3: set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ U3 @ V3 ) @ ( graph_7634368646383411377_set_a @ Edges ) )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ U3 @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) ) @ Vertices ) ) ) ).
% ulgraph.adj_relation_wf
thf(fact_703_ulgraph_Oadj__relation__wf,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,U3: product_prod_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ U3 @ V3 ) @ ( graph_210517301254161018od_a_a @ Edges ) )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ U3 @ ( insert4534936382041156343od_a_a @ V3 @ bot_bo3357376287454694259od_a_a ) ) @ Vertices ) ) ) ).
% ulgraph.adj_relation_wf
thf(fact_704_ulgraph_Oadj__relation__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,U3: a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ V3 ) @ ( graph_8122095853558514513tion_a @ Edges ) )
=> ( ord_less_eq_set_a @ ( insert_a @ U3 @ ( insert_a @ V3 @ bot_bot_set_a ) ) @ Vertices ) ) ) ).
% ulgraph.adj_relation_wf
thf(fact_705_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V3 )
=> ( ( undire7215034953758041409_set_a @ Edges @ V3 )
= ( insert_set_set_a @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_706_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V3: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V3 )
=> ( ( undire1050940535076293677ps_nat @ Edges @ V3 )
= ( insert_set_nat @ ( insert_nat @ V3 @ bot_bot_set_nat ) @ bot_bot_set_set_nat ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_707_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V3 )
=> ( ( undire3049230956220217098od_a_a @ Edges @ V3 )
= ( insert914553114930139863od_a_a @ ( insert4534936382041156343od_a_a @ V3 @ bot_bo3357376287454694259od_a_a ) @ bot_bo777872063958040403od_a_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_708_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( ( undire4753905205749729249oops_a @ Edges @ V3 )
= ( insert_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_709_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_710_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_711_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_712_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_713_singletonD,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_714_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_715_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_716_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_717_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_718_singleton__iff,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_719_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_720_doubleton__eq__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D2: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D2 @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_721_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_722_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_723_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_724_insert__not__empty,axiom,
! [A: set_a,A3: set_set_a] :
( ( insert_set_a @ A @ A3 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_725_insert__not__empty,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ A3 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_726_insert__not__empty,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A @ A3 )
!= bot_bo3357376287454694259od_a_a ) ).
% insert_not_empty
thf(fact_727_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_728_singleton__inject,axiom,
! [A: set_a,B: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_729_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_730_singleton__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_731_Int__emptyI,axiom,
! [A3: set_o,B4: set_o] :
( ! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ~ ( member_o @ X5 @ B4 ) )
=> ( ( inf_inf_set_o @ A3 @ B4 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_732_Int__emptyI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ~ ( member_a @ X5 @ B4 ) )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_733_Int__emptyI,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ~ ( member_set_a @ X5 @ B4 ) )
=> ( ( inf_inf_set_set_a @ A3 @ B4 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_734_Int__emptyI,axiom,
! [A3: set_nat,B4: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ~ ( member_nat @ X5 @ B4 ) )
=> ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_735_Int__emptyI,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ~ ( member1426531477525435216od_a_a @ X5 @ B4 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ B4 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_736_disjoint__iff,axiom,
! [A3: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A3 @ B4 )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ~ ( member_o @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_737_disjoint__iff,axiom,
! [A3: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ~ ( member_a @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_738_disjoint__iff,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B4 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ~ ( member_set_a @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_739_disjoint__iff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ~ ( member_nat @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_740_disjoint__iff,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A3 @ B4 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ~ ( member1426531477525435216od_a_a @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_741_Int__empty__left,axiom,
! [B4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B4 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_742_Int__empty__left,axiom,
! [B4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B4 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_743_Int__empty__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_744_Int__empty__left,axiom,
! [B4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B4 )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_745_Int__empty__right,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_746_Int__empty__right,axiom,
! [A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_747_Int__empty__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_748_Int__empty__right,axiom,
! [A3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_749_disjoint__iff__not__equal,axiom,
! [A3: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B4 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_750_disjoint__iff__not__equal,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B4 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ B4 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_751_disjoint__iff__not__equal,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_752_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A3 @ B4 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ B4 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_753_Int__insert__left,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B4 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B4 @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B4 ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_754_Int__insert__left,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) )
& ( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_755_Int__insert__left,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_756_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B4: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_757_Int__insert__right,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( inf_in8905007599844390133od_a_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_758_Int__insert__right,axiom,
! [A: $o,A3: set_o,B4: set_o] :
( ( ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B4 ) ) ) )
& ( ~ ( member_o @ A @ A3 )
=> ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B4 ) )
= ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_759_Int__insert__right,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B4 ) ) ) )
& ( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_760_Int__insert__right,axiom,
! [A: set_a,A3: set_set_a,B4: set_set_a] :
( ( ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A3 @ B4 ) ) ) )
& ( ~ ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B4 ) )
= ( inf_inf_set_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_761_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_762_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_763_Collect__conv__if,axiom,
! [P: set_a > $o,A: set_a] :
( ( ( P @ A )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if
thf(fact_764_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_765_Collect__conv__if,axiom,
! [P: product_prod_a_a > $o,A: product_prod_a_a] :
( ( ( P @ A )
=> ( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) )
& ( ~ ( P @ A )
=> ( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% Collect_conv_if
thf(fact_766_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_767_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_768_Collect__conv__if2,axiom,
! [P: set_a > $o,A: set_a] :
( ( ( P @ A )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_a
@ ^ [X2: set_a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_769_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_770_Collect__conv__if2,axiom,
! [P: product_prod_a_a > $o,A: product_prod_a_a] :
( ( ( P @ A )
=> ( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) )
& ( ~ ( P @ A )
=> ( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% Collect_conv_if2
thf(fact_771_comp__sgraph_Ois__edge__between__def,axiom,
( undire3044609692436228535ween_o
= ( ^ [X6: set_o,Y5: set_o,E3: set_o] :
? [X2: $o,Y2: $o] :
( ( E3
= ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
& ( member_o @ X2 @ X6 )
& ( member_o @ Y2 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_772_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X6: set_set_a,Y5: set_set_a,E3: set_set_a] :
? [X2: set_a,Y2: set_a] :
( ( E3
= ( insert_set_a @ X2 @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X2 @ X6 )
& ( member_set_a @ Y2 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_773_comp__sgraph_Ois__edge__between__def,axiom,
( undire6814325412647357297en_nat
= ( ^ [X6: set_nat,Y5: set_nat,E3: set_nat] :
? [X2: nat,Y2: nat] :
( ( E3
= ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
& ( member_nat @ X2 @ X6 )
& ( member_nat @ Y2 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_774_comp__sgraph_Ois__edge__between__def,axiom,
( undire7011261089604658374od_a_a
= ( ^ [X6: set_Product_prod_a_a,Y5: set_Product_prod_a_a,E3: set_Product_prod_a_a] :
? [X2: product_prod_a_a,Y2: product_prod_a_a] :
( ( E3
= ( insert4534936382041156343od_a_a @ X2 @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X2 @ X6 )
& ( member1426531477525435216od_a_a @ Y2 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_775_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y5: set_a,E3: set_a] :
? [X2: a,Y2: a] :
( ( E3
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X6 )
& ( member_a @ Y2 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_776_ulgraph_Ois__loop__set__alt,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( collect_set_set_a
@ ^ [Uu: set_set_a] :
? [V: set_a] :
( ( Uu
= ( insert_set_a @ V @ bot_bot_set_set_a ) )
& ( undire5774735625301615776_set_a @ Edges @ V ) ) )
= ( collect_set_set_a
@ ^ [E3: set_set_a] :
( ( member_set_set_a @ E3 @ Edges )
& ( undire3618949687197220622_set_a @ E3 ) ) ) ) ) ).
