TPTP Problem File: ITP176^1.p
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%------------------------------------------------------------------------------
% File : ITP176^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer StandardRules problem prob_305__5389912_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : StandardRules/prob_305__5389912_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 489 ( 184 unt; 130 typ; 0 def)
% Number of atoms : 970 ( 382 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2891 ( 117 ~; 17 |; 75 &;2345 @)
% ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 243 ( 243 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 101 usr; 13 con; 0-3 aty)
% Number of variables : 978 ( 93 ^; 855 !; 30 ?; 978 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:50.745
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
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thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_763305007od_b_b: set_Pr2109276827od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1397930469od_b_b: set_Pr2134561957od_b_b > set_Pr1324126435od_b_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b,type,
image_1764625405_b_b_b: set_Pr1906739389_b_b_b > set_Pr1324126435od_b_b > set_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_532916993od_b_b: set_Pr1071216121od_b_b > set_Product_prod_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1928024979od_b_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b,type,
image_921732011_b_b_b: set_Pr357419743_b_b_b > set_Product_prod_b_b > set_b ).
thf(sy_c_Relation_OImage_001tf__b_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
image_984343321od_b_b: set_Pr2060523537od_b_b > set_b > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_OImage_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
image_1177096571od_b_b: set_Pr621705391od_b_b > set_b > set_Product_prod_b_b ).
thf(sy_c_Relation_OImage_001tf__b_001tf__b,type,
image_b_b: set_Product_prod_b_b > set_b > set_b ).
thf(sy_c_Relation_Oconverse_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
conver1398856763od_b_b: set_Pr1324126435od_b_b > set_Pr306736955tant_a ).
thf(sy_c_Relation_Oconverse_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
conver1943590235tant_a: set_Pr306736955tant_a > set_Pr1324126435od_b_b ).
thf(sy_c_Relation_Oconverse_001tf__b_001tf__b,type,
converse_b_b: set_Product_prod_b_b > set_Product_prod_b_b ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
refl_o1921098222od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
refl_o2134473190od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_Relation_Orefl__on_001tf__b,type,
refl_on_b: set_b > set_Product_prod_b_b > $o ).
thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
total_771970601od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
total_178717291od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_Relation_Ototal__on_001tf__b,type,
total_on_b: set_b > set_Product_prod_b_b > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
collec279084418od_b_b: ( produc1213276845od_b_b > $o ) > set_Pr1324126435od_b_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
collec1481886546od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oinsert_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
insert1909710879tant_a: standard_Constant_a > set_St761939237tant_a > set_St761939237tant_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
insert2037698781od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
insert1098249399od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > set_Pr2007082183od_b_b ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
insert1952693431od_b_b: product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
is_sin1465183929od_b_b: set_Pr1324126435od_b_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
is_sin1713057243od_b_b: set_Product_prod_b_b > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
the_el329196508od_b_b: set_Product_prod_b_b > product_prod_b_b ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_member_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
member1632892294tant_a: standard_Constant_a > set_St761939237tant_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member1516365892od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member1086024932od_b_b: produc970027067od_b_b > set_Pr2109276827od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member942707974od_b_b: produc1990339951od_b_b > set_Pr2134561957od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J,type,
member1417789726_b_b_b: produc276146823_b_b_b > set_Pr1906739389_b_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
member1670214300tant_a: produc1367125253tant_a > set_Pr306736955tant_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member4694106od_b_b: produc1052326083od_b_b > set_Pr1071216121od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J,type,
member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
member599505522od_b_b: produc1605346267od_b_b > set_Pr2060523537od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_G,type,
g: labele1159362096nt_a_b ).
thf(sy_v_P____,type,
p: b > b > $o ).
thf(sy_v_f____,type,
f: b > b ).
thf(sy_v_x____,type,
x: b ).
thf(sy_v_y____,type,
y: b ).
% Relevant facts (355)
thf(fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062,axiom,
( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) )
= ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ).
% \<open>getRel S_Idt G `` {x} = getRel S_Idt G `` {y}\<close>
thf(fact_1_assms_I1_J,axiom,
( g
= ( restri446606278nt_a_b @ g ) ) ).
% assms(1)
thf(fact_2_P__eq,axiom,
! [X: b] :
( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
!= bot_bot_set_b )
=> ( ( hilbert_Eps_b @ ( p @ ( hilbert_Eps_b @ ( p @ X ) ) ) )
= ( hilbert_Eps_b @ ( p @ X ) ) ) ) ).
% P_eq
thf(fact_3_P,axiom,
( p
= ( ^ [X2: b,Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ).
% P
thf(fact_4__092_060open_062_Ix_M_Ay_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
member1285940496od_b_b @ ( product_Pair_b_b @ x @ y ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% \<open>(x, y) \<in> getRel S_Idt G\<close>
thf(fact_5__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
( f
= ( ^ [X2: b] :
( if_b
@ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
= bot_bot_set_b )
@ X2
@ ( hilbert_Eps_b @ ( p @ X2 ) ) ) ) ) ).
% \<open>f \<equiv> \<lambda>x. if getRel S_Idt G `` {x} = {} then x else Eps (P x)\<close>
thf(fact_6_singleton__conv,axiom,
! [A: product_prod_b_b] :
( ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] : X2 = A )
= ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% singleton_conv
thf(fact_7_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X2: b] : X2 = A )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_8_singleton__conv2,axiom,
! [A: product_prod_b_b] :
( ( collec1481886546od_b_b
@ ( ^ [Y2: product_prod_b_b,Z: product_prod_b_b] : Y2 = Z
@ A ) )
= ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% singleton_conv2
thf(fact_9_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y2: b,Z: b] : Y2 = Z
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_10_tuple__disj,axiom,
! [X: product_prod_b_b,Z2: product_prod_b_b] :
( ( collec1481886546od_b_b
@ ^ [Y: product_prod_b_b] :
( ( Y = X )
| ( Y = Z2 ) ) )
= ( insert1952693431od_b_b @ X @ ( insert1952693431od_b_b @ Z2 @ bot_bo1343651123od_b_b ) ) ) ).
% tuple_disj
thf(fact_11_tuple__disj,axiom,
! [X: b,Z2: b] :
( ( collect_b
@ ^ [Y: b] :
( ( Y = X )
| ( Y = Z2 ) ) )
= ( insert_b @ X @ ( insert_b @ Z2 @ bot_bot_set_b ) ) ) ).
% tuple_disj
thf(fact_12_Image__empty1,axiom,
! [X3: set_b] :
( ( image_b_b @ bot_bo1343651123od_b_b @ X3 )
= bot_bot_set_b ) ).
% Image_empty1
thf(fact_13_Image__empty2,axiom,
! [R: set_Product_prod_b_b] :
( ( image_b_b @ R @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Image_empty2
thf(fact_14_Image__empty2,axiom,
! [R: set_Pr621705391od_b_b] :
( ( image_1177096571od_b_b @ R @ bot_bot_set_b )
= bot_bo1343651123od_b_b ) ).
% Image_empty2
thf(fact_15_Image__empty2,axiom,
! [R: set_Pr357419743_b_b_b] :
( ( image_921732011_b_b_b @ R @ bot_bo1343651123od_b_b )
= bot_bot_set_b ) ).
% Image_empty2
thf(fact_16_Image__empty2,axiom,
! [R: set_Pr2007082183od_b_b] :
( ( image_1928024979od_b_b @ R @ bot_bo1343651123od_b_b )
= bot_bo1343651123od_b_b ) ).
% Image_empty2
thf(fact_17_singletonI,axiom,
! [A: produc1213276845od_b_b] : ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) ) ).
% singletonI
thf(fact_18_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_19_singletonI,axiom,
! [A: product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% singletonI
thf(fact_20_some__equality,axiom,
! [P: b > $o,A: b] :
( ( P @ A )
=> ( ! [X4: b] :
( ( P @ X4 )
=> ( X4 = A ) )
=> ( ( hilbert_Eps_b @ P )
= A ) ) ) ).
% some_equality
thf(fact_21_some__eq__trivial,axiom,
! [X: b] :
( ( hilbert_Eps_b
@ ^ [Y: b] : Y = X )
= X ) ).
% some_eq_trivial
thf(fact_22_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_23_empty__Collect__eq,axiom,
! [P: product_prod_b_b > $o] :
( ( bot_bo1343651123od_b_b
= ( collec1481886546od_b_b @ P ) )
= ( ! [X2: product_prod_b_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_24_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_25_Collect__empty__eq,axiom,
! [P: product_prod_b_b > $o] :
( ( ( collec1481886546od_b_b @ P )
= bot_bo1343651123od_b_b )
= ( ! [X2: product_prod_b_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_26_all__not__in__conv,axiom,
! [A2: set_Pr1324126435od_b_b] :
( ( ! [X2: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ X2 @ A2 ) )
= ( A2 = bot_bo1664927607od_b_b ) ) ).
