TPTP Problem File: ITP157^1.p
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%------------------------------------------------------------------------------
% File : ITP157^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_86__6247862_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 : Preferences/prob_86__6247862_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 527 ( 166 unt; 171 typ; 0 def)
% Number of atoms : 982 ( 312 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3250 ( 124 ~; 4 |; 110 &;2654 @)
% ( 0 <=>; 358 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Number of types : 31 ( 30 usr)
% Number of type conns : 530 ( 530 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 141 usr; 10 con; 0-3 aty)
% Number of variables : 1291 ( 171 ^;1096 !; 24 ?;1291 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:43:15.680
%------------------------------------------------------------------------------
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collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_236793797od_a_a: ( produc1572603623od_a_a > produc1572603623od_a_a ) > set_Pr1948701895od_a_a > set_Pr1948701895od_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_734753285od_a_a: ( produc1572603623od_a_a > product_prod_a_a ) > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001tf__a,type,
image_1105229276_a_a_a: ( produc1572603623od_a_a > a ) > set_Pr1948701895od_a_a > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_1765377413od_a_a: ( product_prod_a_a > produc1572603623od_a_a ) > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_794773445od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
image_340790172_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_1763923566od_a_a: ( a > produc1572603623od_a_a ) > set_a > set_Pr1948701895od_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_1725938350od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
lex_pr942790511od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
lex_pr2084296006_a_a_a: set_Pr1948701895od_a_a > set_Product_prod_a_a > set_Pr729215751_a_a_a ).
thf(sy_c_Wellfounded_Olex__prod_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
lex_pr1321960536od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Pr231806599od_a_a ).
thf(sy_c_Wellfounded_Olex__prod_001tf__a_001tf__a,type,
lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
same_f1138477100od_a_a: ( product_prod_a_a > $o ) > ( product_prod_a_a > set_Pr1948701895od_a_a ) > set_Pr1295299783od_a_a ).
thf(sy_c_Wfrec_Osame__fst_001tf__a_001tf__a,type,
same_fst_a_a: ( a > $o ) > ( a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member180906768od_a_a: produc1056976615od_a_a > set_Pr532616391od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member547056400od_a_a: produc1423126247od_a_a > set_Pr2081559751od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1545452880od_a_a: produc47586087od_a_a > set_Pr173461255od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
member575366567_a_a_a: produc1683075710_a_a_a > set_Pr6137822_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
member470802768_a_a_a: produc1120419623_a_a_a > set_Pr729215751_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member1424209616od_a_a: produc1546279079od_a_a > set_Pr660101255od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1361298640od_a_a: produc2010915495od_a_a > set_Pr231806599od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member122808528od_a_a: produc772425383od_a_a > set_Pr1194146439od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
member1403138919_a_a_a: produc1016684094_a_a_a > set_Pr1689873822_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member448091577od_a_a: produc1555800720od_a_a > set_Pr195866352od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member160619001od_a_a: produc1921647824od_a_a > set_Pr681306928od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_Pr,type,
pr: set_Product_prod_a_a ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_z,type,
z: a ).
% Relevant facts (355)
thf(fact_0_assms,axiom,
member_a @ z @ ( prefer161218362good_a @ y @ b @ pr ) ).
% assms
thf(fact_1_at__lst__asgd__ge,axiom,
! [X: produc2010915495od_a_a,Y: produc2010915495od_a_a,B: set_Pr231806599od_a_a,Pr: set_Pr532616391od_a_a] :
( ( member1361298640od_a_a @ X @ ( prefer468349667od_a_a @ Y @ B @ Pr ) )
=> ( member180906768od_a_a @ ( produc539587415od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_2_at__lst__asgd__ge,axiom,
! [X: produc772425383od_a_a,Y: produc772425383od_a_a,B: set_Pr1194146439od_a_a,Pr: set_Pr2081559751od_a_a] :
( ( member122808528od_a_a @ X @ ( prefer1377343203od_a_a @ Y @ B @ Pr ) )
=> ( member547056400od_a_a @ ( produc852390743od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_3_at__lst__asgd__ge,axiom,
! [X: produc1016684094_a_a_a,Y: produc1016684094_a_a_a,B: set_Pr1689873822_a_a_a,Pr: set_Pr729215751_a_a_a] :
( ( member1403138919_a_a_a @ X @ ( prefer1944486522_a_a_a @ Y @ B @ Pr ) )
=> ( member470802768_a_a_a @ ( produc2054708503_a_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_4_at__lst__asgd__ge,axiom,
! [X: produc1555800720od_a_a,Y: produc1555800720od_a_a,B: set_Pr195866352od_a_a,Pr: set_Pr660101255od_a_a] :
( ( member448091577od_a_a @ X @ ( prefer445484748od_a_a @ Y @ B @ Pr ) )
=> ( member1424209616od_a_a @ ( produc1573596055od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_5_at__lst__asgd__ge,axiom,
! [X: produc1921647824od_a_a,Y: produc1921647824od_a_a,B: set_Pr681306928od_a_a,Pr: set_Pr231806599od_a_a] :
( ( member160619001od_a_a @ X @ ( prefer701966604od_a_a @ Y @ B @ Pr ) )
=> ( member1361298640od_a_a @ ( produc563767703od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_6_at__lst__asgd__ge,axiom,
! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_7_at__lst__asgd__ge,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,B: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_8_at__lst__asgd__ge,axiom,
! [X: a,Y: a,B: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ X @ ( prefer161218362good_a @ Y @ B @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_9_prod_Oinject,axiom,
! [X1: produc1921647824od_a_a,X2: produc1921647824od_a_a,Y1: produc1921647824od_a_a,Y2: produc1921647824od_a_a] :
( ( ( produc563767703od_a_a @ X1 @ X2 )
= ( produc563767703od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: product_prod_a_a,X2: produc1572603623od_a_a,Y1: product_prod_a_a,Y2: produc1572603623od_a_a] :
( ( ( produc1555606807od_a_a @ X1 @ X2 )
= ( produc1555606807od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: product_prod_a_a,X2: a,Y1: product_prod_a_a,Y2: a] :
( ( ( produc2061588782_a_a_a @ X1 @ X2 )
= ( produc2061588782_a_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_prod_Oinject,axiom,
! [X1: a,X2: produc1572603623od_a_a,Y1: a,Y2: produc1572603623od_a_a] :
( ( ( produc1095491584od_a_a @ X1 @ X2 )
= ( produc1095491584od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: a,X2: product_prod_a_a,Y1: a,Y2: product_prod_a_a] :
( ( ( produc1299253312od_a_a @ X1 @ X2 )
= ( produc1299253312od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: product_prod_a_a,X2: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ X1 @ X2 )
= ( produc1474507607od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: produc1921647824od_a_a,B2: produc1921647824od_a_a,A2: produc1921647824od_a_a,B3: produc1921647824od_a_a] :
( ( ( produc563767703od_a_a @ A @ B2 )
= ( produc563767703od_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a,B2: produc1572603623od_a_a,A2: product_prod_a_a,B3: produc1572603623od_a_a] :
( ( ( produc1555606807od_a_a @ A @ B2 )
= ( produc1555606807od_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a] :
( ( ( produc2061588782_a_a_a @ A @ B2 )
= ( produc2061588782_a_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A: a,B2: produc1572603623od_a_a,A2: a,B3: produc1572603623od_a_a] :
( ( ( produc1095491584od_a_a @ A @ B2 )
= ( produc1095491584od_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a] :
( ( ( produc1299253312od_a_a @ A @ B2 )
= ( produc1299253312od_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ A @ B2 )
= ( produc1474507607od_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A: a,B2: a,A2: a,B3: a] :
( ( ( product_Pair_a_a @ A @ B2 )
= ( product_Pair_a_a @ A2 @ B3 ) )
= ( ( A = A2 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_23_surj__pair,axiom,