% ulgraph.is_loop_set_alt
thf(fact_777_ulgraph_Ois__loop__set__alt,axiom,
! [Vertices: set_nat,Edges: set_set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( collect_set_nat
@ ^ [Uu: set_nat] :
? [V: nat] :
( ( Uu
= ( insert_nat @ V @ bot_bot_set_nat ) )
& ( undire5005864372999571214op_nat @ Edges @ V ) ) )
= ( collect_set_nat
@ ^ [E3: set_nat] :
( ( member_set_nat @ E3 @ Edges )
& ( undire643512044667278624op_nat @ E3 ) ) ) ) ) ).
% ulgraph.is_loop_set_alt
thf(fact_778_ulgraph_Ois__loop__set__alt,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( collec1673347964119250290od_a_a
@ ^ [Uu: set_Product_prod_a_a] :
? [V: product_prod_a_a] :
( ( Uu
= ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) )
& ( undire7777398424729533289od_a_a @ Edges @ V ) ) )
= ( collec1673347964119250290od_a_a
@ ^ [E3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E3 @ Edges )
& ( undire3428022325429088215od_a_a @ E3 ) ) ) ) ) ).
% ulgraph.is_loop_set_alt
thf(fact_779_ulgraph_Ois__loop__set__alt,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( collect_set_a
@ ^ [Uu: set_a] :
? [V: a] :
( ( Uu
= ( insert_a @ V @ bot_bot_set_a ) )
& ( undire3617971648856834880loop_a @ Edges @ V ) ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ Edges )
& ( undire2905028936066782638loop_a @ E3 ) ) ) ) ) ).
% ulgraph.is_loop_set_alt
thf(fact_780_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_o,Edges: set_set_o,X3: set_o,Y: set_o,E: set_o] :
( ( undire2905056519009681222raph_o @ Vertices @ Edges )
=> ( ( undire3044609692436228535ween_o @ X3 @ Y @ E )
= ( ? [X2: $o,Y2: $o] :
( ( E
= ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
& ( member_o @ X2 @ X3 )
& ( member_o @ Y2 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_781_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X3: set_set_a,Y: set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2578756059399487229_set_a @ X3 @ Y @ E )
= ( ? [X2: set_a,Y2: set_a] :
( ( E
= ( insert_set_a @ X2 @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X2 @ X3 )
& ( member_set_a @ Y2 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_782_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X3: set_nat,Y: set_nat,E: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire6814325412647357297en_nat @ X3 @ Y @ E )
= ( ? [X2: nat,Y2: nat] :
( ( E
= ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
& ( member_nat @ X2 @ X3 )
& ( member_nat @ Y2 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_783_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X3: set_Product_prod_a_a,Y: set_Product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7011261089604658374od_a_a @ X3 @ Y @ E )
= ( ? [X2: product_prod_a_a,Y2: product_prod_a_a] :
( ( E
= ( insert4534936382041156343od_a_a @ X2 @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X2 @ X3 )
& ( member1426531477525435216od_a_a @ Y2 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_784_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X3: set_a,Y: set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8544646567961481629ween_a @ X3 @ Y @ E )
= ( ? [X2: a,Y2: a] :
( ( E
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X3 )
& ( member_a @ Y2 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_785_edges__split__loop__inter__empty,axiom,
( bot_bot_set_set_a
= ( inf_inf_set_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) )
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ) ).
% edges_split_loop_inter_empty
thf(fact_786_is__edge__or__loop,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ( undire2905028936066782638loop_a @ E )
| ( undire4917966558017083288edge_a @ E ) ) ) ).
% is_edge_or_loop
thf(fact_787_induced__is__subgraph,axiom,
! [V5: set_a] :
( ( ord_less_eq_set_a @ V5 @ vertices )
=> ( undire7103218114511261257raph_a @ V5 @ ( undire7777452895879145676dges_a @ edges @ V5 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_788_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_789_inf__bot__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_790_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_791_inf__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_792_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_793_inf__bot__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_794_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_795_inf__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_796_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_797_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_798_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_799_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_800_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_801_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_802_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_803_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_804_le__inf__iff,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_805_le__inf__iff,axiom,
! [X: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y3 @ Z3 ) )
= ( ( ord_less_eq_real @ X @ Y3 )
& ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_806_le__inf__iff,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z3 ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
& ( ord_le3724670747650509150_set_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_807_le__inf__iff,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z3 ) )
= ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
& ( ord_le746702958409616551od_a_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_808_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_809_inf_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_810_inf_Obounded__iff,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B )
& ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_811_inf_Obounded__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C ) )
= ( ( ord_le746702958409616551od_a_a @ A @ B )
& ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_812_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_813_edge__density__ge0,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X3 @ Y ) ) ).
% edge_density_ge0
thf(fact_814_induced__edges__def,axiom,
! [V5: set_a] :
( ( undire7777452895879145676dges_a @ edges @ V5 )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( ord_less_eq_set_a @ E3 @ V5 ) ) ) ) ).
% induced_edges_def
thf(fact_815_inf__right__idem,axiom,
! [X: set_a,Y3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 )
= ( inf_inf_set_a @ X @ Y3 ) ) ).
% inf_right_idem
thf(fact_816_inf__right__idem,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 )
= ( inf_inf_set_set_a @ X @ Y3 ) ) ).
% inf_right_idem
thf(fact_817_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_818_inf_Oright__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_819_inf__left__idem,axiom,
! [X: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y3 ) )
= ( inf_inf_set_a @ X @ Y3 ) ) ).
% inf_left_idem
thf(fact_820_inf__left__idem,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y3 ) )
= ( inf_inf_set_set_a @ X @ Y3 ) ) ).
% inf_left_idem
thf(fact_821_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_822_inf_Oleft__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_823_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_824_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_825_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_826_inf_Oidem,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_827_induced__edges__ss,axiom,
! [V5: set_a] :
( ( ord_less_eq_set_a @ V5 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V5 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_828_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_829_pred__subset__eq2,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_less_eq_a_a_o
@ ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
@ ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ S ) )
= ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_830_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_831_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_832_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_833_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_834_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_835_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_836_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_837_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_838_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_839_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_840_subrelI,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ! [X5: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ S2 ) )
=> ( ord_le746702958409616551od_a_a @ R2 @ S2 ) ) ).