% all_not_in_conv
thf(fact_27_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_28_all__not__in__conv,axiom,
! [A2: set_Product_prod_b_b] :
( ( ! [X2: product_prod_b_b] :
~ ( member1285940496od_b_b @ X2 @ A2 ) )
= ( A2 = bot_bo1343651123od_b_b ) ) ).
% all_not_in_conv
thf(fact_29_empty__iff,axiom,
! [C: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ C @ bot_bo1664927607od_b_b ) ).
% empty_iff
thf(fact_30_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_31_empty__iff,axiom,
! [C: product_prod_b_b] :
~ ( member1285940496od_b_b @ C @ bot_bo1343651123od_b_b ) ).
% empty_iff
thf(fact_32_insert__absorb2,axiom,
! [X: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
= ( insert_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_33_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_34_insert__iff,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A2 ) )
= ( ( A = B )
| ( member1285940496od_b_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_35_insert__iff,axiom,
! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ A2 ) )
= ( ( A = B )
| ( member1516365892od_b_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_36_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_37_insertCI,axiom,
! [A: product_prod_b_b,B2: set_Product_prod_b_b,B: product_prod_b_b] :
( ( ~ ( member1285940496od_b_b @ A @ B2 )
=> ( A = B ) )
=> ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_38_insertCI,axiom,
! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b] :
( ( ~ ( member1516365892od_b_b @ A @ B2 )
=> ( A = B ) )
=> ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_39_some__sym__eq__trivial,axiom,
! [X: b] :
( ( hilbert_Eps_b
@ ( ^ [Y2: b,Z: b] : Y2 = Z
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_40_ImageI,axiom,
! [A: b,B: product_prod_b_b,R2: set_Pr621705391od_b_b,A2: set_b] :
( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A2 )
=> ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_41_ImageI,axiom,
! [A: b,B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A2: set_b] :
( ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A2 )
=> ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_42_ImageI,axiom,
! [A: product_prod_b_b,B: b,R2: set_Pr357419743_b_b_b,A2: set_Product_prod_b_b] :
( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 )
=> ( ( member1285940496od_b_b @ A @ A2 )
=> ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_43_ImageI,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A2: set_Product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
=> ( ( member1285940496od_b_b @ A @ A2 )
=> ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_44_ImageI,axiom,
! [A: product_prod_b_b,B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A2: set_Product_prod_b_b] :
( ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 )
=> ( ( member1285940496od_b_b @ A @ A2 )
=> ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_45_ImageI,axiom,
! [A: produc1213276845od_b_b,B: b,R2: set_Pr1906739389_b_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A @ B ) @ R2 )
=> ( ( member1516365892od_b_b @ A @ A2 )
=> ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_46_ImageI,axiom,
! [A: produc1213276845od_b_b,B: product_prod_b_b,R2: set_Pr2134561957od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A @ B ) @ R2 )
=> ( ( member1516365892od_b_b @ A @ A2 )
=> ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_47_ImageI,axiom,
! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
=> ( ( member1516365892od_b_b @ A @ A2 )
=> ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_48_ImageI,axiom,
! [A: b,B: b,R2: set_Product_prod_b_b,A2: set_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A2 )
=> ( member_b @ B @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_49_ImageI,axiom,
! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
=> ( ( member1632892294tant_a @ A @ A2 )
=> ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_50_f,axiom,
( f
= ( ^ [X2: b] :
( if_b
@ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
= bot_bot_set_b )
@ X2
@ ( hilbert_Eps_b @ ( p @ X2 ) ) ) ) ) ).
% f
thf(fact_51_Image__singleton__iff,axiom,
! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A: standard_Constant_a] :
( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) )
= ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_52_Image__singleton__iff,axiom,
! [B: product_prod_b_b,R2: set_Pr621705391od_b_b,A: b] :
( ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_53_Image__singleton__iff,axiom,
! [B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A: b] :
( ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_54_Image__singleton__iff,axiom,
! [B: b,R2: set_Product_prod_b_b,A: b] :
( ( member_b @ B @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_55_Image__singleton__iff,axiom,
! [B: b,R2: set_Pr357419743_b_b_b,A: product_prod_b_b] :
( ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
= ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_56_Image__singleton__iff,axiom,
! [B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
( ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
= ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_57_Image__singleton__iff,axiom,
! [B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A: product_prod_b_b] :
( ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
= ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_58_pred__equals__eq2,axiom,
! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ( ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R ) )
= ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_59_pred__equals__eq2,axiom,
! [R: set_Pr1324126435od_b_b,S: set_Pr1324126435od_b_b] :
( ( ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ R ) )
= ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_60_rev__ImageI,axiom,
! [A: b,A2: set_b,B: product_prod_b_b,R2: set_Pr621705391od_b_b] :
( ( member_b @ A @ A2 )
=> ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 )
=> ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_61_rev__ImageI,axiom,
! [A: b,A2: set_b,B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b] :
( ( member_b @ A @ A2 )
=> ( ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 )
=> ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_62_rev__ImageI,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: b,R2: set_Pr357419743_b_b_b] :
( ( member1285940496od_b_b @ A @ A2 )
=> ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 )
=> ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_63_rev__ImageI,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b,R2: set_Pr2007082183od_b_b] :
( ( member1285940496od_b_b @ A @ A2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
=> ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_64_rev__ImageI,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b] :
( ( member1285940496od_b_b @ A @ A2 )
=> ( ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 )
=> ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_65_rev__ImageI,axiom,
! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: b,R2: set_Pr1906739389_b_b_b] :
( ( member1516365892od_b_b @ A @ A2 )
=> ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A @ B ) @ R2 )
=> ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_66_rev__ImageI,axiom,
! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: product_prod_b_b,R2: set_Pr2134561957od_b_b] :
( ( member1516365892od_b_b @ A @ A2 )
=> ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A @ B ) @ R2 )
=> ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_67_rev__ImageI,axiom,
! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b] :
( ( member1516365892od_b_b @ A @ A2 )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
=> ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_68_rev__ImageI,axiom,
! [A: b,A2: set_b,B: b,R2: set_Product_prod_b_b] :
( ( member_b @ A @ A2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( member_b @ B @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_69_rev__ImageI,axiom,
! [A: standard_Constant_a,A2: set_St761939237tant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b] :
( ( member1632892294tant_a @ A @ A2 )
=> ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
=> ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_70_Image__iff,axiom,
! [B: b,R2: set_Product_prod_b_b,A2: set_b] :
( ( member_b @ B @ ( image_b_b @ R2 @ A2 ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_71_Image__iff,axiom,
! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) )
= ( ? [X2: standard_Constant_a] :
( ( member1632892294tant_a @ X2 @ A2 )
& ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_72_ImageE,axiom,
! [B: b,R2: set_Pr357419743_b_b_b,A2: set_Product_prod_b_b] :
( ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_73_ImageE,axiom,
! [B: b,R2: set_Pr1906739389_b_b_b,A2: set_Pr1324126435od_b_b] :
( ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_74_ImageE,axiom,
! [B: product_prod_b_b,R2: set_Pr621705391od_b_b,A2: set_b] :
( ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: b] :
( ( member641857272od_b_b @ ( produc1257047359od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_75_ImageE,axiom,
! [B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_76_ImageE,axiom,
! [B: product_prod_b_b,R2: set_Pr2134561957od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member942707974od_b_b @ ( produc1597690145od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_77_ImageE,axiom,
! [B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A2: set_b] :
( ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: b] :
( ( member599505522od_b_b @ ( produc800495189od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_78_ImageE,axiom,
! [B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A2: set_Product_prod_b_b] :
( ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: product_prod_b_b] :
( ( member4694106od_b_b @ ( produc732676669od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_79_ImageE,axiom,
! [B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: produc1213276845od_b_b] :
( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_80_ImageE,axiom,
! [B: b,R2: set_Product_prod_b_b,A2: set_b] :
( ( member_b @ B @ ( image_b_b @ R2 @ A2 ) )
=> ~ ! [X4: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ B ) @ R2 )
=> ~ ( member_b @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_81_ImageE,axiom,
! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) )
=> ~ ! [X4: standard_Constant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ B ) @ R2 )
=> ~ ( member1632892294tant_a @ X4 @ A2 ) ) ) ).
% ImageE
thf(fact_82_Image__singleton,axiom,
! [R2: set_Pr1324126435od_b_b,A: standard_Constant_a] :
( ( image_1916422435od_b_b @ R2 @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) )
= ( collec1481886546od_b_b
@ ^ [B3: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B3 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_83_Image__singleton,axiom,
! [R2: set_Product_prod_b_b,A: b] :
( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( collect_b
@ ^ [B3: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_84_Image__singleton,axiom,
! [R2: set_Pr357419743_b_b_b,A: product_prod_b_b] :
( ( image_921732011_b_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
= ( collect_b
@ ^ [B3: b] : ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B3 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_85_ex__in__conv,axiom,
! [A2: set_Pr1324126435od_b_b] :
( ( ? [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
= ( A2 != bot_bo1664927607od_b_b ) ) ).