! [P: produc2010915495od_a_a] :
? [X3: produc1921647824od_a_a,Y3: produc1921647824od_a_a] :
( P
= ( produc563767703od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_24_surj__pair,axiom,
! [P: produc772425383od_a_a] :
? [X3: product_prod_a_a,Y3: produc1572603623od_a_a] :
( P
= ( produc1555606807od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_25_surj__pair,axiom,
! [P: produc1016684094_a_a_a] :
? [X3: product_prod_a_a,Y3: a] :
( P
= ( produc2061588782_a_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_26_surj__pair,axiom,
! [P: produc1555800720od_a_a] :
? [X3: a,Y3: produc1572603623od_a_a] :
( P
= ( produc1095491584od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_27_surj__pair,axiom,
! [P: produc1921647824od_a_a] :
? [X3: a,Y3: product_prod_a_a] :
( P
= ( produc1299253312od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_28_surj__pair,axiom,
! [P: produc1572603623od_a_a] :
? [X3: product_prod_a_a,Y3: product_prod_a_a] :
( P
= ( produc1474507607od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_29_surj__pair,axiom,
! [P: product_prod_a_a] :
? [X3: a,Y3: a] :
( P
= ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_30_prod__cases,axiom,
! [P2: produc2010915495od_a_a > $o,P: produc2010915495od_a_a] :
( ! [A3: produc1921647824od_a_a,B4: produc1921647824od_a_a] : ( P2 @ ( produc563767703od_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_31_prod__cases,axiom,
! [P2: produc772425383od_a_a > $o,P: produc772425383od_a_a] :
( ! [A3: product_prod_a_a,B4: produc1572603623od_a_a] : ( P2 @ ( produc1555606807od_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_32_prod__cases,axiom,
! [P2: produc1016684094_a_a_a > $o,P: produc1016684094_a_a_a] :
( ! [A3: product_prod_a_a,B4: a] : ( P2 @ ( produc2061588782_a_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_33_prod__cases,axiom,
! [P2: produc1555800720od_a_a > $o,P: produc1555800720od_a_a] :
( ! [A3: a,B4: produc1572603623od_a_a] : ( P2 @ ( produc1095491584od_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_34_prod__cases,axiom,
! [P2: produc1921647824od_a_a > $o,P: produc1921647824od_a_a] :
( ! [A3: a,B4: product_prod_a_a] : ( P2 @ ( produc1299253312od_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_35_prod__cases,axiom,
! [P2: produc1572603623od_a_a > $o,P: produc1572603623od_a_a] :
( ! [A3: product_prod_a_a,B4: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_36_prod__cases,axiom,
! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
( ! [A3: a,B4: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_37_Pair__inject,axiom,
! [A: produc1921647824od_a_a,B2: produc1921647824od_a_a,A2: produc1921647824od_a_a,B3: produc1921647824od_a_a] :
( ( ( produc563767703od_a_a @ A @ B2 )
= ( produc563767703od_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_38_Pair__inject,axiom,
! [A: product_prod_a_a,B2: produc1572603623od_a_a,A2: product_prod_a_a,B3: produc1572603623od_a_a] :
( ( ( produc1555606807od_a_a @ A @ B2 )
= ( produc1555606807od_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_39_Pair__inject,axiom,
! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a] :
( ( ( produc2061588782_a_a_a @ A @ B2 )
= ( produc2061588782_a_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_40_Pair__inject,axiom,
! [A: a,B2: produc1572603623od_a_a,A2: a,B3: produc1572603623od_a_a] :
( ( ( produc1095491584od_a_a @ A @ B2 )
= ( produc1095491584od_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_41_Pair__inject,axiom,
! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a] :
( ( ( produc1299253312od_a_a @ A @ B2 )
= ( produc1299253312od_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_42_Pair__inject,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ A @ B2 )
= ( produc1474507607od_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_43_Pair__inject,axiom,
! [A: a,B2: a,A2: a,B3: a] :
( ( ( product_Pair_a_a @ A @ B2 )
= ( product_Pair_a_a @ A2 @ B3 ) )
=> ~ ( ( A = A2 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_44_prod__cases3,axiom,
! [Y: produc772425383od_a_a] :
~ ! [A3: product_prod_a_a,B4: product_prod_a_a,C: product_prod_a_a] :
( Y
!= ( produc1555606807od_a_a @ A3 @ ( produc1474507607od_a_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_45_prod__cases3,axiom,
! [Y: produc1555800720od_a_a] :
~ ! [A3: a,B4: product_prod_a_a,C: product_prod_a_a] :
( Y
!= ( produc1095491584od_a_a @ A3 @ ( produc1474507607od_a_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_46_prod__cases3,axiom,
! [Y: produc1921647824od_a_a] :
~ ! [A3: a,B4: a,C: a] :
( Y
!= ( produc1299253312od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_47_prod__cases3,axiom,
! [Y: produc1572603623od_a_a] :
~ ! [A3: product_prod_a_a,B4: a,C: a] :
( Y
!= ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_48_prod__induct3,axiom,
! [P2: produc1572603623od_a_a > $o,X: produc1572603623od_a_a] :
( ! [A3: product_prod_a_a,B4: a,C: a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_49_old_Oprod_Oinducts,axiom,
! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A3: a,B4: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_50_old_Oprod_Oinducts,axiom,
! [P2: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
( ! [A3: product_prod_a_a,B4: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_51_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A3: a,B4: a] :
( Y
!= ( product_Pair_a_a @ A3 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [Y: produc1572603623od_a_a] :
~ ! [A3: product_prod_a_a,B4: product_prod_a_a] :
( Y
!= ( produc1474507607od_a_a @ A3 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_53_at__least__as__good__def,axiom,
( prefer55414563od_a_a
= ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
( collec1635618130od_a_a
@ ^ [Y4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ Y4 @ B5 )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% at_least_as_good_def
thf(fact_54_at__least__as__good__def,axiom,
( prefer161218362good_a
= ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B5 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% at_least_as_good_def
thf(fact_55_at__least__as__good__def,axiom,
( prefer477097315od_a_a
= ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
( collec645855634od_a_a
@ ^ [Y4: product_prod_a_a] :
( ( member449909584od_a_a @ Y4 @ B5 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% at_least_as_good_def
thf(fact_56_ssubst__Pair__rhs,axiom,
! [R: a,S: a,R2: set_Product_prod_a_a,S2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_57_ssubst__Pair__rhs,axiom,
! [R: product_prod_a_a,S: product_prod_a_a,R2: set_Pr1948701895od_a_a,S2: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_58_curryI,axiom,
! [F: product_prod_a_a > $o,A: a,B2: a] :
( ( F @ ( product_Pair_a_a @ A @ B2 ) )
=> ( product_curry_a_a_o @ F @ A @ B2 ) ) ).
% curryI
thf(fact_59_curryI,axiom,
! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
( ( F @ ( produc1474507607od_a_a @ A @ B2 ) )
=> ( produc94438221_a_a_o @ F @ A @ B2 ) ) ).
% curryI
thf(fact_60_preference_Oindiff__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_61_preference_Oindiff__trans,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_62_preference_Onot__outside,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1409946857od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
=> ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_63_preference_Onot__outside,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( member_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_64_preference_Onot__outside,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_65_swap__simp,axiom,
! [X: a,Y: a] :
( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
= ( product_Pair_a_a @ Y @ X ) ) ).
% swap_simp
thf(fact_66_swap__simp,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a] :
( ( produc1641853175od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
= ( produc1474507607od_a_a @ Y @ X ) ) ).
% swap_simp
thf(fact_67_curryD,axiom,
! [F: product_prod_a_a > $o,A: a,B2: a] :
( ( product_curry_a_a_o @ F @ A @ B2 )
=> ( F @ ( product_Pair_a_a @ A @ B2 ) ) ) ).
% curryD
thf(fact_68_curryD,axiom,
! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
( ( produc94438221_a_a_o @ F @ A @ B2 )
=> ( F @ ( produc1474507607od_a_a @ A @ B2 ) ) ) ).
% curryD
thf(fact_69_curryE,axiom,
! [F: product_prod_a_a > $o,A: a,B2: a] :
( ( product_curry_a_a_o @ F @ A @ B2 )
=> ( F @ ( product_Pair_a_a @ A @ B2 ) ) ) ).