% subrelI
thf(fact_841_inf__Int__eq2,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_inf_a_a_o
@ ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ R )
@ ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ S ) )
= ( ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( inf_in8905007599844390133od_a_a @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_842_bot__empty__eq2,axiom,
( bot_bot_a_a_o
= ( ^ [X2: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_843_inf__Int__eq,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_in2559554923042384936_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ R )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ S ) )
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_844_inf__Int__eq,axiom,
! [R: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( inf_inf_set_o @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_845_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_846_inf__Int__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( inf_inf_set_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_847_inf__set__def,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( inf_in2559554923042384936_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A5 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_848_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A5: set_o,B5: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A5 )
@ ^ [X2: $o] : ( member_o @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_849_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_850_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A5 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_851_subgraph_Oedges__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ E_H @ E_G ) ) ).
% subgraph.edges_ss
thf(fact_852_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_853_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_854_pred__subset__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ord_le3724670747650509150_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_855_pred__subset__eq,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le1591150415168442102_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ R )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ S ) )
= ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_856_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A5 )
@ ^ [X2: $o] : ( member_o @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_857_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_858_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A5 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_859_less__eq__set__def,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ord_le1591150415168442102_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A5 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_860_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_861_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_862_inf__left__commute,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X @ Z3 ) ) ) ).
% inf_left_commute
thf(fact_863_inf__left__commute,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X @ Z3 ) ) ) ).
% inf_left_commute
thf(fact_864_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_865_inf_Oleft__commute,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A @ C ) )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_866_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_a,K: set_a,B: set_a,A: set_a] :
( ( B4
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_867_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_set_a,K: set_set_a,B: set_set_a,A: set_set_a] :
( ( B4
= ( inf_inf_set_set_a @ K @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B4 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_868_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_869_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_set_a,K: set_set_a,A: set_set_a,B: set_set_a] :
( ( A3
= ( inf_inf_set_set_a @ K @ A ) )
=> ( ( inf_inf_set_set_a @ A3 @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_870_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_871_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_872_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B7: set_a] : ( inf_inf_set_a @ B7 @ A4 ) ) ) ).
% inf.commute
thf(fact_873_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B7: set_set_a] : ( inf_inf_set_set_a @ B7 @ A4 ) ) ) ).
% inf.commute
thf(fact_874_inf__assoc,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Z3 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z3 ) ) ) ).
% inf_assoc
thf(fact_875_inf__assoc,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Z3 )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z3 ) ) ) ).
% inf_assoc
thf(fact_876_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_877_inf_Oassoc,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_878_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_879_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_880_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Z3 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(2)
thf(fact_881_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Z3 )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(2)
thf(fact_882_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X @ Z3 ) ) ) ).
% inf_sup_aci(3)
thf(fact_883_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X @ Z3 ) ) ) ).
% inf_sup_aci(3)
thf(fact_884_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y3 ) )
= ( inf_inf_set_a @ X @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_885_inf__sup__aci_I4_J,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y3 ) )
= ( inf_inf_set_set_a @ X @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_886_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_a,Edges: set_set_a,X3: set_a,Y: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ Edges @ X3 @ Y ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_887_ulgraph_Ois__edge__or__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire2905028936066782638loop_a @ E )
| ( undire4917966558017083288edge_a @ E ) ) ) ) ).
% ulgraph.is_edge_or_loop
thf(fact_888_ulgraph_Oedges__split__loop__inter__empty,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( bot_bot_set_set_a
= ( inf_inf_set_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ Edges )
& ( undire2905028936066782638loop_a @ E3 ) ) )
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ Edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ) ) ).
% ulgraph.edges_split_loop_inter_empty
thf(fact_889_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_890_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_891_inf_OcoboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_892_inf_OcoboundedI2,axiom,
! [B: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_893_inf_OcoboundedI2,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_894_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_895_inf_OcoboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_896_inf_OcoboundedI1,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_897_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_898_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B7: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B7 )
= B7 ) ) ) ).
% inf.absorb_iff2
thf(fact_899_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B7: real,A4: real] :
( ( inf_inf_real @ A4 @ B7 )
= B7 ) ) ) ).
% inf.absorb_iff2
thf(fact_900_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B7: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B7 )
= B7 ) ) ) ).
% inf.absorb_iff2
thf(fact_901_inf_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B7: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B7 )
= B7 ) ) ) ).
% inf.absorb_iff2
thf(fact_902_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B7: set_a] :
( ( inf_inf_set_a @ A4 @ B7 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_903_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B7: real] :
( ( inf_inf_real @ A4 @ B7 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_904_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B7: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B7 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_905_inf_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B7 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_906_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_907_inf_Ocobounded2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_908_inf_Ocobounded2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_909_inf_Ocobounded2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_910_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_911_inf_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_912_inf_Ocobounded1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_913_inf_Ocobounded1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_914_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B7: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B7 ) ) ) ) ).
% inf.order_iff
thf(fact_915_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B7: real] :
( A4
= ( inf_inf_real @ A4 @ B7 ) ) ) ) ).
% inf.order_iff
thf(fact_916_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B7: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B7 ) ) ) ) ).
% inf.order_iff
thf(fact_917_inf_Oorder__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( A4
= ( inf_in8905007599844390133od_a_a @ A4 @ B7 ) ) ) ) ).
% inf.order_iff
thf(fact_918_inf__greatest,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z3 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_919_inf__greatest,axiom,
! [X: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ X @ Z3 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y3 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_920_inf__greatest,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_921_inf__greatest,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y3 )
=> ( ( ord_le746702958409616551od_a_a @ X @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_922_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_923_inf_OboundedI,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_924_inf_OboundedI,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_925_inf_OboundedI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_926_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_927_inf_OboundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_928_inf_OboundedE,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_929_inf_OboundedE,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ~ ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_930_inf__absorb2,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( inf_inf_set_a @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_931_inf__absorb2,axiom,
! [Y3: real,X: real] :
( ( ord_less_eq_real @ Y3 @ X )
=> ( ( inf_inf_real @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_932_inf__absorb2,axiom,
! [Y3: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X )
=> ( ( inf_inf_set_set_a @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_933_inf__absorb2,axiom,
! [Y3: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y3 @ X )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_934_inf__absorb1,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( inf_inf_set_a @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_935_inf__absorb1,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( inf_inf_real @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_936_inf__absorb1,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( inf_inf_set_set_a @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_937_inf__absorb1,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y3 )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_938_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_939_inf_Oabsorb2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( inf_inf_real @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_940_inf_Oabsorb2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_941_inf_Oabsorb2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_942_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_943_inf_Oabsorb1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( inf_inf_real @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_944_inf_Oabsorb1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_945_inf_Oabsorb1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_946_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_947_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( inf_inf_real @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_948_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_949_le__iff__inf,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_950_inf__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y3: set_a] :
( ! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ord_less_eq_set_a @ X5 @ Z )
=> ( ord_less_eq_set_a @ X5 @ ( F2 @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y3 )
= ( F2 @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_951_inf__unique,axiom,
! [F2: real > real > real,X: real,Y3: real] :
( ! [X5: real,Y4: real] : ( ord_less_eq_real @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: real,Y4: real] : ( ord_less_eq_real @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: real,Y4: real,Z: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ( ord_less_eq_real @ X5 @ Z )
=> ( ord_less_eq_real @ X5 @ ( F2 @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_real @ X @ Y3 )
= ( F2 @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_952_inf__unique,axiom,
! [F2: set_set_a > set_set_a > set_set_a,X: set_set_a,Y3: set_set_a] :
( ! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Z )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( F2 @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y3 )
= ( F2 @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_953_inf__unique,axiom,
! [F2: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y4 )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ Z )
=> ( ord_le746702958409616551od_a_a @ X5 @ ( F2 @ Y4 @ Z ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
= ( F2 @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_954_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_955_inf_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( inf_inf_real @ A @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% inf.orderI
thf(fact_956_inf_OorderI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A
= ( inf_inf_set_set_a @ A @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_957_inf_OorderI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A
= ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% inf.orderI
thf(fact_958_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_959_inf_OorderE,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( A
= ( inf_inf_real @ A @ B ) ) ) ).