% ex_in_conv
thf(fact_86_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_87_ex__in__conv,axiom,
! [A2: set_Product_prod_b_b] :
( ( ? [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
= ( A2 != bot_bo1343651123od_b_b ) ) ).
% ex_in_conv
thf(fact_88_equals0I,axiom,
! [A2: set_Pr1324126435od_b_b] :
( ! [Y3: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ Y3 @ A2 )
=> ( A2 = bot_bo1664927607od_b_b ) ) ).
% equals0I
thf(fact_89_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_90_equals0I,axiom,
! [A2: set_Product_prod_b_b] :
( ! [Y3: product_prod_b_b] :
~ ( member1285940496od_b_b @ Y3 @ A2 )
=> ( A2 = bot_bo1343651123od_b_b ) ) ).
% equals0I
thf(fact_91_equals0D,axiom,
! [A2: set_Pr1324126435od_b_b,A: produc1213276845od_b_b] :
( ( A2 = bot_bo1664927607od_b_b )
=> ~ ( member1516365892od_b_b @ A @ A2 ) ) ).
% equals0D
thf(fact_92_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_93_equals0D,axiom,
! [A2: set_Product_prod_b_b,A: product_prod_b_b] :
( ( A2 = bot_bo1343651123od_b_b )
=> ~ ( member1285940496od_b_b @ A @ A2 ) ) ).
% equals0D
thf(fact_94_emptyE,axiom,
! [A: produc1213276845od_b_b] :
~ ( member1516365892od_b_b @ A @ bot_bo1664927607od_b_b ) ).
% emptyE
thf(fact_95_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_96_emptyE,axiom,
! [A: product_prod_b_b] :
~ ( member1285940496od_b_b @ A @ bot_bo1343651123od_b_b ) ).
% emptyE
thf(fact_97_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B4: set_b] :
( ( A2
= ( insert_b @ A @ B4 ) )
& ~ ( member_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_98_mk__disjoint__insert,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ A2 )
=> ? [B4: set_Product_prod_b_b] :
( ( A2
= ( insert1952693431od_b_b @ A @ B4 ) )
& ~ ( member1285940496od_b_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_99_mk__disjoint__insert,axiom,
! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A @ A2 )
=> ? [B4: set_Pr1324126435od_b_b] :
( ( A2
= ( insert2037698781od_b_b @ A @ B4 ) )
& ~ ( member1516365892od_b_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_100_insert__commute,axiom,
! [X: b,Y4: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ Y4 @ A2 ) )
= ( insert_b @ Y4 @ ( insert_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_101_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_b] :
( ( A2
= ( insert_b @ B @ C2 ) )
& ~ ( member_b @ B @ C2 )
& ( B2
= ( insert_b @ A @ C2 ) )
& ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_102_insert__eq__iff,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b,B2: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ A @ A2 )
=> ( ~ ( member1285940496od_b_b @ B @ B2 )
=> ( ( ( insert1952693431od_b_b @ A @ A2 )
= ( insert1952693431od_b_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_Product_prod_b_b] :
( ( A2
= ( insert1952693431od_b_b @ B @ C2 ) )
& ~ ( member1285940496od_b_b @ B @ C2 )
& ( B2
= ( insert1952693431od_b_b @ A @ C2 ) )
& ~ ( member1285940496od_b_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_103_insert__eq__iff,axiom,
! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b] :
( ~ ( member1516365892od_b_b @ A @ A2 )
=> ( ~ ( member1516365892od_b_b @ B @ B2 )
=> ( ( ( insert2037698781od_b_b @ A @ A2 )
= ( insert2037698781od_b_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_Pr1324126435od_b_b] :
( ( A2
= ( insert2037698781od_b_b @ B @ C2 ) )
& ~ ( member1516365892od_b_b @ B @ C2 )
& ( B2
= ( insert2037698781od_b_b @ A @ C2 ) )
& ~ ( member1516365892od_b_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_104_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_105_insert__absorb,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ A2 )
=> ( ( insert1952693431od_b_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_106_insert__absorb,axiom,
! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A @ A2 )
=> ( ( insert2037698781od_b_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_107_insert__ident,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ A2 )
= ( insert_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_108_insert__ident,axiom,
! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ X @ A2 )
=> ( ~ ( member1285940496od_b_b @ X @ B2 )
=> ( ( ( insert1952693431od_b_b @ X @ A2 )
= ( insert1952693431od_b_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_109_insert__ident,axiom,
! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
( ~ ( member1516365892od_b_b @ X @ A2 )
=> ( ~ ( member1516365892od_b_b @ X @ B2 )
=> ( ( ( insert2037698781od_b_b @ X @ A2 )
= ( insert2037698781od_b_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_110_Set_Oset__insert,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ~ ! [B4: set_b] :
( ( A2
= ( insert_b @ X @ B4 ) )
=> ( member_b @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_111_Set_Oset__insert,axiom,
! [X: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ X @ A2 )
=> ~ ! [B4: set_Product_prod_b_b] :
( ( A2
= ( insert1952693431od_b_b @ X @ B4 ) )
=> ( member1285940496od_b_b @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_112_Set_Oset__insert,axiom,
! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ X @ A2 )
=> ~ ! [B4: set_Pr1324126435od_b_b] :
( ( A2
= ( insert2037698781od_b_b @ X @ B4 ) )
=> ( member1516365892od_b_b @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_113_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_114_insertI2,axiom,
! [A: product_prod_b_b,B2: set_Product_prod_b_b,B: product_prod_b_b] :
( ( member1285940496od_b_b @ A @ B2 )
=> ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_115_insertI2,axiom,
! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ A @ B2 )
=> ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_116_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_117_insertI1,axiom,
! [A: product_prod_b_b,B2: set_Product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ B2 ) ) ).
% insertI1
thf(fact_118_insertI1,axiom,
! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b] : ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ A @ B2 ) ) ).
% insertI1
thf(fact_119_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_120_insertE,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member1285940496od_b_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_121_insertE,axiom,
! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member1516365892od_b_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_122_some__eq__imp,axiom,
! [P: b > $o,A: b,B: b] :
( ( ( hilbert_Eps_b @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_123_tfl__some,axiom,
! [P2: b > $o,X5: b] :
( ( P2 @ X5 )
=> ( P2 @ ( hilbert_Eps_b @ P2 ) ) ) ).
% tfl_some
thf(fact_124_mem__Collect__eq,axiom,
! [A: product_prod_b_b,P: product_prod_b_b > $o] :
( ( member1285940496od_b_b @ A @ ( collec1481886546od_b_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A: produc1213276845od_b_b,P: produc1213276845od_b_b > $o] :
( ( member1516365892od_b_b @ A @ ( collec279084418od_b_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A2: set_Product_prod_b_b] :
( ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A2: set_Pr1324126435od_b_b] :
( ( collec279084418od_b_b
@ ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_130_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X4: b] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_131_Eps__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X4: b] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( hilbert_Eps_b @ P )
= ( hilbert_Eps_b @ Q ) ) ) ).
% Eps_cong
thf(fact_132_someI,axiom,
! [P: b > $o,X: b] :
( ( P @ X )
=> ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% someI
thf(fact_133_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X2: b] : $false ) ) ).
% empty_def
thf(fact_134_empty__def,axiom,
( bot_bo1343651123od_b_b
= ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] : $false ) ) ).
% empty_def
thf(fact_135_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U: b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_136_insert__compr,axiom,
( insert1952693431od_b_b
= ( ^ [A3: product_prod_b_b,B5: set_Product_prod_b_b] :
( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] :
( ( X2 = A3 )
| ( member1285940496od_b_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_137_insert__compr,axiom,
( insert2037698781od_b_b
= ( ^ [A3: produc1213276845od_b_b,B5: set_Pr1324126435od_b_b] :
( collec279084418od_b_b
@ ^ [X2: produc1213276845od_b_b] :
( ( X2 = A3 )
| ( member1516365892od_b_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_138_insert__compr,axiom,
( insert_b
= ( ^ [A3: b,B5: set_b] :
( collect_b
@ ^ [X2: b] :
( ( X2 = A3 )
| ( member_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_139_some1__equality,axiom,
! [P: b > $o,A: b] :
( ? [X5: b] :
( ( P @ X5 )
& ! [Y3: b] :
( ( P @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ( P @ A )
=> ( ( hilbert_Eps_b @ P )
= A ) ) ) ).