% curryE
thf(fact_70_curryE,axiom,
! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
( ( produc94438221_a_a_o @ F @ A @ B2 )
=> ( F @ ( produc1474507607od_a_a @ A @ B2 ) ) ) ).
% curryE
thf(fact_71_no__better__than__def,axiom,
( prefer926864306od_a_a
= ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
( collec1635618130od_a_a
@ ^ [Y4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ Y4 @ B5 )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% no_better_than_def
thf(fact_72_no__better__than__def,axiom,
( prefer1676310729than_a
= ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B5 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% no_better_than_def
thf(fact_73_no__better__than__def,axiom,
( prefer1498087410od_a_a
= ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
( collec645855634od_a_a
@ ^ [Y4: product_prod_a_a] :
( ( member449909584od_a_a @ Y4 @ B5 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% no_better_than_def
thf(fact_74_as__good__as__def,axiom,
( prefer1867335523od_a_a
= ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
( collec1635618130od_a_a
@ ^ [Y4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ Y4 @ B5 )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ P3 )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% as_good_as_def
thf(fact_75_as__good__as__def,axiom,
( prefer436369274d_as_a
= ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B5 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ P3 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% as_good_as_def
thf(fact_76_as__good__as__def,axiom,
( prefer74720675od_a_a
= ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
( collec645855634od_a_a
@ ^ [Y4: product_prod_a_a] :
( ( member449909584od_a_a @ Y4 @ B5 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ P3 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% as_good_as_def
thf(fact_77_pred__equals__eq2,axiom,
! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S3 ) ) )
= ( R2 = S3 ) ) ).
% pred_equals_eq2
thf(fact_78_pred__equals__eq2,axiom,
! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
( ( ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ S3 ) ) )
= ( R2 = S3 ) ) ).
% pred_equals_eq2
thf(fact_79_preference__def,axiom,
( prefer1409946857od_a_a
= ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
( ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member2057358096od_a_a @ X4 @ Carrier2 ) )
& ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member2057358096od_a_a @ Y4 @ Carrier2 ) )
& ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_80_preference__def,axiom,
( prefer1292084992ence_a
= ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
( ! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member_a @ X4 @ Carrier2 ) )
& ! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member_a @ Y4 @ Carrier2 ) )
& ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_81_preference__def,axiom,
( prefer1338892073od_a_a
= ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
( ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member449909584od_a_a @ X4 @ Carrier2 ) )
& ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member449909584od_a_a @ Y4 @ Carrier2 ) )
& ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_82_preference_Ointro,axiom,
! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member2057358096od_a_a @ X3 @ Carrier ) )
=> ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member2057358096od_a_a @ Y3 @ Carrier ) )
=> ( ( order_491428350od_a_a @ Carrier @ Relation )
=> ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_83_preference_Ointro,axiom,
! [Relation: set_Product_prod_a_a,Carrier: set_a] :
( ! [X3: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
=> ( member_a @ X3 @ Carrier ) )
=> ( ! [X3: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
=> ( member_a @ Y3 @ Carrier ) )
=> ( ( order_preorder_on_a @ Carrier @ Relation )
=> ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_84_preference_Ointro,axiom,
! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member449909584od_a_a @ X3 @ Carrier ) )
=> ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member449909584od_a_a @ Y3 @ Carrier ) )
=> ( ( order_359341630od_a_a @ Carrier @ Relation )
=> ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_85_pair__in__swap__image,axiom,
! [Y: a,X: a,A4: set_Product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( image_794773445od_a_a @ product_swap_a_a @ A4 ) )
= ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_86_pair__in__swap__image,axiom,
! [Y: product_prod_a_a,X: product_prod_a_a,A4: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ ( image_236793797od_a_a @ produc1641853175od_a_a @ A4 ) )
= ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_87_strict__contour__is__diff,axiom,
! [B: set_Pr1948701895od_a_a,Y: produc1572603623od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( collec1635618130od_a_a
@ ^ [A5: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ A5 @ B )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A5 @ Y ) @ Pr )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ A5 ) @ Pr ) ) )
= ( minus_1034916142od_a_a @ ( prefer55414563od_a_a @ Y @ B @ Pr ) @ ( prefer1867335523od_a_a @ Y @ B @ Pr ) ) ) ).
% strict_contour_is_diff
thf(fact_88_strict__contour__is__diff,axiom,
! [B: set_a,Y: a,Pr: set_Product_prod_a_a] :
( ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ B )
& ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ Y ) @ Pr )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ A5 ) @ Pr ) ) )
= ( minus_minus_set_a @ ( prefer161218362good_a @ Y @ B @ Pr ) @ ( prefer436369274d_as_a @ Y @ B @ Pr ) ) ) ).
% strict_contour_is_diff
thf(fact_89_strict__contour__is__diff,axiom,
! [B: set_Product_prod_a_a,Y: product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( collec645855634od_a_a
@ ^ [A5: product_prod_a_a] :
( ( member449909584od_a_a @ A5 @ B )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A5 @ Y ) @ Pr )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ A5 ) @ Pr ) ) )
= ( minus_1507102574od_a_a @ ( prefer477097315od_a_a @ Y @ B @ Pr ) @ ( prefer74720675od_a_a @ Y @ B @ Pr ) ) ) ).
% strict_contour_is_diff
thf(fact_90_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A: produc1572603623od_a_a,P2: produc1572603623od_a_a > $o] :
( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A4: set_Pr1948701895od_a_a] :
( ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_96_DiffI,axiom,
! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ A4 )
=> ( ~ ( member449909584od_a_a @ C2 @ B )
=> ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_97_DiffI,axiom,
! [C2: a,A4: set_a,B: set_a] :
( ( member_a @ C2 @ A4 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_98_DiffI,axiom,
! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ C2 @ A4 )
=> ( ~ ( member2057358096od_a_a @ C2 @ B )
=> ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_99_image__eqI,axiom,
! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( member449909584od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_100_image__eqI,axiom,
! [B2: a,F: product_prod_a_a > a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( member449909584od_a_a @ X @ A4 )
=> ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_101_image__eqI,axiom,
! [B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( member449909584od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_102_image__eqI,axiom,
! [B2: product_prod_a_a,F: a > product_prod_a_a,X: a,A4: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A4 )
=> ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_103_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A4: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A4 )
=> ( member_a @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_104_image__eqI,axiom,
! [B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a,X: a,A4: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A4 )
=> ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_105_image__eqI,axiom,
! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( member2057358096od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_106_image__eqI,axiom,
! [B2: a,F: produc1572603623od_a_a > a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( member2057358096od_a_a @ X @ A4 )
=> ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_107_image__eqI,axiom,
! [B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( member2057358096od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_108_Diff__iff,axiom,
! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
= ( ( member449909584od_a_a @ C2 @ A4 )
& ~ ( member449909584od_a_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_109_Diff__iff,axiom,
! [C2: a,A4: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
= ( ( member_a @ C2 @ A4 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_110_Diff__iff,axiom,
! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
= ( ( member2057358096od_a_a @ C2 @ A4 )
& ~ ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_111_minus__set__def,axiom,
( minus_1507102574od_a_a
= ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( collec645855634od_a_a
@ ( minus_1797478127_a_a_o
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A6 )
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_112_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B5: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_113_minus__set__def,axiom,
( minus_1034916142od_a_a
= ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
( collec1635618130od_a_a
@ ( minus_107310383_a_a_o
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A6 )
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_114_rev__image__eqI,axiom,
! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member449909584od_a_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_115_rev__image__eqI,axiom,
! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: a,F: product_prod_a_a > a] :
( ( member449909584od_a_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_116_rev__image__eqI,axiom,
! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a] :
( ( member449909584od_a_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_117_rev__image__eqI,axiom,
! [X: a,A4: set_a,B2: product_prod_a_a,F: a > product_prod_a_a] :
( ( member_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_118_rev__image__eqI,axiom,
! [X: a,A4: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_119_rev__image__eqI,axiom,
! [X: a,A4: set_a,B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a] :
( ( member_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_120_rev__image__eqI,axiom,
! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a] :
( ( member2057358096od_a_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_121_rev__image__eqI,axiom,
! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: a,F: produc1572603623od_a_a > a] :
( ( member2057358096od_a_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_122_rev__image__eqI,axiom,
! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_123_imageI,axiom,
! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member449909584od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ ( F @ X ) @ ( image_794773445od_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_124_imageI,axiom,
! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > a] :
( ( member449909584od_a_a @ X @ A4 )
=> ( member_a @ ( F @ X ) @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_125_imageI,axiom,
! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > produc1572603623od_a_a] :