% inf.orderE
thf(fact_960_inf_OorderE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_961_inf_OorderE,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( A
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_962_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_963_le__infI2,axiom,
! [B: real,X: real,A: real] :
( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_964_le__infI2,axiom,
! [B: set_set_a,X: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_965_le__infI2,axiom,
! [B: set_Product_prod_a_a,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_966_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_967_le__infI1,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_968_le__infI1,axiom,
! [A: set_set_a,X: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_969_le__infI1,axiom,
! [A: set_Product_prod_a_a,X: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_970_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_971_inf__mono,axiom,
! [A: real,C: real,B: real,D2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ ( inf_inf_real @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_972_inf__mono,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_973_inf__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ( ord_le746702958409616551od_a_a @ B @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ ( inf_in8905007599844390133od_a_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_974_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_975_le__infI,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) ) ) ) ).
% le_infI
thf(fact_976_le__infI,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X @ B )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_977_le__infI,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ A )
=> ( ( ord_le746702958409616551od_a_a @ X @ B )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_978_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_979_le__infE,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B ) ) ) ).
% le_infE
thf(fact_980_le__infE,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_981_le__infE,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ X @ A )
=> ~ ( ord_le746702958409616551od_a_a @ X @ B ) ) ) ).
% le_infE
thf(fact_982_inf__le2,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_983_inf__le2,axiom,
! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_984_inf__le2,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_985_inf__le2,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_986_inf__le1,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_987_inf__le1,axiom,
! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_988_inf__le1,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_989_inf__le1,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_990_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_991_inf__sup__ord_I1_J,axiom,
! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_992_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_993_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_994_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_995_inf__sup__ord_I2_J,axiom,
! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_996_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_997_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_998_is__isolated__vertex__degree0,axiom,
! [V3: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V3 )
=> ( ( undire8867928226783802224gree_a @ edges @ V3 )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_999_incident__loops__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire4753905205749729249oops_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) ) ) ).
% incident_loops_union
thf(fact_1000_inc_Oadd__point__def,axiom,
! [P2: a] :
( ( design2964366272795260673oint_a @ vertices @ P2 )
= ( insert_a @ P2 @ vertices ) ) ).
% inc.add_point_def
thf(fact_1001_inc_Oblock__complement__inter__empty,axiom,
! [Bl1: set_a,Bl2: set_a] :
( ( ( design6447616907850319326ment_a @ vertices @ Bl1 )
= Bl2 )
=> ( ( inf_inf_set_a @ Bl1 @ Bl2 )
= bot_bot_set_a ) ) ).
% inc.block_complement_inter_empty
thf(fact_1002_edges__split__loop,axiom,
( edges
= ( sup_sup_set_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) )
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ) ).
% edges_split_loop
thf(fact_1003_mk__triangle__set_Osimps,axiom,
! [X: a,Y3: a,Z3: a] :
( ( undire8536760333753235943_set_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Y3 @ Z3 ) ) )
= ( insert_a @ X @ ( insert_a @ Y3 @ ( insert_a @ Z3 @ bot_bot_set_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_1004_mk__triangle__set_Osimps,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( undire4638465864238448455_set_a @ ( produc7299740244201487072_set_a @ X @ ( produc9088192753505129239_set_a @ Y3 @ Z3 ) ) )
= ( insert_set_a @ X @ ( insert_set_a @ Y3 @ ( insert_set_a @ Z3 @ bot_bot_set_set_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_1005_mk__triangle__set_Osimps,axiom,
! [X: nat,Y3: nat,Z3: nat] :
( ( undire4970100481470743719et_nat @ ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y3 @ Z3 ) ) )
= ( insert_nat @ X @ ( insert_nat @ Y3 @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_1006_mk__triangle__set_Osimps,axiom,
! [X: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
( ( undire2459242765783757584od_a_a @ ( produc4925843558922497303od_a_a @ X @ ( produc7886510207707329367od_a_a @ Y3 @ Z3 ) ) )
= ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y3 @ ( insert4534936382041156343od_a_a @ Z3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_1007_mk__triangle__set_Oelims,axiom,
! [X: produc4044097585999906000od_a_a,Y3: set_a] :
( ( ( undire8536760333753235943_set_a @ X )
= Y3 )
=> ~ ! [X5: a,Y4: a,Z: a] :
( ( X
= ( produc431845341423274048od_a_a @ X5 @ ( product_Pair_a_a @ Y4 @ Z ) ) )
=> ( Y3
!= ( insert_a @ X5 @ ( insert_a @ Y4 @ ( insert_a @ Z @ bot_bot_set_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_1008_mk__triangle__set_Oelims,axiom,
! [X: produc3364680560414100336_set_a,Y3: set_set_a] :
( ( ( undire4638465864238448455_set_a @ X )
= Y3 )
=> ~ ! [X5: set_a,Y4: set_a,Z: set_a] :
( ( X
= ( produc7299740244201487072_set_a @ X5 @ ( produc9088192753505129239_set_a @ Y4 @ Z ) ) )
=> ( Y3
!= ( insert_set_a @ X5 @ ( insert_set_a @ Y4 @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_1009_mk__triangle__set_Oelims,axiom,
! [X: produc7248412053542808358at_nat,Y3: set_nat] :
( ( ( undire4970100481470743719et_nat @ X )
= Y3 )
=> ~ ! [X5: nat,Y4: nat,Z: nat] :
( ( X
= ( produc487386426758144856at_nat @ X5 @ ( product_Pair_nat_nat @ Y4 @ Z ) ) )
=> ( Y3
!= ( insert_nat @ X5 @ ( insert_nat @ Y4 @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_1010_mk__triangle__set_Oelims,axiom,
! [X: produc8857593507947890343od_a_a,Y3: set_Product_prod_a_a] :
( ( ( undire2459242765783757584od_a_a @ X )
= Y3 )
=> ~ ! [X5: product_prod_a_a,Y4: product_prod_a_a,Z: product_prod_a_a] :
( ( X
= ( produc4925843558922497303od_a_a @ X5 @ ( produc7886510207707329367od_a_a @ Y4 @ Z ) ) )
=> ( Y3
!= ( insert4534936382041156343od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ Y4 @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_1011_inc_Oblock__comp__elem__alt__left,axiom,
! [X: a,Bl: set_a,Ps: set_a] :
( ( member_a @ X @ Bl )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) )
=> ~ ( member_a @ X @ Ps ) ) ) ).