% some1_equality
thf(fact_140_some__eq__ex,axiom,
! [P: b > $o] :
( ( P @ ( hilbert_Eps_b @ P ) )
= ( ? [X6: b] : ( P @ X6 ) ) ) ).
% some_eq_ex
thf(fact_141_someI2__bex,axiom,
! [A2: set_Product_prod_b_b,P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
( ? [X5: product_prod_b_b] :
( ( member1285940496od_b_b @ X5 @ A2 )
& ( P @ X5 ) )
=> ( ! [X4: product_prod_b_b] :
( ( ( member1285940496od_b_b @ X4 @ A2 )
& ( P @ X4 ) )
=> ( Q @ X4 ) )
=> ( Q
@ ( hilber620807523od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member1285940496od_b_b @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_142_someI2__bex,axiom,
! [A2: set_Pr1324126435od_b_b,P: produc1213276845od_b_b > $o,Q: produc1213276845od_b_b > $o] :
( ? [X5: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X5 @ A2 )
& ( P @ X5 ) )
=> ( ! [X4: produc1213276845od_b_b] :
( ( ( member1516365892od_b_b @ X4 @ A2 )
& ( P @ X4 ) )
=> ( Q @ X4 ) )
=> ( Q
@ ( hilber46179633od_b_b
@ ^ [X2: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_143_someI2__bex,axiom,
! [A2: set_b,P: b > $o,Q: b > $o] :
( ? [X5: b] :
( ( member_b @ X5 @ A2 )
& ( P @ X5 ) )
=> ( ! [X4: b] :
( ( ( member_b @ X4 @ A2 )
& ( P @ X4 ) )
=> ( Q @ X4 ) )
=> ( Q
@ ( hilbert_Eps_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_144_someI2__ex,axiom,
! [P: b > $o,Q: b > $o] :
( ? [X_1: b] : ( P @ X_1 )
=> ( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% someI2_ex
thf(fact_145_someI__ex,axiom,
! [P: b > $o] :
( ? [X_1: b] : ( P @ X_1 )
=> ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% someI_ex
thf(fact_146_someI2,axiom,
! [P: b > $o,A: b,Q: b > $o] :
( ( P @ A )
=> ( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% someI2
thf(fact_147_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_148_singleton__inject,axiom,
! [A: product_prod_b_b,B: product_prod_b_b] :
( ( ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b )
= ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_149_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_150_insert__not__empty,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( insert1952693431od_b_b @ A @ A2 )
!= bot_bo1343651123od_b_b ) ).
% insert_not_empty
thf(fact_151_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_152_doubleton__eq__iff,axiom,
! [A: product_prod_b_b,B: product_prod_b_b,C: product_prod_b_b,D: product_prod_b_b] :
( ( ( insert1952693431od_b_b @ A @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
= ( insert1952693431od_b_b @ C @ ( insert1952693431od_b_b @ D @ bot_bo1343651123od_b_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_153_singleton__iff,axiom,
! [B: produc1213276845od_b_b,A: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ B @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_154_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_155_singleton__iff,axiom,
! [B: product_prod_b_b,A: product_prod_b_b] :
( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_156_singletonD,axiom,
! [B: produc1213276845od_b_b,A: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ B @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_157_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_158_singletonD,axiom,
! [B: product_prod_b_b,A: product_prod_b_b] :
( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_159_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X2: b] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X2: b] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_160_Collect__conv__if2,axiom,
! [P: product_prod_b_b > $o,A: product_prod_b_b] :
( ( ( P @ A )
=> ( ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
& ( ~ ( P @ A )
=> ( ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bo1343651123od_b_b ) ) ) ).
% Collect_conv_if2
thf(fact_161_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X2: b] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X2: b] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_162_Collect__conv__if,axiom,
! [P: product_prod_b_b > $o,A: product_prod_b_b] :
( ( ( P @ A )
=> ( ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
& ( ~ ( P @ A )
=> ( ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bo1343651123od_b_b ) ) ) ).
% Collect_conv_if
thf(fact_163_some__in__eq,axiom,
! [A2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b
@ ( hilber46179633od_b_b
@ ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
@ A2 )
= ( A2 != bot_bo1664927607od_b_b ) ) ).
% some_in_eq
thf(fact_164_some__in__eq,axiom,
! [A2: set_b] :
( ( member_b
@ ( hilbert_Eps_b
@ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_b ) ) ).
% some_in_eq
thf(fact_165_some__in__eq,axiom,
! [A2: set_Product_prod_b_b] :
( ( member1285940496od_b_b
@ ( hilber620807523od_b_b
@ ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
@ A2 )
= ( A2 != bot_bo1343651123od_b_b ) ) ).
% some_in_eq
thf(fact_166_congr_I1_J,axiom,
! [X: b,Y4: b,L: standard_Constant_a] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( getRel904497637nt_a_b @ L @ g ) )
=> ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
@ ^ [V: b] : ( image_b_b @ ( getRel904497637nt_a_b @ L @ g ) @ ( insert_b @ V @ bot_bot_set_b ) ) ) ) ).
% congr(1)
thf(fact_167_vert__P,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ ( hilbert_Eps_b @ ( p @ X ) ) ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% vert_P
thf(fact_168__092_060open_062_IS__Idt_M_Ax_M_Ay_J_A_092_060in_062_Aedges_AG_092_060close_062,axiom,
member1516365892od_b_b @ ( produc1432590431od_b_b @ standard_S_Idt_a @ ( product_Pair_b_b @ x @ y ) ) @ ( labele1741081071nt_a_b @ g ) ).
% \<open>(S_Idt, x, y) \<in> edges G\<close>
thf(fact_169_restrict__idemp,axiom,
! [X: labele1159362096nt_a_b] :
( ( restri446606278nt_a_b @ ( restri446606278nt_a_b @ X ) )
= ( restri446606278nt_a_b @ X ) ) ).
% restrict_idemp
thf(fact_170_r2,axiom,
! [X: b] :
( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
= bot_bot_set_b )
= ( ~ ( member_b @ X @ ( labele1424214014nt_a_b @ g ) ) ) ) ).
% r2
thf(fact_171_r1,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
= ( p @ X @ X ) ) ).
% r1
thf(fact_172_congr_I2_J,axiom,
! [X: b,Y4: b,L: standard_Constant_a] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( getRel904497637nt_a_b @ L @ g ) )
=> ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
@ ^ [V: b] : ( image_b_b @ ( converse_b_b @ ( getRel904497637nt_a_b @ L @ g ) ) @ ( insert_b @ V @ bot_bot_set_b ) ) ) ) ).
% congr(2)
thf(fact_173_the__elem__eq,axiom,
! [X: b] :
( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
= X ) ).
% the_elem_eq
thf(fact_174_the__elem__eq,axiom,
! [X: product_prod_b_b] :
( ( the_el329196508od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
= X ) ).
% the_elem_eq
thf(fact_175_prod_Oinject,axiom,
! [X1: b,X22: b,Y1: b,Y22: b] :
( ( ( product_Pair_b_b @ X1 @ X22 )
= ( product_Pair_b_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_176_prod_Oinject,axiom,
! [X1: standard_Constant_a,X22: product_prod_b_b,Y1: standard_Constant_a,Y22: product_prod_b_b] :
( ( ( produc1432590431od_b_b @ X1 @ X22 )
= ( produc1432590431od_b_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_177_old_Oprod_Oinject,axiom,
! [A: b,B: b,A4: b,B6: b] :
( ( ( product_Pair_b_b @ A @ B )
= ( product_Pair_b_b @ A4 @ B6 ) )
= ( ( A = A4 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_178_old_Oprod_Oinject,axiom,
! [A: standard_Constant_a,B: product_prod_b_b,A4: standard_Constant_a,B6: product_prod_b_b] :
( ( ( produc1432590431od_b_b @ A @ B )
= ( produc1432590431od_b_b @ A4 @ B6 ) )
= ( ( A = A4 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_179_converse__inject,axiom,
! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ( ( converse_b_b @ R2 )
= ( converse_b_b @ S2 ) )
= ( R2 = S2 ) ) ).
% converse_inject
thf(fact_180_converse__converse,axiom,
! [R2: set_Product_prod_b_b] :
( ( converse_b_b @ ( converse_b_b @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_181_converse__iff,axiom,
! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ ( conver1398856763od_b_b @ R2 ) )
= ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ R2 ) ) ).
% converse_iff
thf(fact_182_converse__iff,axiom,
! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr306736955tant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ ( conver1943590235tant_a @ R2 ) )
= ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ R2 ) ) ).
% converse_iff
thf(fact_183_converse__iff,axiom,
! [A: b,B: b,R2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( converse_b_b @ R2 ) )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 ) ) ).