( ( member449909584od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ ( F @ X ) @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_126_imageI,axiom,
! [X: a,A4: set_a,F: a > product_prod_a_a] :
( ( member_a @ X @ A4 )
=> ( member449909584od_a_a @ ( F @ X ) @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_127_imageI,axiom,
! [X: a,A4: set_a,F: a > a] :
( ( member_a @ X @ A4 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_128_imageI,axiom,
! [X: a,A4: set_a,F: a > produc1572603623od_a_a] :
( ( member_a @ X @ A4 )
=> ( member2057358096od_a_a @ ( F @ X ) @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_129_imageI,axiom,
! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > product_prod_a_a] :
( ( member2057358096od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ ( F @ X ) @ ( image_734753285od_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_130_imageI,axiom,
! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > a] :
( ( member2057358096od_a_a @ X @ A4 )
=> ( member_a @ ( F @ X ) @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_131_imageI,axiom,
! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ ( F @ X ) @ ( image_236793797od_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_132_DiffD2,axiom,
! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
=> ~ ( member449909584od_a_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_133_DiffD2,axiom,
! [C2: a,A4: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_134_DiffD2,axiom,
! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
=> ~ ( member2057358096od_a_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_135_DiffD1,axiom,
! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
=> ( member449909584od_a_a @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_136_DiffD1,axiom,
! [C2: a,A4: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
=> ( member_a @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_137_DiffD1,axiom,
! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
=> ( member2057358096od_a_a @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_138_DiffE,axiom,
! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
=> ~ ( ( member449909584od_a_a @ C2 @ A4 )
=> ( member449909584od_a_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_139_DiffE,axiom,
! [C2: a,A4: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
=> ~ ( ( member_a @ C2 @ A4 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_140_DiffE,axiom,
! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
=> ~ ( ( member2057358096od_a_a @ C2 @ A4 )
=> ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_141_imageE,axiom,
! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a] :
( ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) )
=> ~ ! [X3: product_prod_a_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_142_imageE,axiom,
! [B2: product_prod_a_a,F: a > product_prod_a_a,A4: set_a] :
( ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_143_imageE,axiom,
! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) )
=> ~ ! [X3: produc1572603623od_a_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_144_imageE,axiom,
! [B2: a,F: product_prod_a_a > a,A4: set_Product_prod_a_a] :
( ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) )
=> ~ ! [X3: product_prod_a_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_145_imageE,axiom,
! [B2: a,F: a > a,A4: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F @ A4 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_146_imageE,axiom,
! [B2: a,F: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a] :
( ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) )
=> ~ ! [X3: produc1572603623od_a_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_147_imageE,axiom,
! [B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a,A4: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) )
=> ~ ! [X3: product_prod_a_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_148_imageE,axiom,
! [B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a,A4: set_a] :
( ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_149_imageE,axiom,
! [B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) )
=> ~ ! [X3: produc1572603623od_a_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_150_Compr__image__eq,axiom,
! [F: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ ( image_794773445od_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_794773445od_a_a @ F
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_151_Compr__image__eq,axiom,
! [F: a > product_prod_a_a,A4: set_a,P2: product_prod_a_a > $o] :
( ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ ( image_1725938350od_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_1725938350od_a_a @ F
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_152_Compr__image__eq,axiom,
! [F: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a,P2: product_prod_a_a > $o] :
( ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ ( image_734753285od_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_734753285od_a_a @ F
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_153_Compr__image__eq,axiom,
! [F: product_prod_a_a > a,A4: set_Product_prod_a_a,P2: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ ( image_340790172_a_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_340790172_a_a_a @ F
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_154_Compr__image__eq,axiom,
! [F: a > a,A4: set_a,P2: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_155_Compr__image__eq,axiom,
! [F: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a,P2: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ ( image_1105229276_a_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_1105229276_a_a_a @ F
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_156_Compr__image__eq,axiom,
! [F: product_prod_a_a > produc1572603623od_a_a,A4: set_Product_prod_a_a,P2: produc1572603623od_a_a > $o] :
( ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ ( image_1765377413od_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_1765377413od_a_a @ F
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_157_Compr__image__eq,axiom,
! [F: a > produc1572603623od_a_a,A4: set_a,P2: produc1572603623od_a_a > $o] :
( ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ ( image_1763923566od_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_1763923566od_a_a @ F
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_158_Compr__image__eq,axiom,
! [F: produc1572603623od_a_a > produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
( ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ ( image_236793797od_a_a @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_236793797od_a_a @ F
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_159_set__diff__eq,axiom,
( minus_1507102574od_a_a
= ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A6 )
& ~ ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_160_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B5: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A6 )
& ~ ( member_a @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_161_set__diff__eq,axiom,
( minus_1034916142od_a_a
= ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A6 )
& ~ ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_162_UNIV__I,axiom,
! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ top_to398455383od_a_a ) ).
% UNIV_I
thf(fact_163_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_164_UNIV__I,axiom,
! [X: produc1572603623od_a_a] : ( member2057358096od_a_a @ X @ top_to52711959od_a_a ) ).
% UNIV_I
thf(fact_165_UNIV__witness,axiom,
? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ top_to398455383od_a_a ) ).
% UNIV_witness
thf(fact_166_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_167_UNIV__witness,axiom,
? [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ top_to52711959od_a_a ) ).
% UNIV_witness
thf(fact_168_UNIV__eq__I,axiom,
! [A4: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A4 )
=> ( top_to398455383od_a_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_169_UNIV__eq__I,axiom,
! [A4: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A4 )
=> ( top_top_set_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_170_UNIV__eq__I,axiom,
! [A4: set_Pr1948701895od_a_a] :
( ! [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A4 )
=> ( top_to52711959od_a_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_171_iso__tuple__UNIV__I,axiom,
! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ top_to398455383od_a_a ) ).
% iso_tuple_UNIV_I
thf(fact_172_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_173_iso__tuple__UNIV__I,axiom,
! [X: produc1572603623od_a_a] : ( member2057358096od_a_a @ X @ top_to52711959od_a_a ) ).
% iso_tuple_UNIV_I
thf(fact_174_same__fstI,axiom,
! [P2: a > $o,X: a,Y5: a,Y: a,R2: a > set_Product_prod_a_a] :
( ( P2 @ X )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y5 @ Y ) @ ( R2 @ X ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ ( product_Pair_a_a @ X @ Y ) ) @ ( same_fst_a_a @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_175_same__fstI,axiom,
! [P2: product_prod_a_a > $o,X: product_prod_a_a,Y5: product_prod_a_a,Y: product_prod_a_a,R2: product_prod_a_a > set_Pr1948701895od_a_a] :
( ( P2 @ X )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y5 @ Y ) @ ( R2 @ X ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ ( produc1474507607od_a_a @ X @ Y5 ) @ ( produc1474507607od_a_a @ X @ Y ) ) @ ( same_f1138477100od_a_a @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_176_map__prod__simp,axiom,
! [F: a > a,G: a > a,A: a,B2: a] :
( ( produc700022774_a_a_a @ F @ G @ ( product_Pair_a_a @ A @ B2 ) )
= ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_177_map__prod__simp,axiom,
! [F: a > product_prod_a_a,G: a > product_prod_a_a,A: a,B2: a] :
( ( produc2012818120od_a_a @ F @ G @ ( product_Pair_a_a @ A @ B2 ) )
= ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_178_map__prod__simp,axiom,
! [F: product_prod_a_a > a,G: product_prod_a_a > a,A: product_prod_a_a,B2: product_prod_a_a] :
( ( produc363290532_a_a_a @ F @ G @ ( produc1474507607od_a_a @ A @ B2 ) )
= ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_179_map__prod__simp,axiom,
! [F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a,A: product_prod_a_a,B2: product_prod_a_a] :
( ( produc1779756150od_a_a @ F @ G @ ( produc1474507607od_a_a @ A @ B2 ) )
= ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_180_map__prod__imageI,axiom,
! [A: a,B2: a,R2: set_Product_prod_a_a,F: a > a,G: a > a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_181_map__prod__imageI,axiom,
! [A: a,B2: a,R2: set_Product_prod_a_a,F: a > product_prod_a_a,G: a > product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R2 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_1765377413od_a_a @ ( produc2012818120od_a_a @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_182_map__prod__imageI,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,R2: set_Pr1948701895od_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_734753285od_a_a @ ( produc363290532_a_a_a @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_183_map__prod__imageI,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,R2: set_Pr1948701895od_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ R2 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_236793797od_a_a @ ( produc1779756150od_a_a @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_184_top__empty__eq,axiom,
( top_to1483957574_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ top_to398455383od_a_a ) ) ) ).