% inc.block_comp_elem_alt_left
thf(fact_1012_inc_Oblock__comp__elem__alt__right,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ vertices )
=> ( ! [X5: a] :
( ( member_a @ X5 @ Ps )
=> ~ ( member_a @ X5 @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) ) ) ) ).
% inc.block_comp_elem_alt_right
thf(fact_1013_inc_Oblock__complement__elem__iff,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ vertices )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl ) ) ) ) ) ).
% inc.block_complement_elem_iff
thf(fact_1014_inc_Oblock__complement__subset__points,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ vertices ) ) ).
% inc.block_complement_subset_points
thf(fact_1015_sup_Oright__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ B )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_1016_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_1017_sup_Oright__idem,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ B )
= ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_1018_sup__left__idem,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y3 ) )
= ( sup_sup_set_set_a @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_1019_sup__left__idem,axiom,
! [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) )
= ( sup_sup_set_a @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_1020_sup__left__idem,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y3 ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_1021_sup_Oleft__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_1022_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_1023_sup_Oleft__idem,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_1024_sup__idem,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_1025_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_1026_sup__idem,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_1027_sup_Oidem,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_1028_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_1029_sup_Oidem,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_1030_Un__iff,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B4 ) )
= ( ( member_o @ C @ A3 )
| ( member_o @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_1031_Un__iff,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) )
= ( ( member_set_a @ C @ A3 )
| ( member_set_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_1032_Un__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_1033_Un__iff,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) )
= ( ( member1426531477525435216od_a_a @ C @ A3 )
| ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_1034_UnCI,axiom,
! [C: $o,B4: set_o,A3: set_o] :
( ( ~ ( member_o @ C @ B4 )
=> ( member_o @ C @ A3 ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_1035_UnCI,axiom,
! [C: set_a,B4: set_set_a,A3: set_set_a] :
( ( ~ ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ A3 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_1036_UnCI,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_1037_UnCI,axiom,
! [C: product_prod_a_a,B4: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B4 )
=> ( member1426531477525435216od_a_a @ C @ A3 ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_1038_comp__sgraph_Oinc_Oadd__existing__point,axiom,
! [P2: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P2 @ S )
=> ( ( design3431343892158072362od_a_a @ S @ P2 )
= S ) ) ).
% comp_sgraph.inc.add_existing_point
thf(fact_1039_comp__sgraph_Oinc_Oadd__existing__point,axiom,
! [P2: set_a,S: set_set_a] :
( ( member_set_a @ P2 @ S )
=> ( ( design4648949625254728801_set_a @ S @ P2 )
= S ) ) ).
% comp_sgraph.inc.add_existing_point
thf(fact_1040_comp__sgraph_Oinc_Oadd__existing__point,axiom,
! [P2: $o,S: set_o] :
( ( member_o @ P2 @ S )
=> ( ( design7782887785804742939oint_o @ S @ P2 )
= S ) ) ).
% comp_sgraph.inc.add_existing_point
thf(fact_1041_comp__sgraph_Oinc_Oadd__existing__point,axiom,
! [P2: a,S: set_a] :
( ( member_a @ P2 @ S )
=> ( ( design2964366272795260673oint_a @ S @ P2 )
= S ) ) ).
% comp_sgraph.inc.add_existing_point
thf(fact_1042_le__sup__iff,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ Z3 )
= ( ( ord_less_eq_set_a @ X @ Z3 )
& ( ord_less_eq_set_a @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1043_le__sup__iff,axiom,
! [X: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y3 ) @ Z3 )
= ( ( ord_less_eq_real @ X @ Z3 )
& ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1044_le__sup__iff,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y3 ) @ Z3 )
= ( ( ord_le3724670747650509150_set_a @ X @ Z3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1045_le__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y3 ) @ Z3 )
= ( ( ord_le746702958409616551od_a_a @ X @ Z3 )
& ( ord_le746702958409616551od_a_a @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1046_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1047_sup_Obounded__iff,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
= ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1048_sup_Obounded__iff,axiom,
! [B: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B @ A )
& ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1049_sup_Obounded__iff,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A )
= ( ( ord_le746702958409616551od_a_a @ B @ A )
& ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1050_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1051_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1052_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_1053_sup__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1054_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_1055_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_1056_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_1057_sup__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= X ) ).
% sup_bot_right
thf(fact_1058_bot__eq__sup__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y3 ) )
= ( ( X = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1059_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y3 ) )
= ( ( X = bot_bot_set_set_a )
& ( Y3 = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1060_bot__eq__sup__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y3 ) )
= ( ( X = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1061_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X @ Y3 ) )
= ( ( X = bot_bo3357376287454694259od_a_a )
& ( Y3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1062_sup__eq__bot__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( ( sup_sup_set_a @ X @ Y3 )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1063_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y3 )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y3 = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1064_sup__eq__bot__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y3 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1065_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X @ Y3 )
= bot_bo3357376287454694259od_a_a )
= ( ( X = bot_bo3357376287454694259od_a_a )
& ( Y3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1066_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1067_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1068_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1069_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1070_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1071_sup__bot_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1072_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1073_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1074_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1075_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A @ B ) )
= ( ( A = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1076_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1077_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1078_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1079_sup__bot_Oright__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1080_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_1081_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1082_Un__empty,axiom,
! [A3: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1083_Un__empty,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ( sup_sup_set_set_a @ A3 @ B4 )
= bot_bot_set_set_a )
= ( ( A3 = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_1084_Un__empty,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A3 @ B4 )
= bot_bot_set_nat )
= ( ( A3 = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1085_Un__empty,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A3 @ B4 )
= bot_bo3357376287454694259od_a_a )
= ( ( A3 = bot_bo3357376287454694259od_a_a )
& ( B4 = bot_bo3357376287454694259od_a_a ) ) ) ).
% Un_empty
thf(fact_1086_inf__sup__absorb,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y3 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1087_inf__sup__absorb,axiom,
! [X: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1088_inf__sup__absorb,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y3 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1089_sup__inf__absorb,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y3 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1090_sup__inf__absorb,axiom,
! [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y3 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1091_sup__inf__absorb,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1092_Un__subset__iff,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C2 )
= ( ( ord_less_eq_set_a @ A3 @ C2 )
& ( ord_less_eq_set_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1093_Un__subset__iff,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A3 @ B4 ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A3 @ C2 )
& ( ord_le3724670747650509150_set_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1094_Un__subset__iff,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) @ C2 )
= ( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
& ( ord_le746702958409616551od_a_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1095_Un__insert__left,axiom,
! [A: set_a,B4: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1096_Un__insert__left,axiom,
! [A: a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1097_Un__insert__left,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A @ B4 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1098_Un__insert__right,axiom,
! [A3: set_set_a,A: set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_1099_Un__insert__right,axiom,
! [A3: set_a,A: a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_1100_Un__insert__right,axiom,
! [A3: set_Product_prod_a_a,A: product_prod_a_a,B4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A @ B4 ) )
= ( insert4534936382041156343od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_1101_Int__Un__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T2 @ ( inf_inf_set_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1102_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1103_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ T2 @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1104_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1105_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1106_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ S @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1107_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1108_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1109_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1110_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1111_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1112_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1113_Un__Int__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T2 @ ( sup_sup_set_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1114_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1115_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ T2 @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1116_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1117_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1118_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ S @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1119_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1120_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1121_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1122_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1123_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1124_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1125_inc_Oadd__existing__point,axiom,
! [P2: a] :
( ( member_a @ P2 @ vertices )
=> ( ( design2964366272795260673oint_a @ vertices @ P2 )
= vertices ) ) ).