% converse_iff
thf(fact_184_vertices__restrict,axiom,
! [G: labele1159362096nt_a_b] :
( ( labele1424214014nt_a_b @ ( restri446606278nt_a_b @ G ) )
= ( labele1424214014nt_a_b @ G ) ) ).
% vertices_restrict
thf(fact_185_converse__empty,axiom,
( ( converse_b_b @ bot_bo1343651123od_b_b )
= bot_bo1343651123od_b_b ) ).
% converse_empty
thf(fact_186__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
! [X: b] :
( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ X ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% \<open>\<And>x. x \<in> vertices G \<Longrightarrow> (x, x) \<in> getRel S_Idt G\<close>
thf(fact_187_labeled__graph_Oexpand,axiom,
! [Labeled_graph: labele1159362096nt_a_b,Labeled_graph2: labele1159362096nt_a_b] :
( ( ( ( labele1741081071nt_a_b @ Labeled_graph )
= ( labele1741081071nt_a_b @ Labeled_graph2 ) )
& ( ( labele1424214014nt_a_b @ Labeled_graph )
= ( labele1424214014nt_a_b @ Labeled_graph2 ) ) )
=> ( Labeled_graph = Labeled_graph2 ) ) ).
% labeled_graph.expand
thf(fact_188_converse_Oinducts,axiom,
! [X1: product_prod_b_b,X22: standard_Constant_a,R2: set_Pr1324126435od_b_b,P: product_prod_b_b > standard_Constant_a > $o] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ X1 @ X22 ) @ ( conver1398856763od_b_b @ R2 ) )
=> ( ! [A5: standard_Constant_a,B7: product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A5 @ B7 ) @ R2 )
=> ( P @ B7 @ A5 ) )
=> ( P @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_189_converse_Oinducts,axiom,
! [X1: standard_Constant_a,X22: product_prod_b_b,R2: set_Pr306736955tant_a,P: standard_Constant_a > product_prod_b_b > $o] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X1 @ X22 ) @ ( conver1943590235tant_a @ R2 ) )
=> ( ! [A5: product_prod_b_b,B7: standard_Constant_a] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ A5 @ B7 ) @ R2 )
=> ( P @ B7 @ A5 ) )
=> ( P @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_190_converse_Oinducts,axiom,
! [X1: b,X22: b,R2: set_Product_prod_b_b,P: b > b > $o] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X22 ) @ ( converse_b_b @ R2 ) )
=> ( ! [A5: b,B7: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A5 @ B7 ) @ R2 )
=> ( P @ B7 @ A5 ) )
=> ( P @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_191_converse_Ointros,axiom,
! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr306736955tant_a] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ R2 )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ ( conver1943590235tant_a @ R2 ) ) ) ).
% converse.intros
thf(fact_192_converse_Ointros,axiom,
! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
=> ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ ( conver1398856763od_b_b @ R2 ) ) ) ).
% converse.intros
thf(fact_193_converse_Ointros,axiom,
! [A: b,B: b,R2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( converse_b_b @ R2 ) ) ) ).
% converse.intros
thf(fact_194_converse_Osimps,axiom,
! [A1: product_prod_b_b,A22: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ A1 @ A22 ) @ ( conver1398856763od_b_b @ R2 ) )
= ( ? [A3: standard_Constant_a,B3: product_prod_b_b] :
( ( A1 = B3 )
& ( A22 = A3 )
& ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A3 @ B3 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_195_converse_Osimps,axiom,
! [A1: standard_Constant_a,A22: product_prod_b_b,R2: set_Pr306736955tant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( conver1943590235tant_a @ R2 ) )
= ( ? [A3: product_prod_b_b,B3: standard_Constant_a] :
( ( A1 = B3 )
& ( A22 = A3 )
& ( member1670214300tant_a @ ( produc1977323903tant_a @ A3 @ B3 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_196_converse_Osimps,axiom,
! [A1: b,A22: b,R2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( converse_b_b @ R2 ) )
= ( ? [A3: b,B3: b] :
( ( A1 = B3 )
& ( A22 = A3 )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_197_converse_Ocases,axiom,
! [A1: product_prod_b_b,A22: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ A1 @ A22 ) @ ( conver1398856763od_b_b @ R2 ) )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_198_converse_Ocases,axiom,
! [A1: standard_Constant_a,A22: product_prod_b_b,R2: set_Pr306736955tant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( conver1943590235tant_a @ R2 ) )
=> ( member1670214300tant_a @ ( produc1977323903tant_a @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_199_converse_Ocases,axiom,
! [A1: b,A22: b,R2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( converse_b_b @ R2 ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_200_converseE,axiom,
! [Yx: produc1367125253tant_a,R2: set_Pr1324126435od_b_b] :
( ( member1670214300tant_a @ Yx @ ( conver1398856763od_b_b @ R2 ) )
=> ~ ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
( ( Yx
= ( produc1977323903tant_a @ Y3 @ X4 ) )
=> ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R2 ) ) ) ).
% converseE
thf(fact_201_converseE,axiom,
! [Yx: produc1213276845od_b_b,R2: set_Pr306736955tant_a] :
( ( member1516365892od_b_b @ Yx @ ( conver1943590235tant_a @ R2 ) )
=> ~ ! [X4: product_prod_b_b,Y3: standard_Constant_a] :
( ( Yx
= ( produc1432590431od_b_b @ Y3 @ X4 ) )
=> ~ ( member1670214300tant_a @ ( produc1977323903tant_a @ X4 @ Y3 ) @ R2 ) ) ) ).
% converseE
thf(fact_202_converseE,axiom,
! [Yx: product_prod_b_b,R2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ Yx @ ( converse_b_b @ R2 ) )
=> ~ ! [X4: b,Y3: b] :
( ( Yx
= ( product_Pair_b_b @ Y3 @ X4 ) )
=> ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 ) ) ) ).
% converseE
thf(fact_203_converseD,axiom,
! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ ( conver1398856763od_b_b @ R2 ) )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ R2 ) ) ).
% converseD
thf(fact_204_converseD,axiom,
! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr306736955tant_a] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ ( conver1943590235tant_a @ R2 ) )
=> ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ R2 ) ) ).
% converseD
thf(fact_205_converseD,axiom,
! [A: b,B: b,R2: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( converse_b_b @ R2 ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 ) ) ).
% converseD
thf(fact_206_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_207_bot__set__def,axiom,
( bot_bo1343651123od_b_b
= ( collec1481886546od_b_b @ bot_bo1302443370_b_b_o ) ) ).
% bot_set_def
thf(fact_208_bot__empty__eq,axiom,
( bot_bo1167600078_b_b_o
= ( ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ bot_bo1664927607od_b_b ) ) ) ).
% bot_empty_eq
thf(fact_209_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_210_bot__empty__eq,axiom,
( bot_bo1302443370_b_b_o
= ( ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ bot_bo1343651123od_b_b ) ) ) ).
% bot_empty_eq
thf(fact_211_bot__empty__eq2,axiom,
( bot_bo1734638456_b_b_o
= ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ bot_bo1664927607od_b_b ) ) ) ).
% bot_empty_eq2
thf(fact_212_bot__empty__eq2,axiom,
( bot_bot_b_b_o
= ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ bot_bo1343651123od_b_b ) ) ) ).
% bot_empty_eq2
thf(fact_213_old_Oprod_Oinducts,axiom,
! [P: product_prod_b_b > $o,Prod: product_prod_b_b] :
( ! [A5: b,B7: b] : ( P @ ( product_Pair_b_b @ A5 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_214_old_Oprod_Oinducts,axiom,
! [P: produc1213276845od_b_b > $o,Prod: produc1213276845od_b_b] :
( ! [A5: standard_Constant_a,B7: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A5 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_215_old_Oprod_Oexhaust,axiom,
! [Y4: product_prod_b_b] :
~ ! [A5: b,B7: b] :
( Y4
!= ( product_Pair_b_b @ A5 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_216_old_Oprod_Oexhaust,axiom,
! [Y4: produc1213276845od_b_b] :
~ ! [A5: standard_Constant_a,B7: product_prod_b_b] :
( Y4
!= ( produc1432590431od_b_b @ A5 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_217_prod__induct3,axiom,
! [P: produc1213276845od_b_b > $o,X: produc1213276845od_b_b] :
( ! [A5: standard_Constant_a,B7: b,C3: b] : ( P @ ( produc1432590431od_b_b @ A5 @ ( product_Pair_b_b @ B7 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_218_prod__cases3,axiom,
! [Y4: produc1213276845od_b_b] :
~ ! [A5: standard_Constant_a,B7: b,C3: b] :
( Y4
!= ( produc1432590431od_b_b @ A5 @ ( product_Pair_b_b @ B7 @ C3 ) ) ) ).