% top_empty_eq
thf(fact_185_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_186_top__empty__eq,axiom,
( top_to2011536774_a_a_o
= ( ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ top_to52711959od_a_a ) ) ) ).
% top_empty_eq
thf(fact_187_fst__eqD,axiom,
! [X: a,Y: a,A: a] :
( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_188_fst__eqD,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,A: product_prod_a_a] :
( ( ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_189_fst__conv,axiom,
! [X1: a,X2: a] :
( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_190_fst__conv,axiom,
! [X1: product_prod_a_a,X2: product_prod_a_a] :
( ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_191_prod__fun__imageE,axiom,
! [C2: product_prod_a_a,F: a > a,G: a > a,R2: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F @ G ) @ R2 ) )
=> ~ ! [X3: a,Y3: a] :
( ( C2
= ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_192_prod__fun__imageE,axiom,
! [C2: product_prod_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a,R2: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ C2 @ ( image_734753285od_a_a @ ( produc363290532_a_a_a @ F @ G ) @ R2 ) )
=> ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( C2
= ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_193_prod__fun__imageE,axiom,
! [C2: produc1572603623od_a_a,F: a > product_prod_a_a,G: a > product_prod_a_a,R2: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ C2 @ ( image_1765377413od_a_a @ ( produc2012818120od_a_a @ F @ G ) @ R2 ) )
=> ~ ! [X3: a,Y3: a] :
( ( C2
= ( produc1474507607od_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_194_prod__fun__imageE,axiom,
! [C2: produc1572603623od_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a,R2: set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ C2 @ ( image_236793797od_a_a @ ( produc1779756150od_a_a @ F @ G ) @ R2 ) )
=> ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( C2
= ( produc1474507607od_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_195_top__empty__eq2,axiom,
( top_top_a_a_o
= ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ top_to398455383od_a_a ) ) ) ).
% top_empty_eq2
thf(fact_196_top__empty__eq2,axiom,
( top_to993969022_a_a_o
= ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ top_to52711959od_a_a ) ) ) ).
% top_empty_eq2
thf(fact_197_eq__fst__iff,axiom,
! [A: a,P: product_prod_a_a] :
( ( A
= ( product_fst_a_a @ P ) )
= ( ? [B6: a] :
( P
= ( product_Pair_a_a @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_198_eq__fst__iff,axiom,
! [A: product_prod_a_a,P: produc1572603623od_a_a] :
( ( A
= ( produc2115494211od_a_a @ P ) )
= ( ? [B6: product_prod_a_a] :
( P
= ( produc1474507607od_a_a @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_199_fstI,axiom,
! [X: product_prod_a_a,Y: a,Z: a] :
( ( X
= ( product_Pair_a_a @ Y @ Z ) )
=> ( ( product_fst_a_a @ X )
= Y ) ) ).
% fstI
thf(fact_200_fstI,axiom,
! [X: produc1572603623od_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( X
= ( produc1474507607od_a_a @ Y @ Z ) )
=> ( ( produc2115494211od_a_a @ X )
= Y ) ) ).
% fstI
thf(fact_201_prod_Ocollapse,axiom,
! [Prod: product_prod_a_a] :
( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_202_prod_Ocollapse,axiom,
! [Prod: produc1572603623od_a_a] :
( ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ Prod ) @ ( produc368344197od_a_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_203_eq__snd__iff,axiom,
! [B2: a,P: product_prod_a_a] :
( ( B2
= ( product_snd_a_a @ P ) )
= ( ? [A5: a] :
( P
= ( product_Pair_a_a @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_204_eq__snd__iff,axiom,
! [B2: product_prod_a_a,P: produc1572603623od_a_a] :
( ( B2
= ( produc368344197od_a_a @ P ) )
= ( ? [A5: product_prod_a_a] :
( P
= ( produc1474507607od_a_a @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_205_sndI,axiom,
! [X: product_prod_a_a,Y: a,Z: a] :
( ( X
= ( product_Pair_a_a @ Y @ Z ) )
=> ( ( product_snd_a_a @ X )
= Z ) ) ).
% sndI
thf(fact_206_sndI,axiom,
! [X: produc1572603623od_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( X
= ( produc1474507607od_a_a @ Y @ Z ) )
=> ( ( produc368344197od_a_a @ X )
= Z ) ) ).
% sndI
thf(fact_207_snd__eqD,axiom,
! [X: a,Y: a,A: a] :
( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_208_snd__eqD,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,A: product_prod_a_a] :
( ( ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_209_snd__conv,axiom,
! [X1: a,X2: a] :
( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_210_snd__conv,axiom,
! [X1: product_prod_a_a,X2: product_prod_a_a] :
( ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_211_surjective__pairing,axiom,
! [T: product_prod_a_a] :
( T
= ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% surjective_pairing
thf(fact_212_surjective__pairing,axiom,
! [T: produc1572603623od_a_a] :
( T
= ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ T ) @ ( produc368344197od_a_a @ T ) ) ) ).