% inc.add_existing_point
thf(fact_1126_if__image__distrib,axiom,
! [P: $o > $o,F2: $o > a,G2: $o > a,S: set_o] :
( ( image_o_a
@ ^ [X2: $o] : ( if_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_a @ ( image_o_a @ F2 @ ( inf_inf_set_o @ S @ ( collect_o @ P ) ) )
@ ( image_o_a @ G2
@ ( inf_inf_set_o @ S
@ ( collect_o
@ ^ [X2: $o] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1127_if__image__distrib,axiom,
! [P: a > $o,F2: a > a,G2: a > a,S: set_a] :
( ( image_a_a
@ ^ [X2: a] : ( if_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_a @ ( image_a_a @ F2 @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_a @ G2
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1128_if__image__distrib,axiom,
! [P: $o > $o,F2: $o > set_a,G2: $o > set_a,S: set_o] :
( ( image_o_set_a
@ ^ [X2: $o] : ( if_set_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_set_a @ ( image_o_set_a @ F2 @ ( inf_inf_set_o @ S @ ( collect_o @ P ) ) )
@ ( image_o_set_a @ G2
@ ( inf_inf_set_o @ S
@ ( collect_o
@ ^ [X2: $o] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1129_if__image__distrib,axiom,
! [P: a > $o,F2: a > set_a,G2: a > set_a,S: set_a] :
( ( image_a_set_a
@ ^ [X2: a] : ( if_set_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F2 @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_set_a @ G2
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1130_if__image__distrib,axiom,
! [P: set_a > $o,F2: set_a > a,G2: set_a > a,S: set_set_a] :
( ( image_set_a_a
@ ^ [X2: set_a] : ( if_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_a @ ( image_set_a_a @ F2 @ ( inf_inf_set_set_a @ S @ ( collect_set_a @ P ) ) )
@ ( image_set_a_a @ G2
@ ( inf_inf_set_set_a @ S
@ ( collect_set_a
@ ^ [X2: set_a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1131_if__image__distrib,axiom,
! [P: a > $o,F2: a > set_set_a,G2: a > set_set_a,S: set_a] :
( ( image_a_set_set_a
@ ^ [X2: a] : ( if_set_set_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_su2076012971530813682_set_a @ ( image_a_set_set_a @ F2 @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_set_set_a @ G2
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1132_if__image__distrib,axiom,
! [P: set_a > $o,F2: set_a > set_a,G2: set_a > set_a,S: set_set_a] :
( ( image_set_a_set_a
@ ^ [X2: set_a] : ( if_set_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_set_a @ ( image_set_a_set_a @ F2 @ ( inf_inf_set_set_a @ S @ ( collect_set_a @ P ) ) )
@ ( image_set_a_set_a @ G2
@ ( inf_inf_set_set_a @ S
@ ( collect_set_a
@ ^ [X2: set_a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1133_if__image__distrib,axiom,
! [P: product_prod_a_a > $o,F2: product_prod_a_a > a,G2: product_prod_a_a > a,S: set_Product_prod_a_a] :
( ( image_3437945252899457948_a_a_a
@ ^ [X2: product_prod_a_a] : ( if_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_sup_set_a @ ( image_3437945252899457948_a_a_a @ F2 @ ( inf_in8905007599844390133od_a_a @ S @ ( collec3336397797384452498od_a_a @ P ) ) )
@ ( image_3437945252899457948_a_a_a @ G2
@ ( inf_in8905007599844390133od_a_a @ S
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1134_if__image__distrib,axiom,
! [P: $o > $o,F2: $o > product_prod_a_a,G2: $o > product_prod_a_a,S: set_o] :
( ( image_5435475662653987220od_a_a
@ ^ [X2: $o] : ( if_Product_prod_a_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_su3048258781599657691od_a_a @ ( image_5435475662653987220od_a_a @ F2 @ ( inf_inf_set_o @ S @ ( collect_o @ P ) ) )
@ ( image_5435475662653987220od_a_a @ G2
@ ( inf_inf_set_o @ S
@ ( collect_o
@ ^ [X2: $o] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1135_if__image__distrib,axiom,
! [P: a > $o,F2: a > product_prod_a_a,G2: a > product_prod_a_a,S: set_a] :
( ( image_7400625782589995694od_a_a
@ ^ [X2: a] : ( if_Product_prod_a_a @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ S )
= ( sup_su3048258781599657691od_a_a @ ( image_7400625782589995694od_a_a @ F2 @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_7400625782589995694od_a_a @ G2
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1136_degree__none,axiom,
! [V3: a] :
( ~ ( member_a @ V3 @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V3 )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1137_sup__left__commute,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_set_a @ Y3 @ ( sup_sup_set_set_a @ X @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_1138_sup__left__commute,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_1139_sup__left__commute,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y3 @ Z3 ) )
= ( sup_su3048258781599657691od_a_a @ Y3 @ ( sup_su3048258781599657691od_a_a @ X @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_1140_sup_Oleft__commute,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A @ C ) )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1141_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1142_sup_Oleft__commute,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A @ C ) )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1143_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_set_a,K: set_set_a,B: set_set_a,A: set_set_a] :
( ( B4
= ( sup_sup_set_set_a @ K @ B ) )
=> ( ( sup_sup_set_set_a @ A @ B4 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1144_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_a,K: set_a,B: set_a,A: set_a] :
( ( B4
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1145_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_Product_prod_a_a,K: set_Product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( B4
= ( sup_su3048258781599657691od_a_a @ K @ B ) )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B4 )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1146_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_set_a,K: set_set_a,A: set_set_a,B: set_set_a] :
( ( A3
= ( sup_sup_set_set_a @ K @ A ) )
=> ( ( sup_sup_set_set_a @ A3 @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1147_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A3 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1148_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Product_prod_a_a,K: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A3
= ( sup_su3048258781599657691od_a_a @ K @ A ) )
=> ( ( sup_su3048258781599657691od_a_a @ A3 @ B )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1149_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_1150_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_1151_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_1152_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B7: set_set_a] : ( sup_sup_set_set_a @ B7 @ A4 ) ) ) ).
% sup.commute
thf(fact_1153_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B7: set_a] : ( sup_sup_set_a @ B7 @ A4 ) ) ) ).
% sup.commute
thf(fact_1154_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A4: set_Product_prod_a_a,B7: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B7 @ A4 ) ) ) ).