% prod_cases3
thf(fact_219_Pair__inject,axiom,
! [A: b,B: b,A4: b,B6: b] :
( ( ( product_Pair_b_b @ A @ B )
= ( product_Pair_b_b @ A4 @ B6 ) )
=> ~ ( ( A = A4 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_220_Pair__inject,axiom,
! [A: standard_Constant_a,B: product_prod_b_b,A4: standard_Constant_a,B6: product_prod_b_b] :
( ( ( produc1432590431od_b_b @ A @ B )
= ( produc1432590431od_b_b @ A4 @ B6 ) )
=> ~ ( ( A = A4 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_221_prod__cases,axiom,
! [P: product_prod_b_b > $o,P3: product_prod_b_b] :
( ! [A5: b,B7: b] : ( P @ ( product_Pair_b_b @ A5 @ B7 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_222_prod__cases,axiom,
! [P: produc1213276845od_b_b > $o,P3: produc1213276845od_b_b] :
( ! [A5: standard_Constant_a,B7: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A5 @ B7 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_223_surj__pair,axiom,
! [P3: product_prod_b_b] :
? [X4: b,Y3: b] :
( P3
= ( product_Pair_b_b @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_224_surj__pair,axiom,
! [P3: produc1213276845od_b_b] :
? [X4: standard_Constant_a,Y3: product_prod_b_b] :
( P3
= ( produc1432590431od_b_b @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_225_local_Orefl,axiom,
refl_on_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% local.refl
thf(fact_226_equiv,axiom,
equiv_equiv_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% equiv
thf(fact_227_getRel__dom_I1_J,axiom,
! [G: labele1159362096nt_a_b,A: b,B: b,L: standard_Constant_a] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( member_b @ A @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% getRel_dom(1)
thf(fact_228_getRel__dom_I2_J,axiom,
! [G: labele1159362096nt_a_b,A: b,B: b,L: standard_Constant_a] :
( ( G
= ( restri446606278nt_a_b @ G ) )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel904497637nt_a_b @ L @ G ) )
=> ( member_b @ B @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% getRel_dom(2)
thf(fact_229_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A6: set_b] :
( A6
= ( insert_b @ ( the_elem_b @ A6 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_230_is__singleton__the__elem,axiom,
( is_sin1713057243od_b_b
= ( ^ [A6: set_Product_prod_b_b] :
( A6
= ( insert1952693431od_b_b @ ( the_el329196508od_b_b @ A6 ) @ bot_bo1343651123od_b_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_231_refl__on__converse,axiom,
! [A2: set_b,R2: set_Product_prod_b_b] :
( ( refl_on_b @ A2 @ ( converse_b_b @ R2 ) )
= ( refl_on_b @ A2 @ R2 ) ) ).
% refl_on_converse
thf(fact_232_is__singletonI,axiom,
! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_233_is__singletonI,axiom,
! [X: product_prod_b_b] : ( is_sin1713057243od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) ) ).
% is_singletonI
thf(fact_234_refl__onD,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ A @ A2 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_235_refl__onD,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A2 @ R2 )
=> ( ( member1516365892od_b_b @ A @ A2 )
=> ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_236_refl__onD,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,A: b] :
( ( refl_on_b @ A2 @ R2 )
=> ( ( member_b @ A @ A2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_237_refl__onD1,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A2 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
=> ( member1285940496od_b_b @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_238_refl__onD1,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A2 @ R2 )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
=> ( member1516365892od_b_b @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_239_refl__onD1,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
( ( refl_on_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
=> ( member_b @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_240_refl__onD2,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A2 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
=> ( member1285940496od_b_b @ Y4 @ A2 ) ) ) ).
% refl_onD2
thf(fact_241_refl__onD2,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A2 @ R2 )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
=> ( member1516365892od_b_b @ Y4 @ A2 ) ) ) ).
% refl_onD2
thf(fact_242_refl__onD2,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
( ( refl_on_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
=> ( member_b @ Y4 @ A2 ) ) ) ).
% refl_onD2
thf(fact_243_refl__on__empty,axiom,
refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% refl_on_empty
thf(fact_244_refl__on__empty,axiom,
refl_o2134473190od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% refl_on_empty
thf(fact_245_is__singletonI_H,axiom,
! [A2: set_Pr1324126435od_b_b] :
( ( A2 != bot_bo1664927607od_b_b )
=> ( ! [X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X4 @ A2 )
=> ( ( member1516365892od_b_b @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_sin1465183929od_b_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_246_is__singletonI_H,axiom,
! [A2: set_b] :
( ( A2 != bot_bot_set_b )
=> ( ! [X4: b,Y3: b] :
( ( member_b @ X4 @ A2 )
=> ( ( member_b @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_247_is__singletonI_H,axiom,
! [A2: set_Product_prod_b_b] :
( ( A2 != bot_bo1343651123od_b_b )
=> ( ! [X4: product_prod_b_b,Y3: product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A2 )
=> ( ( member1285940496od_b_b @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_sin1713057243od_b_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_248_refl__on__singleton,axiom,
! [X: b] : ( refl_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% refl_on_singleton
thf(fact_249_refl__on__singleton,axiom,
! [X: product_prod_b_b] : ( refl_o2134473190od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% refl_on_singleton
thf(fact_250_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A6: set_b] :
? [X2: b] :
( A6
= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_251_is__singleton__def,axiom,
( is_sin1713057243od_b_b
= ( ^ [A6: set_Product_prod_b_b] :
? [X2: product_prod_b_b] :
( A6
= ( insert1952693431od_b_b @ X2 @ bot_bo1343651123od_b_b ) ) ) ) ).
% is_singleton_def
thf(fact_252_is__singletonE,axiom,
! [A2: set_b] :
( ( is_singleton_b @ A2 )
=> ~ ! [X4: b] :
( A2
!= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_253_is__singletonE,axiom,
! [A2: set_Product_prod_b_b] :
( ( is_sin1713057243od_b_b @ A2 )
=> ~ ! [X4: product_prod_b_b] :
( A2
!= ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) ) ) ).
% is_singletonE
thf(fact_254_equiv__class__eq__iff,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
( ( equiv_2048442231od_b_b @ A2 @ R2 )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
= ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ X @ bot_bo1664927607od_b_b ) )
= ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ Y4 @ bot_bo1664927607od_b_b ) ) )
& ( member1516365892od_b_b @ X @ A2 )
& ( member1516365892od_b_b @ Y4 @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_255_equiv__class__eq__iff,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
( ( equiv_equiv_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
= ( ( ( image_b_b @ R2 @ ( insert_b @ X @ bot_bot_set_b ) )
= ( image_b_b @ R2 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) )
& ( member_b @ X @ A2 )
& ( member_b @ Y4 @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_256_equiv__class__eq__iff,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A2 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
= ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
= ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) ) )
& ( member1285940496od_b_b @ X @ A2 )
& ( member1285940496od_b_b @ Y4 @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_257_eq__equiv__class__iff,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
( ( equiv_2048442231od_b_b @ A2 @ R2 )
=> ( ( member1516365892od_b_b @ X @ A2 )
=> ( ( member1516365892od_b_b @ Y4 @ A2 )
=> ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ X @ bot_bo1664927607od_b_b ) )
= ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ Y4 @ bot_bo1664927607od_b_b ) ) )
= ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_258_eq__equiv__class__iff,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
( ( equiv_equiv_b @ A2 @ R2 )
=> ( ( member_b @ X @ A2 )
=> ( ( member_b @ Y4 @ A2 )
=> ( ( ( image_b_b @ R2 @ ( insert_b @ X @ bot_bot_set_b ) )
= ( image_b_b @ R2 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) )
= ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_259_eq__equiv__class__iff,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ X @ A2 )
=> ( ( member1285940496od_b_b @ Y4 @ A2 )
=> ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
= ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) ) )
= ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_260_equiv__class__eq,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
( ( equiv_equiv_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ).
% equiv_class_eq
thf(fact_261_equiv__class__eq,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A2 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
=> ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
= ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% equiv_class_eq
thf(fact_262_eq__equiv__class,axiom,
! [R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
= ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ B @ bot_bo1664927607od_b_b ) ) )
=> ( ( equiv_2048442231od_b_b @ A2 @ R2 )
=> ( ( member1516365892od_b_b @ B @ A2 )
=> ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_263_eq__equiv__class,axiom,
! [R2: set_Product_prod_b_b,A: b,B: b,A2: set_b] :
( ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) )
=> ( ( equiv_equiv_b @ A2 @ R2 )
=> ( ( member_b @ B @ A2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_264_eq__equiv__class,axiom,
! [R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
= ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) )
=> ( ( equiv_1125628061od_b_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ B @ A2 )
=> ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_265_congruentI,axiom,
! [R2: set_Product_prod_b_b,F: b > set_b] :
( ! [Y3: b,Z3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z3 ) @ R2 )
=> ( ( F @ Y3 )
= ( F @ Z3 ) ) )
=> ( equiv_1821124534_set_b @ R2 @ F ) ) ).