% surjective_pairing
thf(fact_213_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_a] :
( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_214_prod_Oexhaust__sel,axiom,
! [Prod: produc1572603623od_a_a] :
( Prod
= ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ Prod ) @ ( produc368344197od_a_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_215_prod_Oswap__def,axiom,
( product_swap_a_a
= ( ^ [P4: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P4 ) @ ( product_fst_a_a @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_216_prod_Oswap__def,axiom,
( produc1641853175od_a_a
= ( ^ [P4: produc1572603623od_a_a] : ( produc1474507607od_a_a @ ( produc368344197od_a_a @ P4 ) @ ( produc2115494211od_a_a @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_217_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
( ( P2 @ X @ Y )
=> ( ( A
= ( product_Pair_a_a @ X @ Y ) )
=> ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_218_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,Y: product_prod_a_a,A: produc1572603623od_a_a] :
( ( P2 @ X @ Y )
=> ( ( A
= ( produc1474507607od_a_a @ X @ Y ) )
=> ( P2 @ ( produc2115494211od_a_a @ A ) @ ( produc368344197od_a_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_219_conjI__realizer,axiom,
! [P2: a > $o,P: a,Q: a > $o,Q2: a] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) )
& ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_220_conjI__realizer,axiom,
! [P2: product_prod_a_a > $o,P: product_prod_a_a,Q: product_prod_a_a > $o,Q2: product_prod_a_a] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ P @ Q2 ) ) )
& ( Q @ ( produc368344197od_a_a @ ( produc1474507607od_a_a @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_221_exI__realizer,axiom,
! [P2: a > a > $o,Y: a,X: a] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_222_exI__realizer,axiom,
! [P2: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a,X: product_prod_a_a] :
( ( P2 @ Y @ X )
=> ( P2 @ ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X @ Y ) ) @ ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_223_image2__eqI,axiom,
! [B2: a,F: product_prod_a_a > a,X: product_prod_a_a,C2: a,G: product_prod_a_a > a,A4: set_Product_prod_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member449909584od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr1853143117_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_224_image2__eqI,axiom,
! [B2: a,F: a > a,X: a,C2: a,G: a > a,A4: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_a @ X @ A4 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr423046308_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_225_image2__eqI,axiom,
! [B2: a,F: produc1572603623od_a_a > a,X: produc1572603623od_a_a,C2: a,G: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member2057358096od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr356189709_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_226_image2__eqI,axiom,
! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,C2: product_prod_a_a,G: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member449909584od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr200450189od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_227_image2__eqI,axiom,
! [B2: product_prod_a_a,F: a > product_prod_a_a,X: a,C2: product_prod_a_a,G: a > product_prod_a_a,A4: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_a @ X @ A4 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr1715025636od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_228_image2__eqI,axiom,
! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,X: produc1572603623od_a_a,C2: product_prod_a_a,G: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member2057358096od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr533818957od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_229_in__lex__prod,axiom,
! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a,R: set_Product_prod_a_a,S: set_Pr1948701895od_a_a] :
( ( member1361298640od_a_a @ ( produc563767703od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc1299253312od_a_a @ A2 @ B3 ) ) @ ( lex_pr1321960536od_a_a @ R @ S ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_230_in__lex__prod,axiom,
! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a,R: set_Pr1948701895od_a_a,S: set_Product_prod_a_a] :
( ( member470802768_a_a_a @ ( produc2054708503_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc2061588782_a_a_a @ A2 @ B3 ) ) @ ( lex_pr2084296006_a_a_a @ R @ S ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_231_in__lex__prod,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a,R: set_Pr1948701895od_a_a,S: set_Pr1948701895od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc1474507607od_a_a @ A2 @ B3 ) ) @ ( lex_pr942790511od_a_a @ R @ S ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_232_in__lex__prod,axiom,
! [A: a,B2: a,A2: a,B3: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Pair_a_a @ A2 @ B3 ) ) @ ( lex_prod_a_a @ R @ S ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_233_mem__Sigma__iff,axiom,
! [A: product_prod_a_a,B2: a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
( ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) )
= ( ( member449909584od_a_a @ A @ A4 )
& ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_234_mem__Sigma__iff,axiom,
! [A: product_prod_a_a,B2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
( ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) )
= ( ( member449909584od_a_a @ A @ A4 )
& ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_235_mem__Sigma__iff,axiom,
! [A: a,B2: product_prod_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
( ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) )
= ( ( member_a @ A @ A4 )
& ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_236_mem__Sigma__iff,axiom,
! [A: a,B2: produc1572603623od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
( ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) )
= ( ( member_a @ A @ A4 )
& ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_237_mem__Sigma__iff,axiom,
! [A: produc1572603623od_a_a,B2: product_prod_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
( ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) )
= ( ( member2057358096od_a_a @ A @ A4 )
& ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_238_mem__Sigma__iff,axiom,
! [A: produc1572603623od_a_a,B2: a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
( ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) )
= ( ( member2057358096od_a_a @ A @ A4 )
& ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_239_mem__Sigma__iff,axiom,
! [A: produc1572603623od_a_a,B2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) )
= ( ( member2057358096od_a_a @ A @ A4 )
& ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_240_mem__Sigma__iff,axiom,
! [A: a,B2: a,A4: set_a,B: a > set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
= ( ( member_a @ A @ A4 )
& ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_241_mem__Sigma__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
= ( ( member449909584od_a_a @ A @ A4 )
& ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_242_SigmaI,axiom,
! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: a,B: product_prod_a_a > set_a] :
( ( member449909584od_a_a @ A @ A4 )
=> ( ( member_a @ B2 @ ( B @ A ) )
=> ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_243_SigmaI,axiom,
! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: produc1572603623od_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ A @ A4 )
=> ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
=> ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_244_SigmaI,axiom,
! [A: a,A4: set_a,B2: product_prod_a_a,B: a > set_Product_prod_a_a] :
( ( member_a @ A @ A4 )
=> ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
=> ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_245_SigmaI,axiom,
! [A: a,A4: set_a,B2: produc1572603623od_a_a,B: a > set_Pr1948701895od_a_a] :
( ( member_a @ A @ A4 )
=> ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
=> ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_246_SigmaI,axiom,
! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: product_prod_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
( ( member2057358096od_a_a @ A @ A4 )
=> ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
=> ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_247_SigmaI,axiom,
! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: a,B: produc1572603623od_a_a > set_a] :
( ( member2057358096od_a_a @ A @ A4 )
=> ( ( member_a @ B2 @ ( B @ A ) )
=> ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_248_SigmaI,axiom,
! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: produc1572603623od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
( ( member2057358096od_a_a @ A @ A4 )
=> ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_249_SigmaI,axiom,
! [A: a,A4: set_a,B2: a,B: a > set_a] :
( ( member_a @ A @ A4 )
=> ( ( member_a @ B2 @ ( B @ A ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_250_SigmaI,axiom,
! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ A4 )
=> ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) ) ) ) ).
% SigmaI
thf(fact_251_SigmaE2,axiom,
! [A: product_prod_a_a,B2: a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
( ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) )
=> ~ ( ( member449909584od_a_a @ A @ A4 )
=> ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_252_SigmaE2,axiom,
! [A: product_prod_a_a,B2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
( ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) )
=> ~ ( ( member449909584od_a_a @ A @ A4 )
=> ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_253_SigmaE2,axiom,
! [A: a,B2: product_prod_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
( ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) )
=> ~ ( ( member_a @ A @ A4 )
=> ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_254_SigmaE2,axiom,
! [A: a,B2: produc1572603623od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
( ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) )
=> ~ ( ( member_a @ A @ A4 )
=> ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_255_SigmaE2,axiom,
! [A: produc1572603623od_a_a,B2: product_prod_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
( ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) )
=> ~ ( ( member2057358096od_a_a @ A @ A4 )
=> ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_256_SigmaE2,axiom,
! [A: produc1572603623od_a_a,B2: a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
( ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) )
=> ~ ( ( member2057358096od_a_a @ A @ A4 )
=> ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_257_SigmaE2,axiom,
! [A: produc1572603623od_a_a,B2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) )
=> ~ ( ( member2057358096od_a_a @ A @ A4 )
=> ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_258_SigmaE2,axiom,
! [A: a,B2: a,A4: set_a,B: a > set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
=> ~ ( ( member_a @ A @ A4 )
=> ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_259_SigmaE2,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
=> ~ ( ( member449909584od_a_a @ A @ A4 )
=> ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% SigmaE2
thf(fact_260_SigmaD2,axiom,
! [A: a,B2: a,A4: set_a,B: a > set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
=> ( member_a @ B2 @ ( B @ A ) ) ) ).
% SigmaD2
thf(fact_261_SigmaD2,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
=> ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ).
% SigmaD2
thf(fact_262_SigmaD1,axiom,
! [A: a,B2: a,A4: set_a,B: a > set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
=> ( member_a @ A @ A4 ) ) ).
% SigmaD1
thf(fact_263_SigmaD1,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
=> ( member449909584od_a_a @ A @ A4 ) ) ).