% sup.commute
thf(fact_1155_sup__assoc,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y3 ) @ Z3 )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y3 @ Z3 ) ) ) ).
% sup_assoc
thf(fact_1156_sup__assoc,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ Z3 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z3 ) ) ) ).
% sup_assoc
thf(fact_1157_sup__assoc,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y3 ) @ Z3 )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y3 @ Z3 ) ) ) ).
% sup_assoc
thf(fact_1158_sup_Oassoc,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1159_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1160_sup_Oassoc,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1161_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1162_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1163_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1164_inf__sup__aci_I6_J,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y3 ) @ Z3 )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_1165_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ Z3 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_1166_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y3 ) @ Z3 )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y3 @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_1167_inf__sup__aci_I7_J,axiom,
! [X: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_set_a @ Y3 @ ( sup_sup_set_set_a @ X @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_1168_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_1169_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y3 @ Z3 ) )
= ( sup_su3048258781599657691od_a_a @ Y3 @ ( sup_su3048258781599657691od_a_a @ X @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_1170_inf__sup__aci_I8_J,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y3 ) )
= ( sup_sup_set_set_a @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_1171_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) )
= ( sup_sup_set_a @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_1172_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y3 ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_1173_Collect__disj__eq,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1174_Collect__disj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1175_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1176_Collect__disj__eq,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su3048258781599657691od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1177_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A5: set_o,B5: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A5 )
| ( member_o @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1178_Un__def,axiom,
( sup_sup_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
| ( member_set_a @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1179_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
| ( member_a @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1180_Un__def,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A5 )
| ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1181_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_1182_Un__left__commute,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ ( sup_sup_set_set_a @ B4 @ C2 ) )
= ( sup_sup_set_set_a @ B4 @ ( sup_sup_set_set_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1183_Un__left__commute,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C2 ) )
= ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1184_Un__left__commute,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B4 @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ B4 @ ( sup_su3048258781599657691od_a_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1185_Un__left__absorb,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ ( sup_sup_set_set_a @ A3 @ B4 ) )
= ( sup_sup_set_set_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_1186_Un__left__absorb,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_1187_Un__left__absorb,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) )
= ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_1188_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_1189_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_1190_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_1191_Un__absorb,axiom,
! [A3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_1192_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_1193_Un__absorb,axiom,
! [A3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_1194_Un__assoc,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_set_a @ A3 @ ( sup_sup_set_set_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1195_Un__assoc,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1196_Un__assoc,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1197_ball__Un,axiom,
! [A3: set_set_a,B4: set_set_a,P: set_a > $o] :
( ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( sup_sup_set_set_a @ A3 @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1198_ball__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1199_ball__Un,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1200_bex__Un,axiom,
! [A3: set_set_a,B4: set_set_a,P: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( sup_sup_set_set_a @ A3 @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: set_a] :
( ( member_set_a @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1201_bex__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1202_bex__Un,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1203_UnI2,axiom,
! [C: $o,B4: set_o,A3: set_o] :
( ( member_o @ C @ B4 )
=> ( member_o @ C @ ( sup_sup_set_o @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1204_UnI2,axiom,
! [C: set_a,B4: set_set_a,A3: set_set_a] :
( ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1205_UnI2,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1206_UnI2,axiom,
! [C: product_prod_a_a,B4: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B4 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1207_UnI1,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ A3 )
=> ( member_o @ C @ ( sup_sup_set_o @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1208_UnI1,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1209_UnI1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1210_UnI1,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1211_UnE,axiom,
! [C: $o,A3: set_o,B4: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B4 ) )
=> ( ~ ( member_o @ C @ A3 )
=> ( member_o @ C @ B4 ) ) ) ).
% UnE
thf(fact_1212_UnE,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) )
=> ( ~ ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_1213_UnE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_1214_UnE,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B4 ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_1215_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1216_sup_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1217_sup_OcoboundedI2,axiom,
! [C: set_set_a,B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1218_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1219_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1220_sup_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1221_sup_OcoboundedI1,axiom,
! [C: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1222_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ A )
=> ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1223_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B7: set_a] :
( ( sup_sup_set_a @ A4 @ B7 )
= B7 ) ) ) ).
% sup.absorb_iff2
thf(fact_1224_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B7: real] :
( ( sup_sup_real @ A4 @ B7 )
= B7 ) ) ) ).
% sup.absorb_iff2
thf(fact_1225_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B7: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B7 )
= B7 ) ) ) ).
% sup.absorb_iff2
thf(fact_1226_sup_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A4 @ B7 )
= B7 ) ) ) ).
% sup.absorb_iff2
thf(fact_1227_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B7: set_set_a,A4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B7 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1228_sup_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B7: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A4 @ B7 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1229_incident__sedges__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire1270416042309875431dges_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ).
% incident_sedges_union
thf(fact_1230_incident__edges__union,axiom,
! [V3: a] :
( ( undire3231912044278729248dges_a @ edges @ V3 )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V3 ) @ ( undire4753905205749729249oops_a @ edges @ V3 ) ) ) ).
% incident_edges_union
thf(fact_1231_incident__edges__sedges,axiom,
! [V3: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V3 )
=> ( ( undire3231912044278729248dges_a @ edges @ V3 )
= ( undire1270416042309875431dges_a @ edges @ V3 ) ) ) ).
% incident_edges_sedges
thf(fact_1232_incident__sedges__empty,axiom,
! [V3: a] :
( ~ ( member_a @ V3 @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V3 )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1233_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1234_degree0__neighborhood__empt__iff,axiom,
! [V3: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V3 )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V3 )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1235_finite__incident__edges,axiom,
! [V3: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V3 ) ) ) ).
% finite_incident_edges
thf(fact_1236_finite__incident__loops,axiom,
! [V3: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V3 ) ) ).
% finite_incident_loops
thf(fact_1237_finite__inc__sedges,axiom,
! [V3: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V3 ) ) ) ).
% finite_inc_sedges
thf(fact_1238_degree0__inc__edges__empt__iff,axiom,
! [V3: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V3 )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_1239_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1240_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_1241_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_1242_induced__is__graph__sys,axiom,
! [V5: set_a] : ( undire2554140024507503526stem_a @ V5 @ ( undire7777452895879145676dges_a @ edges @ V5 ) ) ).
% induced_is_graph_sys
thf(fact_1243_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_1244_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_1245_inc_Oadd__delete__point__inv,axiom,
! [P2: a] :
( ~ ( member_a @ P2 @ vertices )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ vertices @ P2 ) @ P2 )
= vertices ) ) ).
% inc.add_delete_point_inv
thf(fact_1246_inc_Odel__invalid__point,axiom,
! [P2: a] :
( ~ ( member_a @ P2 @ vertices )
=> ( ( design108908007054065099oint_a @ vertices @ P2 )
= vertices ) ) ).
% inc.del_invalid_point
thf(fact_1247_inc_Odel__point__def,axiom,
! [P2: a] :
( ( design108908007054065099oint_a @ vertices @ P2 )
= ( minus_minus_set_a @ vertices @ ( insert_a @ P2 @ bot_bot_set_a ) ) ) ).