% congruentI
thf(fact_266_congruentD,axiom,
! [R2: set_Product_prod_b_b,F: b > set_b,Y4: b,Z2: b] :
( ( equiv_1821124534_set_b @ R2 @ F )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z2 ) @ R2 )
=> ( ( F @ Y4 )
= ( F @ Z2 ) ) ) ) ).
% congruentD
thf(fact_267_equiv__class__self,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b] :
( ( equiv_2048442231od_b_b @ A2 @ R2 )
=> ( ( member1516365892od_b_b @ A @ A2 )
=> ( member1516365892od_b_b @ A @ ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) ) ) ) ) ).
% equiv_class_self
thf(fact_268_equiv__class__self,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,A: b] :
( ( equiv_equiv_b @ A2 @ R2 )
=> ( ( member_b @ A @ A2 )
=> ( member_b @ A @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ).
% equiv_class_self
thf(fact_269_equiv__class__self,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ A @ A2 )
=> ( member1285940496od_b_b @ A @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% equiv_class_self
thf(fact_270_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_271_Collect__empty__eq__bot,axiom,
! [P: product_prod_b_b > $o] :
( ( ( collec1481886546od_b_b @ P )
= bot_bo1343651123od_b_b )
= ( P = bot_bo1302443370_b_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_272_graph__single,axiom,
! [A: standard_Constant_a,B: b,C: b] :
( ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A @ ( product_Pair_b_b @ B @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B @ ( insert_b @ C @ bot_bot_set_b ) ) )
= ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A @ ( product_Pair_b_b @ B @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B @ ( insert_b @ C @ bot_bot_set_b ) ) ) ) ) ).
% graph_single
thf(fact_273_refl__on__domain,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
( ( refl_o2134473190od_b_b @ A2 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
=> ( ( member1285940496od_b_b @ A @ A2 )
& ( member1285940496od_b_b @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_274_refl__on__domain,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b,B: produc1213276845od_b_b] :
( ( refl_o1921098222od_b_b @ A2 @ R2 )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
=> ( ( member1516365892od_b_b @ A @ A2 )
& ( member1516365892od_b_b @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_275_refl__on__domain,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
( ( refl_on_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A2 )
& ( member_b @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_276_total__on__singleton,axiom,
! [X: b] : ( total_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% total_on_singleton
thf(fact_277_total__on__singleton,axiom,
! [X: product_prod_b_b] : ( total_178717291od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% total_on_singleton
thf(fact_278_total__on__converse,axiom,
! [A2: set_b,R2: set_Product_prod_b_b] :
( ( total_on_b @ A2 @ ( converse_b_b @ R2 ) )
= ( total_on_b @ A2 @ R2 ) ) ).
% total_on_converse
thf(fact_279_labeled__graph_Ocollapse,axiom,
! [Labeled_graph: labele1159362096nt_a_b] :
( ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) )
= Labeled_graph ) ).
% labeled_graph.collapse
thf(fact_280_graph__empty__e,axiom,
! [V2: set_b] :
( ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V2 )
= ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V2 ) ) ) ).
% graph_empty_e
thf(fact_281_total__on__def,axiom,
( total_on_b
= ( ^ [A6: set_b,R3: set_Product_prod_b_b] :
! [X2: b] :
( ( member_b @ X2 @ A6 )
=> ! [Y: b] :
( ( member_b @ Y @ A6 )
=> ( ( X2 != Y )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
| ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_282_total__onI,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
( ! [X4: product_prod_b_b,Y3: product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A2 )
=> ( ( member1285940496od_b_b @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y3 ) @ R2 )
| ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_178717291od_b_b @ A2 @ R2 ) ) ).
% total_onI
thf(fact_283_total__onI,axiom,
! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b] :
( ! [X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X4 @ A2 )
=> ( ( member1516365892od_b_b @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ Y3 ) @ R2 )
| ( member1086024932od_b_b @ ( produc449289715od_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_771970601od_b_b @ A2 @ R2 ) ) ).
% total_onI
thf(fact_284_total__onI,axiom,
! [A2: set_b,R2: set_Product_prod_b_b] :
( ! [X4: b,Y3: b] :
( ( member_b @ X4 @ A2 )
=> ( ( member_b @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 )
| ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_on_b @ A2 @ R2 ) ) ).
% total_onI
thf(fact_285_total__on__empty,axiom,
! [R2: set_Product_prod_b_b] : ( total_on_b @ bot_bot_set_b @ R2 ) ).
% total_on_empty
thf(fact_286_total__on__empty,axiom,
! [R2: set_Pr2007082183od_b_b] : ( total_178717291od_b_b @ bot_bo1343651123od_b_b @ R2 ) ).
% total_on_empty
thf(fact_287_labeled__graph_Osel_I2_J,axiom,
! [X1: set_Pr1324126435od_b_b,X22: set_b] :
( ( labele1424214014nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
= X22 ) ).
% labeled_graph.sel(2)
thf(fact_288_labeled__graph_Osel_I1_J,axiom,
! [X1: set_Pr1324126435od_b_b,X22: set_b] :
( ( labele1741081071nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
= X1 ) ).
% labeled_graph.sel(1)
thf(fact_289_labeled__graph_Oexhaust__sel,axiom,
! [Labeled_graph: labele1159362096nt_a_b] :
( Labeled_graph
= ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) ) ) ).
% labeled_graph.exhaust_sel
thf(fact_290_linear__order__on__singleton,axiom,
! [X: b] : ( order_653178932r_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% linear_order_on_singleton
thf(fact_291_linear__order__on__singleton,axiom,
! [X: product_prod_b_b] : ( order_609163036od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% linear_order_on_singleton
thf(fact_292_graph__homomorphism__empty,axiom,
! [G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
( ( graph_222606102_a_b_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ bot_bot_set_b ) @ G @ F )
= ( ( F = bot_bo1343651123od_b_b )
& ( G
= ( restri446606278nt_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_293_graph__homomorphism__empty,axiom,
! [G: labele1159362096nt_a_b,F: set_Pr357419743_b_b_b] :
( ( graph_792023598_b_b_b @ ( labele170595300od_b_b @ bot_bo1964903287od_b_b @ bot_bo1343651123od_b_b ) @ G @ F )
= ( ( F = bot_bo1124087115_b_b_b )
& ( G
= ( restri446606278nt_a_b @ G ) ) ) ) ).
% graph_homomorphism_empty
thf(fact_294_linear__order__on__converse,axiom,
! [A2: set_b,R2: set_Product_prod_b_b] :
( ( order_653178932r_on_b @ A2 @ ( converse_b_b @ R2 ) )
= ( order_653178932r_on_b @ A2 @ R2 ) ) ).
% linear_order_on_converse
thf(fact_295_edges__def,axiom,
( labele1741081071nt_a_b
= ( labele377702064od_b_b
@ ^ [X12: set_Pr1324126435od_b_b,X23: set_b] : X12 ) ) ).
% edges_def
thf(fact_296_vertices__def,axiom,
( labele1424214014nt_a_b
= ( labele1287464530_set_b
@ ^ [X12: set_Pr1324126435od_b_b,X23: set_b] : X23 ) ) ).
% vertices_def
thf(fact_297_lnear__order__on__empty,axiom,
order_653178932r_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% lnear_order_on_empty
thf(fact_298_lnear__order__on__empty,axiom,
order_609163036od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% lnear_order_on_empty
thf(fact_299_congruent2__implies__congruent,axiom,
! [A2: set_Product_prod_b_b,R1: set_Pr2007082183od_b_b,R22: set_Product_prod_b_b,F: product_prod_b_b > b > set_b,A: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A2 @ R1 )
=> ( ( equiv_1036235914_set_b @ R1 @ R22 @ F )
=> ( ( member1285940496od_b_b @ A @ A2 )
=> ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% congruent2_implies_congruent
thf(fact_300_congruent2__implies__congruent,axiom,
! [A2: set_Pr1324126435od_b_b,R1: set_Pr2109276827od_b_b,R22: set_Product_prod_b_b,F: produc1213276845od_b_b > b > set_b,A: produc1213276845od_b_b] :
( ( equiv_2048442231od_b_b @ A2 @ R1 )
=> ( ( equiv_1629197578_set_b @ R1 @ R22 @ F )
=> ( ( member1516365892od_b_b @ A @ A2 )
=> ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% congruent2_implies_congruent
thf(fact_301_congruent2__implies__congruent,axiom,
! [A2: set_b,R1: set_Product_prod_b_b,R22: set_Product_prod_b_b,F: b > b > set_b,A: b] :
( ( equiv_equiv_b @ A2 @ R1 )
=> ( ( equiv_167626658_set_b @ R1 @ R22 @ F )
=> ( ( member_b @ A @ A2 )
=> ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% congruent2_implies_congruent
thf(fact_302_proj__def,axiom,
( equiv_proj_b_b
= ( ^ [R3: set_Product_prod_b_b,X2: b] : ( image_b_b @ R3 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% proj_def
thf(fact_303_equiv__class__subset,axiom,
! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
( ( equiv_equiv_b @ A2 @ R2 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( ord_less_eq_set_b @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ).