% SigmaD1
thf(fact_264_SigmaE,axiom,
! [C2: produc1016684094_a_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
( ( member1403138919_a_a_a @ C2 @ ( produc1282482655_a_a_a @ A4 @ B ) )
=> ~ ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc2061588782_a_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_265_SigmaE,axiom,
! [C2: produc772425383od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
( ( member122808528od_a_a @ C2 @ ( produc1227942600od_a_a @ A4 @ B ) )
=> ~ ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ! [Y3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc1555606807od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_266_SigmaE,axiom,
! [C2: produc1921647824od_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
( ( member160619001od_a_a @ C2 @ ( produc520147185od_a_a @ A4 @ B ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ! [Y3: product_prod_a_a] :
( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc1299253312od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_267_SigmaE,axiom,
! [C2: produc1555800720od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
( ( member448091577od_a_a @ C2 @ ( produc663418545od_a_a @ A4 @ B ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ! [Y3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc1095491584od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_268_SigmaE,axiom,
! [C2: produc47586087od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
( ( member1545452880od_a_a @ C2 @ ( produc197318472od_a_a @ A4 @ B ) )
=> ~ ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ! [Y3: product_prod_a_a] :
( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc524982679od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_269_SigmaE,axiom,
! [C2: produc1683075710_a_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
( ( member575366567_a_a_a @ C2 @ ( produc4724255_a_a_a @ A4 @ B ) )
=> ~ ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc436797294_a_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_270_SigmaE,axiom,
! [C2: produc1990712039od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
( ( member1899387664od_a_a @ C2 @ ( produc1460477704od_a_a @ A4 @ B ) )
=> ~ ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ! [Y3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc1935643479od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_271_SigmaE,axiom,
! [C2: product_prod_a_a,A4: set_a,B: a > set_a] :
( ( member449909584od_a_a @ C2 @ ( product_Sigma_a_a @ A4 @ B ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( product_Pair_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_272_SigmaE,axiom,
! [C2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
( ( member2057358096od_a_a @ C2 @ ( produc304751368od_a_a @ A4 @ B ) )
=> ~ ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ! [Y3: product_prod_a_a] :
( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc1474507607od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_273_mem__Times__iff,axiom,
! [X: produc1016684094_a_a_a,A4: set_Product_prod_a_a,B: set_a] :
( ( member1403138919_a_a_a @ X
@ ( produc1282482655_a_a_a @ A4
@ ^ [Uu: product_prod_a_a] : B ) )
= ( ( member449909584od_a_a @ ( produc1805832986_a_a_a @ X ) @ A4 )
& ( member_a @ ( produc1954021468_a_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_274_mem__Times__iff,axiom,
! [X: produc772425383od_a_a,A4: set_Product_prod_a_a,B: set_Pr1948701895od_a_a] :
( ( member122808528od_a_a @ X
@ ( produc1227942600od_a_a @ A4
@ ^ [Uu: product_prod_a_a] : B ) )
= ( ( member449909584od_a_a @ ( produc1775747843od_a_a @ X ) @ A4 )
& ( member2057358096od_a_a @ ( produc1055472197od_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_275_mem__Times__iff,axiom,
! [X: produc1921647824od_a_a,A4: set_a,B: set_Product_prod_a_a] :
( ( member160619001od_a_a @ X
@ ( produc520147185od_a_a @ A4
@ ^ [Uu: a] : B ) )
= ( ( member_a @ ( produc1043497516od_a_a @ X ) @ A4 )
& ( member449909584od_a_a @ ( produc1191685998od_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_276_mem__Times__iff,axiom,
! [X: produc1555800720od_a_a,A4: set_a,B: set_Pr1948701895od_a_a] :
( ( member448091577od_a_a @ X
@ ( produc663418545od_a_a @ A4
@ ^ [Uu: a] : B ) )
= ( ( member_a @ ( produc530563564od_a_a @ X ) @ A4 )
& ( member2057358096od_a_a @ ( produc921029422od_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_277_mem__Times__iff,axiom,
! [X: produc47586087od_a_a,A4: set_Pr1948701895od_a_a,B: set_Product_prod_a_a] :
( ( member1545452880od_a_a @ X
@ ( produc197318472od_a_a @ A4
@ ^ [Uu: produc1572603623od_a_a] : B ) )
= ( ( member2057358096od_a_a @ ( produc745123715od_a_a @ X ) @ A4 )
& ( member449909584od_a_a @ ( produc24848069od_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_278_mem__Times__iff,axiom,
! [X: produc1683075710_a_a_a,A4: set_Pr1948701895od_a_a,B: set_a] :
( ( member575366567_a_a_a @ X
@ ( produc4724255_a_a_a @ A4
@ ^ [Uu: produc1572603623od_a_a] : B ) )
= ( ( member2057358096od_a_a @ ( produc2019352922_a_a_a @ X ) @ A4 )
& ( member_a @ ( produc262335132_a_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_279_mem__Times__iff,axiom,
! [X: produc1990712039od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ( member1899387664od_a_a @ X
@ ( produc1460477704od_a_a @ A4
@ ^ [Uu: produc1572603623od_a_a] : B ) )
= ( ( member2057358096od_a_a @ ( produc303632707od_a_a @ X ) @ A4 )
& ( member2057358096od_a_a @ ( produc1441227909od_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_280_mem__Times__iff,axiom,
! [X: product_prod_a_a,A4: set_a,B: set_a] :
( ( member449909584od_a_a @ X
@ ( product_Sigma_a_a @ A4
@ ^ [Uu: a] : B ) )
= ( ( member_a @ ( product_fst_a_a @ X ) @ A4 )
& ( member_a @ ( product_snd_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_281_mem__Times__iff,axiom,
! [X: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ X
@ ( produc304751368od_a_a @ A4
@ ^ [Uu: product_prod_a_a] : B ) )
= ( ( member449909584od_a_a @ ( produc2115494211od_a_a @ X ) @ A4 )
& ( member449909584od_a_a @ ( produc368344197od_a_a @ X ) @ B ) ) ) ).
% mem_Times_iff
thf(fact_282_member__product,axiom,
! [X: product_prod_a_a,A4: set_a,B: set_a] :
( ( member449909584od_a_a @ X @ ( product_product_a_a @ A4 @ B ) )
= ( member449909584od_a_a @ X
@ ( product_Sigma_a_a @ A4
@ ^ [Uu: a] : B ) ) ) ).
% member_product
thf(fact_283_member__product,axiom,
! [X: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ X @ ( produc905674258od_a_a @ A4 @ B ) )
= ( member2057358096od_a_a @ X
@ ( produc304751368od_a_a @ A4
@ ^ [Uu: product_prod_a_a] : B ) ) ) ).
% member_product
thf(fact_284_subsetI,axiom,
! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ( member449909584od_a_a @ X3 @ B ) )
=> ( ord_le1824328871od_a_a @ A4 @ B ) ) ).
% subsetI
thf(fact_285_subsetI,axiom,
! [A4: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ).
% subsetI
thf(fact_286_subsetI,axiom,
! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
( ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ( member2057358096od_a_a @ X3 @ B ) )
=> ( ord_le456379495od_a_a @ A4 @ B ) ) ).