% inc.del_point_def
thf(fact_1248_inc_Oblock__complement__def,axiom,
! [B: set_a] :
( ( design6447616907850319326ment_a @ vertices @ B )
= ( minus_minus_set_a @ vertices @ B ) ) ).
% inc.block_complement_def
thf(fact_1249_edge__density__le1,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X3 @ Y ) @ one_one_real ) ).
% edge_density_le1
thf(fact_1250_all__edges__betw__I,axiom,
! [X: a,X3: set_a,Y3: a,Y: set_a] :
( ( member_a @ X @ X3 )
=> ( ( member_a @ Y3 @ Y )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_1251_finite__all__edges__between,axiom,
! [X3: set_a,Y: set_a] :
( ( finite_finite_a @ X3 )
=> ( ( finite_finite_a @ Y )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) ) ) ) ).
% finite_all_edges_between
thf(fact_1252_all__edges__between__Union1,axiom,
! [X7: set_set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( comple2307003609928055243_set_a @ X7 ) @ Y )
= ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [X6: set_a] : ( undire8383842906760478443ween_a @ edges @ X6 @ Y )
@ X7 ) ) ) ).
% all_edges_between_Union1
thf(fact_1253_all__edges__between__Union2,axiom,
! [X3: set_a,Y7: set_set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ ( comple2307003609928055243_set_a @ Y7 ) )
= ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 ) @ Y7 ) ) ) ).
% all_edges_between_Union2
thf(fact_1254_all__edges__between__mono1,axiom,
! [Y: set_a,Z4: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y @ X3 ) @ ( undire8383842906760478443ween_a @ edges @ Z4 @ X3 ) ) ) ).
% all_edges_between_mono1
thf(fact_1255_all__edges__between__mono2,axiom,
! [Y: set_a,Z4: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X3 @ Z4 ) ) ) ).
% all_edges_between_mono2
thf(fact_1256_all__edges__between__Un1,axiom,
! [X3: set_a,Y: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X3 @ Y ) @ Z4 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Z4 ) @ ( undire8383842906760478443ween_a @ edges @ Y @ Z4 ) ) ) ).
% all_edges_between_Un1
thf(fact_1257_all__edges__between__Un2,axiom,
! [X3: set_a,Y: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X3 @ Z4 ) ) ) ).
% all_edges_between_Un2
thf(fact_1258_all__edges__between__rem__wf,axiom,
! [X3: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X3 @ vertices ) @ ( inf_inf_set_a @ Y @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_1259_edge__density__eq0,axiom,
! [A3: set_a,B4: set_a,X3: set_a,Y: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A3 @ B4 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X3 @ A3 )
=> ( ( ord_less_eq_set_a @ Y @ B4 )
=> ( ( undire297304480579013331sity_a @ edges @ X3 @ Y )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_1260_all__edges__betw__D3,axiom,
! [X: a,Y3: a,X3: set_a,Y: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) )
=> ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_1261_all__edges__between__empty_I2_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(2)
thf(fact_1262_all__edges__between__empty_I1_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(1)
thf(fact_1263_all__edges__between__set,axiom,
! [X3: set_a,Y: set_a] :
( ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) )
= ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: a,Y2: a] :
( ( Uu
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X3 )
& ( member_a @ Y2 @ Y )
& ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges ) ) ) ) ).
% all_edges_between_set
thf(fact_1264_all__edges__between__E__ss,axiom,
! [X3: set_a,Y: set_a] : ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) ) @ edges ) ).
% all_edges_between_E_ss
thf(fact_1265_all__edges__between__subset__times,axiom,
! [X3: set_a,Y: set_a] :
( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
@ ( product_Sigma_a_a @ ( inf_inf_set_a @ X3 @ ( comple2307003609928055243_set_a @ edges ) )
@ ^ [Uu: a] : ( inf_inf_set_a @ Y @ ( comple2307003609928055243_set_a @ edges ) ) ) ) ).
% all_edges_between_subset_times
thf(fact_1266_local_Oinj__on__mk__edge,axiom,
! [X3: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X3 @ Y )
= bot_bot_set_a )
=> ( inj_on4851796814176604264_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y ) ) ) ).
% local.inj_on_mk_edge
thf(fact_1267_all__edges__between__subset,axiom,
! [X3: set_a,Y: set_a] :
( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
@ ( product_Sigma_a_a @ X3
@ ^ [Uu: a] : Y ) ) ).
% all_edges_between_subset
thf(fact_1268_all__edges__betw__sigma__neighbor,axiom,
! [X3: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
= ( product_Sigma_a_a @ X3
@ ^ [X2: a] : ( undire401937927514038589s_ss_a @ edges @ X2 @ Y ) ) ) ).
% all_edges_betw_sigma_neighbor
thf(fact_1269_all__edges__betw__prod__def__neighbors,axiom,
! [X3: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ X3
@ ^ [Uu: a] : Y ) )
& ( undire397441198561214472_adj_a @ edges @ X2 @ Y2 ) ) ) ) ) ).
% all_edges_betw_prod_def_neighbors
thf(fact_1270_all__edges__between__def,axiom,
! [X3: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member_a @ X2 @ X3 )
& ( member_a @ Y2 @ Y )
& ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges ) ) ) ) ) ).
% all_edges_between_def
thf(fact_1271_incident__loops__alt,axiom,
! [V3: a] :
( ( undire4753905205749729249oops_a @ edges @ V3 )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire1521409233611534436dent_a @ V3 @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ) ).
% incident_loops_alt
thf(fact_1272_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E3: set_a] :
( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1273_all__edges__between__swap,axiom,
! [X3: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X3 @ Y )
= ( image_4636654165204879301od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [X2: a,Y2: a] : ( product_Pair_a_a @ Y2 @ X2 ) )
@ ( undire8383842906760478443ween_a @ edges @ Y @ X3 ) ) ) ).
% all_edges_between_swap
thf(fact_1274_card1__incident__imp__vert,axiom,
! [V3: a,E: set_a] :
( ( ( undire1521409233611534436dent_a @ V3 @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y3: a] :
( ( if_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y3: a] :
( ( if_a @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y3: set_a] :
( ( if_set_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y3: set_a] :
( ( if_set_a @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( if_set_set_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X: set_set_a,Y3: set_set_a] :
( ( if_set_set_a @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_T,axiom,
! [X: product_prod_a_a,Y3: product_prod_a_a] :
( ( if_Product_prod_a_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_T,axiom,
! [X: product_prod_a_a,Y3: product_prod_a_a] :
( ( if_Product_prod_a_a @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X5: a,Y4: a] :
( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 )
@ ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [U: a,V: a] :
( ( Uu
= ( product_Pair_a_a @ U @ V ) )
& ( undire397441198561214472_adj_a @ edges @ U @ V ) ) ) )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X5 )
@ ( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [U: a,V: a] :
( ( Uu
= ( product_Pair_a_a @ U @ V ) )
& ( undire397441198561214472_adj_a @ edges @ U @ V ) ) ) ) ) ).
%------------------------------------------------------------------------------