% equiv_class_subset
thf(fact_304_equiv__class__subset,axiom,
! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
( ( equiv_1125628061od_b_b @ A2 @ R2 )
=> ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
=> ( ord_le1036320359od_b_b @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% equiv_class_subset
thf(fact_305_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_b @ X4 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_306_subsetI,axiom,
! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
( ! [X4: product_prod_b_b] :
( ( member1285940496od_b_b @ X4 @ A2 )
=> ( member1285940496od_b_b @ X4 @ B2 ) )
=> ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_307_subsetI,axiom,
! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
( ! [X4: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X4 @ A2 )
=> ( member1516365892od_b_b @ X4 @ B2 ) )
=> ( ord_le123089219od_b_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_308_converse__mono,axiom,
! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ ( converse_b_b @ R2 ) @ ( converse_b_b @ S2 ) )
= ( ord_le1036320359od_b_b @ R2 @ S2 ) ) ).
% converse_mono
thf(fact_309_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_310_empty__subsetI,axiom,
! [A2: set_Product_prod_b_b] : ( ord_le1036320359od_b_b @ bot_bo1343651123od_b_b @ A2 ) ).
% empty_subsetI
thf(fact_311_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_312_subset__empty,axiom,
! [A2: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A2 @ bot_bo1343651123od_b_b )
= ( A2 = bot_bo1343651123od_b_b ) ) ).
% subset_empty
thf(fact_313_insert__subset,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( ( member_b @ X @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_314_insert__subset,axiom,
! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X @ A2 ) @ B2 )
= ( ( member1285940496od_b_b @ X @ B2 )
& ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_315_insert__subset,axiom,
! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
( ( ord_le123089219od_b_b @ ( insert2037698781od_b_b @ X @ A2 ) @ B2 )
= ( ( member1516365892od_b_b @ X @ B2 )
& ( ord_le123089219od_b_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_316_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_317_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b] :
( ( ( insert1952693431od_b_b @ A @ A2 )
= ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
= ( ( A = B )
& ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_318_singleton__insert__inj__eq,axiom,
! [B: b,A: b,A2: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_319_singleton__insert__inj__eq,axiom,
! [B: product_prod_b_b,A: product_prod_b_b,A2: set_Product_prod_b_b] :
( ( ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b )
= ( insert1952693431od_b_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_320_Image__mono,axiom,
! [R4: set_Product_prod_b_b,R2: set_Product_prod_b_b,A7: set_b,A2: set_b] :
( ( ord_le1036320359od_b_b @ R4 @ R2 )
=> ( ( ord_less_eq_set_b @ A7 @ A2 )
=> ( ord_less_eq_set_b @ ( image_b_b @ R4 @ A7 ) @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% Image_mono
thf(fact_321_Collect__subset,axiom,
! [A2: set_Product_prod_b_b,P: product_prod_b_b > $o] :
( ord_le1036320359od_b_b
@ ( collec1481886546od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member1285940496od_b_b @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_322_Collect__subset,axiom,
! [A2: set_Pr1324126435od_b_b,P: produc1213276845od_b_b > $o] :
( ord_le123089219od_b_b
@ ( collec279084418od_b_b
@ ^ [X2: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_323_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_324_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X2: b] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_325_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_326_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
! [T: b] :
( ( member_b @ T @ A6 )
=> ( member_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_327_subset__iff,axiom,
( ord_le1036320359od_b_b
= ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
! [T: product_prod_b_b] :
( ( member1285940496od_b_b @ T @ A6 )
=> ( member1285940496od_b_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_328_subset__iff,axiom,
( ord_le123089219od_b_b
= ( ^ [A6: set_Pr1324126435od_b_b,B5: set_Pr1324126435od_b_b] :
! [T: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ T @ A6 )
=> ( member1516365892od_b_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_329_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A6 )
=> ( member_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_330_subset__eq,axiom,
( ord_le1036320359od_b_b
= ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
! [X2: product_prod_b_b] :
( ( member1285940496od_b_b @ X2 @ A6 )
=> ( member1285940496od_b_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_331_subset__eq,axiom,
( ord_le123089219od_b_b
= ( ^ [A6: set_Pr1324126435od_b_b,B5: set_Pr1324126435od_b_b] :
! [X2: produc1213276845od_b_b] :
( ( member1516365892od_b_b @ X2 @ A6 )
=> ( member1516365892od_b_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_332_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_333_subsetD,axiom,
! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b,C: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A2 @ B2 )
=> ( ( member1285940496od_b_b @ C @ A2 )
=> ( member1285940496od_b_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_334_subsetD,axiom,
! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b,C: produc1213276845od_b_b] :
( ( ord_le123089219od_b_b @ A2 @ B2 )
=> ( ( member1516365892od_b_b @ C @ A2 )
=> ( member1516365892od_b_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_335_in__mono,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_336_in__mono,axiom,
! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b,X: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A2 @ B2 )
=> ( ( member1285940496od_b_b @ X @ A2 )
=> ( member1285940496od_b_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_337_in__mono,axiom,
! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b,X: produc1213276845od_b_b] :
( ( ord_le123089219od_b_b @ A2 @ B2 )
=> ( ( member1516365892od_b_b @ X @ A2 )
=> ( member1516365892od_b_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_338_converse__subset__swap,axiom,
! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ R2 @ ( converse_b_b @ S2 ) )
= ( ord_le1036320359od_b_b @ ( converse_b_b @ R2 ) @ S2 ) ) ).
% converse_subset_swap
thf(fact_339_subrelI,axiom,
! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ! [X4: b,Y3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ S2 ) )
=> ( ord_le1036320359od_b_b @ R2 @ S2 ) ) ).
% subrelI
thf(fact_340_subrelI,axiom,
! [R2: set_Pr1324126435od_b_b,S2: set_Pr1324126435od_b_b] :
( ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R2 )
=> ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ S2 ) )
=> ( ord_le123089219od_b_b @ R2 @ S2 ) ) ).
% subrelI
thf(fact_341_bot_Oextremum__uniqueI,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
=> ( A = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_342_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A @ bot_bo1343651123od_b_b )
=> ( A = bot_bo1343651123od_b_b ) ) ).
% bot.extremum_uniqueI
thf(fact_343_bot_Oextremum__unique,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_344_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_b_b] :
( ( ord_le1036320359od_b_b @ A @ bot_bo1343651123od_b_b )
= ( A = bot_bo1343651123od_b_b ) ) ).
% bot.extremum_unique
thf(fact_345_bot_Oextremum,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% bot.extremum
thf(fact_346_bot_Oextremum,axiom,
! [A: set_Product_prod_b_b] : ( ord_le1036320359od_b_b @ bot_bo1343651123od_b_b @ A ) ).
% bot.extremum
thf(fact_347_insert__mono,axiom,
! [C4: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C4 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C4 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_348_subset__insert,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_349_subset__insert,axiom,
! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
( ~ ( member1285940496od_b_b @ X @ A2 )
=> ( ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ X @ B2 ) )
= ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_350_subset__insert,axiom,
! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
( ~ ( member1516365892od_b_b @ X @ A2 )
=> ( ( ord_le123089219od_b_b @ A2 @ ( insert2037698781od_b_b @ X @ B2 ) )
= ( ord_le123089219od_b_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_351_subset__insertI,axiom,
! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% subset_insertI
thf(fact_352_subset__insertI2,axiom,
! [A2: set_b,B2: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_353_subset__singleton__iff,axiom,
! [X3: set_b,A: b] :
( ( ord_less_eq_set_b @ X3 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X3 = bot_bot_set_b )
| ( X3
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_354_subset__singleton__iff,axiom,
! [X3: set_Product_prod_b_b,A: product_prod_b_b] :
( ( ord_le1036320359od_b_b @ X3 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
= ( ( X3 = bot_bo1343651123od_b_b )
| ( X3
= ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ) ) ).
% subset_singleton_iff
% Helper facts (3)
thf(help_If_3_1_If_001tf__b_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X: b,Y4: b] :
( ( if_b @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X: b,Y4: b] :
( ( if_b @ $true @ X @ Y4 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( hilbert_Eps_b
@ ^ [Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
= ( hilbert_Eps_b
@ ^ [Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ) ) ).
%------------------------------------------------------------------------------