% subsetI
thf(fact_287_image__subsetI,axiom,
! [A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
=> ( ord_le1824328871od_a_a @ ( image_794773445od_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_288_image__subsetI,axiom,
! [A4: set_Product_prod_a_a,F: product_prod_a_a > a,B: set_a] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_340790172_a_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_289_image__subsetI,axiom,
! [A4: set_Product_prod_a_a,F: product_prod_a_a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
=> ( ord_le456379495od_a_a @ ( image_1765377413od_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_290_image__subsetI,axiom,
! [A4: set_a,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
=> ( ord_le1824328871od_a_a @ ( image_1725938350od_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_291_image__subsetI,axiom,
! [A4: set_a,F: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_292_image__subsetI,axiom,
! [A4: set_a,F: a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
=> ( ord_le456379495od_a_a @ ( image_1763923566od_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_293_image__subsetI,axiom,
! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
=> ( ord_le1824328871od_a_a @ ( image_734753285od_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_294_image__subsetI,axiom,
! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > a,B: set_a] :
( ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_1105229276_a_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_295_image__subsetI,axiom,
! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
( ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A4 )
=> ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
=> ( ord_le456379495od_a_a @ ( image_236793797od_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_296_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X3: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le1824328871od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_297_subrelI,axiom,
! [R: set_Pr1948701895od_a_a,S: set_Pr1948701895od_a_a] :
( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le456379495od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_298_subset__iff,axiom,
( ord_le1824328871od_a_a
= ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T2: product_prod_a_a] :
( ( member449909584od_a_a @ T2 @ A6 )
=> ( member449909584od_a_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_299_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_300_subset__iff,axiom,
( ord_le456379495od_a_a
= ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
! [T2: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ T2 @ A6 )
=> ( member2057358096od_a_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_301_subset__eq,axiom,
( ord_le1824328871od_a_a
= ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A6 )
=> ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_302_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_303_subset__eq,axiom,
( ord_le456379495od_a_a
= ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
! [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A6 )
=> ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_304_subsetD,axiom,
! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A4 @ B )
=> ( ( member449909584od_a_a @ C2 @ A4 )
=> ( member449909584od_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_305_subsetD,axiom,
! [A4: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( member_a @ C2 @ A4 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_306_subsetD,axiom,
! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a,C2: produc1572603623od_a_a] :
( ( ord_le456379495od_a_a @ A4 @ B )
=> ( ( member2057358096od_a_a @ C2 @ A4 )
=> ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_307_in__mono,axiom,
! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A4 @ B )
=> ( ( member449909584od_a_a @ X @ A4 )
=> ( member449909584od_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_308_in__mono,axiom,
! [A4: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( member_a @ X @ A4 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_309_in__mono,axiom,
! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a,X: produc1572603623od_a_a] :
( ( ord_le456379495od_a_a @ A4 @ B )
=> ( ( member2057358096od_a_a @ X @ A4 )
=> ( member2057358096od_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_310_prop__restrict,axiom,
! [X: product_prod_a_a,Z2: set_Product_prod_a_a,X5: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member449909584od_a_a @ X @ Z2 )
=> ( ( ord_le1824328871od_a_a @ Z2
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_311_prop__restrict,axiom,
! [X: a,Z2: set_a,X5: set_a,P2: a > $o] :
( ( member_a @ X @ Z2 )
=> ( ( ord_less_eq_set_a @ Z2
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_312_prop__restrict,axiom,
! [X: produc1572603623od_a_a,Z2: set_Pr1948701895od_a_a,X5: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
( ( member2057358096od_a_a @ X @ Z2 )
=> ( ( ord_le456379495od_a_a @ Z2
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_313_Collect__restrict,axiom,
! [X5: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ord_le1824328871od_a_a
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_314_Collect__restrict,axiom,
! [X5: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_315_Collect__restrict,axiom,
! [X5: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
( ord_le456379495od_a_a
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_316_Collect__subset,axiom,
! [A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ord_le1824328871od_a_a
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
& ( P2 @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_317_Collect__subset,axiom,
! [A4: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
& ( P2 @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_318_Collect__subset,axiom,
! [A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
( ord_le456379495od_a_a
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A4 )
& ( P2 @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_319_subset__Collect__iff,axiom,
! [B: set_Product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ B @ A4 )
=> ( ( ord_le1824328871od_a_a @ B
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ B )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_320_subset__Collect__iff,axiom,
! [B: set_a,A4: set_a,P2: a > $o] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_321_subset__Collect__iff,axiom,
! [B: set_Pr1948701895od_a_a,A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
( ( ord_le456379495od_a_a @ B @ A4 )
=> ( ( ord_le456379495od_a_a @ B
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ B )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_322_subset__CollectI,axiom,
! [B: set_Product_prod_a_a,A4: set_Product_prod_a_a,Q: product_prod_a_a > $o,P2: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ B @ A4 )
=> ( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le1824328871od_a_a
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ B )
& ( Q @ X4 ) ) )
@ ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_323_subset__CollectI,axiom,
! [B: set_a,A4: set_a,Q: a > $o,P2: a > $o] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ B )
& ( Q @ X4 ) ) )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_324_subset__CollectI,axiom,
! [B: set_Pr1948701895od_a_a,A4: set_Pr1948701895od_a_a,Q: produc1572603623od_a_a > $o,P2: produc1572603623od_a_a > $o] :
( ( ord_le456379495od_a_a @ B @ A4 )
=> ( ! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le456379495od_a_a
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ B )
& ( Q @ X4 ) ) )
@ ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_325_pred__subset__eq,axiom,
! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ord_le1347718902_a_a_o
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ R2 )
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ S3 ) )
= ( ord_le1824328871od_a_a @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_326_pred__subset__eq,axiom,
! [R2: set_a,S3: set_a] :
( ( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R2 )
@ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
= ( ord_less_eq_set_a @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_327_pred__subset__eq,axiom,
! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
( ( ord_le1525442358_a_a_o
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ R2 )
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ S3 ) )
= ( ord_le456379495od_a_a @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_328_less__eq__set__def,axiom,
( ord_le1824328871od_a_a
= ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ord_le1347718902_a_a_o
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A6 )
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_329_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_330_less__eq__set__def,axiom,
( ord_le456379495od_a_a
= ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
( ord_le1525442358_a_a_o
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A6 )
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_331_pred__subset__eq2,axiom,
! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ord_less_eq_a_a_o
@ ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 )
@ ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S3 ) )
= ( ord_le1824328871od_a_a @ R2 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_332_pred__subset__eq2,axiom,
! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
( ( ord_le252190158_a_a_o
@ ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R2 )
@ ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ S3 ) )
= ( ord_le456379495od_a_a @ R2 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_333_GrD1,axiom,
! [X: a,Fx: a,A4: set_a,F: a > a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Fx ) @ ( bNF_Gr_a_a @ A4 @ F ) )
=> ( member_a @ X @ A4 ) ) ).
% GrD1
thf(fact_334_GrD1,axiom,
! [X: product_prod_a_a,Fx: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Fx ) @ ( bNF_Gr1372473186od_a_a @ A4 @ F ) )
=> ( member449909584od_a_a @ X @ A4 ) ) ).
% GrD1
thf(fact_335_GrD2,axiom,
! [X: a,Fx: a,A4: set_a,F: a > a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Fx ) @ ( bNF_Gr_a_a @ A4 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_336_GrD2,axiom,
! [X: product_prod_a_a,Fx: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Fx ) @ ( bNF_Gr1372473186od_a_a @ A4 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_337_all__not__in__conv,axiom,
! [A4: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member449909584od_a_a @ X4 @ A4 ) )
= ( A4 = bot_bo2131659635od_a_a ) ) ).
% all_not_in_conv
thf(fact_338_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_339_all__not__in__conv,axiom,
! [A4: set_Pr1948701895od_a_a] :
( ( ! [X4: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ X4 @ A4 ) )
= ( A4 = bot_bo1116205875od_a_a ) ) ).
% all_not_in_conv
thf(fact_340_empty__iff,axiom,
! [C2: product_prod_a_a] :
~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% empty_iff
thf(fact_341_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_342_empty__iff,axiom,
! [C2: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ C2 @ bot_bo1116205875od_a_a ) ).
% empty_iff
thf(fact_343_ex__in__conv,axiom,
! [A4: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
= ( A4 != bot_bo2131659635od_a_a ) ) ).
% ex_in_conv
thf(fact_344_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_345_ex__in__conv,axiom,
! [A4: set_Pr1948701895od_a_a] :
( ( ? [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A4 ) )
= ( A4 != bot_bo1116205875od_a_a ) ) ).
% ex_in_conv
thf(fact_346_equals0I,axiom,
! [A4: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member449909584od_a_a @ Y3 @ A4 )
=> ( A4 = bot_bo2131659635od_a_a ) ) ).
% equals0I
thf(fact_347_equals0I,axiom,
! [A4: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_348_equals0I,axiom,
! [A4: set_Pr1948701895od_a_a] :
( ! [Y3: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ Y3 @ A4 )
=> ( A4 = bot_bo1116205875od_a_a ) ) ).
% equals0I
thf(fact_349_equals0D,axiom,
! [A4: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A4 = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ A @ A4 ) ) ).
% equals0D
thf(fact_350_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_351_equals0D,axiom,
! [A4: set_Pr1948701895od_a_a,A: produc1572603623od_a_a] :
( ( A4 = bot_bo1116205875od_a_a )
=> ~ ( member2057358096od_a_a @ A @ A4 ) ) ).
% equals0D
thf(fact_352_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% emptyE
thf(fact_353_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_354_emptyE,axiom,
! [A: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ A @ bot_bo1116205875od_a_a ) ).
% emptyE
% Conjectures (1)
thf(conj_0,conjecture,
member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ).
%------------------------------------------------------------------------------